Repository: duvenaud/phd-thesis Branch: master Commit: 62ff4d61f27e Files: 315 Total size: 3.6 MB Directory structure: gitextract_7shfk872/ ├── .gitignore ├── additive.tex ├── appendix.tex ├── code/ │ ├── additive_extrapolation.m │ ├── data/ │ │ ├── quebec/ │ │ │ ├── process.m │ │ │ ├── quebec-1000.mat │ │ │ ├── quebec-2000.mat │ │ │ ├── quebec-5000.mat │ │ │ ├── quebec-all.mat │ │ │ ├── quebec-births-fixed.csv │ │ │ └── quebec-births-standard.csv │ │ └── r_concrete_500.mat │ ├── decomp_birth.m │ ├── decomp_concrete.m │ ├── draw_2d_additive_kernels.m │ ├── draw_3d_additive_kernels.m │ ├── draw_manifolds.m │ ├── gpml/ │ │ ├── .octaverc │ │ ├── Copyright │ │ ├── README │ │ ├── barrier_fct.m │ │ ├── cov/ │ │ │ ├── covADD.m │ │ │ ├── covConst.m │ │ │ ├── covFITC.m │ │ │ ├── covLIN.m │ │ │ ├── covLINard.m │ │ │ ├── covLINone.m │ │ │ ├── covMask.m │ │ │ ├── covMaterniso.m │ │ │ ├── covNNone.m │ │ │ ├── covNoise.m │ │ │ ├── covPPiso.m │ │ │ ├── covPeriodic.m │ │ │ ├── covPoly.m │ │ │ ├── covProd.m │ │ │ ├── covRQard.m │ │ │ ├── covRQiso.m │ │ │ ├── covSEard.m │ │ │ ├── covSEiso.m │ │ │ ├── covSEisoU.m │ │ │ ├── covSEiso_length.m │ │ │ ├── covSEiso_var.m │ │ │ ├── covScale.m │ │ │ └── covSum.m │ │ ├── covFunctions.m │ │ ├── doc/ │ │ │ ├── README │ │ │ ├── changelog │ │ │ ├── demoClassification.m │ │ │ ├── demoRegression.m │ │ │ ├── gpml_randn.m │ │ │ ├── index.html │ │ │ ├── style.css │ │ │ ├── usageClassification.m │ │ │ ├── usageCov.m │ │ │ ├── usageMean.m │ │ │ └── usageRegression.m │ │ ├── gp.m │ │ ├── inf/ │ │ │ ├── infEP.m │ │ │ ├── infExact.m │ │ │ ├── infFITC.m │ │ │ ├── infLOO.m │ │ │ ├── infLaplace.m │ │ │ └── infVB.m │ │ ├── infMethods.m │ │ ├── lik/ │ │ │ ├── likErf.m │ │ │ ├── likGauss.m │ │ │ ├── likLaplace.m │ │ │ ├── likLogistic.m │ │ │ ├── likSech2.m │ │ │ └── likT.m │ │ ├── likFunctions.m │ │ ├── mean/ │ │ │ ├── meanConst.m │ │ │ ├── meanLinear.m │ │ │ ├── meanMask.m │ │ │ ├── meanOne.m │ │ │ ├── meanPow.m │ │ │ ├── meanProd.m │ │ │ ├── meanScale.m │ │ │ ├── meanSum.m │ │ │ └── meanZero.m │ │ ├── meanFunctions.m │ │ ├── penalized_gp.m │ │ ├── startup.m │ │ └── util/ │ │ ├── binaryEPGP.m │ │ ├── binaryGP.m │ │ ├── binaryLaplaceGP.m │ │ ├── brentmin.m │ │ ├── elsympol.m │ │ ├── elsympol2.m │ │ ├── gpr.m │ │ ├── lbfgsb/ │ │ │ ├── Makefile │ │ │ ├── README.html │ │ │ ├── array.h │ │ │ ├── arrayofmatrices.cpp │ │ │ ├── arrayofmatrices.h │ │ │ ├── lbfgsb.cpp │ │ │ ├── license.txt │ │ │ ├── matlabexception.cpp │ │ │ ├── matlabexception.h │ │ │ ├── matlabmatrix.cpp │ │ │ ├── matlabmatrix.h │ │ │ ├── matlabprogram.cpp │ │ │ ├── matlabprogram.h │ │ │ ├── matlabscalar.cpp │ │ │ ├── matlabscalar.h │ │ │ ├── matlabstring.cpp │ │ │ ├── matlabstring.h │ │ │ ├── program.cpp │ │ │ ├── program.h │ │ │ └── solver.f │ │ ├── lbfgsb.m │ │ ├── make.m │ │ ├── minimize.m │ │ ├── minimize_bfgs.m │ │ ├── minimize_lbfgsb.m │ │ ├── minimize_lbfgsb_gradfun.m │ │ ├── minimize_lbfgsb_objfun.m │ │ ├── minimize_may18.m │ │ ├── minimize_may21.m │ │ ├── minimize_pre_oct26.m │ │ ├── old_minimize.m │ │ ├── rewrap.m │ │ ├── solve_chol.c │ │ ├── solve_chol.m │ │ ├── sq_dist.m │ │ └── unwrap.m │ ├── hypers.mat │ ├── plot_additive_decomp.m │ ├── plot_additive_decomp_cov.m │ ├── plot_oned_gp.m │ ├── plot_oned_gplvm.m │ ├── plot_structure_examples.m │ ├── plot_symmetry_samples.m │ └── utils/ │ ├── colorbrew.m │ ├── coord_to_color.m │ ├── coord_to_image.m │ ├── draw_warped_density.m │ ├── logdet.m │ ├── logmvnpdf.m │ ├── logsumexp.m │ ├── logsumexp2.m │ ├── myaa.m │ ├── parseArgs.m │ ├── plot_little_circles.m │ ├── plot_little_circles_3d.m │ ├── save2pdf.m │ ├── saveeps.m │ ├── savepng.m │ ├── set_fig_units_cm.m │ ├── sq_dist.m │ ├── subaxis.m │ ├── test_coord_to_color.m │ ├── test_coord_to_image.m │ └── tightfig.m ├── common/ │ ├── CUEDbiblio.bst │ ├── PhDThesisPSnPDF.cls │ ├── commenting.tex │ ├── glyphtounicode.tex │ ├── header.tex │ ├── humble.tex │ ├── jmb.bst │ ├── official-preamble.tex │ ├── preamble.tex │ └── thesis-info.tex ├── deeplimits.tex ├── description.tex ├── discussion.tex ├── figures/ │ ├── additive/ │ │ ├── .svn/ │ │ │ ├── all-wcprops │ │ │ ├── entries │ │ │ ├── prop-base/ │ │ │ │ ├── 1st_order_censored_add.pdf.svn-base │ │ │ │ ├── 1st_order_censored_ard.pdf.svn-base │ │ │ │ ├── 1st_order_censored_d1.pdf.svn-base │ │ │ │ ├── 1st_order_censored_d1d2.pdf.svn-base │ │ │ │ ├── 1st_order_censored_d2.pdf.svn-base │ │ │ │ ├── 1st_order_censored_data.pdf.svn-base │ │ │ │ ├── 1st_order_censored_truth.pdf.svn-base │ │ │ │ ├── 3d_add_kernel_1.pdf.svn-base │ │ │ │ ├── 3d_add_kernel_2.pdf.svn-base │ │ │ │ ├── 3d_add_kernel_3.pdf.svn-base │ │ │ │ ├── 3d_add_kernel_32.pdf.svn-base │ │ │ │ ├── 3d_add_kernel_321.pdf.svn-base │ │ │ │ ├── additive_draw.pdf.svn-base │ │ │ │ ├── additive_kernel_2nd_order.pdf.svn-base │ │ │ │ ├── additive_kernel_draw_sum.pdf.svn-base │ │ │ │ ├── additive_kernel_draw_sum_p1.pdf.svn-base │ │ │ │ ├── additive_kernel_draw_sum_p2.pdf.svn-base │ │ │ │ ├── additive_kernel_sum_p1.pdf.svn-base │ │ │ │ ├── additive_kernel_sum_p2.pdf.svn-base │ │ │ │ ├── class_graph.pdf.svn-base │ │ │ │ ├── class_graph_ll.pdf.svn-base │ │ │ │ ├── hkl_add_ard_v1.pdf.svn-base │ │ │ │ ├── housing_order_ll.pdf.svn-base │ │ │ │ ├── housing_order_ll_upto.pdf.svn-base │ │ │ │ ├── housing_order_mse.pdf.svn-base │ │ │ │ ├── housing_order_mse_upto.pdf.svn-base │ │ │ │ ├── hulls.pdf.svn-base │ │ │ │ ├── interpretable_1st_order1.pdf.svn-base │ │ │ │ ├── interpretable_1st_order2.pdf.svn-base │ │ │ │ ├── interpretable_2nd_order1.pdf.svn-base │ │ │ │ ├── pumadyn1.pdf.svn-base │ │ │ │ ├── pumadyn2.pdf.svn-base │ │ │ │ ├── pumadyn3.pdf.svn-base │ │ │ │ ├── pumadyn_1d.pdf.svn-base │ │ │ │ ├── reg_graph.pdf.svn-base │ │ │ │ ├── reg_graph_ll.pdf.svn-base │ │ │ │ ├── sqexp_kernel.pdf.svn-base │ │ │ │ └── synth_plot_1.pdf.svn-base │ │ │ └── text-base/ │ │ │ ├── 1st_order_censored_add.pdf.svn-base │ │ │ ├── 1st_order_censored_ard.pdf.svn-base │ │ │ ├── 1st_order_censored_d1.pdf.svn-base │ │ │ ├── 1st_order_censored_d1d2.pdf.svn-base │ │ │ ├── 1st_order_censored_d2.pdf.svn-base │ │ │ ├── 1st_order_censored_data.pdf.svn-base │ │ │ ├── 1st_order_censored_truth.pdf.svn-base │ │ │ ├── 3d_add_kernel_1.pdf.svn-base │ │ │ ├── 3d_add_kernel_2.pdf.svn-base │ │ │ ├── 3d_add_kernel_3.pdf.svn-base │ │ │ ├── 3d_add_kernel_32.pdf.svn-base │ │ │ ├── 3d_add_kernel_321.pdf.svn-base │ │ │ ├── additive_draw.pdf.svn-base │ │ │ ├── additive_kernel.pdf.svn-base │ │ │ ├── additive_kernel_2nd_order.pdf.svn-base │ │ │ ├── additive_kernel_draw_sum.pdf.svn-base │ │ │ ├── additive_kernel_draw_sum_p1.pdf.svn-base │ │ │ ├── additive_kernel_draw_sum_p2.pdf.svn-base │ │ │ ├── additive_kernel_sum_p1.pdf.svn-base │ │ │ ├── additive_kernel_sum_p2.pdf.svn-base │ │ │ ├── class_graph.pdf.svn-base │ │ │ ├── class_graph_ll.pdf.svn-base │ │ │ ├── hkl_add_ard_v1.pdf.svn-base │ │ │ ├── hkl_add_ard_v1.svg.svn-base │ │ │ ├── housing_order_ll.pdf.svn-base │ │ │ ├── housing_order_ll_upto.pdf.svn-base │ │ │ ├── housing_order_mse.pdf.svn-base │ │ │ ├── housing_order_mse_upto.pdf.svn-base │ │ │ ├── hulls.pdf.svn-base │ │ │ ├── hulls.svg.svn-base │ │ │ ├── hulls3.svg.svn-base │ │ │ ├── interpretable_1st_order1.pdf.svn-base │ │ │ ├── interpretable_1st_order2.pdf.svn-base │ │ │ ├── interpretable_2nd_order1.pdf.svn-base │ │ │ ├── pumadyn1.pdf.svn-base │ │ │ ├── pumadyn2.pdf.svn-base │ │ │ ├── pumadyn3.pdf.svn-base │ │ │ ├── pumadyn_1d.pdf.svn-base │ │ │ ├── reg_graph.pdf.svn-base │ │ │ ├── reg_graph_ll.pdf.svn-base │ │ │ ├── sqexp_draw.pdf.svn-base │ │ │ ├── sqexp_kernel.pdf.svn-base │ │ │ └── synth_plot_1.pdf.svn-base │ │ └── compare_models/ │ │ └── .svn/ │ │ ├── all-wcprops │ │ ├── entries │ │ ├── prop-base/ │ │ │ ├── additive.pdf.svn-base │ │ │ ├── ard.pdf.svn-base │ │ │ ├── gam.pdf.svn-base │ │ │ └── hkl.pdf.svn-base │ │ └── text-base/ │ │ ├── additive.pdf.svn-base │ │ ├── additive.svg.svn-base │ │ ├── ard.pdf.svn-base │ │ ├── ard.svg.svn-base │ │ ├── gam.pdf.svn-base │ │ ├── gam.svg.svn-base │ │ ├── hkl.pdf.svn-base │ │ ├── hkl.svg.svn-base │ │ ├── nodes.svg.svn-base │ │ └── old_version.svg.svn-base │ ├── grammar/ │ │ ├── airlinepages/ │ │ │ └── split_and_crop.sh │ │ ├── extrapolation_curves/ │ │ │ ├── 01-airline-s-ex-curve.fig │ │ │ ├── 01-airline-s-ex-curve_hint.fig │ │ │ ├── 02-solar-s-ex-curve.fig │ │ │ ├── 03-mauna2003-s-MSEs.mat │ │ │ └── 03-mauna2003-s-ex-curve.fig │ │ └── solarpages/ │ │ └── split_and_crop.sh │ └── intro/ │ ├── fuzzy-0 │ ├── fuzzy-1 │ ├── fuzzy-2 │ └── fuzzy-3 ├── grammar.tex ├── intro.tex ├── kernels.tex ├── misc/ │ ├── abstract.tex │ ├── acknowledgement.tex │ ├── declaration.tex │ ├── dedication.tex │ └── original-template/ │ ├── .gitignore │ ├── Abstract/ │ │ └── abstract.tex │ ├── Acknowledgement/ │ │ └── acknowledgement.tex │ ├── Appendix1/ │ │ └── appendix1.tex │ ├── Appendix2/ │ │ └── appendix2.tex │ ├── Chapter1/ │ │ └── chapter1.tex │ ├── Chapter2/ │ │ ├── Figs/ │ │ │ └── Vector/ │ │ │ ├── TomandJerry.eps │ │ │ ├── WallE.eps │ │ │ └── minion.eps │ │ └── chapter2.tex │ ├── Chapter3/ │ │ └── chapter3.tex │ ├── Classes/ │ │ ├── PhDThesisPSnPDF.cls │ │ └── glyphtounicode.tex │ ├── Declaration/ │ │ └── declaration.tex │ ├── Dedication/ │ │ └── dedication.tex │ ├── Figs/ │ │ └── University_Crest.eps │ ├── LICENSE │ ├── Makefile │ ├── Preamble/ │ │ └── preamble.tex │ ├── README.md │ ├── References/ │ │ └── references.bib │ ├── Variables.ini │ ├── compile-thesis.sh │ ├── thesis-info.tex │ └── thesis.tex ├── notation.tex ├── quadrature.tex ├── readme.md ├── references.bib ├── thesis.tex └── warped.tex ================================================ FILE CONTENTS ================================================ ================================================ FILE: .gitignore ================================================ *.acn *.acr *.alg *.aux *.bbl *.blg *.dvi *.fdb_latexmk *.glg *.glo *.gls *.idx *.ilg *.ind *.ist *.lof *.log *.lot *.maf *.mtc *.mtc0 *.nav *.nlo *.out *.pdfsync *.ps *.snm *.synctex.gz *.toc *.vrb *.xdy *.tdo *.swp ================================================ FILE: additive.tex ================================================ \input{common/header.tex} \inbpdocument %\chapter{Dropout in Gaussian processes} \chapter{Additive Gaussian Processes} \label{ch:additive} %\begin{quotation} %``I am in [Gaussian processes] stepped in so far that, should I wade no more, Returning were as tedious as go o'er.'' %\hspace*{\fill}--MacBeth %\end{quotation} \Cref{ch:grammar} showed how to learn the structure of a kernel by building it up piece-by-piece. This chapter presents an alternative approach: starting with many different types of structure in a kernel, adjusting kernel parameters to discard whatever structure is \emph{not} present in the current dataset. The advantage of this approach is that we do not need to run an expensive discrete-and-continuous search in order to build a structured model, and implementation is simpler. This model, which we call \emph{additive Gaussian processes}, is a sum of functions of all possible combinations of input variables. This model can be specified by a weighted sum of all possible products of one-dimensional kernels. There are $2^D$ combinations of $D$ objects, so na\"{i}ve computation of this kernel is intractable. Furthermore, if each term has different kernel parameters, fitting or integrating over so many parameters is difficult. To address these problems, we introduce a restricted parameterization of the kernel which allows efficient evaluation of all interaction terms, while still allowing a different weighting of each order of interaction. Empirically, this model has good predictive performance in regression tasks, and its parameters are relatively interpretable. This model also has an interpretation as an approximation to \emph{dropout}, a recently-introduced regularization method for neural networks. The work in this chapter was done in collaboration with Hannes Nickisch and Carl Rasmussen, who derived and coded up the additive kernel. My role in the project was to examine the properties of the resulting model, clarify the connections to existing methods, to create all figures and run all experiments. That work was published in \citet{duvenaud2011additive11}. The connection to dropout regularization in \cref{sec:dropout-gps} is an independent contribution. \section{Different types of multivariate additive {\mbox structure}} \Cref{ch:kernels} showed how additive structure in a \gp{} prior enabled extrapolation in multivariate regression problems. In general, models of the form % \begin{align} f(\vx) = g \big( f(x_1) + f(x_2) + \dots + f(x_D) \big) \label{eq:general-additive} \end{align} % are widely used in machine learning and statistics, partly for this reason, and partly because they are relatively easy to fit and interpret. Examples include logistic regression, linear regression, generalized linear models~\citep{nelder1972generalized} and generalized additive models~\citep{hastie1990generalized}. %, are typically easy to fit and interpret. %Some extensions of this family, such as smoothing-splines ANOVA \citep{wahba1990spline}, also consider terms depending on more than one variable. %However, such models often become intractable and difficult to fit as the number of terms increases. At the other end of the spectrum are models which allow the response to depend on all input variables simultaneously, without any additive decomposition: % \begin{align} f(\vx) = f(x_1, x_2, \dots, x_D) \label{eq:highest-order} \end{align} % An example would be a \gp{} with an \seard{} kernel. Such models are much more flexible than those having the form \eqref{eq:general-additive}, but this flexibility can make it difficult to generalize to new combinations of input variables. In between these extremes are function classes depending on pairs or triplets of inputs, such as % \begin{align} f(x_1, x_2, x_3) = f_{12}(x_1, x_2) + f_{23}(x_2, x_3) + f_{13}(x_1, x_3) . \label{eq:second-order-additive} \end{align} % We call the number of input variables appearing in each term the \emph{order} of that term. Models containing terms only of intermediate order such as \eqref{eq:second-order-additive} allow more flexibility than models of form \eqref{eq:highest-order} (first-order), but have more structure than those of form \eqref{eq:general-additive} ($D$-th order). Capturing the low-order additive structure present in a function can be expected to improve predictive accuracy. However, if the function being learned depends in some way on an interaction between all input variables, a $D$th-order term is required in order for the model to be consistent. %However, if the function also contains lower-order interactions, capturing that structure will still improve the predioctive performance on finite datasets. %To the extent that a set of variables is relevant to the output, this model must be highly uncertain about any novel combination of those variables. \section{Defining additive kernels} To define the additive kernels introduced in this chapter, we first assign each dimension $i \in \{1 \dots D\}$ a one-dimensional \emph{base kernel} $k_i(x_i, x'_i)$. Then the first order, second order and $n$th order additive kernels are defined as: % \begin{align} k_{add_1}({\bf x, x'}) & = \sigma_1^2 \sum_{i=1}^D k_i(x_i, x_i') \\ k_{add_2}({\bf x, x'}) & = \sigma_2^2 \sum_{i=1}^D \sum_{j = i + 1}^D k_i(x_i, x_i') k_j(x_j, x_j') \\ k_{add_n}({\bf x, x'}) & = \sigma_n^2 \sum_{1 \leq i_1 < i_2 < ... < i_n \leq D} \left[ \prod_{d=1}^n k_{i_d}(x_{i_d}, x_{i_d}') \right] \\ k_{add_D}({\bf x, x'}) & = \sigma_D^2 \sum_{1 \leq i_1 < i_2 < ... < i_D \leq D} \left[ \prod_{d=1}^D k_{i_d}(x_{i_d}, x_{i_d}') \right] = \sigma_D^2 \prod_{d=1}^D k_{d}(x_{d}, x_{d}') \end{align} % where $D$ is the dimension of the input space, and $\sigma_n^2$ is the variance assigned to all $n$th order interactions. The $n$th-order kernel is a sum of ${D \choose n}$ terms. In particular, the $D$th-order additive kernel has ${D \choose D} = 1$ term, a product of each dimension's kernel. In the case where each base kernel is a one-dimensional squared-exponential kernel, the $D$th-order term corresponds to the multivariate squared-exponential kernel, also known as \seard{}: % \begin{align} %k_{add_D}({\bf x, x'}) = %\sigma_D^2 \prod_{d=1}^D k_{d}(x_{d}, x_{d}') = \prod_{d=1}^D \kSE(x_d, x_d') = \prod_{d=1}^D \sigma_d^2\exp \left( -\frac{ ( x_{d} - x_{d}')^2}{2 \ell^2_d} \right) = \sigma_D^2 \exp \left( -\sum_{d=1}^D \frac{ ( x_{d} - x_{d}')^2}{2 \ell^2_d} \right) \end{align} % %also commonly known as the Gaussian kernel. The full additive kernel is a sum of the additive kernels of all orders. %\subsubsection{Parameterization} The only design choice necessary to specify an additive kernel is the selection of a one-dimensional base kernel for each input dimension. Parameters of the base kernels (such as length-scales $\ell_1, \ell_2, \dots, \ell_D$) can be learned as per usual by maximizing the marginal likelihood of the training data. \subsection{Weighting different orders of interaction} In addition to the parameters of each dimension's kernel, additive kernels are equipped with a set of $D$ parameters $\sigma_1^2 \dots \sigma_D^2$. %, controlling how much variance is assigned to each order of interaction. These \emph{order variance} parameters have a useful interpretation: the $d$th order variance parameter specifies how much of the target function's variance comes from interactions of the $d$th order. %\input{\additivetablesdir/r_concrete_500_hypers_table.tex} %Table \ref{tbl:concrete} gives an example of parameters learnt by both a SE-GP, and by those learnt by an additive GP whose base kernels are one-dimensional squared exponential kernels. In this case, the 1st, 2nd and 3rd-order interactions contribute most to the final function. % % \Cref{tbl:all_orders} shows examples of the variance contributed by different orders of interaction, estimated on real datasets. These datasets are described in \cref{sec:additive-datasets}. % --- Automatically generated by hypers_to_latex.m --- % Exported at 03-Jun-2011 00:22:23 \begin{table}[h] \caption[Relative variance contributed by each order of the additive model] {Percentage of variance contributed by each order of interaction of the additive model on different datasets. The maximum order of interaction is set to the input dimension or 10, whichever is smaller. } \begin{center} \begin{tabular}{r | r r r r r r r r r r} \multicolumn{1}{c}{} & \multicolumn{10}{c}{Order of interaction} \\ Dataset & \nth{1} & \nth{2} & \nth{3} & \nth{4} & \nth{5} & \nth{6} & \nth{7} & \nth{8} & \nth{9} & \nth{10} \\ \hline pima & $0.1 $ & $0.1 $ & $0.1 $ & $0.3 $ & $1.5 $ & ${\bf 96.4}$ & $1.4 $ & $0.0 $ & & \\ liver & $0.0 $ & $0.2 $ & ${\bf 99.7 } $ & $0.1 $ & $0.0 $ & $0.0 $ & & & & \\ heart & ${\bf 77.6} $ & $0.0 $ & $0.0 $ & $0.0 $ & $0.1 $ & $0.1 $ & $0.1 $ & $0.1 $ & $0.1 $ & $22.0 $ \\ concrete & ${\bf 70.6 } $ & $13.3 $ & $13.8 $ & $2.3 $ & $0.0 $ & $0.0 $ & $0.0 $ & $0.0 $ & & \\ pumadyn-8nh & $0.0 $ & $0.1 $ & $0.1 $ & $0.1 $ & $0.1 $ & $0.1 $ & $0.1 $ & ${\bf 99.5 } $ & & \\ servo & ${\bf 58.7 }$ & $27.4 $ & $0.0 $ & $13.9 $ & & & & & & \\ housing & $0.1 $ & $0.6 $ & ${\bf 80.6 }$ & $1.4 $ & $1.8 $ & $0.8 $ & $0.7 $ & $0.8 $ & $0.6 $ & $12.7 $ \\ \end{tabular} \end{center} \label{tbl:all_orders} \end{table} % End automatically generated LaTeX On different datasets, the dominant order of interaction estimated by the additive model varies widely. In some cases, the variance is concentrated almost entirely onto a single order of interaction. This may may be a side-effect of using the same lengthscales for all orders of interaction; lengthscales appropriate for low-dimensional regression might not be appropriate for high-dimensional regression. %A re-scaling of lengthscales which enforces similar distances between datapoints might improve the model. %An additive \gp{} with all of its variance coming from the 1st order is equivalent to a sum of one-dimensional functions. %An additive \gp{} with all its variance coming from the $D$th order is equivalent to a \gp{} with an \seard{} kernel. % % We can see that the length-scales learnt are not necessarily the same. This is because, in a normal ARD-style kernel, the length-scale of a dimension is conflated with the variance of the function along that dimension. The additive kernel separates these two parameters. % %This table lets us examine the relative contribution of each order of interaction to the total function variance. Again, the ARD kernel is equivalent to an additive kernel in which all the variance is constrained to arise only from the highest-order interactions. %Because the variance parameters can specify which degrees of interaction are important, the additive \gp{} can capture many different types of structure. %By optimizing these parameters, we are approximately performing weighted model selection over a set of $D$ models. %If the function we are modeling is decomposable into a sum of low-dimensional functions, our model can discover this fact and exploit it. %The marginal likelihood will usually favor using lower orders if possible, since the $|K|^{-\frac{1}{2}}$ term in the \gp{} marginal likelihood (\cref{eq:gp_marg_lik}) will usually be larger for less flexible model classes. %Low-order structure sometimes allows extrapolation, as shown in \cref{sec:additivity-extrapolation}. %If low-dimensional additive structure is not present, the kernel parameters can specify a suitably flexible model, with interactions between as many variables as necessary. %The expected number of zero crossings is $O(k^d)$ for product kernels (sqexp), $O(kd)$ for first-order additive, and $O\left(k {d \choose r}\right)$ for $r$th order additive functions. [Conjecture] %To push the analogy, using a product kernel is analogous to performing polynomial regression when we are forced to use a high-degree polynomial, even when the data do not support it. One could also view the Gaussian kernel as one which makes the pessimistic assumption that the function we are model can vary independently for each new combination of variables observed. \subsection{Efficiently evaluating additive kernels} An additive kernel over $D$ inputs with interactions up to order $n$ has $O(2^n)$ terms. Na\"{i}vely summing these terms is intractable. One can exactly evaluate the sum over all terms in $O(D^2)$, while also weighting each order of interaction separately. To efficiently compute the additive kernel, we exploit the fact that the $n$th order additive kernel corresponds to the $n$th \textit{elementary symmetric polynomial}~\citep{macdonald1998symmetric} of the base kernels, which we denote $e_n$. For example, if $\vx$ has 4 input dimensions ($D = 4$), and if we use the shorthand notation $k_d = k_d(x_d, x_d')$, then % \begin{align} k_{\textnormal{add}_0}({\bf x, x'}) & = e_0( k_1, k_2, k_3, k_4 ) = 1 \\ k_{\textnormal{add}_1}({\bf x, x'}) & = e_1( k_1, k_2, k_3, k_4 ) = k_1 + k_2 + k_3 + k_4 \\ k_{\textnormal{add}_2}({\bf x, x'}) & = e_2( k_1, k_2, k_3, k_4 ) = k_1 k_2 + k_1 k_3 + k_1k_4 + k_2 k_3 + k_2 k_4 + k_3 k_4 \\ k_{\textnormal{add}_3}({\bf x, x'}) & = e_3( k_1, k_2, k_3, k_4 ) = k_1 k_2 k_3 + k_1 k_2 k_4 + k_1 k_3 k_4 + k_2 k_3 k_4 \\ k_{\textnormal{add}_4}({\bf x, x'}) & = e_4( k_1, k_2, k_3, k_4 ) = k_1 k_2 k_3 k_4 \end{align} % The Newton-Girard formulas give an efficient recursive form for computing these polynomials. %If we define $s_k$ to be the $k$th power sum: $s_k(k_1,k_2,\dots,k_D) = \sum_{i=1}^Dk_i^k$, then % \begin{align} k_{\textnormal{add}_n}({\bf x, x'}) = e_n(k_1,k_2,\dots,k_D) = \frac{1}{n} \sum_{a=1}^n (-1)^{(a-1)} e_{n-a}(k_1, k_2, \dots,k_D) \sum_{i=1}^Dk_i^a \end{align} % Each iteration has cost $\mathcal{O}(D)$, given the next-lowest polynomial. %\begin{equation} %$s_k(k_1,k_2,\dots,k_n) = k_1^k + k_2^k + \dots + k_D^k = \sum_{i=1}^Dk_i^k$. %\end{equation} %These formulas have time complexity $\mathcal{O}( D^2 )$, while computing a sum over an exponential number of terms. \subsubsection{Evaluation of derivatives} Conveniently, we can use the same trick to efficiently compute the necessary derivatives of the additive kernel with respect to the base kernels. This can be done by removing the base kernel of interest, $k_j$, from each term of the polynomials: % \begin{align} \frac{\partial k_{add_n}}{\partial k_j} = \frac{\partial e_{n}(k_1,k_2,\dots, k_D)}{\partial k_j} = e_{n-1}(k_1, k_2, \dots,k_{j-1},k_{j+1}, \dots k_D) % & = \frac{1}{n-1} \sum_{k=1}^{n-1} (-1)^{(k-1)} e_{n-k-1}(k_1,\dots,k_{j-1},k_{j+1}, \dots k_D)s_k(k_1,\dots,k_{j-1},k_{j+1}, \dots k_D) \label{eq:additive-derivatives} \end{align} % \Cref{eq:additive-derivatives} gives all terms that $k_j$ is multiplied by in the original polynomial, which are exactly the terms required by the chain rule. These derivatives allow gradient-based optimization of the base kernel parameters with respect to the marginal likelihood. % The final derivative is a sum of multilinear terms, so if only one term depends on the hyperparameter under consideration, we can factorise it out and compute the sum with one degree less. \subsubsection{Computational cost} The computational cost of evaluating the Gram matrix $k(\vX, \vX)$ of a product kernel such as the \seard{} scales as $\mathcal{O}(N^2D)$, while the cost of evaluating the Gram matrix of the additive kernel scales as $\mathcal{O}(N^2DR)$, where $R$ is the maximum degree of interaction allowed (up to $D$). In high dimensions this can be a significant cost, even relative to the $\mathcal{O}(N^3)$ cost of inverting the Gram matrix. However, \cref{tbl:all_orders} shows that sometimes only the first few orders of interaction contribute much variance. In those cases, one may be able to limit the maximum degree of interaction in order to save time, without losing much accuracy. \section{Additive models allow non-local interactions} %The SE-GP model relies on local smoothing to make predictions at novel locations. Commonly-used kernels such as the $\kSE$, $\kRQ$ or Mat\'{e}rn kernels are \emph{local} kernels, depending only on the scaled Euclidean distance between two points, all having the form: \begin{align} k({\bf x, x'}) = g \!\left( \; \sum_{d=1}^D \left( \frac{ x_{d} - x_{d}' }{ \ell_d } \right)^2 \right) \end{align} %\begin{eqnarray} %k_{se}({\bf x, x'}) & = & v_D^2 \exp \left( -\frac{r^2}{2} \right) \\ %k_{\nu}({\bf x, x'}) & = & \frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2\nu}r\right) K_\nu \left(\sqrt{2\nu}r\right) %\end{eqnarray} %Where %\begin{equation} %$r = \sqrt{\sum_{d=1}^D \left( x_{d} - x_{d}' \right)^2 / l_d^2 }$. %\end{equation} for some function $g(\cdot)$. \citet{bengio2006curse} argued that models based on local kernels are particularly susceptible to the \emph{curse of dimensionality}~\citep{bellman1956dynamic}, and are generally unable to extrapolate away from the training data. %They emphasize that the locality of the kernels means that these models cannot capture non-local structure. %They argue that many functions that we care about have such structure. Methods based solely on local kernels sometimes require training examples at exponentially-many combinations of inputs. In contrast, additive kernels can allow extrapolation away from the training data. For example, additive kernels of second order give high covariance between function values at input locations which are similar in any two dimensions. \begin{figure}[h!] \centering \renewcommand{\tabcolsep}{0pt} \begin{tabular}{cccc} 1st-order terms: & 2nd-order terms: & 3rd-order terms: & All interactions: \\ $k_1 + k_2 + k_3$ & $k_1k_2 + k_2k_3 + k_1k_3$ & $k_1k_2k_3$ & \\ & & \seard{} kernel & Additive kernel\\ \includegraphics[trim=1em 0em 1em 3em, clip, width=0.25\textwidth]{\additivefigsdir/3d-kernel/3d_add_kernel_1} & \includegraphics[trim=1em 0em 1em 3em, clip, width=0.25\textwidth]{\additivefigsdir/3d-kernel/3d_add_kernel_2} & \includegraphics[trim=1em 0em 1em 3em, clip, width=0.25\textwidth]{\additivefigsdir/3d-kernel/3d_add_kernel_3} & \includegraphics[trim=1em 0em 1em 3em, clip, width=0.25\textwidth]{\additivefigsdir/3d-kernel/3d_add_kernel_321}\\ $\vx - \vx'$ & $\vx - \vx'$ & $\vx - \vx'$ & $\vx - \vx'$\\[0.5em] \end{tabular} \caption[Isocontours of additive kernels in 3 dimensions] {Isocontours of additive kernels in $D = 3$ dimensions. The $D$th-order kernel only considers nearby points relevant, while lower-order kernels allow the output to depend on distant points, as long as they share one or more input value.} \label{fig:kernels3d} \end{figure} \Cref{fig:kernels3d} provides a geometric comparison between squared-exponential kernels and additive kernels in 3 dimensions. \Cref{sec:additivity-multiple-dimensions} contains an example of how additive kernels extrapolate differently than local kernels. %This is one possible reason why low-order interactions might improve predictive accuracy. \section{Dropout in Gaussian processes} \label{sec:dropout-gps} \emph{Dropout} is a recently-introduced method for regularizing neural networks~\citep{hinton2012improving, srivastava2013improving}. Training with dropout entails independently setting to zero (``dropping'') some proportion $p$ of features or inputs, in order to improve the robustness of the resulting network, by reducing co-dependence between neurons. To maintain similar overall activation levels, the remaining weights are divided by $p$. % at test time. Alternatively, feature activations are divided by $p$ during training. Predictions are made by approximately averaging over all possible ways of dropping out neurons. \citet{baldi2013understanding} and \citet{wang2013fast} analyzed dropout in terms of the effective prior induced by this procedure in several models, such as linear and logistic regression. In this section, we perform a similar analysis for \gp{}s, examining the priors on functions that result from performing dropout in the one-hidden-layer neural network implicitly defined by a \gp{}. Recall from \cref{sec:relating} that some \gp{}s can be derived as infinitely-wide one-hidden-layer neural networks, with fixed activation functions $\feat(\vx)$ and independent random weights $\vw$ having zero mean and finite variance $\sigma^2_{\vw}$: % \begin{align} f(\vx) %= \frac{1}{K}{\mathbf \vnetweights}\tra \hPhi(\vx) = \frac{1}{K} \sum_{i=1}^K \netweights_i \hphi_i(\vx) \implies f \distas{K \to \infty} \GPt{0}{\sigma^2_{\vw} \feat(\vx)\tra\feat(\vx')} . \label{eq:one-layer-gp-two} \end{align} % %Mercer's theorem implies that we can write any \gp{} prior equivalently in this way, %where $k(\vx, \vx') = \sigma^2_{\vw} \feat(\vx)\tra\feat(\vx'$). %Having expressed a \gp{} as a neural network, we can examine the prior obtained by performing dropout in this network. \subsection{Dropout on infinitely-wide hidden layers has no effect} First, we examine the prior obtained by dropping features from $\hPhi(\vx)$ by setting weights in $\vnetweights$ to zero independently with probability $p$. For simplicity, we assume that $\expectargs{}{\vnetweights} = \vzero$. If the weights $w_i$ initially have finite variance $\sigma^2_{\netweights}$ before dropout, then the weights after dropout (denoted by $r_i w_i$, where $r_i$ is a Bernoulli random variable) will have variance: % \begin{align} r_i \simiid \textnormal{Ber}(p) \qquad \varianceargs{}{r_i \netweights_i} = p \sigma_{\netweights}^2 \;. \end{align} % Because \cref{eq:one-layer-gp-two} is a result of the central limit theorem, it does not depend on the exact form of the distribution on $\vnetweights$, but only on its mean and variance. Thus the central limit theorem still applies. Performing dropout on the features of an infinitely-wide \MLP{} does not change the resulting model at all, except to rescale the output variance. %If we were to rescale the weights by a factor other than $p^{-1/2}$, we would only rescale the output variance of the model, leaving all other aspects identical. %Is there a better way to drop out features that would lead to robustness? Indeed, dividing all weights by $\sqrt p$ restores the initial variance: % \begin{align} %\expectargs{}{ \frac{1}{p} r_i \alpha_i } = \frac{p}{p}\mu = \mu, \quad \varianceargs{}{ \frac{1}{p^\frac{1}{2}} r_i \netweights_i} = \frac{p}{p} \sigma_{\netweights}^2 = \sigma_{\netweights}^2 \end{align} % in which case dropout on the hidden units has no effect at all. Intuitively, this is because no individual feature can have more than an infinitesimal contribution to the network output. This result does not hold in neural networks having a finite number of hidden features with Gaussian-distributed weights, another model class that also gives rise to \gp{}s. \subsection{Dropout on inputs gives additive covariance} One can also perform dropout on the $D$ inputs to the \gp{}. For simplicity, consider a stationary product kernel ${k(\vx, \vx') = \prod_{d=1}^D k_d(x_d, x_d')}$ which has been normalized such that $k(\vx, \vx) = 1$, and a dropout probability of $p = \nicefrac{1}{2}$. In this case, the generative model can be written as: % \begin{align} \vr = [r_1, r_2, \dots, r_D], \quad \textnormal{each} \;\; r_i \simiid \textnormal{Ber} \left( \frac{1}{2} \right), \quad f(\vx) | \vr \sim \GP \left( 0, \prod_{d=1}^D k_d(x_d, x_d')^{r_d} \right) \end{align} % This is a mixture of $2^D$ \gp{}s, each depending on a different subset of the inputs: % \begin{align} p \left( f(\vx) \right) = \sum_{\vr} p \left( f(\vx) | \vr \right) p( \vr) = \frac{1}{2^D} \sum_{\vr \in \{0,1\}^D} \GP \left(f(\vx) \,\Big|\, 0, \prod_{d=1}^D k_d(x_d, x_d')^{r_d} \right) \label{eq:dropout-mixture} \end{align} We present two results which might give intuition about this model. First, if the kernel on each dimension has the form ${k_d(x_d, x_d') = g \left( \frac{x_d - x_d'}{\ell_d} \right)}$, as does the \kSE{} kernel, then any input dimension can be dropped out by setting its lengthscale $\ell_d$ to $\infty$. In this case, performing dropout on the inputs of a \gp{} corresponds to putting independent spike-and-slab priors on the lengthscales, with each dimension's distribution independently having ``spikes'' at $\ell_d = \infty$ with probability mass of $\nicefrac{1}{2}$. Another way to understand the resulting prior is to note that the dropout mixture (\cref{eq:dropout-mixture}) has the same covariance as an additive \gp{}, scaled by a factor of $2^{-D}$: \begin{align} %f(\vx) \sim \textnormal{\gp} \left(0, \frac{1}{2^{-2D}} \sum_{\vr \in \{0,1\}^D} \prod_{d=1}^D k_d(x_d, x_d')^{r_d} \right) %f(\vx) \sim \textnormal{\gp} \left(0, \frac{1}{2^{D}} \sum_{\vr \in \{0,1\}^D} \prod_{d=1}^D k_d(x_d, x_d')^{r_d} \right) \cov\left( \colvec{f(\vx)}{f(\vx')} \right) = \frac{1}{2^{D}} \sum_{\vr \in \{0,1\}^D} \prod_{d=1}^D k_d(x_d, x_d')^{r_d} \label{eq:dropout-mixture-covariance} \end{align} %TODO: Make this equation more precise % For dropout rates $p \neq \nicefrac{1}{2}$, the $d$th order terms will be weighted by $p^{(D - d)}(1-p)^d$. Therefore, performing dropout on the inputs of a \gp{} gives a distribution with the same first two moments as an additive \gp{}. This suggests an interpretation of additive \gp{}s as an approximation to a mixture of models where each model only depends on a subset of the input variables. \section{Related work} Since additive models are a relatively natural and easy-to-analyze model class, the literature on similar model classes is extensive. This section attempts to provide a broad overview. \subsubsection{Previous examples of additive \sgp{}s} The additive models considered in this chapter are axis-aligned, but transforming the input space allows one to recover non-axis aligned additivity. This model was explored by \citet{gilboa2013scaling}, who developed a linearly-transformed first-order additive \gp{} model, called projection-pursuit \gp{} regression. They showed that inference in this model was possible in $\mathcal{O}(N)$ time. \citet{durrande2011additive} also examined properties of additive \gp{}s, and proposed a layer-wise optimization strategy for kernel hyperparameters in these models. \citet{plate1999accuracy} constructed an additive \gp{} having only first-order and $D$th-order terms, motivated by the desire to trade off the interpretability of first-order models with the flexibility of full-order models. However, \cref{tbl:all_orders} shows that sometimes the intermediate degrees of interaction contribute most of the variance. \citet{kaufman2010bayesian} used a closely related procedure called Gaussian process \ANOVA{} to perform a Bayesian analysis of meteorological data using 2nd and 3rd-order interactions. They introduced a weighting scheme to ensure that each order's total contribution sums to zero. It is not clear if this weighting scheme permits the use of the Newton-Girard formulas to speed computation of the Gram matrix. %[TODO: read and mention \citep{friedman1981projection} from Hannes] \subsubsection{Hierarchical kernel learning} A similar model class was recently explored by \citet{Bach_HKL} called hierarchical kernel learning (\HKL{}). \HKL{} uses a regularized optimization framework to build a weighted sum of an exponential number of kernels that can be computed in polynomial time. % \begin{figure} \centering \begin{tabular}{c|c} Hierarchical kernel learning & All-orders additive \gp{} \\ \includegraphics[height=0.35\textwidth]{\additivefigsdir/compare_models/hkl.pdf} & \includegraphics[height=0.35\textwidth]{\additivefigsdir/compare_models/additive.pdf} \\ \hline \\ \gp{} with product kernel & First-order additive \gp{} \\ \includegraphics[height=0.35\textwidth]{\additivefigsdir/compare_models/ard.pdf} & \includegraphics[height=0.35\textwidth]{\additivefigsdir/compare_models/gam.pdf} \\ \end{tabular} \caption[A comparison of different additive model classes] {A comparison of different additive model classes of 4-dimensional functions. Circles represent different interaction terms, ranging from first-order to fourth-order interactions. Shaded boxes represent the relative weightings of different terms. \emph{Top left: }\HKL{} can select a hull of interaction terms, but must use a pre-determined weighting over those terms. \emph{Top right:} the additive \gp{} model can weight each order of interaction separately, but weights all terms equally within each order. \emph{Bottom row:} \gp{}s with product kernels (such as the \seard{} kernel) and first-order additive \gp{} models are special cases of the all-orders additive \gp{}, with all variance assigned to a single order of interaction.} \label{hulls-figure} \end{figure} % This method chooses among a \emph{hull} of kernels, defined as a set of terms such that if $\prod_{j \in J} k_j(\vx, \vx')$ is included in the set, then so are all products of strict subsets of the same elements: $\prod_{j \in J / i} k_j(\vx, \vx')$, for all $i \in J$. \HKL{} does not estimate a separate weighting parameter for each order. %\HKL{} computes the sum over all orders in $\mathcal{O}(D)$ time by the formula: % %\begin{align} %k_a(\vx, \vx') = v^2 \prod_{d=1}^D \left(1 + \alpha k_d( x_d, x_d') \right) %\label{eqn:uniform} %\end{align} % %which forces the weight of all $n$th order terms to be weighted by $\alpha^n$. \Cref{hulls-figure} contrasts the \HKL{} model class with the additive \gp{} model. Neither model class encompasses the other. The main difficulty with this approach is that its parameters are hard to set other than by cross-validation. \subsubsection{Support vector machines} \citet{vapnik1998statistical} introduced the support vector \ANOVA{} decomposition, which has the same form as our additive kernel. They recommend approximating the sum over all interactions with only one set of interactions ``of appropriate order'', presumably because of the difficulty of setting the parameters of an \SVM{}. This is an example of a model choice which can be automated in the \gp{} framework. \citet{stitson1999support} performed experiments which favourably compared the predictive accuracy of the support vector \ANOVA{} decomposition against polynomial and spline kernels. They too allowed only one order to be active, and set parameters by cross-validation. % % The order of the kernel, kernel parameters, the allowable regression error $\epsilon$, and the cost hyperparameter $c$ were all set by a lengthy cross-validation process. %\subsection{Smoothing Splines ANOVA} \subsubsection{Other related models} A closely related procedure from \citet{wahba1990spline} is smoothing-splines \ANOVA{} (\SSANOVA{}). An \SSANOVA{} model is a weighted sum of splines along each dimension, splines over all pairs of dimensions, all triplets, etc, with each individual interaction term having a separate weighting parameter. Because the number of terms to consider grows exponentially in the order, only terms of first and second order are usually considered in practice. %Learning in SS-ANOVA is usually done via penalized-maximum likelihood with a fixed sparsity hyperparameter. %\citet{shawe2004kernel} define ANOVA kernels thusly: "The ANOVA kernel of degree d is like the all-subsets kernel except that it is restricted to subsets of the given cardinality $d$." This more general model class, in which each interaction term is estimated separately, is known in the physical sciences as high dimensional model representation (\HDMR{}). \citet{rabitz1999general} review some properties and applications of this model class. The main benefits of the model setup and parameterization proposed in this chapter are the ability to include all $D$ orders of interaction with differing weights, and the ability to learn kernel parameters individually per input dimension, allowing automatic relevance determination to operate. \section{Regression and classification experiments} \label{sec:additive-experiments} \subsubsection{Choosing the base kernel} %A $D$-dimensional \kSE-\ARD{} kernel has $D$ lengthscale parameters and one output variance parameter. %A first-order additive $\kSE$ model has $D$ lengthscale parameters and $D$ output variance parameters. An additive \gp{} using a separate $\kSE$ kernel on each input dimension will have ${3 \kerntimes D}$ effective parameters. Because each additional parameter increases the tendency to overfit, %we recommend allowing only one kernel parameter per input dimension. % %Since there always exists a parameterization of the additive function corresponding to the product kernel, we initialize the hyperparameter search for the additive kernel by first learning parameters for the equivalent product kernel. This ensures that we start our search in a reasonable area of hyperparameter space. % in our experiments we fixed each one-dimensional kernel's output variance to be 1, and only estimated the lengthscale of each kernel. \subsubsection{Methods} We compared six different methods. In the results tables below, \gp{} Additive refers to a \gp{} using the additive kernel with squared-exp base kernels. For speed, we limited the maximum order of interaction to 10. \gp{}-1st denotes an additive \gp{} model with only first-order interactions: a sum of one-dimensional kernels. \gp{} Squared-exp is a \gp{} using an \seard{} kernel. \HKL{} was run using the all-subsets kernel, which corresponds to the same set of interaction terms considered by \gp{} Additive. For all \gp{} models, we fit kernel parameters by the standard method of maximizing training-set marginal likelihood, using \LBFGS{}~\citep{nocedal1980updating} for 500 iterations, allowing five random restarts. In addition to learning kernel parameters, we fit a constant mean function to the data. In the classification experiments, approximate \gp{} inference was performed using expectation propagation~\citep{minka2001expectation}. The regression experiments also compared against the structure search method from \cref{ch:grammar}, run up to depth 10, using only the \SE{} and \RQ{} base kernels. \subsection{Datasets} \label{sec:additive-datasets} We compared the above methods on regression and classification datasets from the \UCI{} repository~\citep{UCI}. Their size and dimension are given in \cref{table:regression-dataset-stats,table:classification-dataset-stats}: \begin{table}[h] \caption{Regression dataset statistics} \label{tbl:Regression Dataset Statistics} \begin{center} \begin{tabular}{l | lllll} Method & bach & concrete & pumadyn & servo & housing \\ \hline Dimension & 8 & 8 & 8 & 4 & 13 \\ Number of datapoints & 200 & 500 & 512 & 167 & 506 \end{tabular} \end{center} \label{table:regression-dataset-stats} \end{table} % \begin{table}[h] \caption{Classification dataset statistics} \label{tbl:Classification Dataset Statistics} \begin{center} \begin{tabular}{l | llllll} Method & breast & pima & sonar & ionosphere & liver & heart\\ \hline Dimension & 9 & 8 & 60 & 32 & 6 & 13 \\ Number of datapoints & 449 & 768 & 208 & 351 & 345 & 297 \end{tabular} \end{center} \label{table:classification-dataset-stats} \end{table} \subsubsection{Bach synthetic dataset} In addition to standard UCI repository datasets, we generated a synthetic dataset using the same recipe as \citet{Bach_HKL}. This dataset was presumably designed to demonstrate the advantages of \HKL{} over a \gp{} using an \seard{} kernel. %From a covariance matrix drawn from a Wishart distribution with 1024 degrees of freedom, we select 8 variables. It is generated by passing correlated Gaussian-distributed inputs $x_1, x_2, \dots, x_8$ through a quadratic function % \begin{align} f(\vx) = \sum_{i=1}^4 \sum_{j=i+1}^4 x_i x_j + \epsilon \qquad \epsilon \sim \Nt{0}{\sigma_\epsilon}. \end{align} % %which sums all 2-way products of the first 4 variables, and adds Gaussian noise $\epsilon$. This dataset will presumably be well-modeled by an additive kernel which includes all two-way interactions over the first 4 variables, but does not depend on the extra 4 correlated nuisance inputs or the higher-order interactions.%, as well as the higher-order interactions. %TODO: Move description to grammar experiments? %If the dataset is large enough, \HKL{} can construct a hull around only the 16 cross-terms optimal for predicting the output. %\gp{}-\SE{}, in contrast, can learn to ignore the noisy copy variables, but cannot learn to ignore the higher-order interactions between dimensions. %However, a \gp{} with an additive kernel can learn both to ignore irrelevant variables, and to ignore missing orders of interaction. %With enough data, the marginal likelihood will favour hyperparameters specifying such a model. %In this example, the additive \gp{} is able to recover the relevant structure. \subsection{Results} \Cref{tbl:Regression Mean Squared Error,tbl:Regression Negative Log Likelihood,tbl:Classification Percent Error,tbl:Classification Negative Log Likelihood} show mean performance across 10 train-test splits. Because \HKL{} does not specify a noise model, it was not included in the likelihood comparisons. % --- Automatically generated by resultsToLatex2.m --- % Exported at 19-Jan-2012 10:55:19 \begin{table} \caption[Comparison of predictive error on regression problems] {Regression mean squared error} \label{tbl:Regression Mean Squared Error} \begin{center} \begin{tabular}{l | r r r r r} Method & \rotatebox{0}{ bach } & \rotatebox{0}{ concrete } & \rotatebox{0}{ pumadyn-8nh } & \rotatebox{0}{ servo } & \rotatebox{0}{ housing } \\ \hline Linear Regression & $1.031$ & $0.404$ & $0.641$ & $0.523$ & $0.289$ \\ \gp{}-1st & $1.259$ & $0.149$ & $0.598$ & $0.281$ & $0.161$ \\ \HKL{} & $\mathbf{0.199}$ & $0.147$ & $0.346$ & $0.199$ & $0.151$ \\ \gp{} Squared-exp & $\mathbf{0.045}$ & $0.157$ & $0.317$ & $0.126$ & $\mathbf{0.092}$ \\ \gp{} Additive & $\mathbf{0.045}$ & $\mathbf{0.089}$ & $\mathbf{0.316}$ & $\mathbf{0.110}$ & $0.102$ \\ Structure Search & $\mathbf{0.044}$ & $\mathbf{0.087}$ & $\mathbf{0.315}$ & $\mathbf{0.102}$ & $\mathbf{0.082}$ \end{tabular} \end{center} %\end{table} % End automatically generated LaTeX % % --- Automatically generated by resultsToLatex2.m --- % Exported at 19-Jan-2012 10:55:19 %\begin{table}[h!] \caption[Comparison of predictive likelihood on regression problems] {Regression negative log-likelihood} \label{tbl:Regression Negative Log Likelihood} \begin{center} \begin{tabular}{l | r r r r r} Method & \rotatebox{0}{ bach } & \rotatebox{0}{ concrete } & \rotatebox{0}{ pumadyn-8nh } & \rotatebox{0}{ servo } & \rotatebox{0}{ housing } \\ \hline Linear Regression & $2.430$ & $1.403$ & $1.881$ & $1.678$ & $1.052$ \\ \gp{}-1st & $1.708$ & $0.467$ & $1.195$ & $0.800$ & $0.457$ \\ GP Squared-exp & $\mathbf{-0.131}$ & $0.398$ & $0.843$ & $0.429$ & $0.207$ \\ GP Additive & $\mathbf{-0.131}$ & $\mathbf{0.114}$ & $\mathbf{0.841}$ & $\mathbf{0.309}$ & $0.194$ \\ Structure Search & $\mathbf{-0.141}$ & $\mathbf{0.065}$ & $\mathbf{0.840}$ & $\mathbf{0.265}$ & $\mathbf{0.059}$ \end{tabular} \end{center} \end{table} % End automatically generated LaTeX % % --- Automatically generated by resultsToLatex2.m --- % Exported at 03-Jun-2011 00:23:25 \begin{table} \caption[Comparison of predictive error on classification problems] {Classification percent error} \label{tbl:Classification Percent Error} \begin{center} \begin{tabular}{l | r r r r r r} Method & \rotatebox{0}{ breast } & \rotatebox{0}{ pima } & \rotatebox{0}{ sonar } & \rotatebox{0}{ ionosphere } & \rotatebox{0}{ liver } & \rotatebox{0}{ heart } \\ \hline Logistic Regression & $7.611$ & $24.392$ & $26.786$ & $16.810$ & $45.060$ & $\mathbf{16.082}$ \\ \gp{}-1st & $\mathbf{5.189}$ & $\mathbf{22.419}$ & $\mathbf{15.786}$ & $\mathbf{8.524}$ & $\mathbf{29.842}$ & $\mathbf{16.839}$ \\ \HKL{} & $\mathbf{5.377}$ & $24.261$ & $\mathbf{21.000}$ & $9.119$ & $\mathbf{27.270}$ & $\mathbf{18.975}$ \\ \gp{} Squared-exp & $\mathbf{4.734}$ & $\mathbf{23.722}$ & $\mathbf{16.357}$ & $\mathbf{6.833}$ & $\mathbf{31.237}$ & $\mathbf{20.642}$ \\ \gp{} Additive & $\mathbf{5.566}$ & $\mathbf{23.076}$ & $\mathbf{15.714}$ & $\mathbf{7.976}$ & $\mathbf{30.060}$ & $\mathbf{18.496}$ \\ \end{tabular} \end{center} %\end{table} % End automatically generated LaTeX % % --- Automatically generated by resultsToLatex2.m --- % Exported at 03-Jun-2011 00:23:28 %\begin{table}[h!] \caption[Comparison of predictive likelihood on classification problems] {Classification negative log-likelihood} \label{tbl:Classification Negative Log Likelihood} \begin{center} \begin{tabular}{l | r r r r r r} Method & \rotatebox{0}{ breast } & \rotatebox{0}{ pima } & \rotatebox{0}{ sonar } & \rotatebox{0}{ ionosphere } & \rotatebox{0}{ liver } & \rotatebox{0}{ heart } \\ \hline Logistic Regression & $0.247$ & $0.560$ & $4.609$ & $0.878$ & $0.864$ & $0.575$ \\ \gp{}-1st & $\mathbf{0.163}$ & $\mathbf{0.461}$ & $\mathbf{0.377}$ & $\mathbf{0.312}$ & $\mathbf{0.569}$ & $\mathbf{0.393}$ \\ \gp{} Squared-exp & $\mathbf{0.146}$ & $0.478$ & $\mathbf{0.425}$ & $\mathbf{0.236}$ & $\mathbf{0.601}$ & $0.480$ \\ \gp{} Additive & $\mathbf{0.150}$ & $\mathbf{0.466}$ & $\mathbf{0.409}$ & $\mathbf{0.295}$ & $\mathbf{0.588}$ & $\mathbf{0.415}$ \\ \end{tabular} \end{center} \end{table} % End automatically generated LaTeX On each dataset, the best performance is in boldface, along with all other performances not significantly different under a paired $t$-test. %The additive model never performs significantly worse than any other model, and sometimes performs significantly better than all other models. The additive and structure search methods usually outperformed the other methods, especially on regression problems. %The difference between all methods is larger in the case of regression experiments. The structure search outperforms the additive \gp{} at the cost of a slower search over kernels. Structure search was on the order of 10 times slower than the additive \gp{}, which was on the order of 10 times slower than \gp{}-\kSE{}. The additive \gp{} performed best on datasets well-explained by low orders of interaction, and approximately as well as the \SE{}-\gp{} model on datasets which were well explained by high orders of interaction (see \cref{tbl:all_orders}). Because the additive \gp{} is a superset of both the \gp{}-1st model and the \gp{}-\kSE{} model, instances where the additive \gp{} performs slightly worse are presumably due to over-fitting, or due to the hyperparameter optimization becoming stuck in a local maximum. % Absence of over-fitting may explain the relatively strong performance of GP-GAM on classification tasks. Performance of all \gp{} models could be expected to benefit from approximately integrating over kernel parameters. The performance of \HKL{} is consistent with the results in \citet{Bach_HKL}, performing competitively but slightly worse than \SE{}-\gp{}. \subsubsection{Source code} %Additive Gaussian processes are particularly appealing in practice because their use requires only the specification of the base kernel; all other aspects of \gp{} inference remain the same. %Note that we are also free to choose a different covariance function along each dimension. All of the experiments in this chapter were performed using the standard \GPML{} toolbox, available at \url{http://wwww.gaussianprocess.org/gpml/code}. The additive kernel described in this chapter is included in \GPML{} as of version 3.2. Code to perform all experiments in this chapter is available at \url{http://www.github.com/duvenaud/additive-gps}. \section{Conclusions} This chapter presented a tractable \gp{} model consisting of a sum of exponentially-many functions, each depending on a different subset of the inputs. Our experiments indicate that, to varying degrees, such additive structure is useful for modeling real datasets. When it is present, modeling this structure allows our model to perform better than standard \gp{} models. In the case where no such structure exists, the higher-order interaction terms present in the kernel can recover arbitrarily flexible models. The additive \gp{} also affords some degree of interpretability: the variance parameters on each order of interaction indicate which sorts of structure are present the data, although they do not indicate which particular interactions explain the dataset. The model class considered in this chapter is a subset of that explored by the structure search presented in \cref{ch:grammar}. Thus additive \gp{}s can be considered a quick-and-dirty structure search, being strictly more limited in the types of structure that it can discover, but much faster and simpler to implement. Closely related model classes have previously been explored, most notably smoothing-splines \ANOVA{} and the support vector \ANOVA{} decomposition. However, these models can be difficult to apply in practice because their kernel parameters, regularization penalties, and the relevant orders of interaction must be set by hand or by cross-validation. This chapter illustrates that the \gp{} framework allows these model choices to be performed automatically. \outbpdocument{ \bibliographystyle{plainnat} \bibliography{references.bib} } ================================================ FILE: appendix.tex ================================================ \input{common/header.tex} \inbpdocument \chapter{Gaussian Conditionals} \label{ch:appendix-gaussians} %\section{Formula for Gaussian Conditionals} A standard result shows how to condition on a subset of dimensions $\vy_B$ of a vector $\vy$ having a multivariate Gaussian distribution. If % \begin{align} \vy = \colvec{\vy_A}{\vy_B} \sim \Nt{\colvec{\vmu_A}{\vmu_B}}{\left[ \begin{array}{cc}\vSigma_{AA} & \vSigma_{AB} \\ \vSigma_{BA} & \vSigma_{BB} \end{array} \right]} %\left[ \begin{array} \vx_A \sim \Nt{\vmu_A}{ %| \vx_B \sim \Nt{\vmu_A + \vSigma_{AB} \vSigma_{BB}\inv \left( \vx_B - \vmu_B \right) } %{\vSigma_{AA} - \vSigma_{AB} \vSigma_{BB}\inv \vSigma_{BA} } %\label{eq:gauss_conditional} \end{align} % then % \begin{align} \vy_A | \vy_B \sim \mathcal{N} \big(\vmu_A + \vSigma_{AB} \vSigma_{BB}\inv \left( \vx_B - \vmu_B \right), \vSigma_{AA} - \vSigma_{AB} \vSigma_{BB}\inv \vSigma_{BA} \big). \label{eq:gauss_conditional} \end{align} This result can be used in the context of Gaussian process regression, where $\vy_B = [f(\vx_1), f(\vx_2), \dots, f(\vx_N)]$ represents a set of function values observed at some subset of locations $[\vx_1, \vx_2, \dots, \vx_N]$, while $\vy_A = [f(\vx_1\star), f(\vx_2\star), \ldots, f(\vx_N\star)]$ represents test points whose predictive distribution we'd like to know. In this case, the necessary covariance matrices are given by: % \begin{align} \vSigma_{AA} & = k(\vX^\star, \vX^\star) \\ \vSigma_{AB} & = k(\vX^\star, \vX) \\ \vSigma_{BA} & = k(\vX, \vX^\star) \\ \vSigma_{BB} & = k(\vX, \vX) \end{align} and similarly for the mean vectors. \chapter{Kernel Definitions} \label{sec:kernel-definitions} %\newcommand{\scalefactor}{\sigma_f^2} \newcommand{\scalefactor}{} This appendix gives the formulas for all one-dimensional base kernels used in the thesis. Each of these formulas is multiplied by a scale factor $\sigma_f^2$, which we omit for clarity. % \begin{align} \kC(\inputVar, \inputVar') & = \scalefactor 1 \\ \kSE(\inputVar, \inputVar') & = \scalefactor \exp\left(-\frac{(\inputVar - \inputVar')^2}{2\ell^2}\right) \label{eq:appendix-se}\\ \kPer(x, x') & = \exp\left(-\frac{2}{\ell^2} \sin^2 \left( \pi \frac{\inputVar - \inputVar'}{p} \right)\right) \label{eq:appendix-periodic}\\ \kLin(\inputVar, \inputVar') & = \scalefactor (\inputVar - c)(\inputVar' - c) \label{eq:appendix-lin} \\ \kRQ(x, x') & = \scalefactor \left( 1 + \frac{(\inputVar - \inputVar')^2}{2\alpha\ell^2}\right)^{-\alpha} \label{eq:appendix-rq}\\ %\kPer(\inputVar, \inputVar') & = \sigma_f^2 \frac{\exp\left(\frac{1}{\ell^2}\cos 2 \pi \frac{(\inputVar - \inputVar')}{p}\right) - I_0\left(\frac{1}{\ell^2}\right)}{\exp\left(\frac{1}{\ell^2}\right) - I_0\left(\frac{1}{\ell^2}\right)} \label{eq:generalized-periodic} \\ %\kPer(\inputVar, \inputVar') & = \scalefactor \exp\left(\frac{1}{\ell^2}\cos 2 \pi \frac{(\inputVar - \inputVar')}{p}\right) - I_0\left(\frac{1}{\ell^2}\right) \\ \cos(x, x') & = \scalefactor \cos\left(\frac{2 \pi (x - x')}{p}\right) \label{eq:appendix-cos} \\ \kWN(\inputVar, \inputVar') & = \scalefactor \delta(\inputVar - \inputVar') \label{eq:appendix-wn}\\ \kCP(\kernel_1, \kernel_2)(x, x') & = \scalefactor \sigma(x) k_1(x,x')\sigma(x') + (1-\sigma(x)) k_2(x,x')(1-\sigma(x')) \\ \boldsymbol\sigma(\inputVar, \inputVar') & = \scalefactor \sigma(x)\sigma(x') \\ \boldsymbol{\bar\sigma}(\inputVar, \inputVar') & = \scalefactor (1-\sigma(x))(1-\sigma(x')) \end{align} % where $\delta_{\inputVar, \inputVar'}$ is the Kronecker delta function, $\{c,\ell,p,\alpha\}$ represent kernel parameters, and ${\sigma(x) = \nicefrac{1}{1 + \exp(-x)}}$. \Cref{eq:appendix-se,eq:appendix-periodic,eq:appendix-lin} are plotted in \cref{fig:basic_kernels}, and \cref{eq:appendix-wn,eq:appendix-rq,eq:appendix-cos} are plotted in \cref{fig:basic_kernels_two}. Draws from \gp{} priors with changepoint kernels are shown in \cref{fig:changepoint_examples}. \subsubsection{The zero-mean periodic kernel} %\citet{lloyd-periodic} James Lloyd (personal communication) showed that the standard periodic kernel due to \citet{mackay1998introduction} can be decomposed into a sum of a periodic and a constant component. He derived the equivalent periodic kernel without any constant component: %, shown in \cref{eq:generalized-periodic}. % \begin{align} \kPerGen(\inputVar, \inputVar') & = \sigma_f^2 \frac{\exp\left(\frac{1}{\ell^2}\cos 2 \pi \frac{(\inputVar - \inputVar')}{p}\right) - I_0\left(\frac{1}{\ell^2}\right)}{\exp\left(\frac{1}{\ell^2}\right) - I_0\left(\frac{1}{\ell^2}\right)} \label{eq:generalized-periodic} \end{align} % where $I_0$ is the modified Bessel function of the first kind of order zero. He further showed that its limit as the lengthscale grows is the cosine kernel: % \begin{equation} \lim_{\ell \to \infty} \kPerGen(x, x') = \cos\left(\frac{2 \pi (x - x')}{p}\right). \end{equation} Separating out the constant component allows us to express negative prior covariance, as well as increasing the interpretability of the resulting models. This covariance function is included in the \GPML{} software package~\citep{GPML}, and its source can be viewed at \url{gaussianprocess.org/gpml/code/matlab/cov/covPeriodicNoDC.m}. \chapter{Search Operators} \label{ch:appendix-search} \label{sec:search-operators} %\subsection{Overview} The model construction phase of \procedurename{} starts with the noise kernel, $\kWN$. New kernel expressions are generated by applying search operators to the current kernel, which replace some part of the existing kernel expression with a new kernel expression. %When new base kernels are proposed by the search operators, their parameters are randomly initialised with several restarts. %Parameters are then optimized by conjugate gradients to maximise the likelihood of the data conditioned on the kernel parameters. %The kernels are then scored by the Bayesian information criterion and the top scoring kernel is selected as the new kernel. %The search then proceeds by applying the search operators to the new kernel \ie this is a greedy search algorithm. %In all experiments, 10 random restarts were used for parameter initialisation and the search was run to a depth of 10. %\subsection{Search operators} The search used in the multidimensional regression experiments in \cref{sec:synthetic,sec:additive-experiments} used only the following search operators: % \begin{eqnarray} \mathcal{S} &\to& \mathcal{S} + \mathcal{B} \\ \mathcal{S} &\to& \mathcal{S} \times \mathcal{B} \label{eq:search-multiply}\\ \mathcal{B} &\to& \mathcal{B'} \end{eqnarray} % where $\mathcal{S}$ represents any kernel sub-expression and $\mathcal{B}$ is any base kernel within a kernel expression. These search operators represent addition, multiplication and replacement. When the multiplication operator is applied to a sub-expression which includes a sum of sub-expressions, parentheses () are introduced. For instance, if rule \eqref{eq:search-multiply} is applied to the sub-expression $k_1 + k_2$, the resulting expression is $(k_1 \kernplus k_2) \kerntimes \mathcal{B}$. Afterwards, we added several more search operators in order to speed up the search. This expanded set of operators was used in the experiments in \cref{sec:time_series,sec:Predictive accuracy on time series,ch:description}. These new operators do not change the set of possible models. To accommodate changepoints and changewindows, we introduced the following additional operators to our search: % \begin{eqnarray} \mathcal{S} &\to& \kCP(\mathcal{S},\mathcal{S}) \\ \mathcal{S} &\to& \kCW(\mathcal{S},\mathcal{S}) \\ \mathcal{S} &\to& \kCW(\mathcal{S},\kC) \\ \mathcal{S} &\to& \kCW(\kC,\mathcal{S}) \end{eqnarray} % where $\kC$ is the constant kernel. The last two operators result in a kernel only applying outside, or within, a certain region. To allow the search to simplify existing expressions, we introduced the following operators: % \begin{eqnarray} \mathcal{S} &\to& \mathcal{B}\\ \mathcal{S} + \mathcal{S'} &\to& \mathcal{S}\\ \mathcal{S} \times \mathcal{S'} &\to& \mathcal{S} \end{eqnarray} % where $\mathcal{S'}$ represents any other kernel expression. %Their introduction is currently not rigorously justified. We also introduced the operator % \begin{eqnarray} \mathcal{S} &\to& \mathcal{S} \times (\mathcal{B} + \kC) \end{eqnarray} % Which allows a new base kernel to be added along with the constant kernel, for cases when multiplying by a base kernel by itself would be overly restrictive. \chapter{Example Automatically Generated Report} \label{ch:example-solar} The following pages of this appendix contain an automatically-generated report, run on a dataset measuring annual solar irradiation data from 1610 to 2011. This dataset was previously analyzed by \citet{lean1995reconstruction}. The structure search was run using the \procedurename-interpretable variant, with base kernels $\kSE$, $\kLin$, $\kC$, $\kPer$, $\vsigma$, and $\kWN$. Other example reports can be found at \url{http://www.mlg.eng.cam.ac.uk/Lloyd/abcdoutput/}, including analyses of wheat prices, temperature records, call centre volumes, radio interference, gas production, unemployment, number of births, and wages over time. \newcommand{\solarreportpage}[1]{\includegraphics[width=\columnwidth]{figures/solarpages/02-solar-seperate-pages-#1}} \clearpage %\solarreportpage{1} \solarreportpage{2} \\ \vspace{1cm} \\ \solarreportpage{3} \\ \vspace{1cm} \\ \solarreportpage{4} \\ \vspace{1cm} \\ \solarreportpage{5} \\ \vspace{1cm} \\ \solarreportpage{6} \\ \vspace{1cm} \\ \solarreportpage{7} \\ \vspace{1cm} \\ \solarreportpage{8} \\ \vspace{1cm} \\ \solarreportpage{9} \\ \vspace{1cm} \\ \solarreportpage{10} \\ \vspace{1cm} \\ \solarreportpage{11} \\ \vspace{1cm} \\ \solarreportpage{12} \\ \vspace{1cm} \\ \solarreportpage{13} %\solarreportpage{14} %\solarreportpage{15} %\solarreportpage{16} %\solarreportpage{17} %\solarreportpage{18} %\solarreportpage{19} %\solarreportpage{20} %\solarreportpage{21} %\solarreportpage{22} \chapter{Inference in the Warped Mixture Model} \label{ch:warped-appendix} \subsubsection{Detailed definition of model} The \iwmm{} assumes that the latent density is an infinite mixture of Gaussians: % \begin{align} p(\vx ) = \sum_{c=1}^{\infty} \lambda_c \, {\cal N}(\vx|\bm{\mu}_{c},\vR_{c}\inv) \end{align} % where $\lambda_{c}$, $\bm{\mu}_{c}$ and $\vR_{c}$ is the mixture weight, mean, and precision matrix of the $c$\textsuperscript{th} mixture component. We place a conjugate Gaussian-Wishart priors on the Gaussian parameters $\{\bm{\mu}_{c},\vR_{c}\}$: % \begin{align} p(\bm{\mu}_{c},\vR_{c}) = {\cal N}(\bm{\mu}_{c}|\vu,(r\vR_{c})\inv) {\cal W}(\vR_{c}|\vS\inv,\nu), \label{eq:normal-wishar-prior} \end{align} % where $\vu$ is the mean of $\vmu_c$, $r$ is the relative precision of $\vmu_c$, $\vS\inv$ is the scale matrix for $\vR_c$, and $\nu$ is the number of degrees of freedom for $\vR_c$. The Wishart distribution is defined as: % \begin{align} {\cal W}(\vR|\vS\inv,\nu)=\frac{1}{G}|\vR|^{\frac{\nu-Q-1}{2}}\exp\left(-\frac{1}{2}{\rm tr}(\vS\vR)\right), \end{align} % where $G$ is the normalizing constant. Because we use conjugate Gaussian-Wishart priors for the parameters of the Gaussian mixture components, we can analytically integrate out those parameters given the assignments of points to components. Let $z_n$ be the assignment of the $n$\textsuperscript{th} point. The prior probability of latent coordinates $\vX$ given latent cluster assignments $\CLAS=(\CLASi_1, \CLASi_2, \ldots, \CLASi_N)$ factorizes over clusters, and can be obtained in closed-form by integrating out the Gaussian parameters $\{\vmu_c, \vR_c\}$ to give: % \begin{align} p(\vX | \CLAS, \vS, \nu, r) = \prod_{c=1}^{\infty} \pi^{-\frac{N_c Q}{2}}\frac{r^{Q/2} \left|\vS \right|^{\nu/2}}{r_c^{Q/2} \left| \vS_c \right|^{\nu_c/2}} \times \prod_{q=1}^Q \frac{\Gamma \left( \frac{\nu_c + 1 - q}{2} \right)}{\Gamma \left( \frac{\nu+1-q}{2} \right)}, \label{eq:px_z} \end{align} % where $N_c$ is the number of data points assigned to the $c$\textsuperscript{th} component, $\Gamma(\cdot)$ is the Gamma function, and % \begin{align} r_{c}=r+N_{c}, \hspace{2em} \nu_{c}=\nu+N_{c}, \hspace{2em} %\nonumber %\end{align} %\begin{align} \vu_c=\frac{\displaystyle r\vu + \sum_{n:z_n = c} \vx_n}{r+N_c}, \\[1em] %\end{align} % %\begin{align} \textnormal{and} \hspace{2em} \vS_{c}=\vS+\sum_{n:z_{n}=c}\vx_{n}\vx_{n}\tra + r\vu\vu\tra - r_{c}\vu_{c}\vu_{c}\tra, \end{align} % are the posterior Gaussian-Wishart parameters of the $c$\textsuperscript{th} component~\citep{murphy2007conjugate}. To model the cluster assignments, we use a Dirichlet process~\citep{maceachern1998estimating} with concentration parameter $\eta$. % for infinite mixture modeling in the latent space. %The probability under a Dirichlet-multinomial prior of observing assignment $\CLAS$ is: Under a Dirichlet process prior, the probability of observing a particular cluster assignment $\CLAS$ depends only on the partition induced, and is given by the Chinese restaurant process: % \begin{align} p(\CLAS|\eta) = \frac{\Gamma(\eta) \eta^C}{\Gamma(\eta + N)} \prod_{c=1}^C \Gamma( N_c ) %= \frac{\eta^{C}\prod_{c=1}^C (N_c-1)!}{\eta(\eta+1)\ldots(\eta+N-1)}, \label{eq:pz} \end{align} % where $C$ is the number of components for which $N_c > 0$, and $N$ is the total number of datapoints. The joint distribution of observed coordinates, latent coordinates, and cluster assignments is given by % \begin{align} p(\vY,\vX,\CLAS|\bm{\theta},\bm{S},\nu,\vu,r,\eta) = p(\vY|\vX,\bm{\theta}) p(\vX|\CLAS,\bm{S},\nu,\vu,r)p(\CLAS|\eta), \label{eq:joint} \end{align} % where the factors in the right hand side can be calculated by equations \eqref{eq:py_x}, \eqref{eq:px_z} and \eqref{eq:pz}, respectively. \iffalse \subsubsection{Generative description of model} The infinite warped mixture model generates observations $\vY$ according to the following generative process: % \begin{enumerate} \item Draw mixture weights $\bm{\lambda} \sim \GEM(\eta)$ \item For each component $c=1, 2, \dots, \infty$ \begin{enumerate} \item Draw precision $\vR_c \sim {\mathcal W}(\vS\inv, \nu)$ \item Draw mean $\vmu_c \sim {\mathcal N}(\vu,(r\vR_c)\inv)$ \end{enumerate} \item For each observed dimension $d=1, 2, \dots, D$ \begin{enumerate} \item Draw function $f_{d}(\vx) \sim {\rm GP}(m(\vx),k(\vx,\vx'))$ \end{enumerate} \item For each observation $n=1, 2, \dots,N$ \begin{enumerate} \item Draw latent assignment $z_n \sim {\rm Mult}(\bm{\lambda})$ \item Draw latent coordinates $\vx_n \sim {\mathcal N}(\vmu_{z_n},\vR_{z_n}\inv)$ \item For each observed dimension $d=1, 2, \dots, D$ \begin{enumerate} \item Draw feature $y_{nd} \sim {\cal N}(f_d(\vx_n), \sigma^2_n)$ \end{enumerate} \end{enumerate} \end{enumerate} % Here, $\GEM(\eta)$ is the stick-breaking process \citep{sethuraman94} that generates mixture weights for a Dirichlet process with parameter $\eta$, %${\rm Mult}(\cdot)$ represents a multinomial distribution, ${\rm Mult}(\bm{\lambda})$ represents a multinomial distribution with parameter $\bm{\lambda}$, $m(\vx)$ is the mean function of the Gaussian process, $\vx,\vx'\in{\mathbb R}^{Q}$ and $\sigma^2_n$ is the noise variance of the \gp{} kernel. \fi \subsubsection{Details of inference} \label{sec:iwmm-inference-details} %By placing conjugate Gaussian-Wishart priors on the parameters of the Gaussian mixture components, we analytically integrate out those parameters given the assignments of points to clusters. After analytically integrating out the parameters of the Gaussian mixture components, the only remaining variables to infer are the latent points $\vX$, the cluster assignments $\CLAS$, and the kernel parameters $\vtheta$. We'll estimate the posterior over these parameters using Markov chain Monte Carlo. In particular, we'll alternate between collapsed Gibbs sampling of each row of $\CLAS$, and Hamiltonian Monte Carlo sampling of $\vX$ and $\vtheta$. First, we explain collapsed Gibbs sampling for the cluster assignments $\CLAS$. Given a sample of $\vX$, $p(\CLAS | \vX, \vS, \nu, \vu, r, \eta)$ does not depend on $\vY$. This lets us resample cluster assignments, integrating out the \iGMM{} likelihood in closed form. Given the current state of all but one latent component $z_n$, a new value for $z_n$ is sampled with the following probability: % \begin{align} p(z_{n}=c|\vX,\CLAS_{\setminus n},\bm{S},\nu,\vu,r,\eta) & \propto\! \left\{ \begin{array}{ll} \!\!N_{c\setminus n}\cdot p(\vx_{n}|\vX_{c\setminus n},\bm{S},\nu,\vu,r) & \text{{\small existing components}}\\ \!\!\eta\cdot p(\vx_{n}|\bm{S},\nu,\vu,r) & \text{{\small a new component}} \end{array} \right. \label{eq:gibbs} \end{align} % where $\vX_{c}=\{\vx_{n}|z_{n}=c\}$ is the set of latent coordinates assigned to the $c$\textsuperscript{th} component, and $\setminus n$ represents the value or set when excluding the $n$\textsuperscript{th} data point. We can analytically calculate $p(\vx_{n}|\vX_{c\setminus n},\bm{S},\nu,\vu,r)$ as follows: % \begin{align} p(\vx_{n}|\vX_{c\setminus n},\bm{S},\nu,\vu,r) =\pi^{-\frac{N_{c\setminus n}Q}{2}} \frac{r_{c\setminus n} ^{Q/2} \left| \vS_{c\setminus n} \right|^{\nu_{c\setminus n}/2}} {r_{c\setminus n}'^{Q/2} \left| \vS_{c\setminus n}' \right|^{\nu_{c\setminus n}'/2}} \times \prod_{d=1}^{Q} \frac{\Gamma \left( \frac{\nu_{c\setminus n}' + 1 - d}{2} \right)} {\Gamma \left( \frac{\nu_{c\setminus n} + 1 - d}{2} \right)}, \label{eq:iwmm-cluster-gibbs} \end{align} % where $r_{c}'$, $\nu_{c}'$, $\vu_{c}'$ and $\vS_{c}'$ represent the posterior on Gaussian-Wishart parameters of the $c$\textsuperscript{th} component when the $n$\textsuperscript{th} data point has been assigned to it. %the $c$\textsuperscript{th} component. We can efficiently calculate the determinant $\left| \vS_{c\setminus n}' \right|$ using the rank-one Cholesky update. A special case of \cref{eq:iwmm-cluster-gibbs} gives the likelihood for a new component, $p(\vx_{n}|\bm{S},\nu,\vu,r)$. \subsubsection{Gradients for Hamiltonian Monte Carlo} Hamiltonian Monte Carlo (\HMC{}) sampling of $\vX$ from posterior ${p(\vX|\CLAS,\vY,\bm{\theta},\bm{S},\nu,\vu,r)}$, requires computing the gradient of the log-unnormalized-posterior with respect to $\vX$: % \begin{align} \pderiv{}{\vX} \big[ \log p(\vY|\vX,\bm{\theta}) + \log p(\vX|\CLAS,\bm{S},\nu,\vu,r) \big] \label{eq:warped-hmc1} \end{align} % The first term of gradient \eqref{eq:warped-hmc1} can be calculated by % \begin{align} \pderiv{\log p(\vY | \vX,\bm{\theta})}{\vX} = \pderiv{\log p(\vY | \vX,\bm{\theta})}{\vK} \pderiv{\vK}{\vX} = \left[ -\frac{1}{2}D\vK\inv+\frac{1}{2}\vK\inv \vY \vY^{T} \vK\inv \right]\left[\pderiv{\vK}{\vX}\right], \label{eq:warped-hmc2} \end{align} % where for an $\kSE + \kWN$ kernel with the same lengthscale $\ell$ on all dimensions, % \begin{align} \pderiv{k(\vx_{n},\vx_{m})}{\vx_n} & = -\frac{\sigma^2_f}{\ell^2} \exp \left( - \frac{1}{2 \ell^2} (\vx_n - \vx_m)\tra (\vx_n - \vx_m) \right) (\vx_n - \vx_m). \label{eq:warped-hmc3} \end{align} % %using the chain rule. The second term of \eqref{eq:warped-hmc1} is given by \begin{align} \frac{\partial \log p(\vX|\CLAS,\bm{S},\nu,\vu,r)}{\partial \vx_{n}} = -\nu_{z_{n}}\bm{S}_{z_{n}}\inv(\vx_{n}-\vu_{z_{n}}). \label{eq:warped-hmc4} \end{align} We also infer kernel parameters $\vtheta$ via \HMC{}, using the gradient of the log unnormalized posterior with respect to the kernel parameters, using an improper uniform prior. \subsubsection{Posterior predictive density} \label{sec:iwmm-predictive-density} In the \gplvm{}, the predictive density of at test point $\vy_\star$ is usually computed by finding the point $\vx_\star$ which has the highest probability of being mapped to $\vy_\star$, then using the density of $p(\vx_\star)$ and the Jacobian of the warping at that point to approximate the density at $\vy_\star$. When inference is done this way, approximating the predictive density only requires solving a single optimization for each $\vy_\star$. For our model, we use approximate integration to estimate $p(\vy_\star)$. This is done for two reasons: First, multiple latent points (possibly from different clusters) can map to the same observed point, meaning the standard method can underestimate $p(\vy_\star)$. Second, because we do not optimize the latent coordinates of training points, but instead sample them, we would need to optimize each $p(\vx_\star)$ separately for each sample in the Markov chain. One advantage of our method is that it gives estimates for all $p(\vy_\star)$ at once. However, it may not be as accurate in very high observed dimensions, when the volume to sample over is relatively large. The posterior density in the observed space given the training data is % \begin{align} p(\vy_\star | \vY) & = \int \!\!\! \int p(\vy_\star,\vx_\star, \vX | \vY)d\vx_\star d\vX \nonumber\\ & = \int\!\!\! \int p(\vy_\star | \vx_\star, \vX, \vY)p(\vx_\star|\vX,\vY)p(\vX|\vY)d\vx_\star d\vX. \label{eq:density} \end{align} % We first approximate $p(\vX | \vY)$ using samples from the Gibbs and Hamiltonian Monte Carlo chain. We then approximate $p(\vx_\star | \vX, \vY)$ by sampling points from the posterior density in the latent space and warping them, using the following procedure: % \begin{enumerate} \item Draw a latent cluster assignment $z_\star \sim {\rm Mult} \left( \frac{N_{1}}{N+\eta}, \frac{N_{2}}{N+\eta}, \cdots,\frac{N_{C}}{N+\eta},\frac{\eta}{N+\eta} \right)$ \item Draw a latent cluster precision matrix $\vR_\star \sim {\cal W}(\vS\inv_{z_\star},\nu_{z_\star})$ \item Draw a latent cluster mean $\vmu_\star \sim {\cal N}(\vu_{z_\star},(r_{z_\star}\vR_\star)\inv)$ \item Draw latent coordinates $\vx_\star \sim {\cal N}(\vmu_\star,\vR_\star\inv)$ \item For each observed dimension $d = 1, 2, \ldots, D$, \\ draw observed coordinates $y^\star_d \sim {\cal N}(\vk_\star\tra \vK\inv \vY_{:,d}, k(\vx_\star,\vx_\star) - \vk_\star\tra \vK\inv \vk_\star)$ \end{enumerate} % If $z_\star$ is assigned to a new component in step 1, the prior Gaussian-Wishart distribution \eqref{eq:normal-wishar-prior} is used for sampling in steps 2 and 3. The density drawn from in step 5 is the predictive distribution of a \gp{}, where % $\vk_\star=[ k(\vx_\star, \vx_1), k(\vx_\star, \vx_2), \cdots, k(\vx_\star, \vx_N)]\tra$ and $\vY_{:,d}$ represents the $d$th column of $\vY$. Each step of this sampling procedure draws from the exact conditional distribution, so the Monte Carlo estimate of the conditional predictive density $p(\vy_\star | \vX, \vY)$ will converge to the true marginal distribution as the number of samples increases. Since the observations $\vy_\star$ are conditionally normally distributed, each one adds a smooth contribution to the empirical Monte Carlo estimate of the posterior density, as opposed to a collection of point masses. \subsubsection{Source code} A reference implementation of the above algorithms is available at\\ \url{http://www.github.com/duvenaud/warped-mixtures}. \outbpdocument{ \bibliographystyle{plainnat} \bibliography{references.bib} } ================================================ FILE: code/additive_extrapolation.m ================================================ function additive_extrapolation(savefigs) % Generate some plots showing additive GP regression on synthetic datasets % with 1st-order interactions and censored data. % % David Duvenaud % March 2014 % =============== addpath(genpath( 'utils' )); addpath(genpath( 'gpml' )); if nargin < 1; savefigs = false; end % fixing the seed of the random generators seed=6; randn('state',seed); rand('state',seed); % Rendering setup n_1d = 100; % Fineness of grid. % Regression problem setup. n = 100; % number of observations s = 2; % number of relevant variables noise_std = .02; % standard deviation of noise X = rand(n,2) * 4 - 2; noiseless_Y = truefunc(X); Y = noiseless_Y + randn(n,1) * noise_std; % add some noise with known standard deviation xlims = [-2.3, 6]; ylims = [-2.3, 2]; % Censor data. censor_threshold = -1.5; trainset = or(X(:,1) < censor_threshold, X(:,2) < censor_threshold); X = X(trainset,:); y = Y(trainset); noiseless_Y = noiseless_Y(trainset); [N,D] = size(X); % Set up model. likfunc = 'likGauss'; sn = 0.1; hyp.lik = log(sn); inference = @infExact; meanfunc = {'meanConst'}; hyp.mean = 0; % Train additive model R = 2; covfunc = { 'covADD',{1:R,'covSEiso'} }; % Construct an additive kernel hyp.cov = [ log(ones(1,2*D)), log(ones(1,R))]; % Set hyperparameters. hyp_add = minimize(hyp, @gp, -20, inference, meanfunc, covfunc, likfunc, X, y); % Train ARD model covfunc = { 'covSEard' }; hyp.cov = [ log(ones(1,D)), log(1)]; % Set hyperparameters. hyp = minimize(hyp, @gp, -20, inference, meanfunc, covfunc, likfunc, X, y); % generate a grid on which to render to predictive surface. range = linspace(xlims(1), xlims(2), n_1d); [a,b] = meshgrid(range, range); xstar = [ a(:), b(:) ]; % Make ARD predictions. predictions_ard = gp(hyp, inference, meanfunc, covfunc, likfunc, X, y, xstar); % Make additive predictions. covfunc = { 'covADD',{1:R,'covSEiso'} }; % Construct an additive kernel. predictions_add = gp(hyp_add, inference, meanfunc, covfunc, likfunc, X, y, xstar); figure(1); clf; nice_plot_surface(a,b,X, predictions_ard,length( range), xlims, ylims); hold on; show_xs(X, repmat(ylims(1), size(X,1), 1), noiseless_Y - 0.1); if savefigs nice_figure_save('1st_order_censored_ard') end figure(2); clf; nice_plot_surface(a,b,X, predictions_add, length( range), xlims, ylims); hold on; show_xs(X, repmat(ylims(1), size(X,1), 1), noiseless_Y - 0.1); if savefigs nice_figure_save('1st_order_censored_add') end figure(3); clf; nice_plot_surface(a,b,X, truefunc(xstar), length( range), xlims, ylims); hold on; if savefigs nice_figure_save('1st_order_censored_truth') end end function nice_plot_surface(a,b,X,Y,l, xlims, ylims) surf(a,b,reshape(Y, l, l), 'EdgeColor','none','LineStyle','none','FaceLighting','phong'); xlim(xlims); ylim(xlims); zlim(ylims); set(gcf, 'color', 'white'); set(get(gca,'XLabel'),'Rotation',0,'Interpreter','tex', 'Fontsize', 12, 'FontName','Times New Roman'); set(get(gca,'YLabel'),'Rotation',0,'Interpreter','tex', 'Fontsize', 12, 'FontName','Times New Roman'); set(get(gca,'ZLabel'),'Rotation',0,'Interpreter','tex', 'Fontsize', 12, 'FontName','Times New Roman'); set( gca, 'xTickLabel', '' ); set( gca, 'yTickLabel', '' ); set( gca, 'zTickLabel', '' ); xlabel('x_1'); ylabel('x_2'); zlabel('f(x) '); view([-30,42]); tightfig set_fig_units_cm(6,5); end function show_xs(X, lowy, uppery) line_width = 2; markersize = 14; plot3(X(:,1), X(:,2), lowy, 'b.', 'Linewidth', line_width, ... 'Markersize', markersize); % show the data for i = 1:size(X,1); line([X(i,1), X(i,1)], [X(i,2), X(i,2)], [lowy(i), uppery(i)], ... 'LineStyle', '--', 'Color', [0.5 0.5 0.5]); end end function y = truefunc(x) y = sin(x(:,1) .* 2 - 1.1) + sin(x(:,2) .* 2 - 1.1); end function nice_figure_save(filename) myaa('mypublish', ['../figures/additive/', filename]); %savepng(gcf, ['../figures/additive/', filename]); end ================================================ FILE: code/data/quebec/process.m ================================================ %% Load data data = csvread('quebec-births-fixed.csv', 1, 0); X_full = data(:,1:4); y_full = data(:,5); %% Plot plot(X_full(:,1), y_full, 'o'); % Set random seed. randn('state',0); rand('state',0); %% Subsample for n = [1000,2000,5000] rp = randperm(length(y_full)); idx = rp(1:n); X = X_full(idx,:); y = y_full(idx,:); save(['quebec-', int2str(n), '.mat'], 'X', 'y'); end X = X_full; y = y_full; save('quebec-all.mat', 'X', 'y'); ================================================ FILE: code/data/quebec/quebec-births-fixed.csv ================================================ Standard 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1979.9746748802,1,0,0,189 1979.977412731,0,0,0,206 1979.9801505818,0,0,0,192 1979.9828884326,0,0,0,221 1979.9856262834,0,0,0,285 1979.9883641342,0,0,0,327 1979.991101985,1,0,0,226 1979.9938398357,1,0,0,213 1979.9965776865,0,0,0,230 1980,0,0,0,208 1980.0027378508,0,0,0,229 1980.0054757016,0,0,0,291 1980.0082135524,0,0,0,273 1980.0109514032,1,0,0,226 1980.0136892539,1,0,0,216 1980.0164271047,0,0,0,225 1980.0191649555,0,0,0,284 1980.0219028063,0,0,0,279 1980.0246406571,0,0,0,241 1980.0273785079,0,0,0,267 1980.0301163587,1,0,0,211 1980.0328542095,1,0,0,202 1980.0355920602,0,0,0,268 1980.038329911,0,0,0,264 1980.0410677618,0,0,0,270 1980.0438056126,0,0,0,302 1980.0465434634,0,0,0,286 1980.0492813142,1,0,0,218 1980.052019165,1,0,0,201 1980.0547570158,0,0,0,276 1980.0574948665,0,0,0,278 1980.0602327173,0,0,0,244 1980.0629705681,0,0,0,243 1980.0657084189,0,0,0,267 1980.0684462697,1,0,0,220 1980.0711841205,1,0,0,212 1980.0739219713,0,0,0,247 1980.076659822,0,0,0,280 1980.0793976728,0,0,0,251 1980.0821355236,0,0,0,259 1980.0848733744,0,0,0,265 1980.0876112252,1,0,0,219 1980.090349076,1,0,0,207 1980.0930869268,0,0,0,282 1980.0958247776,0,0,0,269 1980.0985626283,0,0,0,253 1980.1013004791,0,0,0,306 1980.1040383299,0,0,0,264 1980.1067761807,1,0,0,213 1980.1095140315,1,0,0,207 1980.1122518823,0,0,0,242 1980.1149897331,0,0,0,310 1980.1177275839,0,0,0,246 1980.1204654346,0,0,0,300 1980.1232032854,0,0,0,279 1980.1259411362,1,0,0,220 1980.128678987,1,0,0,234 1980.1314168378,0,0,0,262 1980.1341546886,0,0,0,316 1980.1368925394,0,0,0,285 1980.1396303902,0,0,0,328 1980.1423682409,0,0,0,255 1980.1451060917,1,0,0,223 1980.1478439425,1,0,0,215 1980.1505817933,0,0,0,264 1980.1533196441,0,0,0,314 1980.1560574949,0,0,0,317 1980.1587953457,0,0,0,312 1980.1615331964,0,0,0,255 1980.1642710472,1,0,0,250 1980.167008898,1,0,0,215 1980.1697467488,0,0,0,236 1980.1724845996,0,0,0,284 1980.1752224504,0,0,0,285 1980.1779603012,0,0,0,298 1980.180698152,0,0,0,282 1980.1834360027,1,0,0,243 1980.1861738535,1,0,0,248 1980.1889117043,0,0,0,284 1980.1916495551,0,0,0,289 1980.1943874059,0,0,0,291 1980.1971252567,0,0,0,285 1980.1998631075,0,0,0,271 1980.2026009583,1,0,0,235 1980.205338809,1,0,0,220 1980.2080766598,0,0,0,304 1980.2108145106,0,0,0,297 1980.2135523614,0,0,0,311 1980.2162902122,0,0,0,275 1980.219028063,0,0,0,296 1980.2217659138,1,0,0,261 1980.2245037646,1,0,0,257 1980.2272416153,0,0,0,298 1980.2299794661,0,0,0,300 1980.2327173169,0,0,0,293 1980.2354551677,0,0,0,290 1980.2381930185,0,0,0,273 1980.2409308693,1,0,0,242 1980.2436687201,1,0,0,214 1980.2464065709,0,0,0,278 1980.2491444216,0,0,0,323 1980.2518822724,0,0,0,333 1980.2546201232,0,0,0,328 1980.257357974,0,0,1,251 1980.2600958248,1,0,1,225 1980.2628336756,1,0,1,217 1980.2655715264,0,0,1,248 1980.2683093771,0,0,0,298 1980.2710472279,0,0,0,294 1980.2737850787,0,0,0,326 1980.2765229295,0,0,0,295 1980.2792607803,1,0,0,279 1980.2819986311,1,0,0,227 1980.2847364819,0,0,0,296 1980.2874743327,0,0,0,284 1980.2902121834,0,0,0,315 1980.2929500342,0,0,0,296 1980.295687885,0,0,0,291 1980.2984257358,1,0,0,234 1980.3011635866,1,0,0,220 1980.3039014374,0,0,0,298 1980.3066392882,0,0,0,305 1980.309377139,0,0,0,276 1980.3121149897,0,0,0,329 1980.3148528405,0,0,0,296 1980.3175906913,1,0,0,245 1980.3203285421,1,0,0,232 1980.3230663929,0,0,0,294 1980.3258042437,0,0,0,325 1980.3285420945,0,0,0,325 1980.3312799453,0,0,0,304 1980.334017796,0,0,0,292 1980.3367556468,1,0,0,215 1980.3394934976,1,0,0,237 1980.3422313484,0,0,0,293 1980.3449691992,0,0,0,259 1980.34770705,0,0,0,270 1980.3504449008,0,0,0,300 1980.3531827515,0,0,0,280 1980.3559206023,1,0,0,245 1980.3586584531,1,0,0,241 1980.3613963039,0,0,0,297 1980.3641341547,0,0,0,314 1980.3668720055,0,0,0,324 1980.3696098563,0,0,0,318 1980.3723477071,0,0,0,277 1980.3750855578,1,0,0,213 1980.3778234086,1,0,0,233 1980.3805612594,0,1,0,267 1980.3832991102,0,0,0,306 1980.386036961,0,0,0,319 1980.3887748118,0,0,0,326 1980.3915126626,0,0,0,339 1980.3942505134,1,0,0,250 1980.3969883641,1,0,0,233 1980.3997262149,0,0,0,307 1980.4024640657,0,0,0,284 1980.4052019165,0,0,0,306 1980.4079397673,0,0,0,282 1980.4106776181,0,0,0,291 1980.4134154689,1,0,0,232 1980.4161533197,1,0,0,227 1980.4188911704,0,0,0,312 1980.4216290212,0,0,0,333 1980.424366872,0,0,0,312 1980.4271047228,0,0,0,312 1980.4298425736,0,0,0,291 1980.4325804244,1,0,0,226 1980.4353182752,1,0,0,216 1980.438056126,0,0,0,281 1980.4407939767,0,0,0,301 1980.4435318275,0,0,0,255 1980.4462696783,0,0,0,320 1980.4490075291,0,0,0,246 1980.4517453799,1,0,0,227 1980.4544832307,1,0,0,225 1980.4572210815,0,0,0,279 1980.4599589322,0,0,0,284 1980.462696783,0,0,0,284 1980.4654346338,0,0,0,288 1980.4681724846,0,0,0,301 1980.4709103354,1,0,0,248 1980.4736481862,1,0,0,223 1980.476386037,0,0,0,292 1980.4791238878,0,0,0,251 1980.4818617385,0,0,0,301 1980.4845995893,0,0,0,346 1980.4873374401,0,0,0,339 1980.4900752909,1,0,0,247 1980.4928131417,1,0,0,200 1980.4955509925,0,0,0,259 1980.4982888433,0,0,0,250 1980.5010266941,0,0,0,283 1980.5037645448,0,0,0,329 1980.5065023956,0,0,0,313 1980.5092402464,1,0,0,218 1980.5119780972,1,0,0,244 1980.514715948,0,0,0,253 1980.5174537988,0,0,0,325 1980.5201916496,0,0,0,272 1980.5229295004,0,0,0,328 1980.5256673511,0,0,0,306 1980.5284052019,1,0,0,248 1980.5311430527,1,0,0,194 1980.5338809035,0,0,0,257 1980.5366187543,0,0,0,292 1980.5393566051,0,0,0,258 1980.5420944559,0,0,0,302 1980.5448323066,0,0,0,283 1980.5475701574,1,0,0,240 1980.5503080082,1,0,0,230 1980.553045859,0,0,0,296 1980.5557837098,0,0,0,289 1980.5585215606,0,0,0,280 1980.5612594114,0,0,0,294 1980.5639972622,0,0,0,310 1980.5667351129,1,0,0,223 1980.5694729637,1,0,0,213 1980.5722108145,0,0,0,275 1980.5749486653,0,0,0,260 1980.5776865161,0,0,0,299 1980.5804243669,0,0,0,262 1980.5831622177,0,0,0,323 1980.5859000685,1,0,0,224 1980.5886379192,1,0,0,193 1980.59137577,0,0,0,239 1980.5941136208,0,0,0,301 1980.5968514716,0,0,0,268 1980.5995893224,0,0,0,264 1980.6023271732,0,0,0,261 1980.605065024,1,0,0,230 1980.6078028748,1,0,0,217 1980.6105407255,0,0,0,250 1980.6132785763,0,0,0,328 1980.6160164271,0,0,0,307 1980.6187542779,0,0,0,294 1980.6214921287,0,0,0,302 1980.6242299795,1,0,0,208 1980.6269678303,1,0,0,198 1980.629705681,0,0,0,255 1980.6324435318,0,0,0,302 1980.6351813826,0,0,0,249 1980.6379192334,0,0,0,278 1980.6406570842,0,0,0,314 1980.643394935,1,0,0,265 1980.6461327858,1,0,0,203 1980.6488706366,0,0,0,309 1980.6516084873,0,0,0,290 1980.6543463381,0,0,0,310 1980.6570841889,0,0,0,275 1980.6598220397,0,0,0,285 1980.6625598905,1,0,0,247 1980.6652977413,1,0,0,198 1980.6680355921,0,1,0,224 1980.6707734429,0,0,0,241 1980.6735112936,0,0,0,286 1980.6762491444,0,0,0,277 1980.6789869952,0,0,0,308 1980.681724846,1,0,0,232 1980.6844626968,1,0,0,206 1980.6872005476,0,0,0,248 1980.6899383984,0,0,0,307 1980.6926762492,0,0,0,270 1980.6954140999,0,0,0,305 1980.6981519507,0,0,0,281 1980.7008898015,1,0,0,211 1980.7036276523,1,0,0,260 1980.7063655031,0,0,0,298 1980.7091033539,0,0,0,323 1980.7118412047,0,0,0,350 1980.7145790555,0,0,0,287 1980.7173169062,0,0,0,341 1980.720054757,1,0,0,218 1980.7227926078,1,0,0,246 1980.7255304586,0,0,0,320 1980.7282683094,0,0,0,339 1980.7310061602,0,0,0,336 1980.733744011,0,0,0,334 1980.7364818617,0,0,0,333 1980.7392197125,1,0,0,231 1980.7419575633,1,0,0,222 1980.7446954141,0,0,0,317 1980.7474332649,0,0,0,366 1980.7501711157,0,0,0,283 1980.7529089665,0,0,0,312 1980.7556468173,0,0,0,279 1980.758384668,1,0,0,230 1980.7611225188,1,0,0,195 1980.7638603696,0,0,0,300 1980.7665982204,0,0,0,299 1980.7693360712,0,0,0,268 1980.772073922,0,0,0,313 1980.7748117728,0,0,0,299 1980.7775496236,1,0,0,222 1980.7802874743,1,0,0,215 1980.7830253251,0,1,0,237 1980.7857631759,0,0,0,277 1980.7885010267,0,0,0,288 1980.7912388775,0,0,0,276 1980.7939767283,0,0,0,252 1980.7967145791,1,0,0,221 1980.7994524299,1,0,0,219 1980.8021902806,0,0,0,284 1980.8049281314,0,0,0,332 1980.8076659822,0,0,0,259 1980.810403833,0,0,0,281 1980.8131416838,0,0,0,268 1980.8158795346,1,0,0,208 1980.8186173854,1,0,0,231 1980.8213552361,0,0,0,260 1980.8240930869,0,0,0,282 1980.8268309377,0,0,0,276 1980.8295687885,0,0,0,267 1980.8323066393,0,0,0,231 1980.8350444901,1,0,0,231 1980.8377823409,1,0,0,213 1980.8405201917,0,0,0,216 1980.8432580424,0,0,0,247 1980.8459958932,0,0,0,273 1980.848733744,0,0,0,296 1980.8514715948,0,0,0,251 1980.8542094456,1,0,0,222 1980.8569472964,1,0,0,179 1980.8596851472,0,0,0,233 1980.862422998,0,0,0,285 1980.8651608487,0,0,0,289 1980.8678986995,0,0,0,281 1980.8706365503,0,0,0,254 1980.8733744011,1,0,0,225 1980.8761122519,1,0,0,176 1980.8788501027,0,0,0,279 1980.8815879535,0,0,0,270 1980.8843258043,0,0,0,279 1980.887063655,0,0,0,273 1980.8898015058,0,0,0,273 1980.8925393566,1,0,0,172 1980.8952772074,1,0,0,188 1980.8980150582,0,0,0,267 1980.900752909,0,0,0,261 1980.9034907598,0,0,0,316 1980.9062286105,0,0,0,252 1980.9089664613,0,0,0,257 1980.9117043121,1,0,0,214 1980.9144421629,1,0,0,183 1980.9171800137,0,0,0,270 1980.9199178645,0,0,0,257 1980.9226557153,0,0,0,256 1980.9253935661,0,0,0,252 1980.9281314168,0,0,0,267 1980.9308692676,1,0,0,193 1980.9336071184,1,0,0,173 1980.9363449692,0,0,0,261 1980.93908282,0,0,0,254 1980.9418206708,0,0,0,248 1980.9445585216,0,0,0,273 1980.9472963724,0,0,0,254 1980.9500342231,1,0,0,212 1980.9527720739,1,0,0,188 1980.9555099247,0,0,0,273 1980.9582477755,0,0,0,286 1980.9609856263,0,0,0,268 1980.9637234771,0,0,0,285 1980.9664613279,0,0,0,310 1980.9691991787,1,0,0,205 1980.9719370294,1,0,0,190 1980.9746748802,0,0,0,263 1980.977412731,0,0,0,263 1980.9801505818,0,0,0,232 1980.9828884326,0,0,0,168 1980.9856262834,0,0,0,216 1980.9883641342,1,0,0,207 1980.991101985,1,0,0,213 1980.9938398357,0,0,0,278 1980.9965776865,0,0,0,286 1980.9993155373,0,0,0,281 1981,0,0,0,211 1981.0027378508,0,0,0,254 1981.0054757016,1,0,0,196 1981.0082135524,1,0,0,232 1981.0109514032,0,0,0,282 1981.0136892539,0,0,0,289 1981.0164271047,0,0,0,279 1981.0191649555,0,0,0,242 1981.0219028063,0,0,0,294 1981.0246406571,1,0,0,203 1981.0273785079,1,0,0,208 1981.0301163587,0,0,0,265 1981.0328542095,0,0,0,273 1981.0355920602,0,0,0,247 1981.038329911,0,0,0,251 1981.0410677618,0,0,0,247 1981.0438056126,1,0,0,234 1981.0465434634,1,0,0,218 1981.0492813142,0,0,0,267 1981.052019165,0,0,0,280 1981.0547570158,0,0,0,272 1981.0574948665,0,0,0,260 1981.0602327173,0,0,0,273 1981.0629705681,1,0,0,176 1981.0657084189,1,0,0,204 1981.0684462697,0,0,0,247 1981.0711841205,0,0,0,275 1981.0739219713,0,0,0,285 1981.076659822,0,0,0,279 1981.0793976728,0,0,0,273 1981.0821355236,1,0,0,211 1981.0848733744,1,0,0,183 1981.0876112252,0,0,0,279 1981.090349076,0,0,0,255 1981.0930869268,0,0,0,276 1981.0958247776,0,0,0,260 1981.0985626283,0,0,0,298 1981.1013004791,1,0,0,212 1981.1040383299,1,0,0,203 1981.1067761807,0,0,0,266 1981.1095140315,0,0,0,290 1981.1122518823,0,0,0,277 1981.1149897331,0,0,0,291 1981.1177275839,0,0,0,253 1981.1204654346,1,0,0,185 1981.1232032854,1,0,0,212 1981.1259411362,0,0,0,258 1981.128678987,0,0,0,309 1981.1314168378,0,0,0,281 1981.1341546886,0,0,0,276 1981.1368925394,0,0,0,276 1981.1396303902,1,0,0,214 1981.1423682409,1,0,0,234 1981.1451060917,0,0,0,279 1981.1478439425,0,0,0,300 1981.1505817933,0,0,0,299 1981.1533196441,0,0,0,298 1981.1560574949,0,0,0,278 1981.1587953457,1,0,0,224 1981.1615331964,1,0,0,232 1981.1642710472,0,0,0,276 1981.167008898,0,0,0,301 1981.1697467488,0,0,0,264 1981.1724845996,0,0,0,286 1981.1752224504,0,0,0,318 1981.1779603012,1,0,0,233 1981.180698152,1,0,0,233 1981.1834360027,0,0,0,285 1981.1861738535,0,0,0,301 1981.1889117043,0,0,0,279 1981.1916495551,0,0,0,312 1981.1943874059,0,0,0,305 1981.1971252567,1,0,0,225 1981.1998631075,1,0,0,186 1981.2026009583,0,0,0,287 1981.205338809,0,0,0,306 1981.2080766598,0,0,0,336 1981.2108145106,0,0,0,295 1981.2135523614,0,0,0,290 1981.2162902122,1,0,0,223 1981.219028063,1,0,0,221 1981.2217659138,0,0,0,325 1981.2245037646,0,0,0,298 1981.2272416153,0,0,0,279 1981.2299794661,0,0,0,305 1981.2327173169,0,0,0,324 1981.2354551677,1,0,0,209 1981.2381930185,1,0,0,224 1981.2409308693,0,0,0,298 1981.2436687201,0,0,0,340 1981.2464065709,0,0,0,306 1981.2491444216,0,0,0,298 1981.2518822724,0,0,0,300 1981.2546201232,1,0,0,244 1981.257357974,1,0,0,224 1981.2600958248,0,0,0,317 1981.2628336756,0,0,0,305 1981.2655715264,0,0,0,292 1981.2683093771,0,0,0,311 1981.2710472279,0,0,0,324 1981.2737850787,1,0,0,210 1981.2765229295,1,0,0,206 1981.2792607803,0,0,0,256 1981.2819986311,0,0,0,302 1981.2847364819,0,0,0,291 1981.2874743327,0,0,0,341 1981.2902121834,0,0,1,290 1981.2929500342,1,0,1,222 1981.295687885,1,0,1,207 1981.2984257358,0,0,1,265 1981.3011635866,0,0,0,309 1981.3039014374,0,0,0,303 1981.3066392882,0,0,0,330 1981.309377139,0,0,0,344 1981.3121149897,1,0,0,254 1981.3148528405,1,0,0,230 1981.3175906913,0,0,0,288 1981.3203285421,0,0,0,298 1981.3230663929,0,0,0,308 1981.3258042437,0,0,0,288 1981.3285420945,0,0,0,284 1981.3312799453,1,0,0,210 1981.334017796,1,0,0,233 1981.3367556468,0,0,0,311 1981.3394934976,0,0,0,311 1981.3422313484,0,0,0,311 1981.3449691992,0,0,0,289 1981.34770705,0,0,0,322 1981.3504449008,1,0,0,225 1981.3531827515,1,0,0,228 1981.3559206023,0,0,0,273 1981.3586584531,0,0,0,295 1981.3613963039,0,0,0,283 1981.3641341547,0,0,0,314 1981.3668720055,0,0,0,293 1981.3696098563,1,0,0,198 1981.3723477071,1,0,0,203 1981.3750855578,0,1,0,255 1981.3778234086,0,0,0,303 1981.3805612594,0,0,0,298 1981.3832991102,0,0,0,300 1981.386036961,0,0,0,305 1981.3887748118,1,0,0,250 1981.3915126626,1,0,0,199 1981.3942505134,0,0,0,276 1981.3969883641,0,0,0,276 1981.3997262149,0,0,0,275 1981.4024640657,0,0,0,297 1981.4052019165,0,0,0,287 1981.4079397673,1,0,0,222 1981.4106776181,1,0,0,201 1981.4134154689,0,0,0,272 1981.4161533197,0,0,0,281 1981.4188911704,0,0,0,260 1981.4216290212,0,0,0,299 1981.424366872,0,0,0,265 1981.4271047228,1,0,0,264 1981.4298425736,1,0,0,212 1981.4325804244,0,0,0,296 1981.4353182752,0,0,0,296 1981.4380561259,0,0,0,284 1981.4407939767,0,0,0,277 1981.4435318275,0,0,0,296 1981.4462696783,1,0,0,235 1981.4490075291,1,0,0,204 1981.4517453799,0,0,0,257 1981.4544832307,0,0,0,309 1981.4572210815,0,0,0,313 1981.4599589322,0,0,0,334 1981.462696783,0,0,0,299 1981.4654346338,1,0,0,202 1981.4681724846,1,0,0,221 1981.4709103354,0,0,0,271 1981.4736481862,0,0,0,324 1981.476386037,0,0,0,256 1981.4791238878,0,0,0,284 1981.4818617385,0,0,0,286 1981.4845995893,1,0,0,212 1981.4873374401,1,0,0,209 1981.4900752909,0,0,0,299 1981.4928131417,0,0,0,298 1981.4955509925,0,0,0,269 1981.4982888433,0,0,0,296 1981.5010266941,0,0,0,264 1981.5037645448,1,0,0,215 1981.5065023956,1,0,0,208 1981.5092402464,0,0,0,313 1981.5119780972,0,0,0,308 1981.514715948,0,0,0,303 1981.5174537988,0,0,0,260 1981.5201916496,0,0,0,287 1981.5229295004,1,0,0,197 1981.5256673511,1,0,0,186 1981.5284052019,0,0,0,256 1981.5311430527,0,0,0,304 1981.5338809035,0,0,0,285 1981.5366187543,0,0,0,322 1981.5393566051,0,0,0,295 1981.5420944559,1,0,0,217 1981.5448323066,1,0,0,209 1981.5475701574,0,0,0,280 1981.5503080082,0,0,0,300 1981.553045859,0,0,0,302 1981.5557837098,0,0,0,291 1981.5585215606,0,0,0,282 1981.5612594114,1,0,0,185 1981.5639972622,1,0,0,192 1981.5667351129,0,0,0,285 1981.5694729637,0,0,0,302 1981.5722108145,0,0,0,264 1981.5749486653,0,0,0,303 1981.5776865161,0,0,0,252 1981.5804243669,1,0,0,236 1981.5831622177,1,0,0,203 1981.5859000685,0,0,0,279 1981.5886379192,0,0,0,296 1981.59137577,0,0,0,297 1981.5941136208,0,0,0,292 1981.5968514716,0,0,0,293 1981.5995893224,1,0,0,208 1981.6023271732,1,0,0,192 1981.605065024,0,0,0,290 1981.6078028748,0,0,0,287 1981.6105407255,0,0,0,295 1981.6132785763,0,0,0,283 1981.6160164271,0,0,0,291 1981.6187542779,1,0,0,233 1981.6214921287,1,0,0,212 1981.6242299795,0,0,0,269 1981.6269678303,0,0,0,306 1981.629705681,0,0,0,281 1981.6324435318,0,0,0,265 1981.6351813826,0,0,0,314 1981.6379192334,1,0,0,213 1981.6406570842,1,0,0,209 1981.643394935,0,0,0,272 1981.6461327858,0,0,0,275 1981.6488706366,0,0,0,276 1981.6516084873,0,0,0,271 1981.6543463381,0,0,0,275 1981.6570841889,1,0,0,232 1981.6598220397,1,0,0,203 1981.6625598905,0,0,0,257 1981.6652977413,0,0,0,278 1981.6680355921,0,0,0,290 1981.6707734429,0,0,0,279 1981.6735112936,0,0,0,296 1981.6762491444,1,0,0,232 1981.6789869952,1,0,0,225 1981.681724846,0,1,0,211 1981.6844626968,0,0,0,303 1981.6872005476,0,0,0,298 1981.6899383984,0,0,0,304 1981.6926762492,0,0,0,256 1981.6954140999,1,0,0,212 1981.6981519507,1,0,0,233 1981.7008898015,0,0,0,277 1981.7036276523,0,0,0,319 1981.7063655031,0,0,0,306 1981.7091033539,0,0,0,300 1981.7118412047,0,0,0,318 1981.7145790555,1,0,0,216 1981.7173169062,1,0,0,218 1981.720054757,0,0,0,267 1981.7227926078,0,0,0,301 1981.7255304586,0,0,0,300 1981.7282683094,0,0,0,297 1981.7310061602,0,0,0,298 1981.733744011,1,0,0,239 1981.7364818617,1,0,0,200 1981.7392197125,0,0,0,279 1981.7419575633,0,0,0,267 1981.7446954141,0,0,0,288 1981.7474332649,0,0,0,281 1981.7501711157,0,0,0,296 1981.7529089665,1,0,0,216 1981.7556468173,1,0,0,202 1981.758384668,0,0,0,266 1981.7611225188,0,0,0,300 1981.7638603696,0,0,0,279 1981.7665982204,0,0,0,303 1981.7693360712,0,0,0,282 1981.772073922,1,0,0,214 1981.7748117728,1,0,0,171 1981.7775496236,0,1,0,231 1981.7802874743,0,0,0,277 1981.7830253251,0,0,0,280 1981.7857631759,0,0,0,284 1981.7885010267,0,0,0,264 1981.7912388775,1,0,0,193 1981.7939767283,1,0,0,188 1981.7967145791,0,0,0,277 1981.7994524299,0,0,0,278 1981.8021902806,0,0,0,261 1981.8049281314,0,0,0,270 1981.8076659822,0,0,0,265 1981.810403833,1,0,0,214 1981.8131416838,1,0,0,179 1981.8158795346,0,0,0,248 1981.8186173854,0,0,0,276 1981.8213552361,0,0,0,234 1981.8240930869,0,0,0,280 1981.8268309377,0,0,0,270 1981.8295687885,1,0,0,220 1981.8323066393,1,0,0,197 1981.8350444901,0,0,0,237 1981.8377823409,0,0,0,279 1981.8405201917,0,0,0,266 1981.8432580424,0,0,0,276 1981.8459958932,0,0,0,280 1981.848733744,1,0,0,241 1981.8514715948,1,0,0,195 1981.8542094456,0,0,0,248 1981.8569472964,0,0,0,251 1981.8596851472,0,0,0,228 1981.862422998,0,0,0,251 1981.8651608487,0,0,0,243 1981.8678986995,1,0,0,220 1981.8706365503,1,0,0,166 1981.8733744011,0,0,0,249 1981.8761122519,0,0,0,268 1981.8788501027,0,0,0,236 1981.8815879535,0,0,0,262 1981.8843258043,0,0,0,284 1981.887063655,1,0,0,206 1981.8898015058,1,0,0,185 1981.8925393566,0,0,0,234 1981.8952772074,0,0,0,251 1981.8980150582,0,0,0,266 1981.900752909,0,0,0,258 1981.9034907598,0,0,0,270 1981.9062286105,1,0,0,203 1981.9089664613,1,0,0,198 1981.9117043121,0,0,0,245 1981.9144421629,0,0,0,255 1981.9171800137,0,0,0,278 1981.9199178645,0,0,0,233 1981.9226557153,0,0,0,269 1981.9253935661,1,0,0,190 1981.9281314168,1,0,0,205 1981.9308692676,0,0,0,220 1981.9336071184,0,0,0,252 1981.9363449692,0,0,0,248 1981.93908282,0,0,0,271 1981.9418206708,0,0,0,282 1981.9445585216,1,0,0,214 1981.9472963724,1,0,0,204 1981.9500342231,0,0,0,252 1981.9527720739,0,0,0,249 1981.9555099247,0,0,0,275 1981.9582477755,0,0,0,292 1981.9609856263,0,0,0,247 1981.9637234771,1,0,0,194 1981.9664613279,1,0,0,195 1981.9691991787,0,0,0,270 1981.9719370294,0,0,0,266 1981.9746748802,0,0,0,271 1981.977412731,0,0,0,180 1981.9801505818,0,0,0,155 1981.9828884326,1,0,0,187 1981.9856262834,1,0,0,201 1981.9883641342,0,0,0,251 1981.991101985,0,0,0,271 1981.9938398357,0,0,0,285 1981.9965776865,0,0,0,230 1982,0,0,0,198 1982.0027378508,1,0,0,197 1982.0054757016,1,0,0,215 1982.0082135524,0,0,0,241 1982.0109514032,0,0,0,277 1982.0136892539,0,0,0,280 1982.0164271047,0,0,0,257 1982.0191649555,0,0,0,271 1982.0219028063,1,0,0,222 1982.0246406571,1,0,0,194 1982.0273785079,0,0,0,266 1982.0301163587,0,0,0,254 1982.0328542095,0,0,0,244 1982.0355920602,0,0,0,270 1982.038329911,0,0,0,253 1982.0410677618,1,0,0,202 1982.0438056126,1,0,0,186 1982.0465434634,0,0,0,264 1982.0492813142,0,0,0,256 1982.052019165,0,0,0,236 1982.0547570158,0,0,0,293 1982.0574948665,0,0,0,238 1982.0602327173,1,0,0,200 1982.0629705681,1,0,0,204 1982.0657084189,0,0,0,264 1982.0684462697,0,0,0,281 1982.0711841205,0,0,0,257 1982.0739219713,0,0,0,246 1982.076659822,0,0,0,288 1982.0793976728,1,0,0,191 1982.0821355236,1,0,0,212 1982.0848733744,0,0,0,258 1982.0876112252,0,0,0,276 1982.090349076,0,0,0,267 1982.0930869268,0,0,0,280 1982.0958247776,0,0,0,291 1982.0985626283,1,0,0,189 1982.1013004791,1,0,0,191 1982.1040383299,0,0,0,267 1982.1067761807,0,0,0,293 1982.1095140315,0,0,0,265 1982.1122518823,0,0,0,278 1982.1149897331,0,0,0,254 1982.1177275839,1,0,0,199 1982.1204654346,1,0,0,211 1982.1232032854,0,0,0,272 1982.1259411362,0,0,0,293 1982.128678987,0,0,0,251 1982.1314168378,0,0,0,268 1982.1341546886,0,0,0,291 1982.1368925394,1,0,0,218 1982.1396303902,1,0,0,198 1982.1423682409,0,0,0,285 1982.1451060917,0,0,0,331 1982.1478439425,0,0,0,290 1982.1505817933,0,0,0,296 1982.1533196441,0,0,0,284 1982.1560574949,1,0,0,197 1982.1587953457,1,0,0,198 1982.1615331964,0,0,0,274 1982.1642710472,0,0,0,274 1982.167008898,0,0,0,270 1982.1697467488,0,0,0,253 1982.1724845996,0,0,0,301 1982.1752224504,1,0,0,225 1982.1779603012,1,0,0,194 1982.180698152,0,0,0,247 1982.1834360027,0,0,0,283 1982.1861738535,0,0,0,277 1982.1889117043,0,0,0,275 1982.1916495551,0,0,0,293 1982.1943874059,1,0,0,219 1982.1971252567,1,0,0,220 1982.1998631075,0,0,0,278 1982.2026009583,0,0,0,266 1982.205338809,0,0,0,280 1982.2080766598,0,0,0,261 1982.2108145106,0,0,0,282 1982.2135523614,1,0,0,205 1982.2162902122,1,0,0,213 1982.219028063,0,0,0,279 1982.2217659138,0,0,0,303 1982.2245037646,0,0,0,269 1982.2272416153,0,0,0,272 1982.2299794661,0,0,0,297 1982.2327173169,1,0,0,228 1982.2354551677,1,0,0,204 1982.2381930185,0,0,0,266 1982.2409308693,0,0,0,304 1982.2436687201,0,0,0,286 1982.2464065709,0,0,0,277 1982.2491444216,0,0,0,287 1982.2518822724,1,0,0,243 1982.2546201232,1,0,0,224 1982.257357974,0,0,0,272 1982.2600958248,0,0,0,316 1982.2628336756,0,0,0,301 1982.2655715264,0,0,0,289 1982.2683093771,0,0,1,267 1982.2710472279,1,0,1,242 1982.2737850787,1,0,1,220 1982.2765229295,0,0,1,207 1982.2792607803,0,0,0,295 1982.2819986311,0,0,0,307 1982.2847364819,0,0,0,307 1982.2874743327,0,0,0,310 1982.2902121834,1,0,0,219 1982.2929500342,1,0,0,203 1982.295687885,0,0,0,252 1982.2984257358,0,0,0,305 1982.3011635866,0,0,0,301 1982.3039014374,0,0,0,259 1982.3066392882,0,0,0,280 1982.309377139,1,0,0,207 1982.3121149897,1,0,0,202 1982.3148528405,0,0,0,267 1982.3175906913,0,0,0,300 1982.3203285421,0,0,0,286 1982.3230663929,0,0,0,303 1982.3258042437,0,0,0,276 1982.3285420945,1,0,0,251 1982.3312799453,1,0,0,211 1982.334017796,0,0,0,296 1982.3367556468,0,0,0,280 1982.3394934976,0,0,0,290 1982.3422313484,0,0,0,289 1982.3449691992,0,0,0,289 1982.34770705,1,0,0,232 1982.3504449008,1,0,0,197 1982.3531827515,0,0,0,252 1982.3559206023,0,0,0,277 1982.3586584531,0,0,0,292 1982.3613963039,0,0,0,273 1982.3641341547,0,0,0,297 1982.3668720055,1,0,0,219 1982.3696098563,1,0,0,200 1982.3723477071,0,0,0,274 1982.3750855578,0,0,0,284 1982.3778234086,0,0,0,279 1982.3805612594,0,0,0,280 1982.3832991102,0,0,0,308 1982.386036961,1,0,0,219 1982.3887748118,1,0,0,192 1982.3915126626,0,1,0,249 1982.3942505134,0,0,0,255 1982.3969883641,0,0,0,270 1982.3997262149,0,0,0,303 1982.4024640657,0,0,0,262 1982.4052019165,1,0,0,206 1982.4079397673,1,0,0,221 1982.4106776181,0,0,0,267 1982.4134154689,0,0,0,273 1982.4161533197,0,0,0,252 1982.4188911704,0,0,0,253 1982.4216290212,0,0,0,257 1982.424366872,1,0,0,209 1982.4271047228,1,0,0,216 1982.4298425736,0,0,0,274 1982.4325804244,0,0,0,311 1982.4353182752,0,0,0,252 1982.438056126,0,0,0,321 1982.4407939767,0,0,0,300 1982.4435318275,1,0,0,208 1982.4462696783,1,0,0,191 1982.4490075291,0,0,0,257 1982.4517453799,0,0,0,275 1982.4544832307,0,0,0,284 1982.4572210815,0,0,0,275 1982.4599589322,0,0,0,286 1982.462696783,1,0,0,216 1982.4654346338,1,0,0,193 1982.4681724846,0,0,0,262 1982.4709103354,0,0,0,276 1982.4736481862,0,0,0,260 1982.476386037,0,0,0,226 1982.4791238878,0,0,0,246 1982.4818617385,1,0,0,195 1982.4845995893,1,0,0,197 1982.4873374401,0,0,0,299 1982.4900752909,0,0,0,280 1982.4928131417,0,0,0,281 1982.4955509925,0,0,0,253 1982.4982888433,0,0,0,202 1982.5010266941,1,0,0,203 1982.5037645448,1,0,0,197 1982.5065023956,0,0,0,250 1982.5092402464,0,0,0,250 1982.5119780972,0,0,0,286 1982.514715948,0,0,0,301 1982.5174537988,0,0,0,281 1982.5201916496,1,0,0,199 1982.5229295004,1,0,0,196 1982.5256673511,0,0,0,267 1982.5284052019,0,0,0,262 1982.5311430527,0,0,0,286 1982.5338809035,0,0,0,291 1982.5366187543,0,0,0,262 1982.5393566051,1,0,0,233 1982.5420944559,1,0,0,208 1982.5448323066,0,0,0,264 1982.5475701574,0,0,0,289 1982.5503080082,0,0,0,276 1982.553045859,0,0,0,294 1982.5557837098,0,0,0,267 1982.5585215606,1,0,0,193 1982.5612594114,1,0,0,213 1982.5639972622,0,0,0,257 1982.5667351129,0,0,0,259 1982.5694729637,0,0,0,274 1982.5722108145,0,0,0,279 1982.5749486653,0,0,0,247 1982.5776865161,1,0,0,215 1982.5804243669,1,0,0,148 1982.5831622177,0,0,0,231 1982.5859000685,0,0,0,265 1982.5886379192,0,0,0,235 1982.59137577,0,0,0,254 1982.5941136208,0,0,0,253 1982.5968514716,1,0,0,173 1982.5995893224,1,0,0,193 1982.6023271732,0,0,0,288 1982.605065024,0,0,0,287 1982.6078028748,0,0,0,273 1982.6105407255,0,0,0,268 1982.6132785763,0,0,0,279 1982.6160164271,1,0,0,206 1982.6187542779,1,0,0,183 1982.6214921287,0,0,0,259 1982.6242299795,0,0,0,290 1982.6269678303,0,0,0,251 1982.629705681,0,0,0,290 1982.6324435318,0,0,0,270 1982.6351813826,1,0,0,223 1982.6379192334,1,0,0,184 1982.6406570842,0,0,0,239 1982.643394935,0,0,0,261 1982.6461327858,0,0,0,258 1982.6488706366,0,0,0,247 1982.6516084873,0,0,0,268 1982.6543463381,1,0,0,200 1982.6570841889,1,0,0,200 1982.6598220397,0,0,0,269 1982.6625598905,0,0,0,279 1982.6652977413,0,0,0,237 1982.6680355921,0,0,0,259 1982.6707734429,0,0,0,262 1982.6735112936,1,0,0,213 1982.6762491444,1,0,0,183 1982.6789869952,0,1,0,215 1982.681724846,0,0,0,260 1982.6844626968,0,0,0,265 1982.6872005476,0,0,0,271 1982.6899383984,0,0,0,295 1982.6926762492,1,0,0,223 1982.6954140999,1,0,0,196 1982.6981519507,0,0,0,279 1982.7008898015,0,0,0,279 1982.7036276523,0,0,0,245 1982.7063655031,0,0,0,257 1982.7091033539,0,0,0,262 1982.7118412047,1,0,0,220 1982.7145790555,1,0,0,219 1982.7173169062,0,0,0,301 1982.720054757,0,0,0,272 1982.7227926078,0,0,0,290 1982.7255304586,0,0,0,292 1982.7282683094,0,0,0,295 1982.7310061602,1,0,0,237 1982.733744011,1,0,0,194 1982.7364818617,0,0,0,273 1982.7392197125,0,0,0,300 1982.7419575633,0,0,0,313 1982.7446954141,0,0,0,291 1982.7474332649,0,0,0,295 1982.7501711157,1,0,0,203 1982.7529089665,1,0,0,198 1982.7556468173,0,0,0,240 1982.758384668,0,0,0,275 1982.7611225188,0,0,0,261 1982.7638603696,0,0,0,279 1982.7665982204,0,0,0,279 1982.7693360712,1,0,0,210 1982.772073922,1,0,0,178 1982.7748117728,0,1,0,208 1982.7775496236,0,0,0,280 1982.7802874743,0,0,0,266 1982.7830253251,0,0,0,269 1982.7857631759,0,0,0,234 1982.7885010267,1,0,0,208 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1983.1067761807,0,0,0,267 1983.1095140315,0,0,0,251 1983.1122518823,0,0,0,260 1983.1149897331,1,0,0,208 1983.1177275839,1,0,0,149 1983.1204654346,0,0,0,245 1983.1232032854,0,0,0,268 1983.1259411362,0,0,0,260 1983.128678987,0,0,0,247 1983.1314168378,0,0,0,251 1983.1341546886,1,0,0,205 1983.1368925394,1,0,0,179 1983.1396303902,0,0,0,203 1983.1423682409,0,0,0,275 1983.1451060917,0,0,0,223 1983.1478439425,0,0,0,235 1983.1505817933,0,0,0,263 1983.1533196441,1,0,0,219 1983.1560574949,1,0,0,152 1983.1587953457,0,0,0,227 1983.1615331964,0,0,0,318 1983.1642710472,0,0,0,268 1983.167008898,0,0,0,250 1983.1697467488,0,0,0,245 1983.1724845996,1,0,0,205 1983.1752224504,1,0,0,217 1983.1779603012,0,0,0,266 1983.180698152,0,0,0,267 1983.1834360027,0,0,0,259 1983.1861738535,0,0,0,288 1983.1889117043,0,0,0,284 1983.1916495551,1,0,0,193 1983.1943874059,1,0,0,224 1983.1971252567,0,0,0,288 1983.1998631075,0,0,0,280 1983.2026009583,0,0,0,249 1983.205338809,0,0,0,254 1983.2080766598,0,0,0,241 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1983.4188911704,0,0,0,258 1983.4216290212,1,0,0,213 1983.424366872,1,0,0,194 1983.4271047228,0,0,0,278 1983.4298425736,0,0,0,258 1983.4325804244,0,0,0,267 1983.4353182752,0,0,0,293 1983.4380561259,0,0,0,269 1983.4407939767,1,0,0,208 1983.4435318275,1,0,0,199 1983.4462696783,0,0,0,288 1983.4490075291,0,0,0,278 1983.4517453799,0,0,0,286 1983.4544832307,0,0,0,296 1983.4572210815,0,0,0,286 1983.4599589322,1,0,0,194 1983.462696783,1,0,0,174 1983.4654346338,0,0,0,257 1983.4681724846,0,0,0,305 1983.4709103354,0,0,0,260 1983.4736481862,0,0,0,285 1983.476386037,0,0,0,229 1983.4791238878,1,0,0,197 1983.4818617385,1,0,0,166 1983.4845995893,0,0,0,264 1983.4873374401,0,0,0,272 1983.4900752909,0,0,0,279 1983.4928131417,0,0,0,287 1983.4955509925,0,0,0,235 1983.4982888433,1,0,0,212 1983.5010266941,1,0,0,185 1983.5037645448,0,0,0,267 1983.5065023956,0,0,0,265 1983.5092402464,0,0,0,252 1983.5119780972,0,0,0,294 1983.514715948,0,0,0,263 1983.5174537988,1,0,0,194 1983.5201916496,1,0,0,189 1983.5229295004,0,0,0,256 1983.5256673511,0,0,0,256 1983.5284052019,0,0,0,231 1983.5311430527,0,0,0,274 1983.5338809035,0,0,0,272 1983.5366187543,1,0,0,179 1983.5393566051,1,0,0,203 1983.5420944559,0,0,0,237 1983.5448323066,0,0,0,269 1983.5475701574,0,0,0,244 1983.5503080082,0,0,0,262 1983.553045859,0,0,0,257 1983.5557837098,1,0,0,184 1983.5585215606,1,0,0,182 1983.5612594114,0,0,0,251 1983.5639972622,0,0,0,239 1983.5667351129,0,0,0,265 1983.5694729637,0,0,0,266 1983.5722108145,0,0,0,306 1983.5749486653,1,0,0,184 1983.5776865161,1,0,0,181 1983.5804243669,0,0,0,258 1983.5831622177,0,0,0,261 1983.5859000685,0,0,0,268 1983.5886379192,0,0,0,284 1983.59137577,0,0,0,258 1983.5941136208,1,0,0,198 1983.5968514716,1,0,0,181 1983.5995893224,0,0,0,242 1983.6023271732,0,0,0,259 1983.605065024,0,0,0,265 1983.6078028748,0,0,0,240 1983.6105407255,0,0,0,252 1983.6132785763,1,0,0,203 1983.6160164271,1,0,0,173 1983.6187542779,0,0,0,257 1983.6214921287,0,0,0,265 1983.6242299795,0,0,0,254 1983.6269678303,0,0,0,268 1983.629705681,0,0,0,250 1983.6324435318,1,0,0,189 1983.6351813826,1,0,0,206 1983.6379192334,0,0,0,238 1983.6406570842,0,0,0,269 1983.643394935,0,0,0,252 1983.6461327858,0,0,0,261 1983.6488706366,0,0,0,264 1983.6516084873,1,0,0,208 1983.6543463381,1,0,0,180 1983.6570841889,0,0,0,240 1983.6598220397,0,0,0,251 1983.6625598905,0,0,0,261 1983.6652977413,0,0,0,234 1983.6680355921,0,0,0,252 1983.6707734429,1,0,0,200 1983.6735112936,1,0,0,219 1983.6762491444,0,1,0,199 1983.6789869952,0,0,0,263 1983.681724846,0,0,0,286 1983.6844626968,0,0,0,271 1983.6872005476,0,0,0,276 1983.6899383984,1,0,0,239 1983.6926762492,1,0,0,188 1983.6954140999,0,0,0,271 1983.6981519507,0,0,0,262 1983.7008898015,0,0,0,252 1983.7036276523,0,0,0,293 1983.7063655031,0,0,0,269 1983.7091033539,1,0,0,208 1983.7118412047,1,0,0,194 1983.7145790555,0,0,0,264 1983.7173169062,0,0,0,283 1983.720054757,0,0,0,306 1983.7227926078,0,0,0,281 1983.7255304586,0,0,0,267 1983.7282683094,1,0,0,202 1983.7310061602,1,0,0,176 1983.733744011,0,0,0,248 1983.7364818617,0,0,0,265 1983.7392197125,0,0,0,262 1983.7419575633,0,0,0,297 1983.7446954141,0,0,0,307 1983.7474332649,1,0,0,187 1983.7501711157,1,0,0,198 1983.7529089665,0,0,0,248 1983.7556468173,0,0,0,291 1983.758384668,0,0,0,284 1983.7611225188,0,0,0,245 1983.7638603696,0,0,0,273 1983.7665982204,1,0,0,201 1983.7693360712,1,0,0,189 1983.772073922,0,1,0,188 1983.7748117728,0,0,0,250 1983.7775496236,0,0,0,273 1983.7802874743,0,0,0,269 1983.7830253251,0,0,0,305 1983.7857631759,1,0,0,221 1983.7885010267,1,0,0,189 1983.7912388775,0,0,0,239 1983.7939767283,0,0,0,260 1983.7967145791,0,0,0,230 1983.7994524299,0,0,0,250 1983.8021902806,0,0,0,256 1983.8049281314,1,0,0,204 1983.8076659822,1,0,0,172 1983.810403833,0,0,0,250 1983.8131416838,0,0,0,277 1983.8158795346,0,0,0,253 1983.8186173854,0,0,0,269 1983.8213552361,0,0,0,236 1983.8240930869,1,0,0,183 1983.8268309377,1,0,0,185 1983.8295687885,0,0,0,228 1983.8323066393,0,0,0,256 1983.8350444901,0,0,0,288 1983.8377823409,0,0,0,251 1983.8405201917,0,0,0,281 1983.8432580424,1,0,0,204 1983.8459958932,1,0,0,191 1983.848733744,0,0,0,222 1983.8514715948,0,0,0,270 1983.8542094456,0,0,0,256 1983.8569472964,0,0,0,257 1983.8596851472,0,0,0,269 1983.862422998,1,0,0,203 1983.8651608487,1,0,0,188 1983.8678986995,0,0,0,228 1983.8706365503,0,0,0,269 1983.8733744011,0,0,0,259 1983.8761122519,0,0,0,268 1983.8788501027,0,0,0,229 1983.8815879535,1,0,0,173 1983.8843258043,1,0,0,178 1983.887063655,0,0,0,262 1983.8898015058,0,0,0,270 1983.8925393566,0,0,0,256 1983.8952772074,0,0,0,254 1983.8980150582,0,0,0,279 1983.900752909,1,0,0,197 1983.9034907598,1,0,0,193 1983.9062286105,0,0,0,212 1983.9089664613,0,0,0,259 1983.9117043121,0,0,0,278 1983.9144421629,0,0,0,251 1983.9171800137,0,0,0,234 1983.9199178645,1,0,0,171 1983.9226557153,1,0,0,168 1983.9253935661,0,0,0,223 1983.9281314168,0,0,0,252 1983.9308692676,0,0,0,274 1983.9336071184,0,0,0,285 1983.9363449692,0,0,0,244 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1989.810403833,0,,0,226 1989.8131416838,0,,0,272 1989.8158795346,0,,0,293 1989.8186173854,0,,0,243 1989.8213552361,1,,0,185 1989.8240930869,1,,0,195 1989.8268309377,0,,0,215 1989.8295687885,0,,0,220 1989.8323066393,0,,0,280 1989.8350444901,0,,0,255 1989.8377823409,0,,0,259 1989.8405201917,1,,0,193 1989.8432580424,1,,0,205 1989.8459958932,0,,0,242 1989.848733744,0,,0,280 1989.8514715948,0,,0,262 1989.8542094456,0,,0,272 1989.8569472964,0,,0,265 1989.8596851472,1,,0,201 1989.862422998,1,,0,184 1989.8651608487,0,,0,232 1989.8678986995,0,,0,241 1989.8706365503,0,,0,274 1989.8733744011,0,,0,252 1989.8761122519,0,,0,275 1989.8788501027,1,,0,182 1989.8815879535,1,,0,199 1989.8843258043,0,,0,259 1989.887063655,0,,0,287 1989.8898015058,0,,0,266 1989.8925393566,0,,0,281 1989.8952772074,0,,0,265 1989.8980150582,1,,0,209 1989.900752909,1,,0,187 1989.9034907598,0,,0,250 1989.9062286105,0,,0,269 1989.9089664613,0,,0,292 1989.9117043121,0,,0,246 1989.9144421629,0,,0,259 1989.9171800137,1,,0,187 1989.9199178645,1,,0,196 1989.9226557153,0,,0,267 1989.9253935661,0,,0,258 1989.9281314168,0,,0,249 1989.9308692676,0,,0,277 1989.9336071184,0,,0,265 1989.9363449692,1,,0,200 1989.93908282,1,,0,180 1989.9418206708,0,,0,220 1989.9445585216,0,,0,267 1989.9472963724,0,,0,255 1989.9500342231,0,,0,292 1989.9527720739,0,,0,278 1989.9555099247,1,,0,178 1989.9582477755,1,,0,184 1989.9609856263,0,,0,270 1989.9637234771,0,,0,308 1989.9664613279,0,,0,276 1989.9691991787,0,,0,288 1989.9719370294,0,,0,258 1989.9746748802,1,,0,167 1989.977412731,1,,0,178 1989.9801505818,0,,0,180 1989.9828884326,0,,0,174 1989.9856262834,0,,0,269 1989.9883641342,0,,0,332 1989.991101985,0,,0,309 1989.9938398357,1,,0,213 1989.9965776865,1,,0,189 1990,0,,0,211 1990.0027378508,0,,0,183 1990.0054757016,0,,0,251 1990.0082135524,0,,0,298 1990.0109514032,0,,0,281 1990.0136892539,1,,0,210 1990.0164271047,1,,0,213 1990.0191649555,0,,0,258 1990.0219028063,0,,0,247 1990.0246406571,0,,0,288 1990.0273785079,0,,0,272 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1990.3586584531,1,,0,230 1990.3613963039,1,,0,211 1990.3641341547,0,,0,285 1990.3668720055,0,,0,313 1990.3696098563,0,,0,319 1990.3723477071,0,,0,298 1990.3750855578,0,,0,290 1990.3778234086,1,,0,223 1990.3805612594,1,,0,225 1990.3832991102,0,,0,240 1990.386036961,0,,0,278 1990.3887748118,0,,0,339 1990.3915126626,0,,0,330 1990.3942505134,0,,0,324 1990.3969883641,1,,0,234 1990.3997262149,1,,0,208 1990.4024640657,0,,0,296 1990.4052019165,0,,0,318 1990.4079397673,0,,0,317 1990.4106776181,0,,0,330 1990.4134154689,0,,0,305 1990.4161533197,1,,0,253 1990.4188911704,1,,0,225 1990.4216290212,0,,0,271 1990.424366872,0,,0,286 1990.4271047228,0,,0,319 1990.4298425736,0,,0,287 1990.4325804244,0,,0,295 1990.4353182752,1,,0,247 1990.438056126,1,,0,228 1990.4407939767,0,,0,267 1990.4435318275,0,,0,330 1990.4462696783,0,,0,304 1990.4490075291,0,,0,339 1990.4517453799,0,,0,280 1990.4544832307,1,,0,253 1990.4572210815,1,,0,227 1990.4599589322,0,,0,305 1990.462696783,0,,0,305 1990.4654346338,0,,0,321 1990.4681724846,0,,0,312 1990.4709103354,0,,0,305 1990.4736481862,1,,0,211 1990.476386037,1,,0,215 1990.4791238878,0,,0,215 1990.4818617385,0,,0,287 1990.4845995893,0,,0,311 1990.4873374401,0,,0,318 1990.4900752909,0,,0,310 1990.4928131417,1,,0,216 1990.4955509925,1,,0,239 1990.4982888433,0,,0,223 1990.5010266941,0,,0,290 1990.5037645448,0,,0,295 1990.5065023956,0,,0,337 1990.5092402464,0,,0,278 1990.5119780972,1,,0,201 1990.514715948,1,,0,200 1990.5174537988,0,,0,279 1990.5201916496,0,,0,334 1990.5229295004,0,,0,301 1990.5256673511,0,,0,311 1990.5284052019,0,,0,302 1990.5311430527,1,,0,231 1990.5338809035,1,,0,237 1990.5366187543,0,,0,252 1990.5393566051,0,,0,306 1990.5420944559,0,,0,305 1990.5448323066,0,,0,298 1990.5475701574,0,,0,283 1990.5503080082,1,,0,220 1990.553045859,1,,0,226 1990.5557837098,0,,0,286 1990.5585215606,0,,0,295 1990.5612594114,0,,0,307 1990.5639972622,0,,0,290 1990.5667351129,0,,0,263 1990.5694729637,1,,0,210 1990.5722108145,1,,0,236 1990.5749486653,0,,0,299 1990.5776865161,0,,0,306 1990.5804243669,0,,0,305 1990.5831622177,0,,0,286 1990.5859000685,0,,0,319 1990.5886379192,1,,0,232 1990.59137577,1,,0,216 1990.5941136208,0,,0,245 1990.5968514716,0,,0,305 1990.5995893224,0,,0,289 1990.6023271732,0,,0,314 1990.605065024,0,,0,297 1990.6078028748,1,,0,212 1990.6105407255,1,,0,218 1990.6132785763,0,,0,264 1990.6160164271,0,,0,288 1990.6187542779,0,,0,312 1990.6214921287,0,,0,303 1990.6242299795,0,,0,276 1990.6269678303,1,,0,239 1990.629705681,1,,0,213 1990.6324435318,0,,0,270 1990.6351813826,0,,0,316 1990.6379192334,0,,0,296 1990.6406570842,0,,0,298 1990.643394935,0,,0,286 1990.6461327858,1,,0,264 1990.6488706366,1,,0,218 1990.6516084873,0,,0,268 1990.6543463381,0,,0,282 1990.6570841889,0,,0,318 1990.6598220397,0,,0,280 1990.6625598905,0,,0,302 1990.6652977413,1,,0,224 1990.6680355921,1,,0,175 1990.6707734429,0,,0,218 1990.6735112936,0,,0,284 1990.6762491444,0,,0,303 1990.6789869952,0,,0,293 1990.681724846,0,,0,350 1990.6844626968,1,,0,226 1990.6872005476,1,,0,217 1990.6899383984,0,,0,319 1990.6926762492,0,,0,294 1990.6954140999,0,,0,309 1990.6981519507,0,,0,315 1990.7008898015,0,,0,309 1990.7036276523,1,,0,241 1990.7063655031,1,,0,274 1990.7091033539,0,,0,275 1990.7118412047,0,,0,354 1990.7145790555,0,,0,278 1990.7173169062,0,,0,329 1990.720054757,0,,0,305 1990.7227926078,1,,0,256 1990.7255304586,1,,0,217 1990.7282683094,0,,0,341 1990.7310061602,0,,0,296 1990.733744011,0,,0,344 1990.7364818617,0,,0,331 1990.7392197125,0,,0,317 1990.7419575633,1,,0,229 1990.7446954141,1,,0,257 1990.7474332649,0,,0,285 1990.7501711157,0,,0,279 1990.7529089665,0,,0,301 1990.7556468173,0,,0,290 1990.758384668,0,,0,318 1990.7611225188,1,,0,203 1990.7638603696,1,,0,200 1990.7665982204,0,,0,229 1990.7693360712,0,,0,292 1990.772073922,0,,0,309 1990.7748117728,0,,0,282 1990.7775496236,0,,0,303 1990.7802874743,1,,0,224 1990.7830253251,1,,0,215 1990.7857631759,0,,0,268 1990.7885010267,0,,0,266 1990.7912388775,0,,0,283 1990.7939767283,0,,0,252 1990.7967145791,0,,0,289 1990.7994524299,1,,0,201 1990.8021902806,1,,0,220 1990.8049281314,0,,0,265 1990.8076659822,0,,0,288 1990.810403833,0,,0,284 1990.8131416838,0,,0,299 1990.8158795346,0,,0,289 1990.8186173854,1,,0,206 1990.8213552361,1,,0,205 1990.8240930869,0,,0,255 1990.8268309377,0,,0,282 1990.8295687885,0,,0,274 1990.8323066393,0,,0,296 1990.8350444901,0,,0,293 1990.8377823409,1,,0,209 1990.8405201917,1,,0,197 1990.8432580424,0,,0,262 1990.8459958932,0,,0,308 1990.848733744,0,,0,260 1990.8514715948,0,,0,290 1990.8542094456,0,,0,253 1990.8569472964,1,,0,206 1990.8596851472,1,,0,194 1990.862422998,0,,0,262 1990.8651608487,0,,0,293 1990.8678986995,0,,0,256 1990.8706365503,0,,0,247 1990.8733744011,0,,0,265 1990.8761122519,1,,0,192 1990.8788501027,1,,0,217 1990.8815879535,0,,0,255 1990.8843258043,0,,0,268 1990.887063655,0,,0,266 1990.8898015058,0,,0,265 1990.8925393566,0,,0,242 1990.8952772074,1,,0,187 1990.8980150582,1,,0,186 1990.900752909,0,,0,248 1990.9034907598,0,,0,259 1990.9062286105,0,,0,264 1990.9089664613,0,,0,255 1990.9117043121,0,,0,260 1990.9144421629,1,,0,214 1990.9171800137,1,,0,177 1990.9199178645,0,,0,284 1990.9226557153,0,,0,261 1990.9253935661,0,,0,309 1990.9281314168,0,,0,295 1990.9308692676,0,,0,273 1990.9336071184,1,,0,181 1990.9363449692,1,,0,221 1990.93908282,0,,0,249 1990.9418206708,0,,0,256 1990.9445585216,0,,0,279 1990.9472963724,0,,0,284 1990.9500342231,0,,0,282 1990.9527720739,1,,0,193 1990.9555099247,1,,0,182 1990.9582477755,0,,0,258 1990.9609856263,0,,0,309 1990.9637234771,0,,0,311 1990.9664613279,0,,0,285 1990.9691991787,0,,0,264 1990.9719370294,1,,0,207 1990.9746748802,1,,0,174 1990.977412731,0,,0,181 1990.9801505818,0,,0,188 1990.9828884326,0,,0,221 1990.9856262834,0,,0,316 1990.9883641342,0,,0,335 1990.991101985,1,,0,237 1990.9938398357,1,,0,229 1990.9965776865,0,,0,218 ================================================ FILE: code/decomp_birth.m ================================================ function decomp_birth(savefigs) % % A simple demo script to show how to build a GP model in a realistic scenario. % % David Duvenaud % March 2014 if nargin < 1; savefigs = false; end randn('state',0); rand('state',0); show_samples = false; % Show samples from the posterior. % How to save figure. decompfigsdir = '../figures/worked-example/'; fileprefix = [decompfigsdir 'births']; dataset_name = 'data/quebec/quebec-all.mat'; load(dataset_name); feature_names = {'time', 'Weekend', 'Floating holiday' , 'Easter'}; % Fix dates X(:,1) = linspace(min(X(:,1)),max(X(:,1)), length(X)); X = X(end-2000:end,:); y = y(end-2000:end); % Normalize the data. %X = X - repmat(mean(X), size(X,1), 1 ); %X = X ./ repmat(std(X), size(X,1), 1 ); y = y - mean(y); %y = y / std(y); [N,D] = size(X); % Easy part of model set up: meanfunc = {'meanZero'}; inference = @infExact; likfunc = @likGauss; hyp.mean = []; hyp.lik = ones(1,eval(likfunc())).*log(0.1); longterm_ell = 1.5; longterm_sf = 14; longterm_cov = { 'covMask', { 1, 'covSEiso'}}; longterm_hyp = [log(longterm_ell); log(longterm_sf)]; weekly_ell = 1; weekly_sf = 55; weekly_period = 1/52; weekly_se_ell = 2; weekly_se_sf = 1; %weekly_period_cov = { 'covMask', { 1, 'covPeriodic'}}; weekly_period_cov = { 'covMask', { 1, {'covProd', {'covPeriodic', 'covSEiso'}}}}; weekly_period_hyp = [log(weekly_ell); log(weekly_period); log(weekly_sf); ... log(weekly_se_ell); log(weekly_se_sf)]; yearly_ell = 0.5; yearly_sf = 17.5; yearly_period = 1; yearly_se_ell = 5.7; yearly_se_sf = 1; yearly_period_cov = { 'covMask', { 1, {'covProd', {'covPeriodic', 'covSEiso'}}}}; yearly_period_hyp = [log(yearly_ell); log(yearly_period); log(yearly_sf); ... log(yearly_se_ell); log(yearly_se_sf)]; %yearly_short_ell = 1/365; yearly_short_sf = 1; %yearly_period_short_cov = { 'covMask', { 1, 'covPeriodic'}}; %yearly_period_short_hyp = [log(yearly_short_ell); log(yearly_period); log(yearly_short_sf)]; %medterm_ell = 0.2; medterm_sf = 10; %medterm_cov = { 'covMask', { 1, 'covSEiso'}}; %medterm_hyp = [log(medterm_ell); log(medterm_sf)]; shortterm_ell = 0.04; shortterm_sf = 10; shortterm_cov = { 'covMask', { 1, 'covSEiso'}}; shortterm_hyp = [log(shortterm_ell); log(shortterm_sf)]; % Now construct the kernel. kernel_components = cell(0); kernel_hypers = cell(0); kernel_names = cell(0); kernel_names{end+1} = 'Long-term'; kernel_components{end+1} = longterm_cov; kernel_hypers{end+1} = longterm_hyp; kernel_names{end+1} = 'Weekly'; kernel_components{end+1} = weekly_period_cov; kernel_hypers{end+1} = weekly_period_hyp; kernel_names{end+1} = 'Yearly'; kernel_components{end+1} = yearly_period_cov; kernel_hypers{end+1} = yearly_period_hyp; %kernel_components{end+1} = yearly_period_short_cov; %kernel_hypers{end+1} = yearly_period_short_hyp; %kernel_components{end+1} = medterm_cov; %kernel_hypers{end+1} = medterm_hyp; kernel_names{end+1} = 'Short-term'; kernel_components{end+1} = shortterm_cov; kernel_hypers{end+1} = shortterm_hyp; % concatenate all additive components complete_cov = { 'covSum', kernel_components }; hyp.cov = unwrap(kernel_hypers); subset = randperm(N, 1000); %hyp = minimize(hyp, @gp, -100, inference, meanfunc, complete_cov, likfunc, ... %X(subset, :), y(subset)); load('hypers.mat', 'hyp'); % Pack up hyperparameters again. kernel_hypers = rewrap(kernel_hypers, hyp.cov); long_hypers = kernel_hypers{1}; fprintf('Estimated long-term ell: %f\n', exp(long_hypers(1))); fprintf('Estimated long-term sf: %f\n', exp(long_hypers(2))); fprintf('\n'); week_hypers = kernel_hypers{2}; fprintf('Estimated weekly ell: %f\n', exp(week_hypers(1)) * exp(week_hypers(5))); fprintf('Estimated weekly frequency: %f\n', 1/exp(week_hypers(2))); fprintf('Estimated weekly sf: %f\n', exp(week_hypers(3))); fprintf('Estimated weekly se ell: %f\n', exp(week_hypers(4))); fprintf('\n'); year_hypers = kernel_hypers{3}; fprintf('Estimated yearly ell: %f\n', exp(year_hypers(1)) * exp(year_hypers(5))); fprintf('Estimated yearly period: %f\n', exp(year_hypers(2))); fprintf('Estimated yearly sf: %f\n', exp(year_hypers(3))); fprintf('Estimated yearly se ell: %f\n', exp(year_hypers(4))); fprintf('\n'); %year_period_hypers = kernel_hypers{4}; %fprintf('Estimated yearly period ell: %f\n', exp(year_period_hypers(1))); %fprintf('Estimated yearly period: %f\n', exp(year_period_hypers(2))); %fprintf('Estimated yearly period sf: %f\n', exp(year_period_hypers(3))); %fprintf('\n'); short_hypers = kernel_hypers{4}; fprintf('Estimated short-term ell: %f\n', exp(short_hypers(1))); fprintf('Estimated short-term sf: %f\n', exp(short_hypers(2))); fprintf('\n'); plot_additive_decomp_cov( X, y, kernel_components, kernel_hypers, hyp.lik, ... savefigs, fileprefix, kernel_names ) %plot_additive_decomp_fancy( X, y, kernel_components, kernel_hypers, hyp.lik, ... % show_samples, savefigs, fileprefix ) ================================================ FILE: code/decomp_concrete.m ================================================ function decomp_concrete(savefigs) % % A simple demo script to show the decomposition of a dataset into individual components. % % David Duvenaud % 2011 if nargin < 1; savefigs = false; end show_samples = true; % How to save figure. decompfigsdir = '../figures/decomp/'; fileprefix = [decompfigsdir 'concrete']; dataset_name = 'data/r_concrete_500.mat'; load(dataset_name); feature_names = {'Cement','Slag','Fly Ash','Water','Plasticizer','Coarse','Fine','Age'}; X = X(1:100,:); y = y(1:100); % Normalize the data. X = X - repmat(mean(X), size(X,1), 1 ); X = X ./ repmat(std(X), size(X,1), 1 ); y = y - mean(y); y = y / std(y); [N,D] = size(X); % Easy part of model set up: meanfunc = {'meanZero'}; inference = @infExact; likfunc = @likGauss; hyp.mean = []; hyp.lik = ones(1,eval(likfunc())).*log(0.1); ell = 1; sf = 0.1; % Now construct the kernel. kernel_components = cell(1, D); kernel_hypers = cell(1, D); for i = 1:D kernel_components{i} = { 'covMask', { i, 'covSEiso'}}; kernel_hypers{i} = [log(ell);log(sf)]; end % concatenate all additive components complete_cov = { 'covSum', kernel_components }; hyp.cov = unwrap(kernel_hypers); hyp = minimize(hyp, @gp, -100, inference, meanfunc, complete_cov, likfunc, X, y); % Pack up hyperparameters again. kernel_hypers = rewrap(kernel_hypers, hyp.cov); plot_additive_decomp( X, y, kernel_components, kernel_hypers, hyp.lik, ... show_samples, savefigs, fileprefix ) plot_cov = false; if plot_cov % Remove coarse and fine remove = [6,7]; kernel_components(remove) = []; kernel_hypers(remove) = []; feature_names(remove) = []; plot_additive_decomp_cov( X, y, kernel_components, kernel_hypers, hyp.lik, ... savefigs, fileprefix, feature_names ) end ================================================ FILE: code/draw_2d_additive_kernels.m ================================================ function draw_2d_additive_kernels % Make figures of additive kernels in 3 dimensions % % David Duvenaud % March 2014 % ================= addpath(genpath([pwd '/../'])) addpath('../../utils/'); clear all close all dpi = 100; % generate a grid range = -4:0.1:4; [a,b] = meshgrid(range, range); xstar = [a(:) b(:)]; % Plot a sqexp kernel % ========================== covfunc = {'covSEiso'}; hyp.cov = log([1,1]); K = feval(covfunc{:}, hyp.cov, xstar, [0,0]); figure; h = surf(a,b,reshape(K, length( range), length( range) ), 'EdgeColor','none','LineStyle','none','FaceLighting','phong'); nice_figure_save('sqexp_kernel'); % Plot a draw from a that prior % ================================================ seed=0; randn('state',seed); rand('state',seed); K = feval(covfunc{:}, hyp.cov, xstar); %imagesc(reshape(K, length(xstar), length(xstar))); n = length( xstar); K = K + diag(ones(n,1)).*0.000001; %mu = feval(meanfunc{:}, hyp.mean, x); y = chol(K)'*gpml_randn(rand, n, 1);% + mu;% + exp(hyp.lik)*gpml_randn(0.2, n, 1); figure; h = surf(a,b,reshape(y, length( range), length( range) ), 'EdgeColor','none','LineStyle','none','FaceLighting','phong'); nice_figure_save('sqexp_draw'); % Plot an additive kernel % ========================== covfunc = {'covADD',{1,'covSEiso'}}; hyp.cov = log([1,1,1,1,sqrt(2)/2]); K = feval(covfunc{:}, hyp.cov, xstar, [0,0]); %imagesc(K) figure; h = surf(a,b,reshape(K, length( range), length( range) ), 'EdgeColor','none','LineStyle','none','FaceLighting','phong'); zlim([0 1]) nice_figure_save('additive_kernel'); % Plot a draw from that prior % ================================================ seed=0; randn('state',seed); rand('state',seed); K = feval(covfunc{:}, hyp.cov, xstar); %imagesc(reshape(K, length(xstar), length(xstar))) n = length( xstar); K = K + diag(ones(n,1)).*0.000001; %mu = feval(meanfunc{:}, hyp.mean, x); y = chol(K)'*gpml_randn(rand, n, 1);% + mu;% + exp(hyp.lik)*gpml_randn(0.2, n, 1); figure; h = surf(a,b,reshape(y, length( range), length( range) ), 'EdgeColor','none','LineStyle','none','FaceLighting','phong'); nice_figure_save('additive_draw'); %plot(x, y, '-') % Plot an additive kernel with 2nd order interactions % ========================== covfunc = {'covADD',{[1,2],'covSEiso'}}; hyp.cov = log([1,1,1,1,1,1]); K = feval(covfunc{:}, hyp.cov, xstar, [0,0]); figure; h = surf(a,b,reshape(K, length( range), length( range) ), 'EdgeColor','none','LineStyle','none','FaceLighting','phong'); nice_figure_save('additive_kernel_2nd_order'); % Plot an additive prior as the sum of two 1d kernels % =================================================== covfunc = {'covADD',{1,'covSEiso'}}; hyp.cov = log([1,1,sqrt(1/2)]); xstar1 = [zeros(length(a(:)), 1) b(:)]; xstar2 = [a(:) zeros(length(b(:)), 1)]; K1 = feval(covfunc{:}, hyp.cov, a(:), [0]); K2 = feval(covfunc{:}, hyp.cov, b(:), [0]); %imagesc(K1) figure; h = surf(a,b,reshape(K1, length( range), length( range) ), 'EdgeColor','none','LineStyle','none','FaceLighting','phong'); zlim([0 1]) caxis([0 1]) nice_figure_save('additive_kernel_sum_p1'); figure; h = surf(a,b,reshape(K2, length( range), length( range) ), 'EdgeColor','none','LineStyle','none','FaceLighting','phong'); zlim([0 1]) caxis([0 1]) nice_figure_save('additive_kernel_sum_p2'); figure; h = surf(a,b,reshape(K1 + K2, length( range), length( range) ), 'EdgeColor','none','LineStyle','none','FaceLighting','phong'); zlim([0 1]) caxis([0 1]) % Plot a draw from those priors % ================================================ covfunc = {'covADD',{1,'covSEiso'}}; hyp.cov = log([1,1,1,1,sqrt(1/2)]); seed=5; % 5 is pretty good randn('state',seed); rand('state',seed); n = length( xstar1); K1 = feval(covfunc{:}, hyp.cov, xstar1); K2 = feval(covfunc{:}, hyp.cov, xstar2); K1 = K1 + diag(ones(n,1)).*0.000001; K2 = K2 + diag(ones(n,1)).*0.000001; y1 = chol(K1)'*gpml_randn(rand, n, 1); y2 = chol(K2)'*gpml_randn(rand, n, 1); figure; h = surf(a,b,reshape(y1 + 0.7, length( range), length( range) ), 'EdgeColor','none','LineStyle','none','FaceLighting','phong'); zlim([-1 4]); nice_figure_save('additive_kernel_draw_sum_p1'); figure; h = surf(a,b,reshape(y2 - 0.7, length( range), length( range) ), 'EdgeColor','none','LineStyle','none','FaceLighting','phong'); zlim([-1 4]); nice_figure_save('additive_kernel_draw_sum_p2'); figure; h = surf(a,b,reshape(y1+y2, length( range), length( range) ), 'EdgeColor','none','LineStyle','none','FaceLighting','phong'); nice_figure_save('additive_kernel_draw_sum'); end function nice_figure_save(filename) % axis off set(gcf, 'color', 'white'); set( gca, 'xTickLabel', '' ); set( gca, 'yTickLabel', '' ); set( gca, 'zTickLabel', '' ); set(gca, 'TickDir', 'in') tightfig set_fig_units_cm(12,12); myaa('publish'); savepng(gcf, ['../figures/additive/2d-kernel/', filename]); %filename_eps = ['../figures/additive/3d-kernel/', filename, '.eps'] %filename_pdf = ['../figures/additive/3d-kernel/', filename, '.pdf'] %print -depsc2 filename_eps %eps2pdf( filename_eps, filename_pdf, true); end ================================================ FILE: code/draw_3d_additive_kernels.m ================================================ function draw_3d_additive_kernels % Make figures of additive kernels in 3 dimensions. % % David Duvenaud % March 2014 % ================= addpath(genpath([pwd '/gpml/'])) addpath('utils/'); clear all close all dpi = 200; % X,Y,Z iz the meshgrid and V is my function evaluated at each meshpoint range = -17:.3:17; [X,Y,Z] = meshgrid(range, range, range ); xstar = [X(:) Y(:) Z(:)]; a = .9; first_order_variance = 1.07; second_order_variance = 25; third_order_variance = 1000009; set_fig_units_cm(5,5); covfunc = {'covADD',{[1,2,3],'covSEiso'}}; hyp.cov = log([1,1,1,1,1,1,first_order_variance, second_order_variance, third_order_variance]); V = feval(covfunc{:}, hyp.cov, xstar, [0,0,0]); V = reshape(V, length(range),length(range),length(range)); figure(1); draw_isosurface( X, Y, Z, V, a, range); nice_figure_save('3d_add_kernel_321'); covfunc = {'covADD',{[2,3],'covSEiso'}}; hyp.cov = log([1,1,1,1,1,1, second_order_variance, third_order_variance]); V = feval(covfunc{:}, hyp.cov, xstar, [0,0,0]); V = reshape(V, length(range),length(range),length(range)); figure(2); draw_isosurface( X, Y, Z, V, a, range); nice_figure_save('3d_add_kernel_32'); covfunc = {'covADD',{[3],'covSEiso'}}; hyp.cov = log([1,1,1,1,1,1, third_order_variance]); V = feval(covfunc{:}, hyp.cov, xstar, [0,0,0]); V = reshape(V, length(range),length(range),length(range)); figure(3); draw_isosurface( X, Y, Z, V, a, range); nice_figure_save('3d_add_kernel_3'); covfunc = {'covADD',{[2],'covSEiso'}}; hyp.cov = log([1,1,1,1,1,1, 25]); V = feval(covfunc{:}, hyp.cov, xstar, [0,0,0]); V = reshape(V, length(range),length(range),length(range)); figure(4); draw_isosurface( X, Y, Z, V, a, range); nice_figure_save('3d_add_kernel_2'); covfunc = {'covADD',{[1],'covSEiso'}}; hyp.cov = log([1,1,1,1,1,1,first_order_variance]); V = feval(covfunc{:}, hyp.cov, xstar, [0,0,0]); V = reshape(V, length(range),length(range),length(range)); figure(5); draw_isosurface( X, Y, Z, V, a, range); nice_figure_save('3d_add_kernel_1'); end function draw_isosurface( X, Y, Z, V, a, range) p = patch(isosurface(X,Y,Z,V,a)); % isosurfaces at max(V)/a isonormals(X,Y,Z,V,p); % plot the surfaces set(p,'FaceColor','red','EdgeColor','none'); % set colors camlight; camlight(-80,-10); lighting gouraud; %alpha(.1); % set the transparency for the isosurfaces view(3); daspect([1 1 1]); %axis tight; axis( [ min(range), max(range),min(range), max(range),min(range), max(range)]); %axis off; end function nice_figure_save(filename) % axis off set(gcf, 'color', 'white'); set( gca, 'xTickLabel', '' ); set( gca, 'yTickLabel', '' ); set( gca, 'zTickLabel', '' ); set(gca, 'TickDir', 'in') tightfig set_fig_units_cm(12,12); myaa('publish'); savepng(gcf, ['../figures/additive/3d-kernel/', filename]); %filename_eps = ['../figures/additive/3d-kernel/', filename, '.eps'] %filename_pdf = ['../figures/additive/3d-kernel/', filename, '.pdf'] %print -depsc2 filename_eps %eps2pdf( filename_eps, filename_pdf, true); end ================================================ FILE: code/draw_manifolds.m ================================================ function draw_shapes() % % Show latent surfaces warped into observed spaces using composite kernels. % % David Duvenaud % March 2014 addpath('utils'); shared_color_scale = 7; % Specify coordinates for different shapes torus.name = 'torus'; torus.kernel = @per_kernel_2d; torus.seed = 4; torus.color_scale = shared_color_scale; torus.camera = [0.25 0.4 0.5]; torus.view = [-100,9]; manifold.name = 'manifold'; manifold.kernel = @se_kernel; manifold.seed = 0; manifold.color_scale = shared_color_scale; manifold.camera = [-6.2031 -13.0451 1.7971]; manifold.view = [-57,-5]; cylinder.name = 'cylinder'; cylinder.kernel = @(x,y)(se_kernel(x(1,:), y(1,:)).*per_kernel(x(2,:),y(2,:))); cylinder.seed = 0; cylinder.color_scale = shared_color_scale; cylinder.camera = [-8.4894 8.3373 12.5418]; cylinder.view = [-134,31]; mobius.name = 'mobius'; mobius.kernel = @mobius_kernel; mobius.seed = 0; mobius.color_scale = shared_color_scale; mobius.camera = [-0.0533 0.0835 0.0333]; mobius.view = [75,-9]; draw_shape(cylinder); draw_shape(manifold); draw_shape(torus); draw_shape(mobius); end function draw_shape(shape) % Fix the seed of the random generators. randn('state', shape.seed); rand('state', shape.seed); topologyfigsdir = '../figures/topology'; output_dim = 3; exact_grid_resolution = 20; aux_grid_resulution = 200; xrange = linspace( 0, 1, exact_grid_resolution); [x1, x2] = ndgrid( xrange, xrange ); % Make a grid exactpoints = [ x1(:), x2(:) ]; N = size(exactpoints, 1); xrange_aux = linspace( 0, 1, aux_grid_resulution ); [x1, x2] = ndgrid( xrange_aux, xrange_aux ); %randn(n, D); % Deepest latent layer. x0aux = [ x1(:), x2(:) ]; % Specify color pattern for traingles. circle_colors = coord_to_image([x0aux(:,1).*shape.color_scale, ... x0aux(:,2).*shape.color_scale]); mu = zeros(output_dim, N); sigma = shape.kernel(exactpoints', exactpoints') + eye(N) * 1e-4; k_x_xaux = shape.kernel(exactpoints', x0aux'); Y = mvnrnd( mu, sigma)'; % Sample the observed space. % Work out warping distribution, conditional on the already sampled points. xaux = k_x_xaux' / sigma * Y; % Plot the observed space. figure; clf; tri = delaunay(x0aux(:,1),x0aux(:,2)); trisurf(tri,xaux(:,1),xaux(:,2),xaux(:,3), sum(circle_colors,2), 'EdgeColor', 'none', 'facealpha', 0.5 ); axis off set(gcf, 'color', 'white'); tightfig set_fig_units_cm(10,10); %set(gca, 'projection', 'perspective'); view(shape.view); set(gca, 'CameraPosition', shape.camera ); xlim([min(xaux(:,1)), max(xaux(:,1))]); ylim([min(xaux(:,2)), max(xaux(:,2))]); zlim([min(xaux(:,3)), max(xaux(:,3))]); %pos = get(gca, 'CameraPosition') % Render to image file. filename = sprintf('%s/%s', topologyfigsdir, shape.name ); %myaa('publish'); %savepng(gcf, filename); end function sigma = se_kernel(x, y) if nargin == 0 sigma = 'Normal SE covariance.'; return; end sigma = exp( -0.5.*sq_dist(x, y)); end function sigma = per_kernel(x, y) if nargin == 0 sigma = 'Periodic covariance.'; return; end period = 1; lengthscale = 10; sigma = exp( -2.*sin(pi*(sqrt(sq_dist(x, y)))./period).^2 ./ lengthscale); end function sigma = per_kernel_2d(x, y, period, lengthscale) if nargin == 0; sigma = 'Two-dimensional toroidal covariance.'; return; end if nargin < 3; period = 1; end if nargin < 4; lengthscale = 1000; end; sigma = exp( -2.*sin(pi*(sqrt(sq_dist(x(1,:), y(1,:))))./period).^2 ./ lengthscale) ... .*exp( -2.*sin(pi*(sqrt(sq_dist(x(2,:), y(2,:))))./period).^2 ./ lengthscale); end function sigma = mobius_kernel(x, y, period, lengthscale) if nargin == 0; sigma = 'Two-dimensional mobius covariance.'; return; end if nargin < 3; period = 1; end if nargin < 4; lengthscale = 50; end; sigma = per_kernel_2d(x, y, period, lengthscale) ... + per_kernel_2d([x(2,:); x(1,:)], y, period, lengthscale); end function sigma = cone_kernel(x, y) if nargin == 0 sigma = 'Toroidal covariance.'; return; end nx = size(x, 1); ny = size(y, 1); xmat = repmat(x(:,2), 1, ny); ymat = repmat(y(:,2), 1, nx); period = 1; lengthscale = 10; sigma = exp( -2.*sin(pi*(sqrt(sq_dist(x(:,1)', y(:,1)')))./period).^2 ./ lengthscale) ... .*xmat.*ymat'; end ================================================ FILE: code/gpml/.octaverc ================================================ startup ================================================ FILE: code/gpml/Copyright ================================================ GAUSSIAN PROCESS REGRESSION AND CLASSIFICATION Toolbox version 3.1 for GNU Octave 3.2.x and Matlab 7.x Copyright (c) 2005-2010 Carl Edward Rasmussen & Hannes Nickisch. All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. THIS SOFTWARE IS PROVIDED BY CARL EDWARD RASMUSSEN & HANNES NICKISCH ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL CARL EDWARD RASMUSSEN & HANNES NICKISCH OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. The views and conclusions contained in the software and documentation are those of the authors and should not be interpreted as representing official policies, either expressed or implied, of Carl Edward Rasmussen & Hannes Nickisch. The code and associated documentation is available from http://gaussianprocess.org/gpml/code. ================================================ FILE: code/gpml/README ================================================ GAUSSIAN PROCESS REGRESSION AND CLASSIFICATION Toolbox version 3.1 for GNU Octave 3.2.x and Matlab 7.x Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-27. 0) HOW TO READ ============== If you want to get started without further delay, then read section 1) below and jump right to the examples in doc/index.html. 1) ABOUT THESE PROGRAMS ======================= This collection of matlab programs implements and demonstrates some of the algorithms described in a) the book by Rasmussen and Williams: "Gaussian Processes for Machine Learning", the MIT Press 2006, in b) the article by Nickisch and Rasmussen: "Approximations for Binary Gaussian Process Classification", JMLR 2008 and in c) the article by Candela and Rasmussen: "A Unifying View of Sparse Approximate Gaussian Process Regression", JMLR 2005. There are 6 code subdirectories: cov, doc, inf, lik, mean and util. cov/ contains covariance functions cov*.m => see covFunctions.m doc/ contains an index.html file providing documentation. This information is also available from http://www.GaussianProcess.org/gpml/code. Usage of mean, cov, classification and regression is demonstrated in usage*.m. Further details can be found in the developer documentation manual.pdf. inf/ contains the inference algorithms inf*.m => see infMethods.m lik/ contains the likelihood functions lik*.m => see likFunctions.m mean/ contains the mean functions mean*.m => see meanFunctions.m util/ contains optimisation routines, backward compatibility programs and small auxiliary scripts Before running the demos or any other part of the code, you should execute startup.m to add all necessary directories to your path. In Octave, you can also use the .octaverc file. 2) ABOUT MEX FILES ================== Some of the programs make use of the MEX facility in matlab for more efficient implementation. However, if you don't know about how to compile MEX files, you do not need to worry about this -- the code runs anyway. If you do compile the MEX files, this is automatically detected, and the program will run more efficiently. If you do not have a compiler installed on your system, you might want to consult [2]. Two components of the toolbox can be accelerated by MEX. First, there is an auxiliary functions solve_chol.m having a MEX equivalent. This can be compiled by either executing util/make.m from the Matlab/Octave command line. In addition to that, we provide a Makefile for Matlab in util/ that you can run by simply typing make in your shell after having supplied your Matlab path and your operating system. We recommend the make.m script because it works both under Matlab and Octave. Second, in order to use the L-BFGS minimiser for hyperparameter optimisation as an equivalent to minimize.m, you have to compile Peter Carbonetto's "Matlab interface for L-BFGS-B" [1]. The challenge here is the Fortran 77 code. We provide a Makefile suitable for Linux 32/64 bit and Mac whenever you have 1) g77 or 2) gfortran properly set up. Under Ubuntu, you can achieve this by installing the packages fort77 and gfortran, respectively. Compilation is done by editing util/lbfgsb/Makefile. In any case, you need to provide $MATLAB_HOME which can be found by the commands 'locate matlab' or 'find / -name "matlab"'. You can choose between two compilation modes: a) using the mex utility by Matlab [default] provide $MEX, then type 'make mex' b) without mex utility by Matlab provide $MEX_SUFFIX and $MATLAB_LIB, then type 'make nomex' In Ubuntu 10.04 LTS, the libg2c library needed for both 1)+a) and 1)+b) is not included per default. If 'ls /usr/lib/libg2c.*' does not list anything this is the case on your machine. You then whant to install the packages gcc-3.4-base and libg2c0 e.g. from http://packages.ubuntu.com/hardy/gcc-3.4-base and http://packages.ubuntu.com/hardy/libg2c0. After installation, you have to create a symbolic link by 'cd /usr/lib' and 'ln -s libg2c.so.0 libg2c.so'. [1] http://www.cs.ubc.ca/~pcarbo/lbfgsb-for-matlab.html [2] http://www.mathworks.com/support/compilers/R2010a 3) CURRENT VERSION ================== The current version of the programs is 3.0. Previous versions of the code are available at http://www.gaussianprocess.org/gpml/code/oldcode.html. 4) DIFFERENCES TO PREVIOUS VERSIONS =================================== NEW in version 3.1, 2010-10-10 ------------------------------ - following a suggestion by Ed Snelson we now support FITC regression - cov/covFITC.m and inf/infFITC.m have been added with Ed Snelson's help - the covariance interface was slightly changed to make that possible NEW in version 3.0, 2010-07-23 ------------------------------ A major code reorganisation effort did take place in the current release. First, classification and regression are now done by a single file gp.m which is completely generic in the likelihood. The previous regression program gpr.m corresponds to gp.m with Gaussian likelihood. Several other likelihoods for robust regresssion were added. Further, the code now supports mean functions with a similar specification mechanism as already used by the covariance functions. Previous implementations correspond to using meanZero.m. We merged the covariance functions covMatern3iso.m and covMatern5.iso into a single covMaterniso.m and added a covariance function for additive functions as well as the possibility to use only certain components of the data. Finally, we included covPPiso.m a piecewise polynomial covariance function with compact support. New likelihood functions in lik/: likGauss.m, likLaplace.m, likLogistic.m and likT.m New mean functions in mean/: meanConst.m, meanLinear.m, meanPow.m, meanProd.m, meanSum.m and meanZero.m New covariance functions in cov/: covADD.m, covMask.m, covPPiso.m The gprSRPP.m function which previously provided "Subset of Regressors" and the "Projected Process" approximation has now been removed. NEW in version 2.1, 2007-07-25 ------------------------------ covConst.m: fixed a bug which caused an error in the derivative of the log marginal likelihood for certain combinations of covariance functions and approximation methods. (Thanks to Antonio Eleuteri for reporting the problem) gauher.m: added the function "gauher.m" which was mistakenly missing from the previous release. This caused an error for certain combinations of approximation method and likelihood function. logistic.m: modified the approximation of moments calculation to use a mixture of cumulative Gaussian, rather than Gauss-Hermite quadrature, as the former turns out to be more accurate. NEW in version 2.0, 2007-06-25 ------------------------------ Some code restructuring has taken place for the classification code to make it more modular, to facilitate addition of new likelihood functions and approximations methods. Now, all classification is done using the binaryGP function, which (among other things) takes an approximation method and a likelihood function as an arguments. Thus, binaryGP replaces both binaryEPGP and binaryLapaceGP, although wrapper functions are still provided for backward compatibility. This gives added flexibility: now EP can also be used wth the logistic likelihood function (implemented using Gauss-Hermite quadrature). approxEP.m: New file, containing the Expectation Propagation approximation method, which was previously contained in binaryEPGP.m approxLA.m: New file, containing Laplaces approximation method, which was previously contained in binaryLaplace.m approximations.m: New file, help for the approximation methods. binaryEPGP.m: This file has been replaced by a wrapper (for backward compatibility) which calls the more general binaryGP function. binaryGP.m: New general function to do binary classification. binaryLaplaceGP.m: This file has been replaced by a wrapper (for backward compatibility) which calls the more general binaryGP function. covMatern3iso.m, covMatern5iso.m, covNNone.m, covRQard.m, covRQiso.m, cosSEard, covSEiso: now check more carefully, that persistent variables have the correct sizes, and some variable names have been modified. cumGauss.m: New file, containing code for the cumulative Gaussian likelihood function likelihoods.m: New file, help for likelihood functions logistic.m: New file, logistic likelihood NEW in version 1.3, 2006-09-08 ------------------------------ covRQard.m: bugfix: replaced x with x' and z with z' in line 36 covRQiso.m: bugfix: replaced x with x' and z with z' in line 28 minimize.m: correction: replaced "error()" with "error('')", and made a few cosmetic changes binaryEPGP.m: added the line "lml = -n*log(2);" in line 77. This change should be largely inconsequential, but occationally may save things when the covariance matrix is exceptionally badly conditioned. NEW in version 1.2, 2006-05-10 ------------------------------ added the "erfint" function to "binaryLaplaceGP.m". The erfint function was missing by mistake, preventing the use of the "logistic" likelihood. NEW in version 1.1, 2006-04-12 ------------------------------ added files: "covProd.m" and "covPeriodic.m" changes: "covSEiso.m" was changed slightly to avoid the use of persistent variables NEW in version 1.0, 2006-03-29 ------------------------------ initial version shipped with the book ================================================ FILE: code/gpml/barrier_fct.m ================================================ function [f df] = barrier_fct(lh,covfunc,x,target) % wrapper used during GP training: penalizes unreasonable signal-to-noise % ratios % % Marc Deisenroth, 2010-12-16 scale = 100; % scaling factor maxSNR = 1000; % bound for SNR order = 10; % order of polynomial [f df] = poly_barrier(lh, covfunc, x, target, maxSNR, order); %% function [f df] = poly_barrier(lh, covfunc, x, target, maxSNR, order) % call gpr [f df] = gpr(lh,covfunc,x,target); D = length(lh)-2; % signal-to-noise ratio snr = exp(lh(D+1)-lh(D+2)); % new objective function (penalizes too high SNRs) f = f + (snr/maxSNR).^order; % update corresponding gradients of the NLML df(D+1) = df(D+1) + order*(snr/maxSNR).^(order-1)*snr/maxSNR; df(D+2) = df(D+2) - order*(snr/maxSNR).^(order-1)*snr/maxSNR; % % set max. length-scale ratio % maxLSR = 1e5; % for the squared length-scales % % % look at length-scales now % [minLLS min_idx] = min(lh(1:D)); % smallest log-length-scale % % for i = setdiff(1:D,min_idx) % don't change fct value with lsr = 1 % % lsr = exp(2*lh(i)-2*minLLS); % length-scale-ratio; always > 1 % % % new objective function (penalizes too high LSRs) % f = f + (lsr/maxLSR).^order; % % % update corresponding gradients of the NLML % df(i) = df(i) + 2*order*(lsr/maxLSR).^(order-1)*lsr/maxLSR; % df(min_idx) = df(min_idx) - 2*order*(lsr/maxLSR).^(order-1)*lsr/maxLSR; % % end %% function [f df] = log_barrier(lh, covfunc, x, target, maxSNR, scale) % for hard constraints % call gpr [f df] = gpr(lh,covfunc,x,target); % new objective function f = f - scale*log(maxSNR - exp(lh(end-1))/exp(lh(end))); % derivatives outer = -scale/(maxSNR - exp(lh(end-1))/exp(lh(end))); inner1 = - exp(lh(end-1))/exp(lh(end)); inner2 = exp(lh(end-1))/exp(lh(end)); df(end-1) = df(end-1) + outer*inner1; df(end) = df(end) + outer*inner2; ================================================ FILE: code/gpml/cov/covADD.m ================================================ function K = covADD(cov, hyp, x, z, i) % Additive covariance function using a 1d base covariance function % cov(x^p,x^q;hyp) with individual hyperparameters hyp. % % Note: cov must be a one-dimensional covariance function. % % k(x^p,x^q) = \sum_{r \in R} sf_r \sum_{|I|=r} % \prod_{i \in I} cov(x^p_i,x^q_i;hyp_i) % % hyp = [ hyp_1 % hyp_2 % ... % hyp_D % log(sf_R(1)) % ... % log(sf_R(end)) ] % % where hyp_d are the parameters of the 1d covariance function which are shared % over the different values of R(1) to R(end). % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. R = cov{1}; nh = eval(feval(cov{2})); % number of hypers per individual covariance nr = numel(R); % number of different degrees of interaction if nargin<3 % report number of hyper parameters K = ['D*', int2str(nh), '+', int2str(nr)]; return end if nargin<4, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode [n,D] = size(x); % dimensionality sf2 = exp( 2*hyp(D*nh+(1:nr)) ); % signal variances of individual degrees Kd = Kdim(cov{2},hyp,x,z); % evaluate dimensionwise covariances K if nargin<5 % covariances EE = elsympol2(Kd,max(R)); % Rth elementary symmetric polynomials K = 0; for ii=1:nr, K = K + sf2(ii)*EE(:,:,R(ii)+1); end % sf2 weighted sum else % derivatives if i <= D*nh % individual covariance function parameters j = fix(1+(i-1)/nh); % j is the dimension of the hyperparameter if dg, zj='diag'; else if xeqz, zj=[]; else zj=z(:,j); end, end dKj = feval(cov{2},hyp(nh*(j-1)+(1:nh)),x(:,j),zj,i-(j-1)*nh); % other dK=0 % the final derivative is a sum of multilinear terms, so if only one term % depends on the hyperparameter under consideration, we can factorise it % out and compute the sum with one degree less E = elsympol2(Kd(:,:,[1:j-1,j+1:D]),max(R)-1); % R-1th elementary sym polyn K = 0; for ii=1:nr, K = K + sf2(ii)*E(:,:,R(ii)); end % sf2 weighted sum K = dKj.*K; elseif i <= D*nh+nr EE = elsympol2(Kd,max(R)); % Rth elementary symmetric polynomials j = i-D*nh; K = 2*sf2(j)*EE(:,:,R(j)+1); % rest of the sf2 weighted sum else error('Unknown hyperparameter') end end % evaluate dimensionwise covariances K function K = Kdim(cov,hyp,x,z) [n,D] = size(x); % dimensionality nh = eval(feval(cov)); % number of hypers per individual covariance if nargin<4, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode if dg % allocate memory K = zeros(n,1,D); else if xeqz, K = zeros(n,n,D); else K = zeros(n,size(z,1),D); end end for d=1:D hyp_d = hyp(nh*(d-1)+(1:nh)); % hyperparamter of dimension d if dg K(:,:,d) = feval(cov,hyp_d,x(:,d),'diag'); else if xeqz K(:,:,d) = feval(cov,hyp_d,x(:,d)); else K(:,:,d) = feval(cov,hyp_d,x(:,d),z(:,d)); end end end ================================================ FILE: code/gpml/cov/covConst.m ================================================ function K = covConst(hyp, x, z, i) % covariance function for a constant function. The covariance function is % parameterized as: % % k(x^p,x^q) = s2; % % The scalar hyperparameter is: % % hyp = [ log(sqrt(s2)) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<2, K = '1'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode s2 = exp(2*hyp); % s2 n = size(x,1); if dg % vector kxx K = s2*ones(n,1); else if xeqz % symmetric matrix Kxx K = s2*ones(n); else % cross covariances Kxz K = s2*ones(n,size(z,1)); end end if nargin>3 % derivatives if i==1 K = 2*K; else error('Unknown hyperparameter') end end ================================================ FILE: code/gpml/cov/covFITC.m ================================================ function [K,Kuu,Ku] = covFITC(cov, xu, hyp, x, z, i) % Covariance function to be used together with the FITC approximation. % % The function allows for more than one output argument and does not respect the % interface of a proper covariance function. In fact, it wraps a proper % covariance function such that it can be used together with infFITC.m. % Instead of outputing the full covariance, it returns cross-covariances between % the inputs x, z and the inducing inputs xu as needed by infFITC.m % % Copyright (c) by Ed Snelson, Carl Edward Rasmussen % and Hannes Nickisch, 2010-12-21. % % See also COVFUNCTIONS.M, INFFITC.M. if nargin<4, K = feval(cov{:}); return, end if nargin<5, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode if size(xu,2) ~= size(x,2) error('Dimensionality of inducing inputs must match training inputs'); end if nargin<6 % covariances if dg K = feval(cov{:},hyp,x,'diag'); else if xeqz K = feval(cov{:},hyp,x,'diag'); Kuu = feval(cov{:},hyp,xu); Ku = feval(cov{:},hyp,xu,x); else K = feval(cov{:},hyp,xu,z); end end else % derivatives if dg K = feval(cov{:},hyp,x,'diag',i); else if xeqz K = feval(cov{:},hyp,x,'diag',i); Kuu = feval(cov{:},hyp,xu,[],i); Ku = feval(cov{:},hyp,xu,x,i); else K = feval(cov{:},hyp,xu,z,i); end end end ================================================ FILE: code/gpml/cov/covLIN.m ================================================ function K = covLIN(hyp, x, z, i) % Linear covariance function. The covariance function is parameterized as: % % k(x^p,x^q) = x^p'*x^q % % The are no hyperparameters: % % hyp = [ ] % % Note that there is no bias or scale term; use covConst to add these. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<2, K = '0'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode % compute inner products if dg % vector kxx K = sum(x.*x,2); else if xeqz % symmetric matrix Kxx K = x*x'; else % cross covariances Kxz K = x*z'; end end if nargin>3 % derivatives error('Unknown hyperparameter') end ================================================ FILE: code/gpml/cov/covLINard.m ================================================ function K = covLINard(hyp, x, z, i) % Linear covariance function with Automatic Relevance Determination (ARD). The % covariance function is parameterized as: % % k(x^p,x^q) = x^p'*inv(P)*x^q % % where the P matrix is diagonal with ARD parameters ell_1^2,...,ell_D^2, where % D is the dimension of the input space. The hyperparameters are: % % hyp = [ log(ell_1) % log(ell_2) % .. % log(ell_D) ] % % Note that there is no bias term; use covConst to add a bias. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<2, K = 'D'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode ell = exp(hyp); [n,D] = size(x); x = x*diag(1./ell); % precompute inner products if dg % vector kxx K = sum(x.*x,2); else if xeqz % symmetric matrix Kxx K = x*x'; else % cross covariances Kxz z = z*diag(1./ell); K = x*z'; end end if nargin>3 % derivatives if i<=D if dg K = -2*x(:,i).*x(:,i); else if xeqz K = -2*x(:,i)*x(:,i)'; else K = -2*x(:,i)*z(:,i)'; end end else error('Unknown hyperparameter') end end ================================================ FILE: code/gpml/cov/covLINone.m ================================================ function K = covLINone(hyp, x, z, i) % Linear covariance function with a single hyperparameter. The covariance % function is parameterized as: % % k(x^p,x^q) = (x^p'*x^q + 1)/t2; % % where the P matrix is t2 times the unit matrix. The second term plays the % role of the bias. The hyperparameter is: % % hyp = [ log(sqrt(t2)) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<2, K = '1'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode it2 = exp(-2*hyp); % t2 inverse % precompute inner products if dg % vector kxx K = sum(x.*x,2); else if xeqz % symmetric matrix Kxx K = x*x'; else % cross covariances Kxz K = x*z'; end end if nargin<4 % covariances K = it2*(1+K); else % derivatives if i==1 K = -2*it2*(1+K); else error('Unknown hyperparameter') end end ================================================ FILE: code/gpml/cov/covMask.m ================================================ function K = covMask(cov, hyp, x, z, i) % Apply a covariance function to a subset of the dimensions only. The subset can % either be specified by a 0/1 mask by a boolean mask or by an index set. % % This function doesn't actually compute very much on its own, it merely does % some bookkeeping, and calls another covariance function to do the actual work. % % The function was suggested by Iain Murray, 2010-02-18 and is based on an % earlier implementation of his dating back to 2009-06-16. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. mask = fix(cov{1}(:)); % either a binary mask or an index set cov = cov(2); % covariance function to be masked %%%% Next line is a quick fix by James Lloyd - might be wrong! if iscell(cov{:}), cov = cov{:}; end nh_string = feval(cov{:}); % number of hyperparameters of the full covariance if max(mask)<2 && length(mask)>1, mask = find(mask); end % convert 1/0->index D = length(mask); % masked dimension if nargin<3, K = num2str(eval(nh_string)); return, end % number of parameters if nargin<4, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode if eval(nh_string)~=length(hyp) % check hyperparameters error('number of hyperparameters does not match size of masked data') end if nargin<5 % covariances if dg K = feval(cov{:}, hyp, x(:,mask), 'diag'); else if xeqz K = feval(cov{:}, hyp, x(:,mask)); else K = feval(cov{:}, hyp, x(:,mask), z(:,mask)); end end else % derivatives if i <= eval(nh_string) if dg K = feval(cov{:}, hyp, x(:,mask), 'diag', i); else if xeqz K = feval(cov{:}, hyp, x(:,mask), [], i); else K = feval(cov{:}, hyp, x(:,mask), z(:,mask), i); end end else error('Unknown hyperparameter') end end ================================================ FILE: code/gpml/cov/covMaterniso.m ================================================ function K = covMaterniso(d, hyp, x, z, i) % Matern covariance function with nu = d/2 and isotropic distance measure. For % d=1 the function is also known as the exponential covariance function or the % Ornstein-Uhlenbeck covariance in 1d. The covariance function is: % % k(x^p,x^q) = s2f * f( sqrt(d)*r ) * exp(-sqrt(d)*r) % % with f(t)=1 for d=1, f(t)=1+t for d=3 and f(t)=1+t+t²/3 for d=5. % Here r is the distance sqrt((x^p-x^q)'*inv(P)*(x^p-x^q)), P is ell times % the unit matrix and sf2 is the signal variance. The hyperparameters are: % % hyp = [ log(ell) % log(sqrt(sf2)) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<3, K = '2'; return; end % report number of parameters if nargin<4, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode ell = exp(hyp(1)); sf2 = exp(2*hyp(2)); if all(d~=[1,3,5]), error('only 1, 3 and 5 allowed for d'), end % degree switch d case 1, f = @(t) 1; df = @(t) 1; case 3, f = @(t) 1 + t; df = @(t) t; case 5, f = @(t) 1 + t.*(1+t/3); df = @(t) t.*(1+t)/3; end m = @(t,f) f(t).*exp(-t); dm = @(t,f) df(t).*t.*exp(-t); % precompute distances if dg % vector kxx K = zeros(size(x,1),1); else if xeqz % symmetric matrix Kxx K = sqrt( sq_dist(sqrt(d)*x'/ell) ); else % cross covariances Kxz K = sqrt( sq_dist(sqrt(d)*x'/ell,sqrt(d)*z'/ell) ); end end if nargin<5 % covariances K = sf2*m(K,f); else % derivatives if i==1 K = sf2*dm(K,f); elseif i==2 K = 2*sf2*m(K,f); else error('Unknown hyperparameter') end end ================================================ FILE: code/gpml/cov/covNNone.m ================================================ function K = covNNone(hyp, x, z, i) % Neural network covariance function with a single parameter for the distance % measure. The covariance function is parameterized as: % % k(x^p,x^q) = sf2 * asin(x^p'*P*x^q / sqrt[(1+x^p'*P*x^p)*(1+x^q'*P*x^q)]) % % where the x^p and x^q vectors on the right hand side have an added extra bias % entry with unit value. P is ell^-2 times the unit matrix and sf2 controls the % signal variance. The hyperparameters are: % % hyp = [ log(ell) % log(sqrt(sf2) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<2, K = '2'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode n = size(x,1); ell2 = exp(2*hyp(1)); sf2 = exp(2*hyp(2)); sx = 1 + sum(x.*x,2); if dg % vector kxx K = sx./(sx+ell2); else if xeqz % symmetric matrix Kxx S = 1 + x*x'; K = S./(sqrt(ell2+sx)*sqrt(ell2+sx)'); else % cross covariances Kxz S = 1 + x*z'; sz = 1 + sum(z.*z,2); K = S./(sqrt(ell2+sx)*sqrt(ell2+sz)'); end end if nargin<4 % covariances K = sf2*asin(K); else % derivatives if i==1 % lengthscale if dg V = K; else vx = sx./(ell2+sx); if xeqz V = repmat(vx/2,1,n) + repmat(vx'/2,n,1); else vz = sz./(ell2+sz); nz = size(z,1); V = repmat(vx/2,1,nz) + repmat(vz'/2,n,1); end end K = -2*sf2*(K-K.*V)./sqrt(1-K.*K); elseif i==2 % magnitude K = 2*sf2*asin(K); else error('Unknown hyperparameter') end end ================================================ FILE: code/gpml/cov/covNoise.m ================================================ function K = covNoise(hyp, x, z, i) % Independent covariance function, ie "white noise", with specified variance. % The covariance function is specified as: % % k(x^p,x^q) = s2 * \delta(p,q) % % where s2 is the noise variance and \delta(p,q) is a Kronecker delta function % which is 1 iff p=q and zero otherwise. The hyperparameter is % % hyp = [ log(sqrt(s2)) ] % % For more help on design of covariance functions, try "help covFunctions". % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<2, K = '1'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode n = size(x,1); s2 = exp(2*hyp); % noise variance % precompute raw if dg % vector kxx K = ones(n,1); else if xeqz % symmetric matrix Kxx K = eye(n); else % cross covariances Kxz K = zeros(n,size(z,1)); end end if nargin<4 % covariances K = s2*K; else % derivatives if i==1 K = 2*s2*K; else error('Unknown hyperparameter') end end ================================================ FILE: code/gpml/cov/covPPiso.m ================================================ function K = covPPiso(v, hyp, x, z, i) % Piecewise polynomial covariance function with compact support, v = 0,1,2,3. % The covariance functions are 2v times contin. diff'ble and the corresponding % processes are hence v times mean-square diffble. The covariance function is: % % k(x^p,x^q) = s2f * (1-r)_+.^j * f(r,j) % % where r is the distance sqrt((x^p-x^q)'*inv(P)*(x^p-x^q)), P is ell^2 times % the unit matrix and sf2 is the signal variance. The hyperparameters are: % % hyp = [ log(ell) % log(sqrt(sf2)) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<3, K = '2'; return; end % report number of parameters if nargin<4, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode ell = exp(hyp(1)); sf2 = exp(2*hyp(2)); if all(v~=[0,1,2,3]), error('only 0,1,2 and 3 allowed for v'), end % degree j = floor(size(x,2)/2)+v+1; % exponent switch v case 0, f = @(r,j) 1; df = @(r,j) 0; case 1, f = @(r,j) 1 + (j+1)*r; df = @(r,j) (j+1); case 2, f = @(r,j) 1 + (j+2)*r + ( j^2+ 4*j+ 3)/ 3*r.^2; df = @(r,j) (j+2) + 2*( j^2+ 4*j+ 3)/ 3*r; case 3, f = @(r,j) 1 + (j+3)*r + (6*j^2+36*j+45)/15*r.^2 ... + (j^3+9*j^2+23*j+15)/15*r.^3; df = @(r,j) (j+3) + 2*(6*j^2+36*j+45)/15*r ... + (j^3+9*j^2+23*j+15)/ 5*r.^2; end pp = @(r,j,v,f) max(1-r,0).^(j+v).*f(r,j); dpp = @(r,j,v,f) max(1-r,0).^(j+v-1).*r.*( (j+v)*f(r,j) - max(1-r,0).*df(r,j) ); % precompute squared distances if dg % vector kxx K = zeros(size(x,1),1); else if xeqz % symmetric matrix Kxx K = sqrt( sq_dist(x'/ell) ); else % cross covariances Kxz K = sqrt( sq_dist(x'/ell,z'/ell) ); end end if nargin<5 % covariances K = sf2*pp( K, j, v, f ); else % derivatives if i==1 K = sf2*dpp( K, j, v, f ); elseif i==2 K = 2*sf2*pp( K, j, v, f ); else error('Unknown hyperparameter') end end ================================================ FILE: code/gpml/cov/covPeriodic.m ================================================ function K = covPeriodic(hyp, x, z, i) % Stationary covariance function for a smooth periodic function, with period p: % % k(x,y) = sf2 * exp( -2*sin^2( pi*||x-y||/p )/ell^2 ) % % where the hyperparameters are: % % hyp = [ log(ell) % log(p) % log(sqrt(sf2)) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-01-05. % % See also COVFUNCTIONS.M. if nargin<2, K = '3'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode n = size(x,1); ell = exp(hyp(1)); p = exp(hyp(2)); sf2 = exp(2*hyp(3)); % precompute distances if dg % vector kxx K = zeros(size(x,1),1); else if xeqz % symmetric matrix Kxx K = sqrt(sq_dist(x')); else % cross covariances Kxz K = sqrt(sq_dist(x',z')); end end K = pi*K/p; if nargin<4 % covariances K = sin(K)/ell; K = K.*K; K = sf2*exp(-2*K); else % derivatives if i==1 K = sin(K)/ell; K = K.*K; K = 4*sf2*exp(-2*K).*K; elseif i==2 R = sin(K)/ell; K = 4*sf2/ell*exp(-2*R.*R).*R.*cos(K).*K; elseif i==3 K = sin(K)/ell; K = K.*K; K = 2*sf2*exp(-2*K); else error('Unknown hyperparameter') end end ================================================ FILE: code/gpml/cov/covPoly.m ================================================ function K = covPoly(d, hyp, x, z, i) % Polynomial covariance function. The covariance function is parameterized as: % % k(x^p,x^q) = sf^2 * ( c + (x^p)'*(x^q) )^d % % The hyperparameters are: % % hyp = [ log(c) % log(sf) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<3, K = '2'; return; end % report number of parameters if nargin<4, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode c = exp(hyp(1)); % inhomogeneous offset sf2 = exp(2*hyp(2)); % signal variance if d~=max(1,fix(d)), error('only nonzero integers allowed for d'), end % degree % precompute inner products if dg % vector kxx K = sum(x.*x,2); else if xeqz % symmetric matrix Kxx K = x*x'; else % cross covariances Kxz K = x*z'; end end if nargin<5 % covariances K = sf2*( c + K ).^d; else % derivatives if i==1 K = c*d*sf2*( c + K ).^(d-1); elseif i==2 K = 2*sf2*( c + K ).^d; else error('Unknown hyperparameter') end end ================================================ FILE: code/gpml/cov/covProd.m ================================================ function K = covProd(cov, hyp, x, z, i) % covProd - compose a covariance function as the product of other covariance % functions. This function doesn't actually compute very much on its own, it % merely does some bookkeeping, and calls other covariance functions to do the % actual work. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if numel(cov)==0, error('We require at least one factor.'), end for ii = 1:numel(cov) % iterate over covariance functions f = cov(ii); if iscell(f{:}), f = f{:}; end % expand cell array if necessary j(ii) = cellstr(feval(f{:})); % collect number hypers end if nargin<3 % report number of parameters K = char(j(1)); for ii=2:length(cov), K = [K, '+', char(j(ii))]; end, return end if nargin<4, z = []; end % make sure, z exists [n,D] = size(x); v = []; % v vector indicates to which covariance parameters belong for ii = 1:length(cov), v = [v repmat(ii, 1, eval(char(j(ii))))]; end if nargin<5 % covariances K = 1; if nargin==3, z = []; end % set default for ii = 1:length(cov) % iteration over factor functions f = cov(ii); if iscell(f{:}), f = f{:}; end % expand cell array if necessary K = K .* feval(f{:}, hyp(v==ii), x, z); % accumulate covariances end else % derivatives if i<=length(v) K = 1; vi = v(i); % which covariance function j = sum(v(1:i)==vi); % which parameter in that covariance for ii = 1:length(cov) % iteration over factor functions f = cov(ii); if iscell(f{:}), f = f{:}; end % expand cell if necessary if ii==vi K = K .* feval(f{:}, hyp(v==ii), x, z, j); % accumulate covariances else K = K .* feval(f{:}, hyp(v==ii), x, z); % accumulate covariances end end else error('Unknown hyperparameter') end end ================================================ FILE: code/gpml/cov/covRQard.m ================================================ function K = covRQard(hyp, x, z, i) % Rational Quadratic covariance function with Automatic Relevance Determination % (ARD) distance measure. The covariance function is parameterized as: % % k(x^p,x^q) = sf2 * [1 + (x^p - x^q)'*inv(P)*(x^p - x^q)/(2*alpha)]^(-alpha) % % where the P matrix is diagonal with ARD parameters ell_1^2,...,ell_D^2, where % D is the dimension of the input space, sf2 is the signal variance and alpha % is the shape parameter for the RQ covariance. The hyperparameters are: % % hyp = [ log(ell_1) % log(ell_2) % .. % log(ell_D) % log(sqrt(sf2)) % log(alpha) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-08-04. % % See also COVFUNCTIONS.M. if nargin<2, K = '(D+2)'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode [n,D] = size(x); ell = exp(hyp(1:D)); sf2 = exp(2*hyp(D+1)); alpha = exp(hyp(D+2)); % precompute squared distances if dg % vector kxx D2 = zeros(size(x,1),1); else if xeqz % symmetric matrix Kxx D2 = sq_dist(diag(1./ell)*x'); else % cross covariances Kxz D2 = sq_dist(diag(1./ell)*x',diag(1./ell)*z'); end end if nargin<4 % covariances K = sf2*((1+0.5*D2/alpha).^(-alpha)); else % derivatives if i<=D % length scale parameter if dg K = D2*0; else if xeqz K = sf2*(1+0.5*D2/alpha).^(-alpha-1).*sq_dist(x(:,i)'/ell(i)); else K = sf2*(1+0.5*D2/alpha).^(-alpha-1).*sq_dist(x(:,i)'/ell(i),z(:,i)'/ell(i)); end end elseif i==D+1 % magnitude parameter K = 2*sf2*((1+0.5*D2/alpha).^(-alpha)); elseif i==D+2 K = (1+0.5*D2/alpha); K = sf2*K.^(-alpha).*(0.5*D2./K - alpha*log(K)); else error('Unknown hyperparameter') end end ================================================ FILE: code/gpml/cov/covRQiso.m ================================================ function K = covRQiso(hyp, x, z, i) % Rational Quadratic covariance function with isotropic distance measure. The % covariance function is parameterized as: % % k(x^p,x^q) = sf2 * [1 + (x^p - x^q)'*inv(P)*(x^p - x^q)/(2*alpha)]^(-alpha) % % where the P matrix is ell^2 times the unit matrix, sf2 is the signal % variance and alpha is the shape parameter for the RQ covariance. The % hyperparameters are: % % hyp = [ log(ell) % log(sqrt(sf2)) % log(alpha) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<2, K = '3'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode ell = exp(hyp(1)); sf2 = exp(2*hyp(2)); alpha = exp(hyp(3)); % precompute squared distances if dg % vector kxx D2 = zeros(size(x,1),1); else if xeqz % symmetric matrix Kxx D2 = sq_dist(x'/ell); else % cross covariances Kxz D2 = sq_dist(x'/ell,z'/ell); end end if nargin<4 % covariances K = sf2*((1+0.5*D2/alpha).^(-alpha)); else % derivatives if i==1 % length scale parameter K = sf2*(1+0.5*D2/alpha).^(-alpha-1).*D2; elseif i==2 % magnitude parameter K = 2*sf2*((1+0.5*D2/alpha).^(-alpha)); elseif i==3 K = (1+0.5*D2/alpha); K = sf2*K.^(-alpha).*(0.5*D2./K - alpha*log(K)); else error('Unknown hyperparameter') end end ================================================ FILE: code/gpml/cov/covSEard.m ================================================ function K = covSEard(hyp, x, z, i) % Squared Exponential covariance function with Automatic Relevance Detemination % (ARD) distance measure. The covariance function is parameterized as: % % k(x^p,x^q) = sf2 * exp(-(x^p - x^q)'*inv(P)*(x^p - x^q)/2) % % where the P matrix is diagonal with ARD parameters ell_1^2,...,ell_D^2, where % D is the dimension of the input space and sf2 is the signal variance. The % hyperparameters are: % % hyp = [ log(ell_1) % log(ell_2) % . % log(ell_D) % log(sqrt(sf2)) ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<2, K = '(D+1)'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode [n,D] = size(x); ell = exp(hyp(1:D)); % characteristic length scale sf2 = exp(2*hyp(D+1)); % signal variance % precompute squared distances if dg % vector kxx K = zeros(size(x,1),1); else if xeqz % symmetric matrix Kxx K = sq_dist(diag(1./ell)*x'); else % cross covariances Kxz K = sq_dist(diag(1./ell)*x',diag(1./ell)*z'); end end K = sf2*exp(-K/2); % covariance if nargin>3 % derivatives if i<=D % length scale parameters if dg K = K*0; else if xeqz K = K.*sq_dist(x(:,i)'/ell(i)); else K = K.*sq_dist(x(:,i)'/ell(i),z(:,i)'/ell(i)); end end elseif i==D+1 % magnitude parameter K = 2*K; else error('Unknown hyperparameter') end end ================================================ FILE: code/gpml/cov/covSEiso.m ================================================ function K = covSEiso(hyp, x, z, i) % Squared Exponential covariance function with isotropic distance measure. The % covariance function is parameterized as: % % k(x^p,x^q) = sf^2 * exp(-(x^p - x^q)'*inv(P)*(x^p - x^q)/2) % % where the P matrix is ell^2 times the unit matrix and sf^2 is the signal % variance. The hyperparameters are: % % hyp = [ log(ell) % log(sf) ] % % For more help on design of covariance functions, try "help covFunctions". % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<2, K = '2'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode ell = exp(hyp(1)); % characteristic length scale sf2 = exp(2*hyp(2)); % signal variance % precompute squared distances if dg % vector kxx K = zeros(size(x,1),1); else if xeqz % symmetric matrix Kxx K = sq_dist(x'/ell); else % cross covariances Kxz K = sq_dist(x'/ell,z'/ell); end end if nargin<4 % covariances K = sf2*exp(-K/2); else % derivatives if i==1 K = sf2*exp(-K/2).*K; elseif i==2 K = 2*sf2*exp(-K/2); else error('Unknown hyperparameter') end end ================================================ FILE: code/gpml/cov/covSEisoU.m ================================================ function K = covSEisoU(hyp, x, z, i) % Squared Exponential covariance function with isotropic distance measure with % unit magnitude. The covariance function is parameterized as: % % k(x^p,x^q) = exp(-(x^p - x^q)'*inv(P)*(x^p - x^q)/2) % % where the P matrix is ell^2 times the unit matrix and sf2 is the signal % variance. The hyperparameters are: % % hyp = [ log(ell) ] % % For more help on design of covariance functions, try "help covFunctions". % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<2, K = '1'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode ell = exp(hyp(1)); % characteristic length scale % precompute squared distances if dg % vector kxx K = zeros(size(x,1),1); else if xeqz % symmetric matrix Kxx K = sq_dist(x'/ell); else % cross covariances Kxz K = sq_dist(x'/ell,z'/ell); end end if nargin<4 % covariances K = exp(-K/2); else % derivatives if i==1 K = exp(-K/2).*K; else error('Unknown hyperparameter') end end ================================================ FILE: code/gpml/cov/covSEiso_length.m ================================================ function K = covSEiso_length(hyp, x, z, i) % Squared Exponential covariance function with isotropic distance measure with % unit magnitude. The covariance function is parameterized as: % % k(x^p,x^q) = exp(-(x^p - x^q)'*inv(P)*(x^p - x^q)/2) % % where the P matrix is ell^2 times the unit matrix and sf2 is the signal % variance. The hyperparameters are: % % hyp = [ log(ell) ] % % For more help on design of covariance functions, try "help covFunctions". % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % % See also COVFUNCTIONS.M. if nargin<2, K = '1'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode ell = exp(hyp(1)); % characteristic length scale % precompute squared distances if dg % vector kxx K = zeros(size(x,1),1); else if xeqz % symmetric matrix Kxx K = sq_dist(x'/ell); else % cross covariances Kxz K = sq_dist(x'/ell,z'/ell); end end if nargin<4 % covariances K = exp(-K/2); else % derivatives if i==1 K = exp(-K/2).*K; else error('Unknown hyperparameter') end end ================================================ FILE: code/gpml/cov/covSEiso_var.m ================================================ function K = covSEiso_var(hyp, x, z, i) % Squared Exponential covariance function with isotropic, fixed distance measure. The % covariance function is parameterized as: % % k(x^p,x^q) = sf^2 * exp(-(x^p - x^q)'*(x^p - x^q)/2) % % where the sf^2 is the signal variance. The hyperparameters are: % % hyp = [ log(sf) ] % % For more help on design of covariance functions, try "help covFunctions". % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-10. % Made by David Duvenaud, May 2011 % % See also COVFUNCTIONS.M. if nargin<2, K = '1'; return; end % report number of parameters if nargin<3, z = []; end % make sure, z exists xeqz = numel(z)==0; dg = strcmp(z,'diag') && numel(z)>0; % determine mode sf2 = exp(2*hyp(1)); % signal variance % precompute squared distances if dg % vector kxx K = zeros(size(x,1),1); else if xeqz % symmetric matrix Kxx K = sq_dist(x'); else % cross covariances Kxz K = sq_dist(x',z'); end end if nargin<4 % covariances K = sf2*exp(-K/2); else % derivatives if i==1 K = 2*sf2*exp(-K/2); else error('Unknown hyperparameter') end end ================================================ FILE: code/gpml/cov/covScale.m ================================================ function K = covScale(cov, hyp, x, z, i) % meanScale - compose a mean function as a scaled version of another one. % % k(x^p,x^q) = sf^2 * k_0(x^p,x^q) % % The hyperparameter is: % % hyp = [ log(sf) ] % % This function doesn't actually compute very much on its own, it merely does % some bookkeeping, and calls other mean function to do the actual work. % % Copyright (c) by Carl Edward Rasmussen & Hannes Nickisch 2010-09-10. % % See also MEANFUNCTIONS.M. if nargin<3 % report number of parameters K = [feval(cov{:}),'+1']; return end if nargin<4, z = []; end % make sure, z exists [n,D] = size(x); sf2 = exp(2*hyp(1)); % signal variance if nargin<5 % covariances K = sf2*feval(cov{:},hyp(2:end),x,z); else % derivatives if i==1 K = 2*sf2*feval(cov{:},hyp(2:end),x,z); else K = sf2*feval(cov{:},hyp(2:end),x,z,i-1); end end ================================================ FILE: code/gpml/cov/covSum.m ================================================ function K = covSum(cov, hyp, x, z, i) % covSum - compose a covariance function as the sum of other covariance % functions. This function doesn't actually compute very much on its own, it % merely does some bookkeeping, and calls other covariance functions to do the % actual work. % % Copyright (c) by Carl Edward Rasmussen & Hannes Nickisch 2010-09-10. % % See also COVFUNCTIONS.M. if numel(cov)==0, error('We require at least one summand.'), end for ii = 1:numel(cov) % iterate over covariance functions f = cov(ii); if iscell(f{:}), f = f{:}; end % expand cell array if necessary j(ii) = cellstr(feval(f{:})); % collect number hypers end if nargin<3 % report number of parameters K = char(j(1)); for ii=2:length(cov), K = [K, '+', char(j(ii))]; end, return end if nargin<4, z = []; end % make sure, z exists [n,D] = size(x); v = []; % v vector indicates to which covariance parameters belong for ii = 1:length(cov), v = [v repmat(ii, 1, eval(char(j(ii))))]; end if nargin<5 % covariances K = 0; if nargin==3, z = []; end % set default for ii = 1:length(cov) % iteration over summand functions f = cov(ii); if iscell(f{:}), f = f{:}; end % expand cell array if necessary K = K + feval(f{:}, hyp(v==ii), x, z); % accumulate covariances end else % derivatives if i<=length(v) vi = v(i); % which covariance function j = sum(v(1:i)==vi); % which parameter in that covariance f = cov(vi); if iscell(f{:}), f = f{:}; end % dereference cell array if necessary K = feval(f{:}, hyp(v==vi), x, z, j); % compute derivative else error('Unknown hyperparameter') end end ================================================ FILE: code/gpml/covFunctions.m ================================================ % covariance functions to be use by Gaussian process functions. There are two % different kinds of covariance functions: simple and composite: % % simple covariance functions: % covConst - covariance for constant functions % covLIN - linear covariance function % covLINard - linear covariance function with ARD % covLINone - linear covariance function with bias % covMaterniso - Matern covariance function with nu=1/2, 3/2 or 5/2 % covNNone - neural network covariance function % covNoise - independent covariance function (i.e. white noise) % covPeriodic - smooth periodic covariance function (1d) with unit period % covPoly - polynomial covariance function % covPPiso - piecewise polynomial covariance function (compact support) % covRQard - rational quadratic covariance function with ARD % covRQiso - isotropic rational quadratic covariance function % covSEard - squared exponential covariance function with ARD % covSEiso - isotropic squared exponential covariance function % covSEisoU - as above but without latent scale % % composite (meta) covariance functions (see explanation at the bottom): % covScale - scaled version of a covariance function % covProd - products of covariance functions % covSum - sums of covariance functions % covADD - additive covariance function % covMask - mask some dimensions of the data % % special purpose (wrapper) covariance functions % covFITC - to be used in conjunction with infFITC for large scale % regression problems; any covariance can be wrapped by % covFITC such that the FITC approximation is applicable % % Naming convention: all covariance functions are named "cov/cov*.m". A trailing % "iso" means isotropic, "ard" means Automatic Relevance Determination, and % "one" means that the distance measure is parameterized by a single parameter. % % The covariance functions are written according to a special convention where % the exact behaviour depends on the number of input and output arguments % passed to the function. If you want to add new covariance functions, you % should follow this convention if you want them to work with the function gp. % There are four different ways of calling the covariance functions: % % 1) With no (or one) input argument(s): % % s = covNAME % % The covariance function returns a string s telling how many hyperparameters it % expects, using the convention that "D" is the dimension of the input space. % For example, calling "covRQard" returns the string '(D+2)'. % % 2) With two input arguments: % % K = covNAME(hyp, x) equivalent to K = covNAME(hyp, x, []) % % The function computes and returns the covariance matrix where hyp are % the hyperparameters and x is an n by D matrix of cases, where % D is the dimension of the input space. The returned covariance matrix is of % size n by n. % % 3) With three input arguments: % % Ks = covNAME(hyp, x, xs) % kss = covNAME(hyp, xs, 'diag') % % The function computes test set covariances; kss is a vector of self covariances % for the test cases in xs (of length ns) and Ks is an (n by ns) matrix of cross % covariances between training cases x and test cases xs. % % 4) With four input arguments: % % dKi = covNAME(hyp, x, [], i) % dKsi = covNAME(hyp, x, xs, i) % dkssi = covNAME(hyp, xs, 'diag', i) % % The function computes and returns the partial derivatives of the % covariance matrices with respect to hyp(i), i.e. with % respect to the hyperparameter number i. % % Covariance functions can be specified in two ways: either as a string % containing the name of the covariance function or using a cell array. For % example: % % cov = 'covRQard'; % cov = {'covRQard'}; % cov = {@covRQard}; % % are supported. Only the second and third form using the cell array can be used % for specifying composite covariance functions, made up of several % contributions. For example: % % cov = {'covScale', {'covRQiso'}}; % cov = {'covSum', {'covRQiso','covSEard','covNoise'}}; % cov = {'covProd',{'covRQiso','covSEard','covNoise'}}; % cov = {'covMask',{mask,'covSEiso'}} % q=1; cov = {'covPPiso',q}; % d=3; cov = {'covPoly',d}; % cov = {'covADD',{[1,2],'covSEiso'}}; % cov = {@covFITC, {@covSEiso}, u}; where u are the inducing inputs % % specifies a covariance function which is the sum of three contributions. To % find out how many hyperparameters this covariance function requires, we do: % % feval(cov{:}) % % which returns the string '3+(D+1)+1' (i.e. the 'covRQiso' contribution uses % 3 parameters, the 'covSEard' uses D+1 and 'covNoise' a single parameter). % % See also doc/usageCov.m. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18 ================================================ FILE: code/gpml/doc/README ================================================ GAUSSIAN PROCESS REGRESSION AND CLASSIFICATION Toolbox version 3.1 for GNU Octave 3.2.x and Matlab 7.x Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-09-27. 0) HOW TO READ ============== If you want to get started without further delay, then read section 1) below and jump right to the examples in doc/index.html. 1) ABOUT THESE PROGRAMS ======================= This collection of matlab programs implements and demonstrates some of the algorithms described in a) the book by Rasmussen and Williams: "Gaussian Processes for Machine Learning", the MIT Press 2006, in b) the article by Nickisch and Rasmussen: "Approximations for Binary Gaussian Process Classification", JMLR 2008 and in c) the article by Candela and Rasmussen: "A Unifying View of Sparse Approximate Gaussian Process Regression", JMLR 2005. There are 6 code subdirectories: cov, doc, inf, lik, mean and util. cov/ contains covariance functions cov*.m => see covFunctions.m doc/ contains an index.html file providing documentation. This information is also available from http://www.GaussianProcess.org/gpml/code. Usage of mean, cov, classification and regression is demonstrated in usage*.m. Further details can be found in the developer documentation manual.pdf. inf/ contains the inference algorithms inf*.m => see infMethods.m lik/ contains the likelihood functions lik*.m => see likFunctions.m mean/ contains the mean functions mean*.m => see meanFunctions.m util/ contains optimisation routines, backward compatibility programs and small auxiliary scripts Before running the demos or any other part of the code, you should execute startup.m to add all necessary directories to your path. In Octave, you can also use the .octaverc file. 2) ABOUT MEX FILES ================== Some of the programs make use of the MEX facility in matlab for more efficient implementation. However, if you don't know about how to compile MEX files, you do not need to worry about this -- the code runs anyway. If you do compile the MEX files, this is automatically detected, and the program will run more efficiently. If you do not have a compiler installed on your system, you might want to consult [2]. Two components of the toolbox can be accelerated by MEX. First, there is an auxiliary functions solve_chol.m having a MEX equivalent. This can be compiled by either executing util/make.m from the Matlab/Octave command line. In addition to that, we provide a Makefile for Matlab in util/ that you can run by simply typing make in your shell after having supplied your Matlab path and your operating system. We recommend the make.m script because it works both under Matlab and Octave. Second, in order to use the L-BFGS minimiser for hyperparameter optimisation as an equivalent to minimize.m, you have to compile Peter Carbonetto's "Matlab interface for L-BFGS-B" [1]. The challenge here is the Fortran 77 code. We provide a Makefile suitable for Linux 32/64 bit and Mac whenever you have 1) g77 or 2) gfortran properly set up. Under Ubuntu, you can achieve this by installing the packages fort77 and gfortran, respectively. Compilation is done by editing util/lbfgsb/Makefile. In any case, you need to provide $MATLAB_HOME which can be found by the commands 'locate matlab' or 'find / -name "matlab"'. You can choose between two compilation modes: a) using the mex utility by Matlab [default] provide $MEX, then type 'make mex' b) without mex utility by Matlab provide $MEX_SUFFIX and $MATLAB_LIB, then type 'make nomex' In Ubuntu 10.04 LTS, the libg2c library needed for both 1)+a) and 1)+b) is not included per default. If 'ls /usr/lib/libg2c.*' does not list anything this is the case on your machine. You then whant to install the packages gcc-3.4-base and libg2c0 e.g. from http://packages.ubuntu.com/hardy/gcc-3.4-base and http://packages.ubuntu.com/hardy/libg2c0. After installation, you have to create a symbolic link by 'cd /usr/lib' and 'ln -s libg2c.so.0 libg2c.so'. [1] http://www.cs.ubc.ca/~pcarbo/lbfgsb-for-matlab.html [2] http://www.mathworks.com/support/compilers/R2010a 3) CURRENT VERSION ================== The current version of the programs is 3.0. Previous versions of the code are available at http://www.gaussianprocess.org/gpml/code/oldcode.html. 4) DIFFERENCES TO PREVIOUS VERSIONS =================================== NEW in version 3.1, 2010-10-10 ------------------------------ - following a suggestion by Ed Snelson we now support FITC regression - cov/covFITC.m and inf/infFITC.m have been added with Ed Snelson's help - the covariance interface was slightly changed to make that possible NEW in version 3.0, 2010-07-23 ------------------------------ A major code reorganisation effort did take place in the current release. First, classification and regression are now done by a single file gp.m which is completely generic in the likelihood. The previous regression program gpr.m corresponds to gp.m with Gaussian likelihood. Several other likelihoods for robust regresssion were added. Further, the code now supports mean functions with a similar specification mechanism as already used by the covariance functions. Previous implementations correspond to using meanZero.m. We merged the covariance functions covMatern3iso.m and covMatern5.iso into a single covMaterniso.m and added a covariance function for additive functions as well as the possibility to use only certain components of the data. Finally, we included covPPiso.m a piecewise polynomial covariance function with compact support. New likelihood functions in lik/: likGauss.m, likLaplace.m, likLogistic.m and likT.m New mean functions in mean/: meanConst.m, meanLinear.m, meanPow.m, meanProd.m, meanSum.m and meanZero.m New covariance functions in cov/: covADD.m, covMask.m, covPPiso.m The gprSRPP.m function which previously provided "Subset of Regressors" and the "Projected Process" approximation has now been removed. NEW in version 2.1, 2007-07-25 ------------------------------ covConst.m: fixed a bug which caused an error in the derivative of the log marginal likelihood for certain combinations of covariance functions and approximation methods. (Thanks to Antonio Eleuteri for reporting the problem) gauher.m: added the function "gauher.m" which was mistakenly missing from the previous release. This caused an error for certain combinations of approximation method and likelihood function. logistic.m: modified the approximation of moments calculation to use a mixture of cumulative Gaussian, rather than Gauss-Hermite quadrature, as the former turns out to be more accurate. NEW in version 2.0, 2007-06-25 ------------------------------ Some code restructuring has taken place for the classification code to make it more modular, to facilitate addition of new likelihood functions and approximations methods. Now, all classification is done using the binaryGP function, which (among other things) takes an approximation method and a likelihood function as an arguments. Thus, binaryGP replaces both binaryEPGP and binaryLapaceGP, although wrapper functions are still provided for backward compatibility. This gives added flexibility: now EP can also be used wth the logistic likelihood function (implemented using Gauss-Hermite quadrature). approxEP.m: New file, containing the Expectation Propagation approximation method, which was previously contained in binaryEPGP.m approxLA.m: New file, containing Laplaces approximation method, which was previously contained in binaryLaplace.m approximations.m: New file, help for the approximation methods. binaryEPGP.m: This file has been replaced by a wrapper (for backward compatibility) which calls the more general binaryGP function. binaryGP.m: New general function to do binary classification. binaryLaplaceGP.m: This file has been replaced by a wrapper (for backward compatibility) which calls the more general binaryGP function. covMatern3iso.m, covMatern5iso.m, covNNone.m, covRQard.m, covRQiso.m, cosSEard, covSEiso: now check more carefully, that persistent variables have the correct sizes, and some variable names have been modified. cumGauss.m: New file, containing code for the cumulative Gaussian likelihood function likelihoods.m: New file, help for likelihood functions logistic.m: New file, logistic likelihood NEW in version 1.3, 2006-09-08 ------------------------------ covRQard.m: bugfix: replaced x with x' and z with z' in line 36 covRQiso.m: bugfix: replaced x with x' and z with z' in line 28 minimize.m: correction: replaced "error()" with "error('')", and made a few cosmetic changes binaryEPGP.m: added the line "lml = -n*log(2);" in line 77. This change should be largely inconsequential, but occationally may save things when the covariance matrix is exceptionally badly conditioned. NEW in version 1.2, 2006-05-10 ------------------------------ added the "erfint" function to "binaryLaplaceGP.m". The erfint function was missing by mistake, preventing the use of the "logistic" likelihood. NEW in version 1.1, 2006-04-12 ------------------------------ added files: "covProd.m" and "covPeriodic.m" changes: "covSEiso.m" was changed slightly to avoid the use of persistent variables NEW in version 1.0, 2006-03-29 ------------------------------ initial version shipped with the book ================================================ FILE: code/gpml/doc/changelog ================================================ GAUSSIAN PROCESS REGRESSION AND CLASSIFICATION Toolbox version 3.1 for GNU Octave 3.2.x and Matlab 7.x Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-15. Changes and small incremental bug fixes applied since last release. 2011-02-15, hn - added gfortran support to util/lbfgsb/Makefile (thanks to Ernst Kloppenburg) 2011-01-05 - changed parametrisation in cov/covPeriodic.m (thanks to Philipp Hennig) 2010-12-21 - fixed a stupid bug in cov/covFITC.m (thanks to Krzysztof Chalupka) 2010-12-16 - changed scalars in util/solve_chol.c to long to provide 64bit compatibility (thanks to Ernst Kloppenburg) 2010-12-14 - added infLOO to allow pseudo-likelihood optimisation instead of the marginal likelihood 2010-12-13 - fixed a performance issue in util/sq_dist.m (thanks to Krzysztof Chalupka) 2010-11-26 - fixed a bug causing numerical problems in infFITC and covFITC whenever Kuu is close to singular (thanks to Joris M. Mooij) 2010-11-24 - lbfgs compilation requires some packages not in jaunty but in hardy - updated the documentation to account for this ================================================ FILE: code/gpml/doc/demoClassification.m ================================================ disp(' '); disp('clear all, close all'); clear all, close all disp(' ') disp('n1 = 80; n2 = 40; % number of data points from each class'); n1 = 80; n2 = 40; disp('S1 = eye(2); S2 = [1 0.95; 0.95 1]; % the two covariance matrices'); S1 = eye(2); S2 = [1 0.95; 0.95 1]; disp('m1 = [0.75; 0]; m2 = [-0.75; 0]; % the two means'); m1 = [0.75; 0]; m2 = [-0.75; 0]; disp(' ') disp('x1 = bsxfun(@plus, chol(S1)''*gpml_randn(0.2, 2, n1), m1);') x1 = bsxfun(@plus, chol(S1)'*gpml_randn(0.2, 2, n1), m1); disp('x2 = bsxfun(@plus, chol(S2)''*gpml_randn(0.3, 2, n2), m2);') x2 = bsxfun(@plus, chol(S2)'*gpml_randn(0.3, 2, n2), m2); disp(' '); disp('x = [x1 x2]''; y = [-ones(1,n1) ones(1,n2)]'';'); x = [x1 x2]'; y = [-ones(1,n1) ones(1,n2)]'; disp('plot(x1(1,:), x1(2,:), ''b+''); hold on;'); plot(x1(1,:), x1(2,:), 'b+', 'MarkerSize', 12); hold on; disp('plot(x2(1,:), x2(2,:), ''r+'');'); plot(x2(1,:), x2(2,:), 'r+', 'MarkerSize', 12); disp(' '); disp('[t1 t2] = meshgrid(-4:0.1:4,-4:0.1:4);'); [t1 t2] = meshgrid(-4:0.1:4,-4:0.1:4); disp('t = [t1(:) t2(:)]; n = length(t); % these are the test inputs'); t = [t1(:) t2(:)]; n = length(t); % these are the test inputs disp('tmm = bsxfun(@minus, t, m1'');'); tmm = bsxfun(@minus, t, m1'); disp('p1 = n1*exp(-sum(tmm*inv(S1).*tmm/2,2))/sqrt(det(S1));'); p1 = n1*exp(-sum(tmm*inv(S1).*tmm/2,2))/sqrt(det(S1)); disp('tmm = bsxfun(@minus, t, m2'');'); tmm = bsxfun(@minus, t, m2'); disp('p2 = n2*exp(-sum(tmm*inv(S2).*tmm/2,2))/sqrt(det(S2));'); p2 = n2*exp(-sum(tmm*inv(S2).*tmm/2,2))/sqrt(det(S2)); set(gca, 'FontSize', 24); disp('contour(t1, t2, reshape(p2./(p1+p2), size(t1)), [0.1:0.1:0.9]);'); contour(t1, t2, reshape(p2./(p1+p2), size(t1)), [0.1:0.1:0.9]); [c h] = contour(t1, t2, reshape(p2./(p1+p2), size(t1)), [0.5 0.5]); set(h, 'LineWidth', 2) colorbar grid axis([-4 4 -4 4]) %print -depsc f5.eps disp(' '); disp('Hit any key to continue...'); pause disp(' '); disp('meanfunc = @meanConst; hyp.mean = 0;'); meanfunc = @meanConst; hyp.mean = 0; disp('covfunc = @covSEard; hyp.cov = log([1 1 1]);'); covfunc = @covSEard; hyp.cov = log([1 1 1]); disp('likfunc = @likErf;'); likfunc = @likErf; disp(' '); disp('hyp = minimize(hyp, @gp, -40, @infEP, meanfunc, covfunc, likfunc, x, y);'); hyp = minimize(hyp, @gp, -40, @infEP, meanfunc, covfunc, likfunc, x, y); disp('[a b c d lp] = gp(hyp, @infEP, meanfunc, covfunc, likfunc, x, y, t, ones(n, 1));'); [a b c d lp] = gp(hyp, @infEP, meanfunc, covfunc, likfunc, x, y, t, ones(n, 1)); disp(' '); figure set(gca, 'FontSize', 24); disp('plot(x1(1,:), x1(2,:), ''b+''); hold on;'); plot(x1(1,:), x1(2,:), 'b+', 'MarkerSize', 12); hold on; disp('plot(x2(1,:), x2(2,:), ''r+'')'); plot(x2(1,:), x2(2,:), 'r+', 'MarkerSize', 12) disp('contour(t1, t2, reshape(exp(lp), size(t1)), [0.1:0.1:0.9]);'); contour(t1, t2, reshape(exp(lp), size(t1)), [0.1:0.1:0.9]); [c h] = contour(t1, t2, reshape(exp(lp), size(t1)), [0.5 0.5]); set(h, 'LineWidth', 2) colorbar grid axis([-4 4 -4 4]) %print -depsc f6.eps disp(' '); ================================================ FILE: code/gpml/doc/demoRegression.m ================================================ disp(' '); disp('clear all, close all'); clear all, close all disp(' '); disp('meanfunc = {@meanSum, {@meanLinear, @meanConst}}; hyp.mean = [0.5; 1];'); meanfunc = {@meanSum, {@meanLinear, @meanConst}}; hyp.mean = [0.5; 1]; disp('covfunc = {@covMaterniso, 3}; ell = 1/4; sf = 1; hyp.cov = log([ell; sf]);'); covfunc = {@covMaterniso, 3}; ell = 1/4; sf = 1; hyp.cov = log([ell; sf]); disp('likfunc = @likGauss; sn = 0.1; hyp.lik = log(sn);'); likfunc = @likGauss; sn = 0.1; hyp.lik = log(sn); disp(' '); disp('n = 20;'); n = 20; disp('x = gpml_randn(0.3, n, 1);'); x = gpml_randn(0.3, n, 1); disp('K = feval(covfunc{:}, hyp.cov, x);'); K = feval(covfunc{:}, hyp.cov, x); disp('mu = feval(meanfunc{:}, hyp.mean, x);'); mu = feval(meanfunc{:}, hyp.mean, x); disp('y = chol(K)''*gpml_randn(0.15, n, 1) + mu + exp(hyp.lik)*gpml_randn(0.2, n, 1);'); y = chol(K)'*gpml_randn(0.15, n, 1) + mu + exp(hyp.lik)*gpml_randn(0.2, n, 1); figure(1) set(gca, 'FontSize', 24) disp(' '); disp('plot(x, y, ''+'')'); plot(x, y, '+', 'MarkerSize', 12) axis([-1.9 1.9 -0.9 3.9]) grid on xlabel('input, x') ylabel('output, y') %print -depsc f0.eps disp(' '); disp('Hit any key to continue...'); pause disp(' '); disp('nlml = gp(hyp, @infExact, meanfunc, covfunc, likfunc, x, y)'); nlml = gp(hyp, @infExact, meanfunc, covfunc, likfunc, x, y) disp(' ') disp('z = linspace(-1.9, 1.9, 101)'';'); z = linspace(-1.9, 1.9, 101)'; disp('[m s2] = gp(hyp, @infExact, meanfunc, covfunc, likfunc, x, y, z);'); [m s2] = gp(hyp, @infExact, meanfunc, covfunc, likfunc, x, y, z); figure(2) set(gca, 'FontSize', 24); disp(' '); disp('f = [m+2*sqrt(s2); flipdim(m-2*sqrt(s2),1)];'); f = [m+2*sqrt(s2); flipdim(m-2*sqrt(s2),1)]; disp('fill([z; flipdim(z,1)], f, [7 7 7]/8);'); fill([z; flipdim(z,1)], f, [7 7 7]/8); disp('hold on; plot(z, m); plot(x, y, ''+'')') hold on; plot(z, m, 'LineWidth', 2); plot(x, y, '+', 'MarkerSize', 12) axis([-1.9 1.9 -0.9 3.9]) grid on xlabel('input, x') ylabel('output, y') %print -depsc f1.eps disp(' '); disp('Hit any key to continue...'); pause disp(' '); disp('covfunc = @covSEiso; hyp2.cov = [0; 0]; hyp2.lik = log(0.1);'); covfunc = @covSEiso; hyp2.cov = [0; 0]; hyp2.lik = log(0.1); disp('hyp2 = minimize(hyp2, @gp, -100, @infExact, [], covfunc, likfunc, x, y)'); hyp2 = minimize(hyp2, @gp, -100, @infExact, [], covfunc, likfunc, x, y); disp(' '); disp('exp(hyp2.lik)'); exp(hyp2.lik) disp('nlml2 = gp(hyp2, @infExact, [], covfunc, likfunc, x, y)'); nlml2 = gp(hyp2, @infExact, [], covfunc, likfunc, x, y) disp('[m s2] = gp(hyp2, @infExact, [], covfunc, likfunc, x, y, z);'); [m s2] = gp(hyp2, @infExact, [], covfunc, likfunc, x, y, z); disp(' ') figure(3) set(gca, 'FontSize', 24) f = [m+2*sqrt(s2); flipdim(m-2*sqrt(s2),1)]; disp('fill([z; flipdim(z,1)], f, [7 7 7]/8)'); fill([z; flipdim(z,1)], f, [7 7 7]/8) disp('hold on; plot(z, m); plot(x, y, ''+'')'); hold on; plot(z, m, 'LineWidth', 2); plot(x, y, '+', 'MarkerSize', 12) grid on xlabel('input, x') ylabel('output, y') axis([-1.9 1.9 -0.9 3.9]) %print -depsc f2.eps disp(' '); disp('Hit any key to continue...'); pause disp(' '); disp('hyp.cov = [0; 0]; hyp.mean = [0; 0]; hyp.lik = log(0.1);'); hyp.cov = [0; 0]; hyp.mean = [0; 0]; hyp.lik = log(0.1); disp('hyp = minimize(hyp, @gp, -100, @infExact, meanfunc, covfunc, likfunc, x, y);'); hyp = minimize(hyp, @gp, -100, @infExact, meanfunc, covfunc, likfunc, x, y); disp('[m s2] = gp(hyp, @infExact, meanfunc, covfunc, likfunc, x, y, z);'); [m s2] = gp(hyp, @infExact, meanfunc, covfunc, likfunc, x, y, z); figure(4) set(gca, 'FontSize', 24) disp(' '); disp('f = [m+2*sqrt(s2); flipdim(m-2*sqrt(s2),1)];'); f = [m+2*sqrt(s2); flipdim(m-2*sqrt(s2),1)]; disp('fill([z; flipdim(z,1)], f, [7 7 7]/8)'); fill([z; flipdim(z,1)], f, [7 7 7]/8) disp('hold on; plot(z, m); plot(x, y, ''+'');'); hold on; plot(z, m, 'LineWidth', 2); plot(x, y, '+', 'MarkerSize', 12) grid on xlabel('input, x') ylabel('output, y') axis([-1.9 1.9 -0.9 3.9]) %print -depsc f3.eps disp(' '); disp('Hit any key to continue...'); pause disp('large scale regression using the FITC approximation') disp('nu = fix(n/2); u = linspace(-1.3,1.3,nu)'';') nu = fix(n/2); u = linspace(-1.3,1.3,nu)'; disp('covfuncF = {@covFITC, {covfunc}, u};') covfuncF = {@covFITC, {covfunc}, u}; disp('[mF s2F] = gp(hyp, @infFITC, meanfunc, covfuncF, likfunc, x, y, z);') [mF s2F] = gp(hyp, @infFITC, meanfunc, covfuncF, likfunc, x, y, z); figure(5) set(gca, 'FontSize', 24) disp(' '); disp('f = [mF+2*sqrt(s2F); flipdim(mF-2*sqrt(s2F),1)];'); f = [mF+2*sqrt(s2F); flipdim(mF-2*sqrt(s2F),1)]; disp('fill([z; flipdim(z,1)], f, [7 7 7]/8)'); fill([z; flipdim(z,1)], f, [7 7 7]/8) disp('hold on; plot(z, mF); plot(x, y, ''+'');'); hold on; plot(z, mF, 'LineWidth', 2); plot(x, y, '+', 'MarkerSize', 12) disp('plot(u,1,''o'')') plot(u,1,'ko', 'MarkerSize', 12) grid on xlabel('input, x') ylabel('output, y') axis([-1.9 1.9 -0.9 3.9]) print -depsc f4.eps disp(' '); ================================================ FILE: code/gpml/doc/gpml_randn.m ================================================ function x = gpml_randn(seed,varargin) % Generate pseudo-random numbers in a quick and dirty way. % The function makes sure, we obtain the same random numbers using Octave and % Matlab for the demo scripts. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch 2010-07-07. if nargin==2 sz = varargin{1}; else sz = zeros(1,nargin-1); for i=1:nargin-1 sz(i) = varargin{i}; end end n = prod(sz); N = ceil(n/2)*2; % minimal uniform random number generator for uniform deviates from [0,1] % by Park and Miller a = 7^5; m = 2^31-1; % using Schrage's algorithm q = fix(m/a); r = mod(m,a); % m = a*q+r u = zeros(N+1,1); u(1) = fix(seed*2^31); for i=2:N+1 % Schrage's algorithm for mod(a*u(i),m) u(i) = a*mod(u(i-1),q) - r*fix(u(i-1)/q); if u(i)<0, u(i) = u(i)+m; end end u = u(2:N+1)/2^31; % Box-Muller transform: Numerical Recipies, 2nd Edition, $7.2.8 % http://en.wikipedia.org/wiki/Box-Muller_transform w = sqrt(- 2*log(u(1:N/2))); % split into two groups x = [w.*cos(2*pi*u(N/2+1:N)); w.*sin(2*pi*u(N/2+1:N))]; x = reshape(x(1:n),sz); ================================================ FILE: code/gpml/doc/index.html ================================================ Documentation for GPML Matlab Code

Documentation for GPML Matlab Code version 3.1

What?

The code provided here originally demonstrated the main algorithms from Rasmussen and Williams: Gaussian Processes for Machine Learning. It has since grown to allow more likelihood functions, further inference methods and a flexible framework for specifying GPs. It does not currently contain inference methods for sparse approximations. Other GP packages can be found here.

The code is written by Carl Edward Rasmussen and Hannes Nickisch; it runs on both Octave 3.2.x and Matlab® 7.x. The code is based on previous versions written by Carl Edward Rasmussen and Chris Williams.

Download, Install and Documentation

All the code including demonstrations and html documentation can be downloaded in a tar or zip archive file.

Minor changes and incremental bugfixes are documented in the changelog.

Please read the copyright notice.

After unpacking the tar or zip file you will find 6 subdirectories: cov, doc, inf, lik, mean and util. It is not necessary to install anything to get started, just run the startup.m script to set your path.

Details about the directory contents and on how to compile mex files can be found in the README. The getting started guide is the remainder of the html file you are currently reading (also available at http://gaussianprocess.org/gpml/code/matlab/doc). A Developer's Guide containing technical documentation is found in manual.pdf, but for the casual user, the guide is below.

Theory

Gaussian Processes (GPs) can conveniently be used for Bayesian supervised learning, such as regression and classification. In its simplest form, GP inference can be implemented in a few lines of code. However, in practice, things typically get a little more complicated: you might want to use complicated covariance functions and mean functions, learn good values for hyperparameters, use non-Gaussian likelihood functions (rendering exact inference intractable), use approximate inference algorithms, or combinations of many or all of the above. This is what the GPML software package does.

Before going straight to the examples, just a brief note about the organization of the package. There are four types of objects which you need to know about:

Gaussian Process
A Gaussian Process is fully specified by a mean function and a covariance function. These functions are specified separately, and consist of a specification of a functional form as well as a set of parameters called hyperparameters, see below.
Mean functions
Several mean functions are available, all start with the four letters mean and reside in the mean directory. An overview is provided by the meanFunctions help function (type help meanFunctions to get help), and an example is the meanLinear function.
Covariance functions
There are many covariance functions available, all start with the three letters cov and reside in the cov directory. An overview is provided by the covFunctions help function (type help covFunctions to get help), and an example is the covSEard "Squared Exponential with Automatic Relevance Determination" covariance function.
For both mean functions and covariance functions, two types exist: simple and composite. Whereas simple types are specified by the function name (or function pointer), composite functions join together several components using cell arrays. Composite functions can be composed of other composite functions, allowing for very flexible and interesting structures. Examples are given below and in the usageMean and usageCov functions.
Hyperparameters
GPs are typically specified using mean and covariance functions which have free parameters called hyperparameters. Also likelihood functions may have such parameters. These are encoded in a struct with the fields mean, cov and lik (some of which may be empty). When specifying hyperparameters, it is important that the number of elements in each of these struct fields, precisely match the number of parameters expected by the mean function, the covariance function and the likelihood functions respectively (otherwise an error will result). Hyperparameters whose natural domain is positive are represented by their logarithms.
Likelihood Functions
The likelihood function specifies the probability of the observations given the latent function, i.e. the GP (and the hyperparameters). All likelihood functions begin with the three letters lik and reside in the lik directory. An overview is provided by the likFunctions help function (type help likFunctions to get help). Some examples are likGauss the Gaussian likelihood and likLogistic the logistic function used in classification (a.k.a. logistic regression).
Inference Methods
The inference methods specify how to compute with the model, i.e. how to infer the (approximate) posterior process, how to find hyperparameters, evaluate the log marginal likelihood and how to make predictions. Inference methods all begin with the three letters inf and reside in the inf directory. An overview is provided by the infMethods help file (type help infMethods to get help). Some examples are infExact for exact inference or infEP for the Expectation Propagation algorithm. Further usage examples are provided for both regression and classification. However, not all combinations of likelihood function and inference method are possible (e.g. you cannot do exact inference with a Laplace likelihood). An exhaustive compatibility matrix between likelihoods (rows) and inference methods (columns) is given in the table below:
Likelihood \ Inference
 		  
GPML name		 Exact
infExact		 FITC
infFITC		 EP
infEP		 Laplace
infLaplace		 Variational Bayes
infVB		 LOO
infLOO		 type, output domain
 		 alternate names
 
Gaussian likGauss $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ regression, IR  
Sech-squared likSech2     $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ regression, IR  
Laplacian likLaplace     $\checkmark$   $\checkmark$ $\checkmark$ regression, IR double exponential
Student's t likT       $\checkmark$ $\checkmark$ $\checkmark$ regression, IR  
Error function likErf     $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ classification, ±1 probit regression
Logistic function likLogistic     $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ classification, ±1 logistic regression
logit regression

All of the objects described above are written in a modular way, so you can add functionality if you feel constrained despite the considerable flexibility provided. Details about how to do this are provided in the developer documentation.

Practice

Using the GPML package is simple. There is only one single function to call: gp, it does posterior inference, learns hyperparameters, computes the marginal likelihood and makes predictions. Generally, the gp function takes the following arguments: a hyperparameter struct, an inference method, a mean function, a covariance function, a likelihood function, training inputs, training targets, and possibly test cases. The exact computations done by the function is controlled by the number of input and output arguments in the call. Here is part of the help message for the gp function (follow the link to see the whole thing):

  function [varargout] = gp(hyp, inf, mean, cov, lik, x, y, xs, ys)

  [ ... snip ...]

  Two modes are possible: training or prediction: if no test cases are
  supplied, then the negative log marginal likelihood and its partial
  derivatives wrt the hyperparameters is computed; this mode is used to fit the
  hyperparameters. If test cases are given, then the test set predictive
  probabilities are returned. Usage:

    training: [nlZ dnlZ          ] = gp(hyp, inf, mean, cov, lik, x, y);
  prediction: [ymu ys2 fmu fs2   ] = gp(hyp, inf, mean, cov, lik, x, y, xs);
          or: [ymu ys2 fmu fs2 lp] = gp(hyp, inf, mean, cov, lik, x, y, xs, ys);

  [ .. snip ...]

Here x and y are training inputs and outputs, and xs and ys are test set inputs and outputs, nlZ is the negative log marginal likelihood and dnlZ its partial derivatives wrt the hyperparameters (which are used for training the hyperparameters). The prediction outputs are ymu and ys2 for test output mean and covariance, and fmu and fs2 are the equivalent quenteties for the corresponding latent variables. Finally, lp are the test output log probabilities.

Instead of exhaustively explaining all the possibilities, we will give two illustrative examples to give you the idea; one for regression and one for classification. You can either follow the example here on this page, or using the two scripts demoRegression and demoClassification (using the scripts, you still need to follow the explanation on this page).

Regression

You can either follow the example here on this page, or use the script demoRegression.

This is a simple example, where we first generate n=20 data points from a GP, where the inputs are scalar (so that it is easy to plot what is going on). We then use various other GPs to make inferences about the underlying function.

First, generate some data from a Gaussian process (it is not essential to understand the details of this):

  clear all, close all
 
  meanfunc = {@meanSum, {@meanLinear, @meanConst}}; hyp.mean = [0.5; 1];
  covfunc = {@covMaterniso, 3}; ell = 1/4; sf = 1; hyp.cov = log([ell; sf]);
  likfunc = @likGauss; sn = 0.1; hyp.lik = log(sn);
 
  n = 20;
  x = gpml_randn(0.3, n, 1);
  K = feval(covfunc{:}, hyp.cov, x);
  mu = feval(meanfunc{:}, hyp.mean, x);
  y = chol(K)'*gpml_randn(0.15, n, 1) + mu + exp(hyp.lik)*gpml_randn(0.2, n, 1);

  plot(x, y, '+')

Above, we first specify the mean function meanfunc, covariance function covfunc of a GP and a likelihood function, likfunc. The corresponding hyperparameters are specified in the hyp structure:

The mean function is composite, adding (using meanSum function) a linear (meanLinear) and a constant (meanConst) to get an affine function. Note, how the different components are composed using cell arrays. The hyperparameters for the mean are given in hyp.mean and consists of a single (because the input will one dimensional, i.e. D=1) slope (set to 0.5) and an off-set (set to 1). The number and the order of these hyperparameters conform to the mean function specification. You can find out how many hyperparameters a mean (or covariance or likelihood function) expects by calling it without arguments, such as feval(@meanfunc{:}). For more information on mean functions see meanFunctions and the directory mean/.

The covariance function is of the Matérn form with isotropic distance measure covMaterniso. This covariance function is also composite, as it takes a constant (related to the smoothness of the GP), which in this case is set to 3. The covariance function takes two hyperparameters, a characteristic length-scale ell and the standard deviation of the signal sf. Note, that these positive parameters are represented in hyp.cov using their logarithms. For more information on covariance functions see covFunctions and cov/.

Finally, the likelihood function is specified to be Gaussian. The standard deviation of the noise sn is set to 0.1. Again, the representation in the hyp.lik is given in terms of its logarithm. For more information about likelihood functions, see likFunctions and lik/.

Then, we generate a dataset with n=20 examples. The inputs x are drawn from a unit Gaussian (using the gpml_randn utility, which generates unit Gaussian pseudo random numbers with a specified seed). We then evaluate the covariance matrix K and the mean vector m by calling the corresponding functions with the hyperparameters and the input locations x. Finally, the targets y are computed by drawing randomly from a Gaussian with the desired covariance and mean and adding Gaussian noise with standard deviation exp(hyp.lik). The above code is a bit special because we explicitly call the mean and covariance functions (in order to generate samples from a GP); ordinarily, we would only directly call the gp function.

f0.gif

Let's ask the model to compute the (joint) negative log probability (density) nlml (also called marginal likelihood or evidence) and to generalize from the training data to other (test) inputs z:

  nlml = gp(hyp, @infExact, meanfunc, covfunc, likfunc, x, y)

  z = linspace(-1.9, 1.9, 101)';
  [m s2] = gp(hyp, @infExact, meanfunc, covfunc, likfunc, x, y, z);

  f = [m+2*sqrt(s2); flipdim(m-2*sqrt(s2),1)]; 
  fill([z; flipdim(z,1)], f, [7 7 7]/8)
  hold on; plot(z, m); plot(x, y, '+')

The gp function is called with a struct of hyperparameters hyp, and inference method, in this case @infExact for exact inference and the mean, covariance and likelihood functions, as well as the inputs and outputs of the training data. With no test inputs, gp returns the negative log probability of the training data, in this example nlml=11.97.

To compute the predictions at test locations we add the test inputs z as a final argument, and gp returns the mean m variance s2 at the test location. The program is using algorithm 2.1 from the GPML book. Plotting the mean function plus/minus two standard deviations (corresponding to a 95% confidence interval):

f1.gif

Typically, we would not a priori know the values of the hyperparameters hyp, let alone the form of the mean, covariance or likelihood functions. So, let's pretend we didn't know any of this. We assume a particular structure and learn suitable hyperparameters:

  covfunc = @covSEiso; hyp2.cov = [0; 0]; hyp2.lik = log(0.1);

  hyp2 = minimize(hyp2, @gp, -100, @infExact, [], covfunc, likfunc, x, y);
  exp(hyp2.lik)
  nlml2 = gp(hyp2, @infExact, [], covfunc, likfunc, x, y)

  [m s2] = gp(hyp2, @infExact, [], covfunc, likfunc, x, y, z);
  f = [m+2*sqrt(s2); flipdim(m-2*sqrt(s2),1)];
  fill([z; flipdim(z,1)], f, [7 7 7]/8)
  hold on; plot(z, m); plot(x, y, '+')

First, we guess that a squared exponential covariance function covSEiso may be suitable. This covariance function takes two hyperparameters: a characteristic length-scale and a signal standard deviation (magnitude). These hyperparameters are non-negative and represented by their logarithms; thus, initializing hyp2.cov to zero, correspond to unit characteristic length-scale and unit signal standard deviation. The likelihood hyperparameter in hyp2.lik is also initialized. We assume that the mean function is zero, so we simply ignore it (and when in the following we call gp, we give an empty argument for the mean function).

In the following line, we optimize over the hyperparameters, by minimizing the negative log marginal likelihood w.r.t. the hyperparameters. The third parameter in the call to minimize limits the number of function evaluations to a maximum of 100. The inferred noise standard deviation is exp(hyp2.lik)=0.15, somewhat larger than the one used to generate the data (0.1). The final negative log marginal likelihood is nlml2=14.13, showing that the joint probability (density) of the training data is about exp(14.13-11.97)=8.7 times smaller than for the setup actually generating the data. Finally, we plot the predictive distribution.

f2.gif

This plot shows clearly, that the model is indeed quite different from the generating process. This is due to the different specifications of both the mean and covariance functions. Below we'll try to do a better job, by allowing more flexibility in the specification.

Note that the confidence interval in this plot is the confidence for the distribution of the (noisy) data. If instead you want the confidence region for the underlying function, you should use the 3rd and 4th output arguments from gp as these refer to the latent process, rather than the data points.

  hyp.cov = [0; 0]; hyp.mean = [0; 0]; hyp.lik = log(0.1);
  hyp = minimize(hyp, @gp, -100, @infExact, meanfunc, covfunc, likfunc, x, y);
  [m s2] = gp(hyp, @infExact, meanfunc, covfunc, likfunc, x, y, z);
 
  f = [m+2*sqrt(s2); flipdim(m-2*sqrt(s2),1)];
  fill([z; flipdim(z,1)], f, [7 7 7]/8)
  hold on; plot(z, m); plot(x, y, '+');

Here, we have changed the specification by adding the affine mean function. All the hyperparameters are learnt by optimizing the marginal likelihood.

f3.gif

This shows that a much better fit is achieved when allowing a mean function (although the covariance function is still different from that of the generating process).

Exercises for the reader

Inference Methods
Try using Expectation Propagation instead of exact inference in the above, by exchanging @infExact with @infEP. You get exactly identical results, why?
Mean or Covariance
Try training a GP where the affine part of the function is captured by the covariance function instead of the mean function. That is, use a GP with no explicit mean function, but further additive contributions to the covariance. How would you expect the marginal likelihood to compare to the previous case?

Large scale regression

In case the number of training inputs x exceeds a few thousands, exact inference using infExact.m takes too long. We offer the FITC approximation based on a low-rank plus diagonal approximation to the exact covariance to deal with these cases. The general idea is to use inducing points u and to base the computations on cross-covariances between training, test and inducing points only.

Using the FITC approximation is very simple, we just have to wrap the covariance function covfunc into covFITC.m and call gp.m with the inference method infFITC.m as demonstrated by the following lines of code. 

nu = fix(n/2); u = linspace(-1.3,1.3,nu)';
covfuncF = {@covFITC, {covfunc}, u};
[mF s2F] = gp(hyp, @infFITC, meanfunc, covfuncF, likfunc, x, y, z);

We define equispaced inducing points u that are shown in the figure as black circles. Note that the predictive variance is overestimated outside the support of the inducing inputs. In a multivariate example where densely sampled inducing inputs are infeasible, one can simply use a random subset of the training points. 

nu = fix(n/2); iu = randperm(n); iu = iu(1:nu); u = x(iu,:);
f4.gif

Classification

You can either follow the example here on this page, or use the script demoClassification.

The difference between regression and classification isn't of fundamental nature. We can use a Gaussian process latent function in essentially the same way, it is just that the Gaussian likelihood function often used for regression is inappropriate for classification. And since exact inference is only possible for Gaussian likelihood, we also need an alternative, approximate, inference method.

Here, we will demonstrate binary classification, using two partially overlapping Gaussian sources of data in two dimensions. First we generate the data:

  clear all, close all
 
  n1 = 80; n2 = 40;                   % number of data points from each class
  S1 = eye(2); S2 = [1 0.95; 0.95 1];           % the two covariance matrices
  m1 = [0.75; 0]; m2 = [-0.75; 0];                            % the two means
 
  x1 = bsxfun(@plus, chol(S1)'*gpml_randn(0.2, 2, n1), m1);
  x2 = bsxfun(@plus, chol(S2)'*gpml_randn(0.3, 2, n2), m2);
 
  x = [x1 x2]'; y = [-ones(1,n1) ones(1,n2)]';
  plot(x1(1,:), x1(2,:), 'b+'); hold on;
  plot(x2(1,:), x2(2,:), 'r+');

120 data points are generated from two Gaussians with different means and covariances. One Gaussian is isotropic and contains 2/3 of the data (blue), the other is highly correlated and contains 1/3 of the points (red). Note, that the labels for the targets are ±1 (and not 0/1).

In the plot, we superimpose the data points with the posterior equi-probability contour lines for the probability of class two given complete information about the generating mechanism

  [t1 t2] = meshgrid(-4:0.1:4,-4:0.1:4);
  t = [t1(:) t2(:)]; n = length(t);                 % these are the test inputs
  tmm = bsxfun(@minus, t, m1');
  p1 = n1*exp(-sum(tmm*inv(S1).*tmm/2,2))/sqrt(det(S1));
  tmm = bsxfun(@minus, t, m2');
  p2 = n2*exp(-sum(tmm*inv(S2).*tmm/2,2))/sqrt(det(S2));
  contour(t1, t2, reshape(p2./(p1+p2), size(t1)), [0.1:0.1:0.9]);
f5.gif

We specify a Gaussian process model as follows: a constant mean function, with initial parameter set to 0, a squared exponential with automatic relevance determination (ARD) covariance function covSEard. This covariance function has one characteristic length-scale parameter for each dimension of the input space, and a signal magnitude parameter, for a total of 3 parameters (as the input dimension is D=2). ARD with separate length-scales for each input dimension is a very powerful tool to learn which inputs are important for predictions: if length-scales are short, inputs are very important, and when they grow very long (compared to the spread of the data), the corresponding inputs will be largely ignored. Both length-scales and the signal magnitude are initialized to 1 (and represented in the log space). Finally, the likelihood function likErf has the shape of the error-function (or cumulative Gaussian), which doesn't take any hyperparameters (so hyp.lik does not exist).

  meanfunc = @meanConst; hyp.mean = 0;
  covfunc = @covSEard; ell = 1.0; sf = 1.0; hyp.cov = log([ell ell sf]);
  likfunc = @likErf;

  hyp = minimize(hyp, @gp, -40, @infEP, meanfunc, covfunc, likfunc, x, y);
  [a b c d lp] = gp(hyp, @infEP, meanfunc, covfunc, likfunc, x, y, t, ones(n, 1));

  plot(x1(1,:), x1(2,:), 'b+'); hold on; plot(x2(1,:), x2(2,:), 'r+')
  contour(t1, t2, reshape(exp(lp), size(t1)), [0.1:0.1:0.9]);

We train the hyperparameters using minimize, to minimize the negative log marginal likelihood. We allow for 40 function evaluations, and specify that inference should be done with the Expectation Propagation (EP) inference method @infEP, and pass the usual parameters. Training is done using algorithm 3.5 and 5.2 from the gpml book. When computing test probabilities, we call gp with additional test inputs, and as the last argument a vector of targets for which the log probabilities lp should be computed. The fist four output arguments of the function are mean and variance for the targets and corresponding latent variables respectively. The test set predictions are computed using algorithm 3.6 from the gpml book. The contour plot for the predictive distribution is shown below. Note, that the predictive probability is fairly close to the probabilities of the generating process in regions of high data density. Note also, that as you move away from the data, the probability approaches 1/3, the overall class probability.

f6.gif

Examining the two ARD characteristic length-scale parameters after learning, you will find that they are fairly similar, reflecting the fact that for this data set, both inputs important.

Exercise for the reader

Inference Methods
Use the Laplace Approximation for inference @infLaplace, and compare the approximate marginal likelihood for the two methods. Which approximation is best?
Covariance Function
Try using the squared exponential with isotropic distance measure covSEiso instead of ARD distance measure covSEard. Which is best?

Acknowledgements

Innumerable colleagues have helped to improve this software. Some of these are: John Cunningham, Máté Lengyel, Joris Mooij, Iain Murray and Chris Williams. Especially Ed Snelson helped to improve the code and to include sparse approximations.

Last modified: February 15th 2011 ================================================ FILE: code/gpml/doc/style.css ================================================ body {font-family: sans-serif; font-size: 16px} table {font-size: inherit;} ================================================ FILE: code/gpml/doc/usageClassification.m ================================================ % demonstrate usage of classification % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch 2010-06-21. clear all, close all %% SAY WHICH CODE WE WISH TO EXERCISE % id = [1,1]; % use Gauss/Exact id = [1,2; 2,2; 3,2]; % compare Laplace % id = [1,3; 2,3; 3,3]; % compare EP % id = [1,4; 3,4]; % compare VB seed = 943; randn('seed',seed), rand('seed',seed) ntr = 50; nte = 1e4; % number of training and test points xtr = 10*sort(rand(ntr,1)); % sample dataset p = @(x) 1./(1+exp(-5*sin(x))); % "true" underlying probability ytr = 2*(p(xtr)>rand(ntr,1))-1; % draw labels +1/-1 i = randperm(ntr); nout = 3; % add outliers ytr(i(1:nout)) = -ytr(i(1:nout)); xte = linspace(0,10,1e4)'; % support, we test our function on cov = {@covSEiso}; sf = 1; ell = 0.7; % setup the GP hyp0.cov = log([ell;sf]); mean = {@meanZero}; % m(x) = 0 hyp0.mean = []; lik_list = {'likGauss','likErf','likLogistic'}; % allowable likelihoods inf_list = {'infExact','infLaplace','infEP','infVB'}; % possible inference algs Ncg = 50; % number of conjugate gradient steps sdscale = 0.5; % how many sd wide should the error bars become? col = {'k',[.8,0,0],[0,.5,0],'b',[0,.75,.75],[.7,0,.5]}; % colors ymu{1} = 2*p(xte)-1; ys2{1} = 0; for i=1:size(id,1) lik = lik_list{id(i,1)}; % setup the likelihood if strcmp(lik,'likGauss') sn = .2; hyp0.lik = log(sn); else hyp0.lik = []; end inf = inf_list{id(i,2)}; fprintf('OPT: %s/%s\n',lik_list{id(i,1)},inf_list{id(i,2)}) hyp = minimize(hyp0,'gp', -Ncg, inf, mean, cov, lik, xtr, ytr); % opt hypers [ymu{i+1}, ys2{i+1}] = gp(hyp, inf, mean, cov, lik, xtr, ytr, xte); % predict [nlZ(i+1)] = gp(hyp, inf, mean, cov, lik, xtr, ytr); end figure, hold on for i=1:size(id,1)+1 plot(xte,ymu{i},'Color',col{i},'LineWidth',2) if i==1 leg = {'function'}; else leg{end+1} = sprintf('%s/%s -lZ=%1.2f',... lik_list{id(i-1,1)},inf_list{id(i-1,2)},nlZ(i)); end end for i=1:size(id,1)+1 ysd = sdscale*sqrt(ys2{i}); fill([xte;flipud(xte)],[ymu{i}+ysd;flipud(ymu{i}-ysd)],... col{i},'EdgeColor',col{i},'FaceAlpha',0.1,'EdgeAlpha',0.3); end for i=1:size(id,1)+1, plot(xte,ymu{i},'Color',col{i},'LineWidth',2), end plot(xtr,ytr,'k+'), plot(xtr,ytr,'ko'), legend(leg) ================================================ FILE: code/gpml/doc/usageCov.m ================================================ % demonstrate usage of covariance functions % % See also covFunctions.m. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18 clear all, close all n = 5; D = 3; x = randn(n,D); xs = randn(3,D); % create a data set % set up simple covariance functions cn = {'covNoise'}; sn = .1; hypn = log(sn); % one hyperparameter cc = {@covConst}; sf = 2; hypc = log(sf); % function handles OK cl = {@covLIN}; hypl = []; % linear is parameter-free cla = {'covLINard'}; L = rand(D,1); hypla = log(L); % linear (ARD) clo = {@covLINone}; ell = .9; hyplo = log(ell); % linear with bias cp = {@covPoly,3}; c = 2; hypp = log([c;sf]); % third order poly cga = {@covSEard}; hypga = log([L;sf]); % Gaussian with ARD cgi = {'covSEiso'}; hypgi = log([ell;sf]); % isotropic Gaussian cgu = {'covSEisoU'}; hypgu = log(ell); % isotropic Gauss no scale cra = {'covRQard'}; al = 2; hypra = log([L;sf;al]); % ration. quad. cri = {@covRQiso}; hypri = log([ell;sf;al]); % isotropic cm = {'covMaterniso',3}; hypm = log([ell;sf]); % Matern class q=3 cnn = {'covNNone'}; hypnn = log([L;sf]); % neural network cpe = {'covPeriodic'}; om = 2; hyppe = log([ell;om;sf]); % periodic ccc = {'covPPiso',2}; hypcc = hypm; % compact support poly degree 2 % set up composite covariance functions csc = {'covScale',{cgu}}; hypsc = [log(3); hypgu]; % scale by 9 csu = {'covSum',{cn,cc,cl}}; hypsu = [hypn; hypc; hypl]; % sum cpr = {@covProd,{cc,ccc}}; hyppr = [hypc; hypcc]; % product mask = [0,1,0]; % binary mask excluding all but the 2nd component cma = {'covMask',{mask,cgi{:}}}; hypma = hypgi; % additive based on SEiso using unary and pairwise interactions cad = {'covADD',{[1,2],'covSEiso'}}; % 0) specify covariance function cov = cma; hyp = hypma; % 1) query the number of parameters feval(cov{:}) % 2) evaluate the function on x feval(cov{:},hyp,x) % 3) evaluate the function on x and xs to get cross-terms kss = feval(cov{:},hyp,xs,'diag') Ks = feval(cov{:},hyp,x,xs) % 4) compute the derivatives w.r.t. to hyperparameter i i = 1; feval(cov{:},hyp,x,[],i) ================================================ FILE: code/gpml/doc/usageMean.m ================================================ % demonstrate usage of mean functions % % See also meanFunctions.m. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18 clear all, close all n = 5; D = 2; x = randn(n,D); % create a random data set % set up simple mean functions m0 = {'meanZero'}; hyp0 = []; % no hyperparameters are needed m1 = {'meanOne'}; hyp1 = []; % no hyperparameters are needed mc = {@meanConst}; hypc = 2; % also function handles are possible ml = {@meanLinear}; hypl = [2;3]; % m(x) = 2*x1 + 3*x2 % set up composite mean functions msc = {'meanScale',{m1}}; hypsc = [3; hyp1]; % scale by 3 msu = {'meanSum',{m0,mc,ml}}; hypsu = [hyp0; hypc; hypl]; % sum mpr = {@meanProd,{mc,ml}}; hyppr = [hypc; hypl]; % product mpo = {'meanPow',{3,msu}}; hyppo = hypsu; % third power mask = [0,1,0]; % binary mask excluding all but the 2nd component mma = {'meanMask',{mask,mpo{:}}}; hypma = hyppo; % 0) specify mean function % mean = m0; hyp = hyp0; % mean = msu; hyp = hypsu; % mean = mpr; hyp = hyppr; mean = mpo; hyp = hyppo; % 1) query the number of parameters feval(mean{:}) % 2) evaluate the function on x feval(mean{:},hyp,x) % 3) compute the derivatives w.r.t. to hyperparameter i i = 2; feval(mean{:},hyp,x,i) ================================================ FILE: code/gpml/doc/usageRegression.m ================================================ % demonstrate usage of regression % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch 2010-07-21. clear all, close all %% SAY WHICH CODE WE WISH TO EXERCISE id = [1,1]; % use Exact % id = [1,2; 3,2; 4,2]; % compare Laplace id = [1,3; 2,3; 3,3]; % compare EP % id = [1,4; 2,4; 3,4; 4,4]; % compare VB seed = 197; randn('seed',seed), rand('seed',seed) ntr = 50; nte = 1e4; % number of training and test points xtr = 10*sort(rand(ntr,1)); % sample dataset f = @(x) sin(x)+sqrt(x); % "true" underlying function sn = 0.2; ytr = f(xtr) + randn(ntr,1)*sn; % add Gaussian noise i = randperm(ntr); nout = 3; % add outliers ytr(i(1:nout)) = 5; xte = linspace(0,10,1e4)'; % support, we test our function on cov = {@covSEiso}; sf = 1; ell = 0.4; % setup the GP hyp0.cov = log([ell;sf]); mean = {@meanSum,{@meanLinear,@meanConst}}; a = 1/5; b = 1; % m(x) = a*x+b hyp0.mean = [a;b]; lik_list = {'likGauss','likLaplace','likSech2','likT'}; % possible likelihoods inf_list = {'infExact','infLaplace','infEP','infVB'}; % allowable inference algs Ncg = 50; % number of conjugate gradient steps sdscale = 0.5; % how many sd wide should the error bars become? col = {'k',[.8,0,0],[0,.5,0],'b',[0,.75,.75],[.7,0,.5]}; % colors ymu{1} = f(xte); ys2{1} = sn^2; nlZ(1) = -Inf; for i=1:size(id,1) lik = lik_list{id(i,1)}; % setup the likelihood if strcmp(lik,'likT') nu = 4; hyp0.lik = log([nu-1;sqrt((nu-2)/nu)*sn]); else hyp0.lik = log(sn); end inf = inf_list{id(i,2)}; fprintf('OPT: %s/%s\n',lik_list{id(i,1)},inf_list{id(i,2)}) if Ncg==0 hyp = hyp0; else hyp = minimize(hyp0,'gp', -Ncg, inf, mean, cov, lik, xtr, ytr); % opt hypers end [ymu{i+1}, ys2{i+1}] = gp(hyp, inf, mean, cov, lik, xtr, ytr, xte); % predict [nlZ(i+1)] = gp(hyp, inf, mean, cov, lik, xtr, ytr); end figure, hold on for i=1:size(id,1)+1 plot(xte,ymu{i},'Color',col{i},'LineWidth',2) if i==1 leg = {'function'}; else leg{end+1} = sprintf('%s/%s -lZ=%1.2f',... lik_list{id(i-1,1)},inf_list{id(i-1,2)},nlZ(i)); end end for i=1:size(id,1)+1 ysd = sdscale*sqrt(ys2{i}); fill([xte;flipud(xte)],[ymu{i}+ysd;flipud(ymu{i}-ysd)],... col{i},'EdgeColor',col{i},'FaceAlpha',0.1,'EdgeAlpha',0.3); end for i=1:size(id,1)+1, plot(xte,ymu{i},'Color',col{i},'LineWidth',2), end plot(xtr,ytr,'k+'), plot(xtr,ytr,'ko'), legend(leg) ================================================ FILE: code/gpml/gp.m ================================================ function [varargout] = gp(hyp, inf, mean, cov, lik, x, y, xs, ys) % Gaussian Process inference and prediction. The gp function provides a % flexible framework for Bayesian inference and prediction with Gaussian % processes for scalar targets, i.e. both regression and binary % classification. The prior is Gaussian process, defined through specification % of its mean and covariance function. The likelihood function is also % specified. Both the prior and the likelihood may have hyperparameters % associated with them. % % Two modes are possible: training or prediction: if no test cases are % supplied, then the negative log marginal likelihood and its partial % derivatives w.r.t. the hyperparameters is computed; this mode is used to fit % the hyperparameters. If test cases are given, then the test set predictive % probabilities are returned. Usage: % % training: [nlZ dnlZ ] = gp(hyp, inf, mean, cov, lik, x, y); % prediction: [ymu ys2 fmu fs2 ] = gp(hyp, inf, mean, cov, lik, x, y, xs); % or: [ymu ys2 fmu fs2 lp] = gp(hyp, inf, mean, cov, lik, x, y, xs, ys); % % where: % % hyp column vector of hyperparameters % inf function specifying the inference method % cov prior covariance function (see below) % mean prior mean function % lik likelihood function % x n by D matrix of training inputs % y column vector of length n of training targets % xs ns by D matrix of test inputs % ys column vector of length nn of test targets % % nlZ returned value of the negative log marginal likelihood % dnlZ column vector of partial derivatives of the negative % log marginal likelihood w.r.t. each hyperparameter % ymu column vector (of length ns) of predictive output means % ys2 column vector (of length ns) of predictive output variances % fmu column vector (of length ns) of predictive latent means % fs2 column vector (of length ns) of predictive latent variances % lp column vector (of length ns) of log predictive probabilities % % post struct representation of the (approximate) posterior % 3rd output in training mode and 6th output in prediction mode % % See also covFunctions.m, infMethods.m, likFunctions.m, meanFunctions.m. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18 if nargin<7 || nargin>9 disp('Usage: [nlZ dnlZ ] = gp(hyp, inf, mean, cov, lik, x, y);') disp(' or: [ymu ys2 fmu fs2 ] = gp(hyp, inf, mean, cov, lik, x, y, xs);') disp(' or: [ymu ys2 fmu fs2 lp] = gp(hyp, inf, mean, cov, lik, x, y, xs, ys);') return end if isempty(inf), inf = @infExact; else % set default inf if iscell(inf), inf = inf{1}; end % cell input is allowed if ischar(inf), inf = str2func(inf); end % convert into function handle end if isempty(mean), mean = {@meanZero}; end % set default mean if ischar(mean) || isa(mean, 'function_handle'), mean = {mean}; end % make cell if isempty(cov), error('Covariance function cannot be empty'); end % no default if ischar(cov) || isa(cov, 'function_handle'), cov = {cov}; end % make cell cov1 = cov{1}; if isa(cov1, 'function_handle'), cov1 = func2str(cov1); end if strcmp(cov1,'covFITC'); inf = @infFITC; end % only one possible inf alg if isempty(lik), lik = @likGauss; else % set default lik if iscell(lik), lik = lik{1}; end % cell input is allowed if ischar(lik), lik = str2func(lik); end % convert into function handle end D = size(x,2); if ~isfield(hyp,'mean'), hyp.mean = []; end % check the hyp specification if eval(feval(mean{:})) ~= numel(hyp.mean) error('Number of mean function hyperparameters disagree with mean function') end if ~isfield(hyp,'cov'), hyp.cov = []; end if eval(feval(cov{:})) ~= numel(hyp.cov) error('Number of cov function hyperparameters disagree with cov function') end if ~isfield(hyp,'lik'), hyp.lik = []; end if eval(feval(lik)) ~= numel(hyp.lik) error('Number of lik function hyperparameters disagree with lik function') end try % call the inference method % issue a warning if a classification likelihood is used in conjunction with % labels different from +1 and -1 if strcmp(func2str(lik),'likErf') || strcmp(func2str(lik),'likLogistic') uy = unique(y); if any( uy~=+1 & uy~=-1 ) warning('You attempt classification using labels different from {+1,-1}\n') end end if nargin>7 % compute marginal likelihood and its derivatives only if needed post = inf(hyp, mean, cov, lik, x, y); else if nargout==1 [post nlZ] = inf(hyp, mean, cov, lik, x, y); dnlZ = {}; else [post nlZ dnlZ] = inf(hyp, mean, cov, lik, x, y); end end catch msgstr = lasterr; if nargin > 7, error('Inference method failed [%s]', msgstr); else warning('Inference method failed [%s] .. attempting to continue',msgstr) dnlZ = struct('cov',0*hyp.cov, 'mean',0*hyp.mean, 'lik',0*hyp.lik); varargout = {NaN, dnlZ}; return % continue with a warning end end if nargin==7 % if no test cases are provided varargout = {nlZ, dnlZ, post}; % report -log marg lik, derivatives and post else alpha = post.alpha; L = post.L; sW = post.sW; if issparse(alpha) % handle things for sparse representations nz = alpha ~= 0; % determine nonzero indices if issparse(L), L = full(L(nz,nz)); end % convert L and sW if necessary if issparse(sW), sW = full(sW(nz)); end else nz = true(size(alpha)); end % non-sparse representation if numel(L)==0 % in case L is not provided, we compute it K = feval(cov{:}, hyp.cov, x(nz,:)); L = chol(eye(sum(nz))+sW*sW'.*K); end Ltril = all(all(tril(L,-1)==0)); % is L an upper triangular matrix? ns = size(xs,1); % number of data points nperbatch = 1000; % number of data points per mini batch nact = 0; % number of already processed test data points ymu = zeros(ns,1); ys2 = ymu; fmu = ymu; fs2 = ymu; lp = ymu; % allocate mem while nact use Cholesky parameters (alpha,sW,L) V = L'\(repmat(sW,1,length(id)).*Ks); fs2(id) = kss - sum(V.*V,1)'; % predictive variances else % L is not triangular => use alternative parametrisation fs2(id) = kss + sum(Ks.*(L*Ks),1)'; % predictive variances end fs2(id) = max(fs2(id),0); % remove numerical noise i.e. negative variances if nargin<9 [lp(id) ymu(id) ys2(id)] = lik(hyp.lik, [], fmu(id), fs2(id)); else [lp(id) ymu(id) ys2(id)] = lik(hyp.lik, ys(id), fmu(id), fs2(id)); end nact = id(end); % set counter to index of last processed data point end if nargin<9 varargout = {ymu, ys2, fmu, fs2, [], post}; % assign output arguments else varargout = {ymu, ys2, fmu, fs2, lp, post}; end end ================================================ FILE: code/gpml/inf/infEP.m ================================================ function [post nlZ dnlZ] = infEP(hyp, mean, cov, lik, x, y) % Expectation Propagation approximation to the posterior Gaussian Process. % The function takes a specified covariance function (see covFunction.m) and % likelihood function (see likFunction.m), and is designed to be used with % gp.m. See also infFunctions.m. In the EP algorithm, the sites are % updated in random order, for better performance when cases are ordered % according to the targets. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch 2010-02-25. % % See also INFMETHODS.M. persistent last_ttau last_tnu % keep tilde parameters between calls n = size(x,1); tol = 1e-6 * n; % thanks yunus % tolerance to stop EP iterations max_sweep = 100; % Changed by David Duvenaud from 10 to 100. min_sweep = 2; if ~ischar(lik), lik = func2str(lik); end % make likelihood variable a string inf = 'infEP'; K = feval(cov{:}, hyp.cov, x); % evaluate the covariance matrix m = feval(mean{:}, hyp.mean, x); % evaluate the mean vector % A note on naming: variables are given short but descriptive names in % accordance with Rasmussen & Williams "GPs for Machine Learning" (2006): mu % and s2 are mean and variance, nu and tau are natural parameters. A leading t % means tilde, a subscript _ni means "not i" (for cavity parameters), or _n % for a vector of cavity parameters. % marginal likelihood for ttau = tnu = zeros(n,1); equals n*log(2) for likCum* nlZ0 = -sum(feval(lik, hyp.lik, y, m, diag(K), inf)); if any(size(last_ttau) ~= [n 1]) % find starting point for tilde parameters ttau = zeros(n,1); % initialize to zero if we have no better guess tnu = zeros(n,1); Sigma = K; % initialize Sigma and mu, the parameters of .. mu = zeros(n,1); % .. the Gaussian posterior approximation nlZ = nlZ0; else ttau = last_ttau; % try the tilde values from previous call tnu = last_tnu; [Sigma, mu, nlZ, L] = epComputeParams(K, y, ttau, tnu, lik, hyp, m, inf); if nlZ > nlZ0 % if zero is better .. ttau = zeros(n,1); % .. then initialize with zero instead tnu = zeros(n,1); Sigma = K; % initialize Sigma and mu, the parameters of .. mu = zeros(n,1); % .. the Gaussian posterior approximation nlZ = nlZ0; end end nlZ_old = Inf; sweep = 0; % converged, max. sweeps or min. sweeps? while (abs(nlZ-nlZ_old) > tol && sweep < max_sweep) || sweep2 % do we want derivatives? dnlZ = hyp; % allocate space for derivatives ssi = sqrt(ttau); V = L'\(repmat(ssi,1,n).*K); Sigma = K - V'*V; mu = Sigma*tnu; tau_n = 1./diag(Sigma)-ttau; % compute the log marginal likelihood nu_n = mu./diag(Sigma)-tnu; % vectors of cavity parameters F = alpha*alpha'-repmat(sW,1,n).*solve_chol(L,diag(sW)); % covariance hypers for j=1:length(hyp.cov) dK = feval(cov{:}, hyp.cov, x, [], j); dnlZ.cov(j) = -sum(sum(F.*dK))/2; end for i = 1:numel(hyp.lik) % likelihood hypers dlik = feval(lik, hyp.lik, y, nu_n./tau_n+m, 1./tau_n, inf, i); dnlZ.lik(i) = -sum(dlik); end [junk,dlZ] = feval(lik, hyp.lik, y, nu_n./tau_n+m, 1./tau_n, inf); % mean hyps for i = 1:numel(hyp.mean) dm = feval(mean{:}, hyp.mean, x, i); dnlZ.mean(i) = -dlZ'*dm; end end % function to compute the parameters of the Gaussian approximation, Sigma and % mu, and the negative log marginal likelihood, nlZ, from the current site % parameters, ttau and tnu. Also returns L (useful for predictions). function [Sigma mu nlZ L] = epComputeParams(K, y, ttau, tnu, lik, hyp, m, inf) n = length(y); % number of training cases ssi = sqrt(ttau); % compute Sigma and mu L = chol(eye(n)+ssi*ssi'.*K); % L'*L=B=eye(n)+sW*K*sW V = L'\(repmat(ssi,1,n).*K); Sigma = K - V'*V; mu = Sigma*tnu; tau_n = 1./diag(Sigma)-ttau; % compute the log marginal likelihood nu_n = mu./diag(Sigma)-tnu+m.*tau_n; % vectors of cavity parameters lZ = feval(lik, hyp.lik, y, nu_n./tau_n, 1./tau_n, inf); nlZ = sum(log(diag(L))) -sum(lZ) -tnu'*Sigma*tnu/2 ... -(nu_n-m.*tau_n)'*((ttau./tau_n.*(nu_n-m.*tau_n)-2*tnu)./(ttau+tau_n))/2 ... +sum(tnu.^2./(tau_n+ttau))/2-sum(log(1+ttau./tau_n))/2; ================================================ FILE: code/gpml/inf/infExact.m ================================================ function [post nlZ dnlZ] = infExact(hyp, mean, cov, lik, x, y) % Exact inference for a GP with Gaussian likelihood. Compute a parametrization % of the posterior, the negative log marginal likelihood and its derivatives % w.r.t. the hyperparameters. See also "help infMethods". % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18 % % See also INFMETHODS.M. likstr = lik; if ~ischar(lik), likstr = func2str(lik); end if ~strcmp(likstr,'likGauss') % NOTE: no explicit call to likGauss error('Exact inference only possible with Gaussian likelihood'); end [n, D] = size(x); K = feval(cov{:}, hyp.cov, x); % evaluate covariance matrix m = feval(mean{:}, hyp.mean, x); % evaluate mean vector sn2 = exp(2*hyp.lik); % noise variance of likGauss L = chol(K/sn2+eye(n)); % Cholesky factor of covariance with noise alpha = solve_chol(L,y-m)/sn2; post.alpha = alpha; % return the posterior parameters post.sW = ones(n,1)/sqrt(sn2); % sqrt of noise precision vector post.L = L; % L = chol(eye(n)+sW*sW'.*K) if nargout>1 % do we want the marginal likelihood? nlZ = (y-m)'*alpha/2 + sum(log(diag(L))) + n*log(2*pi*sn2)/2; % -log marg lik if nargout>2 % do we want derivatives? dnlZ = hyp; % allocate space for derivatives Q = solve_chol(L,eye(n))/sn2 - alpha*alpha'; % precompute for convenience for i = 1:numel(hyp.cov) dnlZ.cov(i) = sum(sum(Q.*feval(cov{:}, hyp.cov, x, [], i)))/2; end dnlZ.lik = sn2*trace(Q); for i = 1:numel(hyp.mean), dnlZ.mean(i) = -feval(mean{:}, hyp.mean, x, i)'*alpha; end end end ================================================ FILE: code/gpml/inf/infFITC.m ================================================ function [post nlZ dnlZ] = infFITC(hyp, mean, cov, lik, x, y) % FITC approximation to the posterior Gaussian process. The function is % equivalent to infExact with the covariance function: % Kt = Q + G; G = diag(diag(K-Q); Q = Ku'*inv(Kuu + snu2*eye(nu))*Ku; % where Ku and Kuu are covariances w.r.t. to inducing inputs xu and % snu2 = sn2/1e6 is the noise of the inducing inputs. We fixed the standard % deviation of the inducing inputs to be a one per mil of the measurement noise % standard deviation. % The implementation exploits the Woodbury matrix identity % inv(Kt) = inv(G) - inv(G)*Ku'*inv(Kuu+Ku*inv(G)*Ku')*Ku*inv(G) % in order to be applicable to large datasets. The computational complexity % is O(n nu^2) where n is the number of data points x and nu the number of % inducing inputs in xu. % The function takes a specified covariance function (see covFunction.m) and % likelihood function (see likFunction.m), and is designed to be used with % gp.m and in conjunction with covFITC and likGauss. % % Copyright (c) by Ed Snelson, Carl Edward Rasmussen % and Hannes Nickisch, 2010-11-26. % % See also INFMETHODS.M, COVFITC.M. likstr = lik; if ~ischar(lik), likstr = func2str(lik); end if ~strcmp(likstr,'likGauss') % NOTE: no explicit call to likGauss error('FITC inference only possible with Gaussian likelihood'); end cov1 = cov{1}; if isa(cov1, 'function_handle'), cov1 = func2str(cov1); end if ~strcmp(cov1,'covFITC'); error('Only covFITC supported.'), end % check cov [diagK,Kuu,Ku] = feval(cov{:}, hyp.cov, x); % evaluate covariance matrix m = feval(mean{:}, hyp.mean, x); % evaluate mean vector [n, D] = size(x); nu = size(Kuu,1); sn2 = exp(2*hyp.lik); % noise variance of likGauss snu2 = 1e-6*sn2; % hard coded inducing inputs noise Luu = chol(Kuu + snu2*eye(nu)); % Kuu + snu2*I = Luu'*Luu V = Luu'\Ku; % V = inv(Luu')*Ku => V'*V = Q dg = diagK + sn2 - sum(V.*V,1)'; % D + sn2*eye(n) = diag(K) + sn2 - diag(Q) V = V./repmat(sqrt(dg)',nu,1); Lu = chol(eye(nu) + V*V'); r = (y-m)./sqrt(dg); be = Lu'\(V*r); iKuu = solve_chol(Luu,eye(nu)); % inv(Kuu + snu2*I) = iKuu post.alpha = Luu\(Lu\be); % return the posterior parameters post.L = solve_chol(Lu*Luu,eye(nu)) - iKuu; % Sigma-inv(Kuu) post.sW = []; % unused for FITC if nargout>1 % do we want the marginal likelihood nlZ = sum(log(diag(Lu))) + (sum(log(dg)) + n*log(2*pi) + r'*r - be'*be)/2; if nargout>2 % do we want derivatives? dnlZ = hyp; % allocate space for derivatives W = Ku./repmat(sqrt(dg)',nu,1); W = chol(Kuu+W*W'+snu2*eye(nu))'\Ku; % inv(K) = inv(G) - inv(G)*W'*W*inv(G); al = (y-m - W'*(W*((y-m)./dg)))./dg; B = iKuu*Ku; Wdg = W./repmat(dg',nu,1); w = B*al; for i = 1:numel(hyp.cov) [ddiagKi,dKuui,dKui] = feval(cov{:}, hyp.cov, x, [], i); % eval cov deriv R = 2*dKui-dKuui*B; v = ddiagKi - sum(R.*B,1)'; % diag part of cov deriv dnlZ.cov(i) = (ddiagKi'*(1./dg) +w'*(dKuui*w-2*(dKui*al)) -al'*(v.*al) ... - sum(Wdg.*Wdg,1)*v - sum(sum((R*Wdg').*(B*Wdg'))) )/2; end dnlZ.lik = sn2*(sum(1./dg) -sum(sum(W.*W,1)'./(dg.*dg)) -al'*al); % since snu2 is a fixed fraction of sn2, there is a covariance-like term in % the derivative as well dKuui = 2*snu2; R = -dKuui*B; v = -sum(R.*B,1)'; % diag part of cov deriv dnlZ.lik = dnlZ.lik + (w'*dKuui*w -al'*(v.*al)... - sum(Wdg.*Wdg,1)*v - sum(sum((R*Wdg').*(B*Wdg'))) )/2; for i = 1:numel(hyp.mean) dnlZ.mean(i) = -feval(mean{:}, hyp.mean, x, i)'*al; end end end ================================================ FILE: code/gpml/inf/infLOO.m ================================================ function [post nloo dnloo loo] = infLOO(hyp, mean, cov, lik, x, y) % Leave-One-Out - Perform Least-Squares GP predictions in the style of the % Pseudo Likelihood in §5.4.2 p. 117 and the Probabilistic Least-squares % Classifier of §6.5 p. 146 of the GPML book. % For Gaussian likelihood, the prediction is the same as with infExact, for % lik{Erf,Logistic} the result corresponds to probabilistic least-squares % classification. In any case, the objective returned is not the marginal % likelihood but the predictive log probability. % The standard deviation of the noise either comes from the sn paramtere of the % likelihood function lik{Gauss,Laplace,Sech2} or from the covariance function % by means of using covSum and covNoise i.e. for lik{Erf,Logistic}. % We then compute the negative leave-one-out predictive probability and its % derivatives w.r.t. the hyperparameters. See also "help infMethods". % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-12-14 % % See also INFMETHODS.M. [n, D] = size(x); K = feval(cov{:}, hyp.cov, x); % evaluate covariance matrix m = feval(mean{:}, hyp.mean, x); % evaluate mean vector if numel(hyp.lik)==0 cov1 = cov{1}; if isa(cov1, 'function_handle'), cov1 = func2str(cov1); end if ~strcmp(cov1,'covSum'); error('Only covSum supported.'), end % check cov cov2 = cov{2}; npar = 0; for i=1:length(cov2) cov2i = cov2{i}; if numel(cov2i)==1 if isa(cov2i, 'function_handle'), cov2i = func2str(cov2i); end if strcmp(cov2i,'covNoise'), isn = npar+1; sn2 = exp(2*hyp.cov(isn)); end end npar = npar + eval(feval(cov2i)); end if ~exist('sn2','var'), error('We need covNoise in the covSum.'), end else % Gauss, Laplace, Sech2 sn2 = exp(2*hyp.lik(end)); isn = 0; K = K + sn2*eye(n); end L = chol(K)/sqrt(sn2); % Cholesky factor of covariance with noise alpha = solve_chol(L,y-m)/sn2; post.alpha = alpha; % return the posterior parameters post.sW = ones(n,1)/sqrt(sn2); % sqrt of noise precision vector post.L = L; % L = chol(eye(n)+sW*sW'.*K) if nargout>1 % do we want the marginal likelihood? sdeff = 1./post.sW; % effective Gaussian width mneff = K*alpha + m; % alpha = inv(K)*(mneff-m), effective Gaussian center % all loo marginal means and variances at once V = post.L'\diag(1./sdeff); r = sum(V.*V,1)'; % r=diag( inv(K) ) loo.fmu = mneff - alpha./r; % GPML book, eq. 5.12 loo.fs2 = 1./r - sn2; % GPML book, eq. 5.12 [loo.lp, loo.ymu, loo.ys2] = feval(lik,hyp.lik,y,loo.fmu,loo.fs2); nloo = -sum(loo.lp); % negative leave-one-out predictive probability if nargout>2 % do we want derivatives? [lZ,dlZmn,d2lZmn] = feval(lik,hyp.lik,y,loo.fmu,loo.fs2,'infEP'); dlZva = (d2lZmn+dlZmn.*dlZmn)/2; % derivative w.r.t. variance dnloo = hyp; iK = solve_chol(L,eye(n))/sn2; for j=1:numel(hyp.cov) dKj = feval(cov{:}, hyp.cov, x, [], j); Zj = solve_chol(L,dKj)/sn2; % GPML book, eq. 5.13 dva = sum(Zj.*iK,2)./(r.*r); % GPML book, eq. 5.13 dmn = (Zj*alpha)./r - alpha.*dva; % GPML book, eq. 5.13 if j==isn, dva = dva - 2*sn2; end % additional part for covNoise dnloo.cov(j) = -dlZmn'*dmn -dlZva'*dva; end for j=1:numel(hyp.lik) dnloo.lik(j) = -sum( feval(lik,hyp.lik,y,loo.fmu,loo.fs2,'infEP',j) ); if j==numel(hyp.lik) dva = 2*sn2*sum(iK.*iK,2)./(r.*r); % GPML book, eq. 5.13 dmn = 2*solve_chol(L,alpha)./r - alpha.*dva; % GPML book, eq. 5.13 dva = dva - 2*sn2; dnloo.lik(j) = dnloo.lik(j) -dlZmn'*dmn -dlZva'*dva; end end for j=1:numel(hyp.mean) dmj = feval(mean{:}, hyp.mean, x, j); dmn = solve_chol(L,dmj)./(sn2*r); dnloo.mean(j) = -dlZmn'*dmn; end end end ================================================ FILE: code/gpml/inf/infLaplace.m ================================================ function [post nlZ dnlZ] = infLaplace(hyp, mean, cov, lik, x, y) % Laplace approximation to the posterior Gaussian process. % The function takes a specified covariance function (see covFunction.m) and % likelihood function (see likFunction.m), and is designed to be used with % gp.m. See also infFunctions.m. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch 2010-02-23. % % See also INFMETHODS.M. persistent last_alpha % copy of the last alpha tol = 1e-6; % tolerance for when to stop the Newton iterations smax = 2; Nline = 10; thr = 1e-4; % line search parameters maxit = 20; % max number of Newton steps in f if ~ischar(lik), lik = func2str(lik); end % make likelihood variable a string inf = 'infLaplace'; n = size(x,1); K = feval(cov{:}, hyp.cov, x); % evaluate the covariance matrix m = feval(mean{:}, hyp.mean, x); % evaluate the mean vector Psi_old = Inf; % make sure while loop starts by the largest old objective val if any(size(last_alpha)~=[n,1]) % find a good starting point for alpha and f alpha = zeros(n,1); f = K*alpha+m; % start at mean if sizes do not match [lp,dlp,d2lp] = feval(lik,hyp.lik,y,f,[],inf); W = -d2lp; Psi_new = -lp; else alpha = last_alpha; f = K*alpha+m; % try last one [lp,dlp,d2lp] = feval(lik,hyp.lik,y,f,[],inf); W = -d2lp; Psi_new = alpha'*(f-m)/2 - lp; % objective for last alpha Psi_def = -feval(lik,hyp.lik,y,m,[],inf); % objective for default init f==m if Psi_def < Psi_new % if default is better, we use it alpha = zeros(n,1); f = K*alpha+m; [lp,dlp,d2lp] = feval(lik,hyp.lik,y,f,[],inf); W = -d2lp; Psi_new = -lp; end end isWneg = any(W<0); % flag indicating whether we found negative values of W it = 0; % this happens for the Student's t likelihood while Psi_old - Psi_new > tol && it2 % do we want derivatives? dnlZ = hyp; % allocate space for derivatives if isWneg % switch between Cholesky and LU decomposition mode Z = -post.L; % inv(K+inv(W)) g = sum(iA.*K,2)/2; % deriv. of ln|B| wrt W; g = diag(inv(inv(K)+diag(W)))/2 else Z = repmat(sW,1,n).*solve_chol(post.L,diag(sW)); %sW*inv(B)*sW=inv(K+inv(W)) C = post.L'\(repmat(sW,1,n).*K); % deriv. of ln|B| wrt W g = (diag(K)-sum(C.^2,1)')/2; % g = diag(inv(inv(K)+W))/2 end dfhat = g.*d3lp; % deriv. of nlZ wrt. fhat: dfhat=diag(inv(inv(K)+W)).*d3lp/2 for j=1:length(hyp.cov) % covariance hypers dK = feval(cov{:}, hyp.cov, x, [], j); dnlZ.cov(j) = sum(sum(Z.*dK))/2 - alpha'*dK*alpha/2; % explicit part b = dK*dlp; % b-K*(Z*b) = inv(eye(n)+K*diag(W))*b dnlZ.cov(j) = dnlZ.cov(j) - dfhat'*( b-K*(Z*b) ); % implicit part end for j=1:length(hyp.lik) % likelihood hypers [lp_dhyp,d2lp_dhyp] = feval(lik,hyp.lik,y,f,[],inf,j); dnlZ.lik(j) = -g'*d2lp_dhyp - sum(lp_dhyp); % explicit part, implicit is 0 end for j=1:length(hyp.mean) % mean hypers dm = feval(mean{:}, hyp.mean, x, j); dnlZ.mean(j) = -alpha'*dm; % explicit part dnlZ.mean(j) = dnlZ.mean(j) - dfhat'*(dm-K*(Z*dm)); % implicit part end end % criterion Psi at alpha + s*dalpha for line search function [Psi,alpha,f,dlp,W] = Psi_line(s,dalpha,alpha,hyp,K,m,lik,y,inf) alpha = alpha + s*dalpha; f = K*alpha+m; [lp,dlp,d2lp] = feval(lik,hyp.lik,y,f,[],inf); W = -d2lp; Psi = alpha'*(f-m)/2 - lp; % Compute the log determinant ldA and the inverse iA of a square nxn matrix % A = eye(n) + K*diag(w) from its LU decomposition; for negative definite A, we % return ldA = Inf. We also return mwiA = -diag(w)*inv(A). function [ldA,iA,mwiA] = logdetA(K,w) [m,n] = size(K); if m~=n, error('K has to be nxn'), end A = eye(n)+K.*repmat(w',n,1); [L,U,P] = lu(A); u = diag(U); % compute LU decomposition, A = P'*L*U signU = prod(sign(u)); % sign of U detP = 1; % compute sign (and det) of the permutation matrix P p = P*(1:n)'; for i=1:n if i~=p(i) detP = -detP; j = find(p==i); p([i,j]) = p([j,i]); % swap entries end end if signU~=detP % log becomes complex for negative values, encoded by infinity ldA = Inf; else % det(L) = 1 and U triangular => det(A) = det(P)*prod(diag(U)) ldA = sum(log(abs(u))); end if nargout>1, iA = inv(U)*inv(L)*P; end % return the inverse, as well if nargout>2, mwiA = -repmat(w,1,n).*iA; end ================================================ FILE: code/gpml/inf/infVB.m ================================================ function [post, nlZ, dnlZ] = infVB(hyp, mean, cov, lik, x, y) % Variational approximation to the posterior Gaussian process with MKL % covariance function hyperparameter optimisation. % The function takes a likelihood function (see likFunction.m), and is designed % to be used with gp.m. See also infFunctions.m. % % Minimisation of an upper bound on the negative marginal likelihood % \int N(f|0,K) p(y|f) df \ge Psi(ga,theta) with hyperparameters % theta=cov.hyp and likelihood lik(y,f) = p(y|f). % % Psi(ga,theta) = nlZ = % = (ln|K+ga|-ln|ga| + h(ga) - b'*inv(inv(K)+inv(ga)))*b)/2 % = (ln|A| + ln|K| + h(ga) - b'*inv(A)*b)/2 with A = inv(K)+inv(ga) % % The map (ga,K) |-> Psi(ga,theta) - ln|K| is jointly convex and ln|K| is % concave in its linear parameters theta. % % We optimise the convex Psi(ga,theta)-ln|K|+z'*theta/2 + B(t,theta) criterion % in the outer loop using the barrier function B(t,theta) = -sum(log(theta))/t % to enforce positivity w.r.t. theta. % The linear term z'*theta/2 is an upper bound on the concave ln|K| term. % We use an interleaved optimisation with inner loops doing Newton steps in ga % and an outer loop doing joint Newton updates in (ga,theta). Line searches are % done using derivative-free 'Brent's minimum' search. % % Copyright (c) by Hannes Nickisch 2010-07-22. % % See also INFMETHODS.M. maxitinner = 20; % number of inner Newton steps in ga % some less important parameters ep = 1e-8; % small constant for the ridge added to the stab. Newton direction smax = 5; Nline = 15; thr = 1e-4; % line search parameters tol = 1e-7; % tolerance for when to stop the Newton iterations inf = 'infVB'; [n,D] = size(x); K = feval(cov{:}, hyp.cov, x); % evaluate the covariance matrix m = feval(mean{:}, hyp.mean, x); % evaluate the mean vector if ~ischar(lik), lik = func2str(lik); end if (norm(m)>1e-10 || numel(hyp.mean)>0) ... && (strcmp(lik,'likErf')||strcmp(lik,'likLogistic')) error('only meanZero implemented for classification') end y = y-m; % no we have either zero mean or non-classification likelihoods if strcmp(lik,'likGauss') ga = exp(2*hyp.lik)*ones(n,1); % best ga is known for Gaussian likelihood elseif strcmp(lik,'likErf') ga = ones(n,1); % use a fixed ga for errf likelihood due to asymptotics else % INNER compute the Newton direction of ga ga = ones(n,1); % initial values itinner = 0; nlZ_new = 1e100; nlZ_old = Inf; % make sure while loop starts while nlZ_old-nlZ_new>tol && itinner0 Psi_s = @(s,ddga,ga,K,inf,hyp,lik,y) Psi(ga+s*ddga,K,inf,hyp,lik,y); [s,nlZ_new] = brentmin(0,s,Nline,thr, Psi_s,0,ddga,ga,K,inf,hyp,lik,y); ga = abs(ga + s*ddga); % update variational parameters end end [nlZ,dnlZ,d2nlZ,b] = Psi(ga,K,inf,hyp,lik,y); % upp bd on neg log marg lik W = 1./ga; sW = sqrt(W); % return the posterior parameters L = chol(eye(n)+sW*sW'.*K); % L'*L = B =eye(n)+sW*K*sW iKtil = repmat(sW,1,n).*solve_chol(L,diag(sW)); % sW*B^-1*sW=inv(K+inv(W)) alpha = b - iKtil*(K*b); post.alpha = alpha; post.sW = sW; post.L = L; if nargout>2 % do we want derivatives? dnlZ = hyp; % allocate space for derivatives for j=1:length(hyp.cov) % covariance hypers dK = feval(cov{:}, hyp.cov, x, [], j); if j==1, v = iKtil*(b./W); end dnlZ.cov(j) = sum(sum(iKtil.*dK))/2 - (v'*dK*v)/2; % implicit derivative = 0 end if ~strcmp(lik,'likGauss') % likelihood hypers for j=1:length(hyp.lik) dhhyp = feval(lik,hyp.lik,y,[],ga,inf,j); dnlZ.lik(j) = sum(dhhyp)/2; % implicit derivative = 0 end else % special treatment for the Gaussian case dnlZ.lik = sum(sum( (L'\eye(n)).^2 )) - exp(2*hyp.lik)*(alpha'*alpha); end for j=1:length(hyp.mean) % mean hypers dm = feval(mean{:}, hyp.mean, x, j); dnlZ.mean(j) = -alpha'*dm; % implicit derivative = 0 end end % variational lower bound along with derivatives w.r.t. ga and theta=hyp.cov % psi = (ln|K+ga|-ln|ga| + h(ga) - b'*inv(inv(K)+inv(ga)))*b)/2 % = (ln|A| + ln|K| + h(ga) - b'*inv(A)*b)/2 with A = inv(K)+inv(ga) % the code is numerically stable function [nlZ,dga,d2ga,b] = Psi(ga,K,inf,hyp,lik,y) n = size(K,1); [h,b,dh,db,d2h,d2b] = feval(lik,hyp.lik,y,[],ga,inf); W = 1./ga; sW = sqrt(W); L = chol(eye(n)+sW*sW'.*K); % sum(log(diag(L))) = log|B|/2 C = L'\(repmat(sW,1,n).*K); t = C*b; % t'*t-b'*K*b = -b'*inv(inv(K)+diag(1./ga))*b nlZ = sum(log(diag(L))) + ( sum(h) + t'*t-b'*K*b )/2; if nargout>1 % Hessian w.r.t. variational parameters gamma iKtil = repmat(sW,1,n).*solve_chol(L,diag(sW)); % sW*inv(B)*sW=inv(K+inv(W)) Khat = K-C'*C; v = Khat*b; % K-K*sW*inv(B)*sW*K=inv(inv(K)+W) dga = ( diag(iKtil)-1./ga + dh - (v./ga).^2 - 2*v.*db )/2; if nargout>2 % gradient w.r.t. variational parameters gamma w = v./ga.^2; d2ga = ( -iKtil.^2+diag(1./ga.^2) + diag(d2h) )/2 ... -Khat.*(w*(w+2*db)') -Khat.*(db*db') +diag(w.^2.*ga)-diag(v.*d2b); d2ga = (d2ga+d2ga')/2; % symmetrise end end ================================================ FILE: code/gpml/infMethods.m ================================================ % Inference methods: Compute the (approximate) posterior for a Gaussian process. % Methods currently implemented include: % % infExact Exact inference (only possible with Gaussian likelihood) % infFITC Large scale regression with approximate covariance matrix % infLaplace Laplace's Approximation % infEP Expectation Propagation % infVB Variational Bayes % % infLOO Leave-One-Out predictive probability and Least-Squares Approxim. % % The interface to the approximation methods is the following: % % function [post nlZ dnlZ] = inf..(hyp, cov, lik, x, y) % % where: % % hyp is a struct of hyperparameters % cov is the name of the covariance function (see covFunctions.m) % lik is the name of the likelihood function (see likFunctions.m) % x is a n by D matrix of training inputs % y is a (column) vector (of size n) of targets % % nlZ is the returned value of the negative log marginal likelihood % dnlZ is a (column) vector of partial derivatives of the negative % log marginal likelihood w.r.t. each hyperparameter % post struct representation of the (approximate) posterior containing % alpha is a (sparse or full column vector) containing inv(K)*m, where K % is the prior covariance matrix and m the approx posterior mean % sW is a (sparse or full column) vector containing diagonal of sqrt(W) % the approximate posterior covariance matrix is inv(inv(K)+W) % L is a (sparse or full) matrix, L = chol(sW*K*sW+eye(n)) % % Usually, the approximate posterior to be returned admits the form % N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is diagonal; % if not, then L contains instead -inv(K+inv(W)), and sW is unused. % % For more information on the individual approximation methods and their % implementations, see the separate inf??.m files. See also gp.m % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18 ================================================ FILE: code/gpml/lik/likErf.m ================================================ function [varargout] = likErf(hyp, y, mu, s2, inf, i) % likErf - Error function or cumulative Gaussian likelihood function for binary % classification or probit regression. The expression for the likelihood is % likErf(t) = (1+erf(t/sqrt(2)))/2 = normcdf(t). % % Several modes are provided, for computing likelihoods, derivatives and moments % respectively, see likelihoods.m for the details. In general, care is taken % to avoid numerical issues when the arguments are extreme. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-07-22. % % See also likFunctions.m. if nargin<2, varargout = {'0'}; return; end % report number of hyperparameters if nargin>1, y = sign(y); y(y==0) = 1; else y = 1; end % allow only +/- 1 values if numel(y)==0, y = 1; end if nargin<5 % prediction mode if inf is not present y = y.*ones(size(mu)); % make y a vector s2zero = 1; if nargin>3, if norm(s2)>0, s2zero = 0; end, end % s2==0 ? if s2zero % log probability evaluation [p,lp] = cumGauss(y,mu); else % prediction lp = likErf(hyp, y, mu, s2, 'infEP'); p = exp(lp); end ymu = {}; ys2 = {}; if nargout>1 ymu = 2*p-1; % first y moment if nargout>2 ys2 = 4*p.*(1-p); % second y moment end end varargout = {lp,ymu,ys2}; else % inference mode switch inf case 'infLaplace' if nargin<6 % no derivative mode f = mu; yf = y.*f; % product latents and labels dlp = {}; d2lp = {}; d3lp = {}; % return arguments [p,lp] = cumGauss(y,f); if nargout>1 % derivative of log likelihood n_p = gauOverCumGauss(yf,p); dlp = y.*n_p; % derivative of log likelihood if nargout>2 % 2nd derivative of log likelihood d2lp = -n_p.^2 - yf.*n_p; if nargout>3 % 3rd derivative of log likelihood d3lp = 2*y.*n_p.^3 +3*f.*n_p.^2 +y.*(f.^2-1).*n_p; end end end varargout = {sum(lp),dlp,d2lp,d3lp}; else % derivative mode varargout = {[]}; % derivative w.r.t. hypers end case 'infEP' if nargin<6 % no derivative mode z = mu./sqrt(1+s2); dlZ = {}; d2lZ = {}; [junk,lZ] = cumGauss(y,z); % log part function if numel(y)>0, z=z.*y; end if nargout>1 if numel(y)==0, y=1; end n_p = gauOverCumGauss(z,exp(lZ)); dlZ = y.*n_p./sqrt(1+s2); % 1st derivative wrt mean if nargout>2 d2lZ = -n_p.*(z+n_p)./(1+s2); % 2nd derivative wrt mean end end varargout = {lZ,dlZ,d2lZ}; else % derivative mode varargout = {[]}; % deriv. wrt hyp.lik end case 'infVB' % naive variational lower site bound based on asymptotical properties of lik % normcdf(t) -> -(t²-2dt+c)/2 for t->-oo (tight lower bound) d = 0.158482605320942; c = -1.785873318175113; ga = s2; n = numel(ga); b = d*y.*ones(n,1); db = zeros(n,1); d2b = db; h = -2*c*ones(n,1); h(ga>1) = Inf; dh = zeros(n,1); d2h = dh; varargout = {h,b,dh,db,d2h,d2b}; end end function [p,lp] = cumGauss(y,f) if numel(y)>0, yf = y.*f; else yf = f; end % product of latents and labels p = (1+erf(yf/sqrt(2)))/2; % likelihood if nargout>1, lp = logphi(yf,p); end % log likelihood % safe implementation of the log of phi(x) = \int_{-\infty}^x N(f|0,1) df % logphi(z) = log(normcdf(z)) function lp = logphi(z,p) lp = zeros(size(z)); % allocate memory zmin = -6.2; zmax = -5.5; ok = z>zmax; % safe evaluation for large values bd = z-5; % naive evaluation for large values of f n_p(ok) = (exp(-f(ok).^2/2)/sqrt(2*pi)) ./ p(ok); bd = f<-6; % tight upper bound evaluation n_p(bd) = sqrt(f(bd).^2/4+1)-f(bd)/2; interp = ~ok & ~bd; % linearly interpolate between both of them tmp = f(interp); lam = -5-f(interp); n_p(interp) = (1-lam).*(exp(-tmp.^2/2)/sqrt(2*pi))./p(interp) + ... lam .*(sqrt(tmp.^2/4+1)-tmp/2); ================================================ FILE: code/gpml/lik/likGauss.m ================================================ function [varargout] = likGauss(hyp, y, mu, s2, inf, i) % likGauss - Gaussian likelihood function for regression. The expression for the % likelihood is % likGauss(t) = exp(-(t-y)^2/2*sn^2) / sqrt(2*pi*sn^2), % where y is the mean and sn is the standard deviation. % % The hyperparameters are: % % hyp = [ log(sn) ] % % Several modes are provided, for computing likelihoods, derivatives and moments % respectively, see likelihoods.m for the details. In general, care is taken % to avoid numerical issues when the arguments are extreme. % % See also likFunctions.m. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18 if nargin<2, varargout = {'1'}; return; end % report number of hyperparameters sn2 = exp(2*hyp); if nargin<5 % prediction mode if inf is not present if numel(y)==0, y = zeros(size(mu)); end s2zero = 1; if nargin>3, if norm(s2)>0, s2zero = 0; end, end % s2==0 ? if s2zero % log probability lp = -(y-mu).^2./sn2/2-log(2*pi*sn2)/2; s2 = 0; else lp = likGauss(hyp, y, mu, s2, 'infEP'); % prediction end ymu = {}; ys2 = {}; if nargout>1 ymu = mu; % first y moment if nargout>2 ys2 = s2 + sn2; % second y moment end end varargout = {lp,ymu,ys2}; else switch inf case 'infLaplace' if nargin<6 % no derivative mode if numel(y)==0, y=0; end ymmu = y-mu; dlp = {}; d2lp = {}; d3lp = {}; lp = -ymmu.^2/(2*sn2) - log(2*pi*sn2)/2; if nargout>1 dlp = ymmu/sn2; % dlp, derivative of log likelihood if nargout>2 % d2lp, 2nd derivative of log likelihood d2lp = -ones(size(ymmu))/sn2; if nargout>3 % d3lp, 3rd derivative of log likelihood d3lp = zeros(size(ymmu)); end end end varargout = {sum(lp),dlp,d2lp,d3lp}; else % derivative mode lp_dhyp = (y-mu).^2/sn2 - 1; % derivative of log likelihood w.r.t. hypers d2lp_dhyp = 2*ones(size(mu))/sn2; % and also of the second mu derivative varargout = {lp_dhyp,d2lp_dhyp}; end case 'infEP' if nargin<6 % no derivative mode lZ = -(y-mu).^2./(sn2+s2)/2 - log(2*pi*(sn2+s2))/2; % log part function dlZ = {}; d2lZ = {}; if nargout>1 dlZ = (y-mu)./(sn2+s2); % 1st derivative w.r.t. mean if nargout>2 d2lZ = -1./(sn2+s2); % 2nd derivative w.r.t. mean end end varargout = {lZ,dlZ,d2lZ}; else % derivative mode dlZhyp = ((y-mu).^2./(sn2+s2)-1) ./ (1+s2./sn2); % deriv. w.r.t. hyp.lik varargout = {dlZhyp}; end case 'infVB' if nargin<6 % variational lower site bound % t(s) = exp(-(y-s)^2/2sn2)/sqrt(2*pi*sn2) % the bound has the form: b*s - s.^2/(2*ga) - h(ga)/2 with b=y/ga ga = s2; n = numel(ga); b = y./ga; y = y.*ones(n,1); db = -y./ga.^2; d2b = 2*y./ga.^3; h = zeros(n,1); dh = h; d2h = h; % allocate memory for return args id = ga(:)<=sn2+1e-8; % OK below noise variance h(id) = y(id).^2./ga(id) + log(2*pi*sn2); h(~id) = Inf; dh(id) = -y(id).^2./ga(id).^2; d2h(id) = 2*y(id).^2./ga(id).^3; id = ga<0; h(id) = Inf; dh(id) = 0; d2h(id) = 0; % neg. var. treatment varargout = {h,b,dh,db,d2h,d2b}; else ga = s2; n = numel(ga); dhhyp = zeros(n,1); dhhyp(ga(:)<=sn2) = 2; dhhyp(ga<0) = 0; % negative variances get a special treatment varargout = {dhhyp}; % deriv. w.r.t. hyp.lik end end end ================================================ FILE: code/gpml/lik/likLaplace.m ================================================ function [varargout] = likLaplace(hyp, y, mu, s2, inf, i) % likLaplace - Laplacian likelihood function for regression. % The expression for the likelihood is % likLaplace(t) = exp(-|t-y|/b)/(2*b) with b = sn/sqrt(2), % where y is the mean and sn^2 is the variance. % % The hyperparameters are: % % hyp = [ log(sn) ] % % Several modes are provided, for computing likelihoods, derivatives and moments % respectively, see likelihoods.m for the details. In general, care is taken % to avoid numerical issues when the arguments are extreme. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-07-21. % % See also likFunctions.m. if nargin<2, varargout = {'1'}; return; end % report number of hyperparameters sn = exp(hyp); b = sn/sqrt(2); if nargin<5 % prediction mode if inf is not present if numel(y)==0, y = zeros(size(mu)); end s2zero = 1; if nargin>3, if norm(s2)>0, s2zero = 0; end, end % s2==0 ? if s2zero % log probability evaluation lp = -abs(y-mu)./b -log(2*b); s2 = 0; else % prediction lp = likLaplace(hyp, y, mu, s2, 'infEP'); end ymu = {}; ys2 = {}; if nargout>1 ymu = mu; % first y moment if nargout>2 ys2 = s2 + sn.^2; % second y moment end end varargout = {lp,ymu,ys2}; else % inference mode switch inf case 'infLaplace' if nargin<6 % no derivative mode if numel(y)==0, y=0; end ymmu = y-mu; dlp = {}; d2lp = {}; d3lp = {}; lp = -abs(ymmu)/b -log(2*b); if nargout>1 dlp = sign(ymmu)/b; % dlp, derivative of log likelihood if nargout>2 % d2lp, 2nd derivative of log likelihood d2lp = zeros(size(ymmu)); if nargout>3 % d3lp, 3rd derivative of log likelihood d3lp = zeros(size(ymmu)); end end end varargout = {sum(lp),dlp,d2lp,d3lp}; else % derivative mode lp_dhyp = abs(y-mu)/b - 1; % derivative of log likelihood w.r.t. hypers d2lp_dhyp = zeros(size(mu)); % and also of the second mu derivative varargout = {lp_dhyp,d2lp_dhyp}; end case 'infEP' n = max([length(y),length(mu),length(s2),length(sn)]); on = ones(n,1); y = y.*on; mu = mu.*on; s2 = s2.*on; sn = sn.*on; % vectors only fac = 1e3; % factor between the widths of the two distributions ... % ... from when one considered a delta peak, we use 3 orders of magnitude idlik = fac*sn1 % 1st moment % M1p = sqrt(tvar).*( zp.*exp(lM0p) - exp(-zp.*zp)/sqrt(pi) ); % M1m = sqrt(tvar).*( zm.*exp(lM0m) - exp(-zm.*zm)/sqrt(pi) ); % lZmax = max([vp-lZ(id),vm-lZ(id)],[],2); % m1m0 = exp(lZmax).*( exp(vp-lZ(id)-lZmax).*M1p ... % -exp(vm-lZ(id)-lZmax).*M1m )/2 + y(id); % dlZ(id) = (m1m0-mu(id))./s2(id); % if nargout>2 % 2nd moment % M2p = 2*tvar.*exp(vp-lZ(id)+lM0p)/2+(2*tvar+sqrt(2)*tmu) ... % .*M1p.*exp(vp-lZ(id)); % M2m = 2*tvar.*exp(vm-lZ(id)+lM0m)/2+(2*tvar-sqrt(2)*tmu) ... % .*M1m.*exp(vm-lZ(id)); % m2m0 = (M2p+M2m).*sn(id)/sqrt(8) + 2*y(id).*m1m0 - y(id).^2; % d2lZ(id) = (m2m0 - m1m0.^2)./s2(id).^2 - 1./s2(id); % end % end % an implementation based on logphi(t) = log(normcdf(t)) zp = (tmu+sqrt(2)*tvar)./sqrt(tvar); zm = (tmu-sqrt(2)*tvar)./sqrt(tvar); ap = logphi(-zp)+sqrt(2)*tmu; am = logphi( zm)-sqrt(2)*tmu; lZ(id) = logsum2exp([ap,am]) + tvar - log(sn(id)*sqrt(2)); if nargout>1 lqp = -zp.^2/2 - log(2*pi)/2 - logphi(-zp); % log( N(z)/Phi(z) ) lqm = -zm.^2/2 - log(2*pi)/2 - logphi( zm); dap = -exp(lqp-log(s2(id))/2) + sqrt(2)./sn(id); dam = exp(lqm-log(s2(id))/2) - sqrt(2)./sn(id); % ( exp(ap).*dap + exp(am).*dam )./( exp(ap) + exp(am) ) dlZ(id) = expABz_expAx([ap,am],[1;1],[dap,dam],[1;1]); if nargout>2 a = sqrt(8)./sn(id)./sqrt(s2(id)); bp = 2./sn(id).^2 - (a - zp./s2(id)).*exp(lqp); bm = 2./sn(id).^2 - (a + zm./s2(id)).*exp(lqm); % d2lZ(id) = ( exp(ap).*bp + exp(am).*bm )./( exp(ap) + exp(am) ) ... % - dlZ(id).^2; d2lZ(id) = expABz_expAx([ap,am],[1;1],[bp,bm],[1;1]) - dlZ(id).^2; end end end varargout = {lZ,dlZ,d2lZ}; else % derivative mode dlZhyp = zeros(n,1); if any(idlik) dlZhyp(idlik) = 0; end if any(idgau) dlZhyp(idgau) = ... likLaplace(log(sn(idgau)), mu(idgau), y(idgau), 'infLaplace', 1); end if any(id) % substitution to obtain unit variance, zero mean Laplacian tmu = (mu(id)-y(id))./sn(id); tvar = s2(id)./sn(id).^2; zp = (tvar+tmu/sqrt(2))./sqrt(tvar); vp = tvar+sqrt(2)*tmu; zm = (tvar-tmu/sqrt(2))./sqrt(tvar); vm = tvar-sqrt(2)*tmu; dzp = (-s2(id)./sn(id)+tmu.*sn(id)/sqrt(2)) ./ sqrt(s2(id)); dvp = -2*tvar - sqrt(2)*tmu; dzm = (-s2(id)./sn(id)-tmu.*sn(id)/sqrt(2)) ./ sqrt(s2(id)); dvm = -2*tvar + sqrt(2)*tmu; lezp = lerfc(zp); % ap = exp(vp).*ezp lezm = lerfc(zm); % am = exp(vm).*ezm vmax = max([vp+lezp,vm+lezm],[],2); % subtract max to avoid numerical pb ep = exp(vp+lezp-vmax); em = exp(vm+lezm-vmax); dap = ep.*(dvp - 2/sqrt(pi)*exp(-zp.^2-lezp).*dzp); dam = em.*(dvm - 2/sqrt(pi)*exp(-zm.^2-lezm).*dzm); dlZhyp(id) = (dap+dam)./(ep+em) - 1; end varargout = {dlZhyp}; % deriv. wrt hypers end case 'infVB' if nargin<6 % variational lower site bound % t(s) = exp(-sqrt(2)|y-s|/sn) / sqrt(2*sn²) % the bound has the form: b*s - s.^2/(2*ga) - h(ga)/2 with b=y/ga!! ga = s2; n = numel(ga); b = y./ga; y = y.*ones(n,1); db = -y./ga.^2; d2b = 2*y./ga.^3; h = 2*ga/sn^2 + log(2*sn^2) + y.^2./ga; dh = 2/sn^2 - y.^2./ga.^2; d2h = 2*y.^2./ga.^3; id = ga<0; h(id) = Inf; dh(id) = 0; d2h(id) = 0; % neg. var. treatment varargout = {h,b,dh,db,d2h,d2b}; else ga = s2; dhhyp = -4*ga/sn^2 + 2; dhhyp(ga<0) = 0; % negative variances get a special treatment varargout = {dhhyp}; % deriv. wrt hyp.lik end end end % computes y = log( sum(exp(x),2) ) in a numerically safe way by subtracting % the row maximum to avoid cancelation after taking the exp % the sum is done along the rows function [y,x] = logsum2exp(logx) N = size(logx,2); max_logx = max(logx,[],2); % we have all values in the log domain, and want to calculate a sum x = exp(logx-max_logx*ones(1,N)); y = log(sum(x,2)) + max_logx; % numerically safe implementation of f(t) = log(1-erf(t)) = log(erfc(t)) function f = lerfc(t) f = zeros(size(t)); tmin = 20; tmax = 25; ok = ttmax; % evaluate tight bound interp = ~ok & ~bd; % linearly interpolate between both of them f(~ok) = log(2/sqrt(pi)) -t(~ok).^2 -log(t(~ok)+sqrt( t(~ok).^2+4/pi )); lam = 1./(1+exp( 12*(1/2-(t(interp)-tmin)/(tmax-tmin)) )); % interp. weights f(interp) = lam.*f(interp) + (1-lam).*log(erfc( t(interp) )); f(ok) = f(ok) + log(erfc( t(ok) )); % safe eval % computes y = ( (exp(A).*B)*z ) ./ ( exp(A)*x ) in a numerically safe way % The function is not general in the sense that it yields correct values for % all types of inputs. We assume that the values are close together. function y = expABz_expAx(A,x,B,z) N = size(A,2); maxA = max(A,[],2); % number of columns, max over columns A = A-maxA*ones(1,N); % subtract maximum value y = ( (exp(A).*B)*z ) ./ ( exp(A)*x ); % safe implementation of the log of phi(x) = \int_{-\infty}^x N(f|0,1) df % logphi(z) = log(normcdf(z)) function lp = logphi(z) lp = zeros(size(z)); % allocate memory zmin = -6.2; zmax = -5.5; ok = z>zmax; % safe evaluation for large values bd = z1, y = sign(y); y(y==0) = 1; else y = 1; end % allow only +/- 1 values if numel(y)==0, y = 1; end if nargin<5 % prediction mode if inf is not present y = y.*ones(size(mu)); % make y a vector s2zero = 1; if nargin>3, if norm(s2)>0, s2zero = 0; end, end % s2==0 ? if s2zero % log probability evaluation yf = y.*mu; % product latents and labels lp = yf; ok = -351 p = exp(lp); ymu = 2*p-1; % first y moment if nargout>2 ys2 = 4*p.*(1-p); % second y moment end end varargout = {lp,ymu,ys2}; else % inference mode switch inf case 'infLaplace' if nargin<6 % no derivative mode f = mu; yf = y.*f; s = -yf; % product latents and labels dlp = {}; d2lp = {}; d3lp = {}; % return arguments ps = max(0,s); lp = -(ps+log(exp(-ps)+exp(s-ps))); % lp = -(log(1+exp(s))) if nargout>1 % first derivatives s = min(0,f); p = exp(s)./(exp(s)+exp(s-f)); % p = 1./(1+exp(-f)) dlp = (y+1)/2-p; % derivative of log likelihood if nargout>2 % 2nd derivative of log likelihood d2lp = -exp(2*s-f)./(exp(s)+exp(s-f)).^2; if nargout>3 % 3rd derivative of log likelihood d3lp = 2*d2lp.*(0.5-p); end end end varargout = {sum(lp),dlp,d2lp,d3lp}; else % derivative mode varargout = {[]}; % derivative w.r.t. hypers end case 'infEP' if nargin<6 % no derivative mode y = y.*ones(size(mu)); % make y a vector % likLogistic(t) \approx 1/2 + \sum_{i=1}^5 (c_i/2) erf(lam_i/sqrt(2)t) lam = sqrt(2)*[0.44 0.41 0.40 0.39 0.36]; % approx coeffs lam_i and c_i c = [1.146480988574439e+02; -1.508871030070582e+03; 2.676085036831241e+03; -1.356294962039222e+03; 7.543285642111850e+01 ]; [lZc,dlZc,d2lZc] = likErf([], y*ones(1,5), mu*lam, s2*(lam.^2), inf); lZ = log_expA_x(lZc,c); % A=lZc, B=dlZc, d=c.*lam', lZ=log(exp(A)*c) dlZ = expABz_expAx(lZc, c, dlZc, c.*lam'); % ((exp(A).*B)*d)./(exp(A)*c) % d2lZ = ((exp(A).*Z)*e)./(exp(A)*c) - dlZ.^2 where e = c.*(lam.^2)' d2lZ = expABz_expAx(lZc, c, dlZc.^2+d2lZc, c.*(lam.^2)') - dlZ.^2; % The scale mixture approximation does not capture the correct asymptotic % behavior; we have linear decay instead of quadratic decay as suggested % by the scale mixture approximation. By observing that for large values % of -f*y ln(p(y|f)) of likLogistic is linear in f with slope y, we are % able to analytically integrate the tail region; there is no contribution % to the second derivative val = abs(mu)-196/200*s2-4; % empirically determined bound at val==0 lam = 1./(1+exp(-10*val)); % interpolation weights lZtail = min(s2/2-abs(mu),-.1); % apply the same to p(y|f) = 1 - p(-y|f) dlZtail = -sign(mu); id = y.*mu>0; lZtail(id) = log(1-exp(lZtail(id))); % label and mean agree dlZtail(id) = 0; lZ = (1-lam).* lZ + lam.* lZtail; % interpolate between scale mixture .. dlZ = (1-lam).*dlZ + lam.*dlZtail; % .. and tail approximation varargout = {lZ,dlZ,d2lZ}; else % derivative mode varargout = {[]}; % deriv. wrt hyp.lik end case 'infVB' % variational lower site bound % using -log(1+exp(-s)) = s/2 -log( 2*cosh(s/2) ); % the bound has the form: b*s - s.^2/(2*ga) - h(ga)/2 with b=1/2 ga = s2; n = numel(ga); b = (y/2).*ones(n,1); db = zeros(n,1); d2b = db; h = zeros(n,1); dh = h; d2h = h; % allocate memory for return argument idlo = ga(:)<=4; h(idlo) = 2*log(2); % constant value below 4 idup = 50<=ga(:); zn = zeros(n,1); % linear behavior for large gammas h(idup) = ga(idup)/4; dh(idup) = zn(idup)+1/4; d2h(idup) = zn(idup); id = ~idlo & ~idup; % interesting zone is in between [g,dg,d2g] = inv_xcothx(1/4*ga(id)); thg = tanh(g); h(id) = 2*logcosh(g) +2*log(2) -g.*thg; g1mt2 = g.*(1-thg.^2); dh(id) = dg.*( thg - g1mt2 )/4; d2h(id) = ( g1mt2.*(2*dg.^2.*thg-d2g) + d2g.*thg )/16; varargout = {h,b,dh,db,d2h,d2b}; end end % numerically safe version of log(cosh(x)) = log(exp(x)+exp(-x))-log(2) function f = logcosh(x) f = abs(x) + log(1+exp(-2*abs(x))) - log(2); % Invert f(x) = x.*coth(x) = y, return the positive value % uses Newton's method to minimize (y-f(x))² w.r.t. x function [x,dx,d2x] = inv_xcothx(y) % f = @(x) x.*coth(x); % df = @(x) x + (1-f(x)).*coth(x); % d2f = @(x) 2*(1-f(x)).*(1-coth(x).^2); if numel(y)==0, x=[]; dx=[]; d2x=[]; return, end x = sqrt(y.^2-1); i = 1; % init ep = eps./(1+abs(x)); th=tanh(x); fx=(x+ep)./(th+ep); % function r = fx-y; % init while i==1 || (i<10 && max(abs(r))>1e-12) dfx = x + (1-fx)./(th+ep); % first derivative d2fx = 2*( 1-fx + (x-th+ep/3)./(th.*th.*th+ep) ); % second derivative x = x - r.*dfx./(dfx.^2+r.*d2fx+ep); % Newton step ep = eps./(1+abs(x)); th=tanh(x); fx=(x+ep)./(th+ep); r = fx-y; i = i+1; end % first derivative dx/dy % derivatives of inverse functions are reciprocals dx = 1./dfx; % second derivative d2x/dy2 % quotient rule and chaine rule d2x = -d2fx.*dx./dfx./dfx; % computes y = log( exp(A)*x ) in a numerically safe way by subtracting the % maximal value in each row to avoid cancelation after taking the exp function y = log_expA_x(A,x) N = size(A,2); maxA = max(A,[],2); % number of columns, max over columns y = log(exp(A-maxA*ones(1,N))*x) + maxA; % exp(A) = exp(A-max(A))*exp(max(A)) % computes y = ( (exp(A).*B)*z ) ./ ( exp(A)*x ) in a numerically safe way % The function is not general in the sense that it yields correct values for % all types of inputs. We assume that the values are close together. function y = expABz_expAx(A,x,B,z) N = size(A,2); maxA = max(A,[],2); % number of columns, max over columns A = A-maxA*ones(1,N); % subtract maximum value y = ( (exp(A).*B)*z ) ./ ( exp(A)*x ); ================================================ FILE: code/gpml/lik/likSech2.m ================================================ function [varargout] = likSech2(hyp, y, mu, s2, inf, i) % likSech2 - sech-square likelihood function for regression. Often, the sech- % square distribution is also referred to as the logistic distribution not to be % confused with the logistic function for classification. The expression for the % likelihood is % likSech2(t) = Z / cosh(tau*(y-t))^2 where % tau = pi/(2*sqrt(3)*sn) and Z = tau/2 % and y is the mean and sn^2 is the variance. % % hyp = [ log(sn) ] % % Several modes are provided, for computing likelihoods, derivatives and moments % respectively, see likelihoods.m for the details. In general, care is taken % to avoid numerical issues when the arguments are extreme. The moments % \int f^k likSech2(y,f) N(f|mu,var) df are calculated via a Gaussian % scale mixture approximation. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-07-21. % % See also likFunctions.m, likLogistic.m. if nargin<2, varargout = {'1'}; return; end % report number of hyperparameters sn = exp(hyp); tau = pi/(2*sqrt(3)*sn); lZ = log(pi) - log(sn) - log(4*sqrt(3)); if nargin<5 % prediction mode if inf is not present if numel(y)==0, y = zeros(size(mu)); end s2zero = 1; if nargin>3, if norm(s2)>0, s2zero = 0; end, end % s2==0 ? if s2zero % log probability evaluation lp = lZ - 2*logcosh(tau*(y-mu)); s2 = 0; else % prediction lp = likSech2(hyp, y, mu, s2, 'infEP'); end ymu = {}; ys2 = {}; if nargout>1 ymu = mu; % first y moment if nargout>2 ys2 = s2 + sn.^2; % second y moment end end varargout = {lp,ymu,ys2}; else % inference mode switch inf case 'infLaplace' r = y-mu; [g,dg,d2g,d3g] = logcosh(tau.*r); % precompute derivatives if nargin<6 % no derivative mode dlp = {}; d2lp = {}; d3lp = {}; lp = lZ - 2*g; if nargout>1 % first derivatives dlp = 2*tau.*dg; if nargout>2 % 2nd derivative of log likelihood d2lp = -2*tau.^2.*d2g; if nargout>3 % 3rd derivative of log likelihood d3lp = 2*tau.^3.*d3g; end end end varargout = {sum(lp),dlp,d2lp,d3lp}; else % derivative mode lp_dhyp = 2*tau.*r.*dg - 1; % derivative w.r.t. sn d2lp_dhyp = 2*tau.^2.*(2*d2g + tau.*r.*d3g); varargout = {lp_dhyp,d2lp_dhyp}; end case 'infEP' n = max([length(y),length(mu),length(s2),length(sn)]); on = ones(n,1); y = y.*on; mu = mu.*on; s2 = s2.*on; sn = sn.*on; % vectors only fac = 1e3; % factor between the widths of the two distributions ... % ... from when one considered a delta peak, we use 3 orders of magnitude idlik = fac*sn1/(2*tau^2) & ~idup; % interesting zone is in between [g,dg,d2g] = inv_xcothx(2*tau^2*ga(id)); thg = tanh(g); h(id) = 4*logcosh(g) -2*g.*thg; h = h + y.^2./ga - 2*lZ; g1mt2 = g.*(1-thg.^2); % first derivatives dh(idup) = 4*tau^2; dh(id) = 4*tau^2*dg.*( thg - g1mt2 ); dh = dh - (y./ga).^2; d2h(id) = 8*tau^4*( g1mt2.*(2*dg.^2.*thg-d2g) +d2g.*thg ); % second derivs d2h = d2h + 2*y.^2./ga.^3; id = ga<0; h(id) = Inf; dh(id) = 0; d2h(id) = 0; % neg. var. treatment varargout = {h,b,dh,db,d2h,d2b}; else ga = s2; n = numel(ga); dhhyp = zeros(n,1); idup = 50<=ga(:); % asymptotic behavior for large gammas dhhyp(idup) = -2*pi^2/3/sn^2*ga(idup); % tau = pi/(2*sqrt(3)*sn) id = ga(:)>1/(2*tau^2) & ~idup; % interesting zone is in between [g,dg] = inv_xcothx(2*tau^2*ga(id)); thg = tanh(g); [f,df] = logcosh(g); % h = 4*f -2*g.*tanh(g) if any(id) dhhyp(id) = 8*tau^2*ga(id).*dg.*( thg - 2*df + g.*(1-thg.^2) ); end dhhyp = dhhyp + 2; % from lZ = log(pi) - log(sn) - log(4*sqrt(3)) dhhyp(ga<0) = 0; % negative variances get a special treatment varargout = {dhhyp}; % deriv. wrt hyp.lik end end end % Invert f(x) = x.*coth(x) = y, return the positive value % uses Newton's method to minimize (y-f(x))² w.r.t. x function [x,dx,d2x] = inv_xcothx(y) % f = @(x) x.*coth(x); % df = @(x) x + (1-f(x)).*coth(x); % d2f = @(x) 2*(1-f(x)).*(1-coth(x).^2); if numel(y)==0, x=[]; dx=[]; d2x=[]; return, end x = sqrt(y.^2-1); i = 1; % init ep = eps./(1+abs(x)); th=tanh(x); fx=(x+ep)./(th+ep); % function r = fx-y; % init while i==1 || (i<10 && max(abs(r))>1e-12) dfx = x + (1-fx)./(th+ep); % first derivative d2fx = 2*( 1-fx + (x-th+ep/3)./(th.*th.*th+ep) ); % second derivative x = x - r.*dfx./(dfx.^2+r.*d2fx+ep); % Newton step ep = eps./(1+abs(x)); th=tanh(x); fx=(x+ep)./(th+ep); r = fx-y; i = i+1; end % first derivative dx/dy; derivatives of inverse functions are reciprocals dx = 1./dfx; % second derivative d2x/dy2; quotient rule and chaine rule d2x = -d2fx.*dx./dfx./dfx; % numerically safe version of log(cosh(x)) = log(exp(x)+exp(-x))-log(2) function [f,df,d2f,d3f] = logcosh(x) a = exp(-2*abs(x)); % always between 0 and 1 and therefore safe to evaluate f = abs(x) + log(1+a) - log(2); df = sign(x).*( 1 - 2*a./(1+a) ); d2f = 4*a./(1+a).^2; d3f = -8*sign(x).*a.*(1-a)./(1+a).^3; % computes y = log( exp(A)*x ) in a numerically safe way by subtracting the % maximal value in each row to avoid cancelation after taking the exp function y = log_expA_x(A,x) N = size(A,2); maxA = max(A,[],2); % number of columns, max over columns y = log(exp(A-maxA*ones(1,N))*x) + maxA; % exp(A) = exp(A-max(A))*exp(max(A)) % computes y = ( (exp(A).*B)*z ) ./ ( exp(A)*x ) in a numerically safe way % The function is not general in the sense that it yields correct values for % all types of inputs. We assume that the values are close together. function y = expABz_expAx(A,x,B,z) N = size(A,2); maxA = max(A,[],2); % number of columns, max over columns A = A-maxA*ones(1,N); % subtract maximum value y = ( (exp(A).*B)*z ) ./ ( exp(A)*x ); ================================================ FILE: code/gpml/lik/likT.m ================================================ function [varargout] = likT(hyp, y, mu, s2, inf, i) % likT - Student's t likelihood function for regression. % The expression for the likelihood is % likT(t) = Z * ( 1 + (t-y)^2/(nu*sn^2) ).^(-(nu+1)/2), % where Z = gamma((nu+1)/2) / (gamma(nu/2)*sqrt(nu*pi)*sn) % and y is the mean (for nu>1) and nu*sn^2/(nu-2) is the variance (for nu>2). % % The hyperparameters are: % % hyp = [ log(nu-1) % log(sn) ] % % Note that the parametrisation guarantees nu>1, thus the mean always exists. % % Several modes are provided, for computing likelihoods, derivatives and moments % respectively, see likelihoods.m for the details. In general, care is taken % to avoid numerical issues when the arguments are extreme. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-07-21. % % See also likFunctions.m. if nargin<2, varargout = {'2'}; return; end % report number of hyperparameters numin = 1; % minimum value of nu nu = exp(hyp(1))+numin; sn2 = exp(2*hyp(2)); % extract hyperparameters lZ = loggamma(nu/2+1/2) - loggamma(nu/2) - log(nu*pi*sn2)/2; if nargin<5 % prediction mode if inf is not present if numel(y)==0, y = zeros(size(mu)); end s2zero = 1; if nargin>3, if norm(s2)>0, s2zero = 0; end, end % s2==0 ? if s2zero % log probability evaluation lp = lZ - (nu+1)*log( 1+(y-mu).^2./(nu.*sn2) )/2; else % prediction lp = likT(hyp, y, mu, s2, 'infEP'); end ymu = {}; ys2 = {}; if nargout>1 ymu = mu; % first y moment; for nu<=1 we this is the mode if nargout>2 if nu<=2 ys2 = Inf(size(s2)); else ys2 = s2 + nu*sn2/(nu-2); % second y moment end end end varargout = {lp,ymu,ys2}; else switch inf case 'infLaplace' r = y-mu; r2 = r.^2; if nargin<6 % no derivative mode dlp = {}; d2lp = {}; d3lp = {}; lp = lZ - (nu+1)*log( 1+r2./(nu.*sn2) )/2; if nargout>1 a = r2+nu*sn2; dlp = (nu+1)*r./a; % dlp, derivative of log likelihood if nargout>2 % d2lp, 2nd derivative of log likelihood d2lp = (nu+1)*(r2-nu*sn2)./a.^2; if nargout>3 % d3lp, 3rd derivative of log likelihood d3lp = (nu+1)*2*r.*(r2-3*nu*sn2)./a.^3; end end end varargout = {sum(lp),dlp,d2lp,d3lp}; else % derivative mode a3 = (r2+nu*sn2).^3; if i==1 % derivative w.r.t. nu lp_dhyp = nu*( dloggamma(nu/2+1/2)-dloggamma(nu/2) )/2 - 1/2 ... -nu*log(1+r2/(nu*sn2))/2 +(nu/2+1/2)*r2./(nu*sn2+r2); lp_dhyp = (1-numin/nu)*lp_dhyp; % correct for lower bound on nu d2lp_dhyp = nu*( r2.*(r2-3*sn2*(1+nu)) + nu*sn2^2 )./a3; d2lp_dhyp = (1-numin/nu)*d2lp_dhyp; % correct for lower bound on nu else % derivative w.r.t. sn lp_dhyp = (nu+1)*r2./(r2+nu*sn2) - 1; d2lp_dhyp = (nu+1)*2*nu*sn2*(nu*sn2-3*r2)./a3; end varargout = {lp_dhyp,d2lp_dhyp}; end case 'infEP' if nargout>1 error('infEP not supported since likT is not log-concave') end n = max([length(y),length(mu),length(s2)]); on = ones(n,1); y = y(:).*on; mu = mu(:).*on; sig = sqrt(s2(:)).*on; % vectors only % since we are not aware of an analytical expression of the integral, % we use Gaussian-Hermite quadrature N = 20; [t,w] = gauher(N); oN = ones(1,N); lZ = likT(hyp, y*oN, sig*t'+mu*oN, []); lZ = log_expA_x(lZ,w); % log( exp(lZ)*w ) varargout = {lZ}; case 'infVB' if nargin<6 % variational lower site bound % t(s) \propto (1+(s-y)^2/(nu*s2))^(-nu/2+1/2) % the bound has the form: b*s - s.^2/(2*ga) - h(ga)/2 with b=y/ga!! ga = s2; n = numel(ga); b = y./ga; y = y.*ones(n,1); db = -y./ga.^2; d2b = 2*y./ga.^3; id = ga<=sn2*nu/(nu+1); h = (nu+1)*( log(ga*(1+1/nu)/sn2) - 1 ) + (nu*sn2+y.^2)./ga; h(id) = y(id).^2./ga(id); h = h - 2*lZ; dh = (nu+1)./ga - (nu*sn2+y.^2)./ga.^2; dh(id) = -y(id).^2./ga(id).^2; d2h = -(nu+1)./ga.^2 + 2*(nu*sn2+y.^2)./ga.^3; d2h(id) = 2*y(id).^2./ga(id).^3; id = ga<0; h(id) = Inf; dh(id) = 0; d2h(id) = 0; % neg. var. treatment varargout = {h,b,dh,db,d2h,d2b}; else ga = s2; n = numel(ga); dhhyp = zeros(n,1); id = ga>sn2*nu/(nu+1); % dhhyp(~id) = 0 if i==1 % log(nu) % h = (nu+1)*log(1+1/nu) - nu + nu*sn2./ga; dhhyp(id) = nu*log(ga(id)*(1+1/nu)/sn2) - 1 - nu + nu*sn2./ga(id); % lZ = loggamma(nu/2+1/2) - loggamma(nu/2) - log(nu)/2 dhhyp = dhhyp - nu*dloggamma(nu/2+1/2) + nu*dloggamma(nu/2) + 1; % -2*lZ dhhyp = (1-numin/nu)*dhhyp; % correct for lower bound on nu else % log(sn) % h = (nu+1)*log(1/sn2) + nu*sn2./ga; dhhyp(id) = -2*(nu+1) + 2*nu*sn2./ga(id); % lZ = - log(sn2)/2 dhhyp = dhhyp + 2; % -2*lZ end dhhyp(ga<0) = 0; % negative variances get a special treatment varargout = {dhhyp}; % deriv. wrt hyp.lik end end end % Returns the log of the gamma function. Source: Pike, M.C., and % I.D. Hill, Algorithm 291, Communications of the ACM, 9,9:p.684 (Sept, 1966). % Accuracy to 10 decimal places. Uses Sterling's formula. % Derivative from Abromowitz and Stegun, formula 6.3.18 with recurrence 6.3.5. function f = loggamma(x) x = x+6; f = 1./(x.*x); f = (((-0.000595238095238*f+0.000793650793651).*f-1/360).*f+1/12)./x; f = (x-0.5).*log(x)-x+0.918938533204673+f; f = f-log(x-1)-log(x-2)-log(x-3)-log(x-4)-log(x-5)-log(x-6); function df = dloggamma(x) x = x+6; df = 1./(x.*x); df = (((df/240-0.003968253986254).*df+1/120).*df-1/120).*df; df = df+log(x)-0.5./x-1./(x-1)-1./(x-2)-1./(x-3)-1./(x-4)-1./(x-5)-1./(x-6); % computes y = log( exp(A)*x ) in a numerically safe way by subtracting the % maximal value in each row to avoid cancelation after taking the exp function y = log_expA_x(A,x) N = size(A,2); maxA = max(A,[],2); % number of columns, max over columns y = log(exp(A-maxA*ones(1,N))*x) + maxA; % exp(A) = exp(A-max(A))*exp(max(A)) % compute abscissas and weight factors for Gaussian-Hermite quadrature % % CALL: [x,w]=gauher(N) % % x = base points (abscissas) % w = weight factors % N = number of base points (abscissas) (integrates a (2N-1)th order % polynomial exactly) % % p(x)=exp(-x^2/2)/sqrt(2*pi), a =-Inf, b = Inf % % The Gaussian Quadrature integrates a (2n-1)th order % polynomial exactly and the integral is of the form % b N % Int ( p(x)* F(x) ) dx = Sum ( w_j* F( x_j ) ) % a j=1 % % this procedure uses the coefficients a(j), b(j) of the % recurrence relation % % b p (x) = (x - a ) p (x) - b p (x) % j j j j-1 j-1 j-2 % % for the various classical (normalized) orthogonal polynomials, % and the zero-th moment % % 1 = integral w(x) dx % % of the given polynomial's weight function w(x). Since the % polynomials are orthonormalized, the tridiagonal matrix is % guaranteed to be symmetric. function [x,w]=gauher(N) if N==20 % return precalculated values x=[ -7.619048541679757;-6.510590157013656;-5.578738805893203; -4.734581334046057;-3.943967350657318;-3.18901481655339 ; -2.458663611172367;-1.745247320814127;-1.042945348802751; -0.346964157081356; 0.346964157081356; 1.042945348802751; 1.745247320814127; 2.458663611172367; 3.18901481655339 ; 3.943967350657316; 4.734581334046057; 5.578738805893202; 6.510590157013653; 7.619048541679757]; w=[ 0.000000000000126; 0.000000000248206; 0.000000061274903; 0.00000440212109 ; 0.000128826279962; 0.00183010313108 ; 0.013997837447101; 0.061506372063977; 0.161739333984 ; 0.260793063449555; 0.260793063449555; 0.161739333984 ; 0.061506372063977; 0.013997837447101; 0.00183010313108 ; 0.000128826279962; 0.00000440212109 ; 0.000000061274903; 0.000000000248206; 0.000000000000126 ]; else b = sqrt( (1:N-1)/2 )'; [V,D] = eig( diag(b,1) + diag(b,-1) ); w = V(1,:)'.^2; x = sqrt(2)*diag(D); end ================================================ FILE: code/gpml/likFunctions.m ================================================ % likelihood functions are provided to be used by the gp.m function: % % likErf (Error function, classification, probit regression) % likLogistic (Logistic, classification, logit regression) % % likGauss (Gaussian, regression) % likLaplace (Laplacian or double exponential, regression) % likSech2 (Sech-square, regression) % likT (Student's t, regression) % % The likelihood functions have three possible modes, the mode being selected % as follows (where "lik" stands for any likelihood function in "lik/lik*.m".): % % 1) With one or no input arguments: [REPORT NUMBER OF HYPERPARAMETERS] % % s = lik OR s = lik(hyp) % % The likelihood function returns a string telling how many hyperparameters it % expects, using the convention that "D" is the dimension of the input space. % For example, calling "likLogistic" returns the string '0'. % % % 2) With three or four input arguments: [PREDICTION MODE] % % lp = lik(hyp, y, mu) OR [lp, ymu, ys2] = lik(hyp, y, mu, s2) % % This allows to evaluate the predictive distribution. Let p(y_*|f_*) be the % likelihood of a test point and N(f_*|mu,s2) an approximation to the posterior % marginal p(f_*|x_*,x,y) as returned by an inference method. The predictive % distribution p(y_*|x_*,x,y) is approximated by. % q(y_*) = \int N(f_*|mu,s2) p(y_*|f_*) df_* % % lp = log( q(y) ) for a particular value of y, if s2 is [] or 0, this % corresponds to log( p(y|mu) ) % ymu and ys2 the mean and variance of the predictive marginal q(y) % note that these two numbers do not depend on a particular % value of y % All vectors have the same size. % % % 3) With five or six input arguments, the fifth being a string [INFERENCE MODE] % % [varargout] = lik(hyp, y, mu, s2, inf) OR % [varargout] = lik(hyp, y, mu, s2, inf, i) % % There are three cases for inf, namely a) infLaplace, b) infEP and c) infVB. % The last input i, refers to derivatives w.r.t. the ith hyperparameter. % % a1) [sum(lp),dlp,d2lp,d3lp] = lik(hyp, y, f, [], 'infLaplace') % lp, dlp, d2lp and d3lp correspond to derivatives of the log likelihood % log(p(y|f)) w.r.t. to the latent location f. % lp = log( p(y|f) ) % dlp = d log( p(y|f) ) / df % d2lp = d^2 log( p(y|f) ) / df^2 % d3lp = d^3 log( p(y|f) ) / df^3 % % a2) [lp_dhyp,d2lp_dhyp] = lik(hyp, y, f, [], 'infLaplace', i) % returns derivatives w.r.t. to the ith hyperparameter % lp_dhyp = d log( p(y|f) ) / (df dhyp_i) % d2lp_dhyp = d^3 log( p(y|f) ) / (df^2 dhyp_i) % % % b1) [lZ,dlZ,d2lZ] = lik(hyp, y, mu, s2, 'infEP') % let Z = \int p(y|f) N(f|mu,s2) df then % lZ = log(Z) % dlZ = d log(Z) / dmu % d2lZ = d^2 log(Z) / dmu^2 % % b2) [dlZhyp] = lik(hyp, y, mu, s2, 'infEP', i) % returns derivatives w.r.t. to the ith hyperparameter % dlZhyp = d log(Z) / dhyp_i % % % c1) [h,b,dh,db,d2h,d2b] = lik(hyp, y, [], ga, 'infVB') % ga is the variance of a Gaussian lower bound to the likelihood p(y|f). % p(y|f) \ge exp( b*f - f.^2/(2*ga) - h(ga)/2 ) \propto N(f|b*ga,ga) % The function returns the linear part b and the "scaling function" h(ga) and % derivatives dh = d h/dga, db = d b/dga, d2h = d^2 h/dga and d2b = d^2 b/dga. % % c2) [dhhyp] = lik(hyp, y, [], ga, 'infVB', i) % dhhyp = dh / dhyp_i, the derivative w.r.t. the ith hyperparameter % % Cumulative likelihoods are designed for binary classification. Therefore, they % only look at the sign of the targets y; zero values are treated as +1. % % See the help for the individual likelihood for the computations specific to % each likelihood function. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18 ================================================ FILE: code/gpml/mean/meanConst.m ================================================ function A = meanConst(hyp, x, i) % Constant mean function. The mean function is parameterized as: % % m(x) = c % % The hyperparameter is: % % hyp = [ c ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-08-04. % % See also MEANFUNCTIONS.M. if nargin<2, A = '1'; return; end % report number of hyperparameters if numel(hyp)~=1, error('Exactly one hyperparameter needed.'), end c = hyp; if nargin==2 A = c*ones(size(x,1),1); % evaluate mean else if i==1 A = ones(size(x,1),1); % derivative else A = zeros(size(x,1),1); end end ================================================ FILE: code/gpml/mean/meanLinear.m ================================================ function A = meanLinear(hyp, x, i) % Linear mean function. The mean function is parameterized as: % % m(x) = sum_i a_i * x_i; % % The hyperparameter is: % % hyp = [ a_1 % a_2 % .. % a_D ] % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-01-10. % % See also MEANFUNCTIONS.M. if nargin<2, A = 'D'; return; end % report number of hyperparameters [n,D] = size(x); if any(size(hyp)~=[D,1]), error('Exactly D hyperparameters needed.'), end a = hyp; if nargin==2 A = x*a; % evaluate mean else if i<=D A = x(:,i); % derivative else A = zeros(n,1); end end ================================================ FILE: code/gpml/mean/meanMask.m ================================================ function A = meanMask(mean, hyp, x, i) % Apply a mean function to a subset of the dimensions only. The subset can % either be specified by a 0/1 mask by a boolean mask or by an index set. % % This function doesn't actually compute very much on its own, it merely does % some bookkeeping, and calls another mean function to do the actual work. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-07-15. % % See also MEANFUNCTIONS.M. mask = mean{1}(:); % either a binary mask or an index set mean = mean(2); % mean function to be masked nh_string = feval(mean{:}); % number of hyperparameters of the full mean if max(mask)<2 && length(mask)>1, mask = boolean(mask); end % convert 1/0->T/F if islogical(mask), D = sum(mask); else D = length(mask); end % masked dimension if nargin<3, A = num2str(eval(nh_string)); return; end % number of parameters if eval(nh_string)~=length(hyp) % check hyperparameters error('number of hyperparameters does not match size of masked data') end if nargin==3 A = feval(mean{:}, hyp, x(:,mask)); else % compute derivative A = feval(mean{:}, hyp, x(:,mask), i); end ================================================ FILE: code/gpml/mean/meanOne.m ================================================ function A = meanOne(hyp, x, i) % One mean function. The mean function does not have any parameters. % % m(x) = 1 % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-08-04. % % See also MEANFUNCTIONS.M. if nargin<2, A = '0'; return; end % report number of hyperparameters if nargin==2 A = ones(size(x,1),1); % evaluate mean else A = zeros(size(x,1),1); % derivative end ================================================ FILE: code/gpml/mean/meanPow.m ================================================ function A = meanPow(mean, hyp, x, i) % meanPow - compose a mean function as the power of another mean function. % % The degree d has to be a strictly positive integer. % % m(x) = m_0(x) ^ d % % This function doesn't actually compute very much on its own, it merely does % some bookkeeping, and calls other mean function to do the actual work. % % Copyright (c) by Carl Edward Rasmussen & Hannes Nickisch 2010-06-18. % % See also MEANFUNCTIONS.M. d = mean{1}; d = abs(floor(d)); d = max(d,1); % positive integer degree mean = mean{2}; if ~iscell(mean), mean = {mean}; end if nargin<3 % report number of parameters A = feval(mean{:}); return end [n,D] = size(x); if nargin==3 % compute mean vector A = feval(mean{:},hyp,x).^d; else % compute derivative vector A = ( d*feval(mean{:},hyp,x).^(d-1) ).* feval(mean{:},hyp,x,i); end ================================================ FILE: code/gpml/mean/meanProd.m ================================================ function A = meanProd(mean, hyp, x, i) % meanProd - compose a mean function as the product of other mean functions. % This function doesn't actually compute very much on its own, it merely does % some bookkeeping, and calls other mean functions to do the actual work. % % m(x) = \prod_i m_i(x) % % Copyright (c) by Carl Edward Rasmussen & Hannes Nickisch 2010-08-04. % % See also MEANFUNCTIONS.M. for ii = 1:numel(mean) % iterate over mean functions f = mean(ii); if iscell(f{:}), f = f{:}; end % expand cell array if necessary j(ii) = cellstr(feval(f{:})); % collect number hypers end if nargin<3 % report number of parameters A = char(j(1)); for ii=2:length(mean), A = [A, '+', char(j(ii))]; end; return end [n,D] = size(x); v = []; % v vector indicates to which mean parameters belong for ii = 1:length(mean), v = [v repmat(ii, 1, eval(char(j(ii))))]; end A = ones(n,1); % allocate space if nargin==3 % compute mean vector for ii = 1:length(mean) % iteration over summand functions f = mean(ii); if iscell(f{:}), f = f{:}; end % expand cell array if needed A = A.*feval(f{:}, hyp(v==ii), x); % accumulate means end else % compute derivative vector if i<=length(v) ii = v(i); % which mean function j = sum(v(1:i)==ii); % which parameter in that mean for jj = 1:length(mean) f = mean(jj); if iscell(f{:}), f = f{:}; end % dereference cell array if necessary if jj==ii A = A .* feval(f{:}, hyp(v==jj), x, j); % multiply derivative else A = A .* feval(f{:}, hyp(v==jj), x); % multiply mean end end else A = zeros(n,1); end end ================================================ FILE: code/gpml/mean/meanScale.m ================================================ function A = meanScale(mean, hyp, x, i) % meanScale - compose a mean function as a scaled version of another one. % % m(x) = a * m_0(x) % % The hyperparameter is: % % hyp = [ a ] % % This function doesn't actually compute very much on its own, it merely does % some bookkeeping, and calls other mean function to do the actual work. % % Copyright (c) by Carl Edward Rasmussen & Hannes Nickisch 2010-07-15. % % See also MEANFUNCTIONS.M. if nargin<3 % report number of parameters A = [feval(mean{:}),'+1']; return end [n,D] = size(x); a = hyp(1); if nargin==3 % compute mean vector A = a*feval(mean{:},hyp(2:end),x); else % compute derivative vector if i==1 A = feval(mean{:},hyp(2:end),x); else A = a*feval(mean{:},hyp(2:end),x,i-1); end end ================================================ FILE: code/gpml/mean/meanSum.m ================================================ function A = meanSum(mean, hyp, x, i) % meanSum - compose a mean function as the sum of other mean functions. % This function doesn't actually compute very much on its own, it merely does % some bookkeeping, and calls other mean functions to do the actual work. % % m(x) = \sum_i m_i(x) % % Copyright (c) by Carl Edward Rasmussen & Hannes Nickisch 2010-08-04. % % See also MEANFUNCTIONS.M. for ii = 1:numel(mean) % iterate over mean functions f = mean(ii); if iscell(f{:}), f = f{:}; end % expand cell array if necessary j(ii) = cellstr(feval(f{:})); % collect number hypers end if nargin<3 % report number of parameters A = char(j(1)); for ii=2:length(mean), A = [A, '+', char(j(ii))]; end; return end [n,D] = size(x); v = []; % v vector indicates to which mean parameters belong for ii = 1:length(mean), v = [v repmat(ii, 1, eval(char(j(ii))))]; end if nargin==3 % compute mean vector A = zeros(n,1); % allocate space for ii = 1:length(mean) % iteration over summand functions f = mean(ii); if iscell(f{:}), f = f{:}; end % expand cell array if needed A = A + feval(f{:}, hyp(v==ii), x); % accumulate means end else % compute derivative vector if i<=length(v) ii = v(i); % which mean function j = sum(v(1:i)==ii); % which parameter in that mean f = mean(ii); if iscell(f{:}), f = f{:}; end % dereference cell array if necessary A = feval(f{:}, hyp(v==ii), x, j); % compute derivative else A = zeros(n,1); end end ================================================ FILE: code/gpml/mean/meanZero.m ================================================ function A = meanZero(hyp, x, i) % Zero mean function. The mean function does not have any parameters. % % m(x) = 0 % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-01-10. % % See also MEANFUNCTIONS.M. if nargin<2, A = '0'; return; end % report number of hyperparameters A = zeros(size(x,1),1); % derivative and mean ================================================ FILE: code/gpml/meanFunctions.m ================================================ % mean functions to be use by Gaussian process functions. There are two % different kinds of mean functions: simple and composite: % % simple mean functions: % % meanZero - zero mean function % meanOne - one mean function % meanConst - constant mean function % meanLinear - linear mean function % % composite covariance functions (see explanation at the bottom): % % meanScale - scaled version of a mean function % meanPow - power of a mean function % meanProd - products of mean functions % meanSum - sums of mean functions % meanMask - mask some dimensions of the data % % Naming convention: all mean functions are named "mean/mean*.m". % % % 1) With no or only a single input argument: % % s = meanNAME or s = meanNAME(hyp) % % The mean function returns a string s telling how many hyperparameters hyp it % expects, using the convention that "D" is the dimension of the input space. % For example, calling "meanLinear" returns the string 'D'. % % 2) With two input arguments: % % m = meanNAME(hyp, x) % % The function computes and returns the mean vector where hyp are the % hyperparameters and x is an n by D matrix of cases, where D is the dimension % of the input space. The returned mean vector is of size n by 1. % % 3) With three input arguments: % % dm = meanNAME(hyp, x, i) % % The function computes and returns the n by 1 vector of partial derivatives % of the mean vector w.r.t. hyp(i) i.e. hyperparameter number i. % % See also doc/usageMean.m. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2011-02-18 ================================================ FILE: code/gpml/penalized_gp.m ================================================ % A silly function to make sure that hyperparameters don't go off to % infinity. % % Should probably be handled somewhere else in GP inference. % % % David Duvenaud % May 2011 % ================================ function [nlz, dnlz] = penalized_gp(hyp, varargin) [nlz, dnlz] = gp(hyp, varargin{:}); hyp = unwrap(hyp); dnlz2 = unwrap(dnlz); scale = 10; polynomial = 10; for i = 1:length(hyp) nlz = nlz + (hyp(i)/scale)^polynomial; dnlz2(i) = dnlz2(i) + polynomial*((hyp(i)/scale)^(polynomial-1))/scale; end dnlz = rewrap( dnlz, dnlz2); end function [s v] = rewrap(s, v) % map elements of v (vector) onto s (any type) if isnumeric(s) if numel(v) < numel(s) error('The vector for conversion contains too few elements') end s = reshape(v(1:numel(s)), size(s)); % numeric values are reshaped v = v(numel(s)+1:end); % remaining arguments passed on elseif isstruct(s) [s p] = orderfields(s); p(p) = 1:numel(p); % alphabetize, store ordering [t v] = rewrap(struct2cell(s), v); % convert to cell, recurse s = orderfields(cell2struct(t,fieldnames(s),1),p); % conv to struct, reorder elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially [s{i} v] = rewrap(s{i}, v); end end end function v = unwrap(s) % extract num elements of s (any type) into v (vector) v = []; if isnumeric(s) v = s(:); % numeric values are recast to column vector elseif isstruct(s) v = unwrap(struct2cell(orderfields(s))); % alphabetize, conv to cell, recurse elseif iscell(s) % cell array elements are for i = 1:numel(s), v = [v; unwrap(s{i})]; end % handled sequentially end % other types are ignored end ================================================ FILE: code/gpml/startup.m ================================================ % startup script to make Octave/Matlab aware of the GPML package % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch 2010-07-05. disp(['executing gpml startup script...']); OCT = exist('OCTAVE_VERSION') ~= 0; % check if we run Matlab or Octave me = mfilename; % what is my filename mydir = which(me); mydir = mydir(1:end-2-numel(me)); % where am I located if OCT && numel(mydir)==2 if strcmp(mydir,'./'), mydir = [pwd,mydir(2:end)]; end end % OCTAVE 3.0.x relative, MATLAB and newer have absolute path addpath(mydir(1:end-1)) addpath([mydir,'cov']) addpath([mydir,'doc']) addpath([mydir,'inf']) addpath([mydir,'lik']) addpath([mydir,'mean']) addpath([mydir,'util']) clear me mydir ================================================ FILE: code/gpml/util/binaryEPGP.m ================================================ function varargout = binaryEPGP(hyper, covfunc, x, y, xstar) % binaryEPGP - The Expectation Propagation approximation for binary Gaussian % process classification. Two modes are possible: training or testing: if no % test cases are supplied, then the approximate negative log marginal % likelihood and its partial derivatives wrt the hyperparameters is computed; % this mode is used to fit the hyperparameters. If test cases are given, then % the test set predictive probabilities are returned. The program is flexible % in allowing a multitude of covariance functions. % % usage: [nlZ, dnlZ ] = binaryEPGP(hyper, covfunc, x, y); % or: [p, mu, s2, nlZ] = binaryEPGP(hyper, covfunc, x, y, xstar); % % where: % % hyper is a (column) vector of hyperparameters % covfunc is the name of the covariance function (see below) % lik is the name of the likelihood function (see below) % x is a n by D matrix of training inputs % y is a (column) vector (of size n) of binary +1/-1 targets % xstar is a nn by D matrix of test inputs % nlZ is the returned value of the negative log marginal likelihood % dnlZ is a (column) vector of partial derivatives of the negative % log marginal likelihood wrt each log hyperparameter % p is a (column) vector (of length nn) of predictive probabilities % mu is a (column) vector (of length nn) of predictive latent means % s2 is a (column) vector (of length nn) of predictive latent variances % % The length of the vector of hyperparameters depends on the covariance % function, as specified by the "covfunc" input to the function, specifying the % name of a covariance function. A number of different covariance function are % implemented, and it is not difficult to add new ones. See "help covFunctions" % for the details % % The function can conveniently be used with the "minimize" function to train % a Gaussian process, eg: % % [hyper, fX, i] = minimize(hyper, 'binaryEPGP', length, 'covSEiso', % 'logistic', x, y); % % Copyright (c) 2004, 2005, 2006, 2007 Carl Edward Rasmussen, 2010-07-21. if nargin<4 || nargin>5 disp('Usage: [nlZ, dnlZ ] = binaryEPGP(hyper, covfunc, x, y);') disp(' or: [p, mu, s2, nlZ] = binaryEPGP(hyper, covfunc, x, y, xstar);') return end % Note, this function is just a wrapper provided for backward compatibility, % the functionality is now provided by the more general gp function. lik = @likErf; inf = @infEP; mean = @meanZero; varargout = cell(nargout, 1); % allocate the right number of output arguments hyp.cov = hyper; if nargin==4 [varargout{:}] = gp(hyp,inf,mean,covfunc,lik,x,y); if nargout>1, varargout{2} = varargout{2}.cov; end else [ymu,ys2,fmu,fs2] = gp(hyp,inf,mean,covfunc,lik,x,y,xstar); if nargout>0, varargout{1} = (1+ymu)/2; end if nargout>1, varargout{2} = fmu; end if nargout>2, varargout{3} = fs2; end if nargout>3, varargout{4} = gp(hyp,inf,mean,covfunc,lik,x,y); end % nlZ end ================================================ FILE: code/gpml/util/binaryGP.m ================================================ function varargout = binaryGP(hyper, approx, covfunc, lik, x, y, xstar) % Approximate binary Gaussian Process classification. Two modes are possible: % training or testing: if no test cases are supplied, then the approximate % negative log marginal likelihood and its partial derivatives wrt the % hyperparameters is computed; this mode is used to fit the hyperparameters. If % test cases are given, then the test set predictive probabilities are % returned. Exact inference is intractible, the function uses a specified % approximation method (see approximations.m), flexible covariance functions % (see covFunctions.m) and likelihood functions (see likelihoods.m). % % usage: [nlZ, dnlZ ] = binaryGP(hyper, approx, covfunc, lik, x, y); % or: [p,mu,s2,nlZ] = binaryGP(hyper, approx, covfunc, lik, x, y, xstar); % % where: % % hyper is a column vector of hyperparameters % approx is a function specifying an approximation method for inference % covfunc is the name of the covariance function (see below) % lik is the name of the likelihood function % x is a n by D matrix of training inputs % y is a (column) vector (of size n) of binary +1/-1 targets % xstar is a nn by D matrix of test inputs % nlZ is the returned value of the negative log marginal likelihood % dnlZ is a (column) vector of partial derivatives of the negative % log marginal likelihood wrt each hyperparameter % p is a (column) vector (of length nn) of predictive probabilities % mu is a (column) vector (of length nn) of predictive latent means % s2 is a (column) vector (of length nn) of predictive latent variances % % The length of the vector of hyperparameters depends on the covariance % function, as specified by the "covfunc" input to the function, specifying the % name of a covariance function. A number of different covariance function are % implemented, and it is not difficult to add new ones. See covFunctions.m for % the details. % % The "lik" input argument specifies the name of the likelihood function (see % likelihoods.m). % % The "approx" input argument to the function specifies an approximation method % (see approximations.m). An approximation method returns a representation of % the approximate Gaussian posterior. Usually, the approximate posterior admits % the form N(m=K*alpha, V=inv(inv(K)+W)), where alpha is a vector and W is % diagonal. The approximation method returns: % % alpha is a (sparse or full column vector) containing inv(K)*m, where K % is the prior covariance matrix and m the approx posterior mean % sW is a (sparse or full column) vector containing diagonal of sqrt(W) % the approximate posterior covariance matrix is inv(inv(K)+W) % L is a (sparse or full) matrix, L = chol(sW*K*sW+eye(n)) % % In cases where the approximate posterior variance does not admit the form % V=inv(inv(K)+W) with diagonal W, L contains instead -inv(K+inv(W)), and sW % is unused. % % The alpha parameter may be sparse. In that case sW and L can either be sparse % or full (retaining only the non-zero rows and columns, as indicated by the % sparsity structure of alpha). The L paramter is allowed to be empty, in % which case it will be computed. % % The function can conveniently be used with the "minimize" function to train % a Gaussian Process, eg: % % [hyper, fX, i] = minimize(hyper, 'binaryGP', length, 'approxEP', 'covSEiso', 'logistic', x, y); % % where "length" gives the length of the run: if it is positive, it gives the % maximum number of line searches, if negative its absolute gives the maximum % allowed number of function evaluations. % % Copyright (c) 2007 Carl Edward Rasmussen and Hannes Nickisch, 2010-07-21. if nargin<6 || nargin>7 disp('Usage: [nlZ, dnlZ ] = binaryGP(hyper,approx,covfunc,lik,x,y);') disp(' or: [p,mu,s2,nlZ] = binaryGP(hyper,approx,covfunc,lik,x,y,xstar);') return end % Note, this function is just a wrapper provided for backward compatibility, % the functionality is now provided by the more general gp function. if strcmp(lik,'logistic') lik = @likLogistic; elseif strcmpi(lik,'cumgauss') lik = @likErf; else error('Allowable likelihoods: logistic and cumGauss.'); end if strcmp(approx,'approxLA') inf = @infLaplace; elseif strcmp(approx,'approxEP') inf = @infEP; else error('Allowable approximations: approxLA and approxEP.'); end mean = @meanZero; varargout = cell(nargout, 1); % allocate the right number of output arguments hyp.cov = hyper; if nargin==6 [varargout{:}] = gp(hyp,inf,mean,covfunc,lik,x,y); if nargout>1, varargout{2} = varargout{2}.cov; end else [ymu,ys2,fmu,fs2] = gp(hyp,inf,mean,covfunc,lik,x,y,xstar); if nargout>0, varargout{1} = (1+ymu)/2; end if nargout>1, varargout{2} = fmu; end if nargout>2, varargout{3} = fs2; end if nargout>3, varargout{4} = gp(hyp,inf,mean,covfunc,lik,x,y); end % nlZ end ================================================ FILE: code/gpml/util/binaryLaplaceGP.m ================================================ function varargout = binaryLaplaceGP(hyper, covfunc, lik, x, y, xstar) % binaryLaplaceGP - Laplace's approximation for binary Gaussian process % classification. Two modes are possible: training or testing: if no test % cases are supplied, then the approximate negative log marginal likelihood % and its partial derivatives wrt the hyperparameters is computed; this mode is % used to fit the hyperparameters. If test cases are given, then the test set % predictive probabilities are returned. The program is flexible in allowing % several different likelihood functions and a multitude of covariance % functions. % % usage: [nlZ, dnlZ ] = binaryLaplaceGP(hyper, covfunc, lik, x, y); % or: [p, mu, s2, nlZ] = binaryLaplaceGP(hyper, covfunc, lik, x, y, xstar); % % where: % % hyper is a (column) vector of hyperparameters % covfunc is the name of the covariance function (see below) % lik is the name of the likelihood function (see below) % x is a n by D matrix of training inputs % y is a (column) vector (of size n) of binary +1/-1 targets % xstar is a nn by D matrix of test inputs % nlZ is the returned value of the negative log marginal likelihood % dnlZ is a (column) vector of partial derivatives of the negative % log marginal likelihood wrt each log hyperparameter % p is a (column) vector (of length nn) of predictive probabilities % mu is a (column) vector (of length nn) of predictive latent means % s2 is a (column) vector (of length nn) of predictive latent variances % % The length of the vector of log hyperparameters depends on the covariance % function, as specified by the "covfunc" input to the function, specifying the % name of a covariance function. A number of different covariance function are % implemented, and it is not difficult to add new ones. See "help covFunctions" % for the details. % % The shape of the likelihood function is given by the "lik" input to the % function, specifying the name of the likelihood function. The two implemented % likelihood functions are: % % logistic the logistic function: 1/(1+exp(-x)) % cumGauss the cumulative Gaussian (error function) % % The function can conveniently be used with the "minimize" function to train % a Gaussian process, eg: % % [hyper, fX, i] = minimize(hyper, 'binaryLaplaceGP', length, 'covSEiso', % 'logistic', x, y); % % Copyright (c) 2004, 2005, 2006, 2007 Carl Edward Rasmussen, 2010-07-21. if nargin<5 || nargin>6 disp('Usage: [nlZ, dnlZ ] = binaryLaplaceGP(hyper, covfunc, lik, x, y);') disp(' or: [p, mu, s2, nlZ] = binaryLaplaceGP(hyper, covfunc, lik, x, y, xstar);') return end % Note, this function is just a wrapper provided for backward compatibility, % the functionality is now provided by the more general gp function. if strcmp(lik,'logistic') lik = @likLogistic; elseif strcmpi(lik,'cumgauss') lik = @likErf; else error('Allowable likelihoods: logistic and cumGauss.'); end inf = @infLaplace; mean = @meanZero; varargout = cell(nargout, 1); % allocate the right number of output arguments hyp.cov = hyper; if nargin==5 [varargout{:}] = gp(hyp,inf,mean,covfunc,lik,x,y); if nargout>1, varargout{2} = varargout{2}.cov; end else [ymu,ys2,fmu,fs2] = gp(hyp,inf,mean,covfunc,lik,x,y,xstar); if nargout>0, varargout{1} = (1+ymu)/2; end if nargout>1, varargout{2} = fmu; end if nargout>2, varargout{3} = fs2; end if nargout>3, varargout{4} = gp(hyp,inf,mean,covfunc,lik,x,y); end % nlZ end ================================================ FILE: code/gpml/util/brentmin.m ================================================ %% BRENTMIN: Brent's minimization method in one dimension function [xmin,fmin,funccount,varargout] = ... brentmin(xlow,xupp,Nitmax,tol,f,nout,varargin) % code taken from % § 10.2 Parabolic Interpolation and Brent's Method in One Dimension % Press, Teukolsky, Vetterling & Flannery % Numerical Recipes in C, Cambridge University Press, 2002 % % [xmin,fmin,funccout,varargout] = BRENTMIN(xlow,xupp,Nit,tol,f,nout,varargin) % Given a function f, and given a search interval this routine isolates % the minimum of fractional precision of about tol using Brent's method. % % INPUT % ----- % xlow,xupp: search interval such that xlow<=xmin<=xupp % Nitmax: maximum number of function evaluations made by the routine % tol: fractional precision % f: [y,varargout{:}] = f(x,varargin{:}) is the function % nout: no. of outputs of f (in varargout) in addition to the y value % % OUTPUT % ------ % fmin: minimal function value % xmin: corresponding abscissa-value % funccount: number of function evaluations made % varargout: additional outputs of f at optimum % % Copyright (c) by Hannes Nickisch 2010-01-10. if nargin<6, nout = 0; end varargout = cell(nout,1); % tolerance is no smaller than machine's floating point precision tol = max(tol,eps); % Evaluate endpoints fa = f(xlow,varargin{:}); fb = f(xupp,varargin{:}); funccount = 2; % number of function evaluations % Compute the start point seps = sqrt(eps); c = 0.5*(3.0 - sqrt(5.0));% golden ratio a = xlow; b = xupp; v = a + c*(b-a); w = v; xf = v; d = 0.0; e = 0.0; x = xf; [fx,varargout{:}] = f(x,varargin{:}); funccount = funccount + 1; fv = fx; fw = fx; xm = 0.5*(a+b); tol1 = seps*abs(xf) + tol/3.0; tol2 = 2.0*tol1; % Main loop while ( abs(xf-xm) > (tol2 - 0.5*(b-a)) ) gs = 1; % Is a parabolic fit possible if abs(e) > tol1 % Yes, so fit parabola gs = 0; r = (xf-w)*(fx-fv); q = (xf-v)*(fx-fw); p = (xf-v)*q-(xf-w)*r; q = 2.0*(q-r); if q > 0.0, p = -p; end q = abs(q); r = e; e = d; % Is the parabola acceptable if ( (abs(p)q*(a-xf)) && (p= xm, e = a-xf; else e = b-xf; end d = c*e; end % The function must not be evaluated too close to xf si = sign(d) + (d == 0); x = xf + si * max( abs(d), tol1 ); [fu,varargout{:}] = f(x,varargin{:}); funccount = funccount + 1; % Update a, b, v, w, x, xm, tol1, tol2 if fu <= fx if x >= xf, a = xf; else b = xf; end v = w; fv = fw; w = xf; fw = fx; xf = x; fx = fu; else % fu > fx if x < xf, a = x; else b = x; end if ( (fu <= fw) || (w == xf) ) v = w; fv = fw; w = x; fw = fu; elseif ( (fu <= fv) || (v == xf) || (v == w) ) v = x; fv = fu; end end xm = 0.5*(a+b); tol1 = seps*abs(xf) + tol/3.0; tol2 = 2.0*tol1; if funccount >= Nitmax % typically we should not get here % warning(sprintf(['Maximum number of iterations (%d) exceeded:', ... % 'precision is not guaranteed'],Nitmax)) % fprintf('[%1.3f,%1.3f,%1.3f]\n',xlow,xf,xupp) break end end % while % check that endpoints are less than the minimum found if ( (fa < fx) && (fa <= fb) ) xf = xlow; fx = fa; elseif fb < fx xf = xupp; fx = fb; end fmin = fx; xmin = xf; ================================================ FILE: code/gpml/util/elsympol.m ================================================ % Evaluate the order R elementary symmetric polynomial Newton's identity aka % the Newton–Girard formulae: http://en.wikipedia.org/wiki/Newton's_identities % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-01-10. function E = elsympol(Z,R) % evaluate 'power sums' of the individual terms in Z sz = size(Z); P = zeros([sz(1:2),R]); for r=1:R, P(:,:,r) = sum(Z.^r,3); end E = zeros([sz(1:2),R+1]); % E(:,:,r+1) yields polynomial r E(:,:,1) = ones(sz(1:2)); if R==0, return, end % init recursion E(:,:,2) = P(:,:,1); if R==1, return, end % init recursion for r=2:R for i=1:r E(:,:,r+1) = E(:,:,r+1) + P(:,:,i).*E(:,:,r+1-i)*(-1)^(i-1)/r; end end ================================================ FILE: code/gpml/util/elsympol2.m ================================================ % Evaluate the order R elementary symmetric polynomial Newton's identity aka % the Newton–Girard formulae: http://en.wikipedia.org/wiki/Newton's_identities % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-01-10. % % Modified by David Duvenaud to be more numerically stable, 2011-04-28. function E = elsympol2(Z,R) % evaluate 'power sums' of the individual terms in Z sz = size(Z); P = zeros([sz(1:2),R]); Z_power = Z; for r=1:R P(:,:,r) = sum(Z_power,3); Z_power = Z_power .* Z; % Evaluate the power incrementally. end E = zeros([sz(1:2),R+1]); % E(:,:,r+1) yields polynomial r E(:,:,1) = ones(sz(1:2)); if R==0, return, end % init recursion E(:,:,2) = P(:,:,1); if R==1, return, end % init recursion for r=2:R for i=1:r E(:,:,r+1) = E(:,:,r+1) + (P(:,:,i).*E(:,:,r+1-i))*((-1)^(i-1)/r); end end ================================================ FILE: code/gpml/util/gpr.m ================================================ function varargout = gpr(hyper, cov, x, y, xs) % gpr - Gaussian process regression, with a named covariance function. Two % modes are possible: training and prediction: if no test data are given, the % function returns minus the log likelihood and its partial derivatives with % respect to the hyperparameters; this mode is used to fit the hyperparameters. % If test data are given, then (marginal) Gaussian predictions are computed, % whose mean and variance are returned. Note that in cases where the covariance % function has noise contributions, the variance returned in s2 is for noisy % test targets; if you want the variance of the noise-free latent function, you % must substract the noise variance. % % usage: [nlZ dnlZ] = gpr(hyp, cov, x, y) % or: [mu s2] = gpr(hyp, cov, x, y, xs) % % where: % % hyp is a (column) vector of log hyperparameters % cov is the covariance function % x is a n by D matrix of training inputs % y is a (column) vector (of size n) of targets % xs is a ns by D matrix of test inputs % nlZ is the returned value of the negative log marginal likelihood % dnlZ is a (column) vector of partial derivatives of the negative % log marginal likelihood wrt each log hyperparameter % mu is a (column) vector (of size nn) of prediced means % s2 is a (column) vector (of size nn) of predicted variances % % For more help on covariance functions, see "help covFunctions". % % Copyright (c) 2010 Carl Edward Rasmussen and Hannes Nickisch 2010-06-18. err = 'we need cov = {''covSum'', {cov1, ..,''covNoise'', .., covP} }; to map to gp.m'; if ~strcmp(cov{1},'covSum'), error(err), end id = 0; nhyp = zeros(length(cov{2}),1); for i=1:length(cov{2}) if strcmp(cov{2}{i},'covNoise'), id = i; break, end nhyp(i) = eval(feval(cov{2}{i})); end if id==0, error(err), else nhyp = sum(nhyp(1:id-1)); end cov{2} = cov{2}([1:id-1,id+1:end]); hyp.lik = hyper(nhyp+1); hyp.cov = hyper([1:nhyp,nhyp+2:end]); % Note, this function is just a wrapper provided for backward compatibility, % the functionality is now provided by the more general gp function. lik = @likGauss; inf = @infExact; mean = @meanZero; varargout = cell(nargout, 1); % allocate the right number of output arguments if nargin==4 [varargout{:}] = gp(hyp,inf,mean,cov,lik,x,y); if nargout>1, varargout{2} = [varargout{2}.lik; varargout{2}.cov]; end else [varargout{:}] = gp(hyp,inf,mean,cov,lik,x,y,xs); end ================================================ FILE: code/gpml/util/lbfgsb/Makefile ================================================ # path to your Matlab installation # you can find it by the commands 'locate matlab' or 'find / -name "matlab"' MATLAB_HOME=/agbs/share/sw/matlab #MATLAB_HOME=/usr/local/share/matlabR2008b # for 'make mex' you need to provide the variable MEX and have mex set up # for 'make nomex' you need to provide the variables MEX_SUFFIX and MATLAB_LIB # Choose Operating System ###################################################### # For 32 bit Linux, uncomment the following three lines.------------------------ MEX = $(MATLAB_HOME)/bin/mex MEX_SUFFIX = mexglx MATLAB_LIB = -L$(MATLAB_HOME)/bin/glnx86 -lmex # For 64 bit Linux, uncomment the following three lines.------------------------ #MEX = $(MATLAB_HOME)/bin/mex #MEX_SUFFIX = mexa64 #MATLAB_LIB = -L$(MATLAB_HOME)/bin/glnxa64 -lmex # For Mac OS X, uncomment the following three lines.---------------------------- #MEX = /Applications/MATLAB7/bin/mex #MEX_SUFFIX = mexmac #MATLAB_LIB = -L$(MATLAB_HOME)/bin/maci -lmex # Choose FORTRAN compiler ###################################################### # To work with f77, uncomment the following two lines.-------------------------- F77 = f77 FORTRAN_LIB = g2c # To use gfortran, uncomment the following two lines.--------------------------- #F77 = gfortran #FORTRAN_LIB = gfortran # Do not edit below ############################################################ CXX = g++ CFLAGS = -O3 -ffast-math -fomit-frame-pointer -fPIC -Werror -pthread -Wall -ansi FFLAGS = -O3 -fPIC -fexceptions MATLAB_INCLUDE=-I$(MATLAB_HOME)/extern/include TARGET = lbfgsb OBJS = solver.o matlabexception.o matlabscalar.o matlabstring.o \ matlabmatrix.o arrayofmatrices.o program.o matlabprogram.o \ lbfgsb.o %.o: %.cpp $(CXX) $(CFLAGS) $(MATLAB_INCLUDE) -o $@ -c $^ %.o: %.f $(F77) $(FFLAGS) -o $@ -c $^ mex: $(TARGET) clean nomex: $(TARGET)_nomex clean $(TARGET): $(OBJS) $(MEX) -cxx CXX=$(CXX) CC=$(CXX) FC=$(FCC) LD=$(CXX) -l$(FORTRAN_LIB) -lm \ -O -output $@ $^ $(TARGET)_nomex: $(OBJS) $(CXX) $^ -shared -o $(TARGET).$(MEX_SUFFIX) $(MATLAB_LIB) -l$(FORTRAN_LIB) -lm clean: rm -f *.o cp $(TARGET).$(MEX_SUFFIX) ../$(TARGET).mex mv $(TARGET).$(MEX_SUFFIX) .. ================================================ FILE: code/gpml/util/lbfgsb/README.html ================================================ MATLAB interface for L-BFGS-B

A MATLAB interface for L-BFGS-B

by Peter Carbonetto
Dept. of Computer Science
University of British Columbia

L-BFGS-B is a collection of Fortran 77 routines for solving large-scale nonlinear optimization problems with bound constraints on the variables. One of the key features of the nonlinear solver is that knowledge of the Hessian is not required; the solver computes search directions by keeping track of a quadratic model of the objective function with a limited-memory BFGS (Broyden-Fletcher-Goldfarb-Shanno) approximation to the Hessian.1 The algorithm was developed by Ciyou Zhu, Richard Byrd and Jorge Nocedal. For more information, go to the original distribution site for the L-BFGS-B software package.

I've designed an interface to the L-BFGS-B solver so that it can be called like any other function in MATLAB.2 See the text below for more information on installing and calling it in MATLAB. Along the way, I've also developed a C++ class that encapsulates all the "messy" details in executing the L-BFGS-B code. See below for instructions on how to use this class for your own C++ code.

If you have any questions, praise, or comments, or would like to report a bug, do not hesitate to contact the author. I've tested this software in MATLAB version 7.2 on both the Mac OS X and Linux operating systems.

Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Canada License. Please cite the source as Peter Carbonetto, Department of Computer Science, University of British Columbia.

Installation

These installation instructions assume you have a UNIX-based operating system, such as Mac OS X or Linux. It is surely possible to install this software on Windows, I'm just not sure how. These instructions also assume you have GNU make installed.

Download the source. First, download and unpack the tar archive. It contains some MATLAB files (ending in .m), some C++ source and header files (ending in .h and .cpp), a single Fortran 77 source file solver.f containing the L-BFGS-B routines, and a Makefile. I've included a version of the Fortran routines that is more recent than what is available for download at the distribution site at Northwestern University.

What we will do is create a MEX file, which is basically a file that contains a routine that can be called from MATLAB as if it were a built-it function. To learn about MEX files, I refer you to this document at the MathWorks website.

Install the C++ and Fortran 77 compilers. In order to build the MEX file, you will need both a C++ compiler and a Fortran 77 compiler. Unfortunately, you can't just use any compiler. You have to use the precise one supported by MATLAB. For instance, if you are running Mac OS X 7.2 on a Linux operating system, you will need to install the GNU compiler collection (GCC) version 3.4.4. Even if you already have a compiler installed on Linux, it may be the wrong version, and if you use the wrong version things could go horribly wrong! It is important that you use the correct version of the compiler, otherwise you will encounter linking errors. Refer to this document to find out which compiler is supported by your version of MATLAB.

Configure MATLAB. Next, you need to set up and configure MATLAB to build MEX Files. This is explained quite nicely in this MathWorks support document.

Modify the Makefile. You are almost ready to build the MEX file. But before you do so, you need to edit the Makefile to coincide with your system setup. At the top of the file are several variables you may need to modify. The variable MEX is the executable called to build the MEX file. The variables CXX and F77 must be the names of your C++ and Fortran 77 compilers. MEXSUFFIX is the MEX file extension specific to your platform. (For instance, the extension for Linux is mexglx.) The variable MATLAB_HOME must be the base directory of your MATLAB installation. Finally, CFLAGS and FFLAGS are options passed to the C++ and Fortran compilers, respectively. Often these flags coincide with your MEX options file (see here for more information on the options file). But often the settings in the MEX option file are incorrect, and so the options must be set manually. MATLAB requires some special compilation flags for various reasons, one being that it requires position-independent code. These instructions are vague (I apologize), and this step may require a bit of trial and error until before you get it right.

On my laptop running Mac OS X 10.3.9 with MATLAB 7.2, my settings for the Makefile are

  MEX         = mex
  MEXSUFFIX   = mexmac
  MATLAB_HOME = /Applications/MATLAB72
  CXX         = g++
  F77         = g77 
  CFLAGS      = -O3 -fPIC -fno-common -fexceptions \
                -no-cpp-precomp 
  FFLAGS      = -O3 -x f77-cpp-input -fPIC -fno-common

On my Linux machine, I set the variables in the Makefile like so:

  MEX         = mex
  MEXSUFFIX   = mexglx
  MATLAB_HOME = /cs/local/generic/lib/pkg/matlab-7.2
  CXX         = g++-3.4.5
  F77         = g77-3.4.5
  CFLAGS      = -O3 -fPIC -pthread 
  FFLAGS      = -O3 -fPIC -fexceptions

It may be helpful to look at the GCC documentation in order to understand what these various compiler flags mean.

Build the MEX file. If you are in the directory containing all the source files, typing make in the command prompt will first compile the Fortran and C++ source files into object code (.o files). After that, the make program calls the MEX script, which in turn links all the object files together into a single MEX file. If you didn't get any errors, then you are ready to try out the bound-constrained solver in MATLAB. Note that even if you didn't get any errors, there's still a possibility that you didn't link the MEX file properly, in which case executing the MEX file will cause MATLAB to crash. But there's only one way to find out: the hard way.

A brief tutorial

I've written a short script examplehs038.m which demonstrates how to call lbfgsb for solving a small optimization problem, the Hock & Schittkowski test problem no. 38.3 The optimization problem has four variables. They are bounded from below by -10 and bounded from above by +10. We've set the starting point to (-3, -1, -3, -1). We've written two MATLAB functions for computing the objective function and the gradient of the objective function at a given point. These functions can be found in the files computeObjectiveHS038.m and computeGradientHS038.m. If you run the script, the solver should very quickly progress toward the point (1, 1, 1, 1). The objective is 0 at this point.

The basic MATLAB function call is

  lbfgsb(x0,lb,ub,objfunc,gradfunc)

The first input argument x0 declares the starting point for the solver. The second and third input arguments define the lower and upper bounds on the variables, respectively. If an entry of lb is set to -Inf, then this is equivalent to no lower bound for that entry (the same applies to upper bounds, except that it must be +Inf instead). The next two input arguments must be the names of MATLAB routines (M-files). The first routine must calculate the objective function at the current point. The output must be a scalar representing the objective evaluated at the current point. The second function must compute the gradient of the objective at the current point. The input is the same as objfunc, but it must return as many outputs as there are inputs. For a complete guide on using the MATLAB interface, type help lbfgsb in the MATLAB prompt.

The script exampleldaimages.m is a more complicated example. It uses the L-BFGS-B solver to compute some statistics of a posterior distribution for a text analysis model called latent Dirichlet allication (LDA). The model is used to estimate the topics for a collection of documents. The optimization problem is to compute an approximation to the posterior, since it is not possible to compute the posterior exactly. In this case, the optimization problem corresponds to minimizing the distance between the true posterior distribution and an approximate distribution. For more information, see the paper of Blei, Ng and Jordan (2003).4 In order to run this example, you will need to install the lightspeed toolbox for MATLAB developed by Tom Minka at Microsoft Research.

The script starts by generating a bunch of canonical images that represent topics; if a pixel is white then a word at that position is more likely to appear in that topic. Note that the order of the pixels in the image is unimportant, since LDA is a "bag of words" model. In another figure we show a small sample of the data generated from the topics. Each image represents a document, and the pixels in the image depict the word proportions in that document.5

Next, the script runs the L-BFGS-B solver to find a local minimum to the variational objective. Buried in the M-file mflda.m is the call to the solver: 

  [xi gamma Phi] = lbfgsb({xi gamma Phi},...
      {repmat(lb,W,K)  repmat(lb,K,D)  repmat(lb,K,M)},...
      {repmat(inf,W,K) repmat(inf,K,D) repmat(inf,K,M)},...
      'computeObjectiveMFLDA','computeGradientMFLDA',...
      {nu eta L w lnZconst},'callbackMFLDA',...
      'm',m_lbfgs,'factr',1e-12,'pgtol',...
      tolerance,'maxiter',maxiter);

This example will give us the opportunity to demonstrate some of the more complicated aspects of the our MATLAB interface. The first input to lbfgsb sets the initial iterate of the optimization algorithm. Notice that we've passed a cell array with entries that are matrices. Indeed, it is possible to pass either a matrix or a cell array. Whatever structure is passed, the other input arguments (and outputs) must also abide by this structure. The lower and upper bounds on variables are also cell arrays. All the upper bounds are set to infinity, which means that the variables are only bounded from below.

It is instructive to examine the callback routines. They look like: 

  f = computeObjectiveMFLDA(xi,gamma,Phi,auxdata)

  [dxi, dgamma, dPhi] = ...
      computeGradientMFLDA(xi,gamma,Phi,auxdata)

Notice that each entry to the cell array is its own input argument to the callback routines. The same applies for the output arguments of the gradient callback function.

The sixth input argument is a cell array that contains auxiliary data. This data will be passed to the MATLAB functions that evaluate the objective and gradient. The seventh input argument specifies a callback routine that is called exactly once for every iteration. This callback routine is most useful for examining the progress of the solver. The remaining input arguments are label/value pairs that override some of the default settings of L-BFGS-B.

After converging to a solution, we display topic samples drawn from the variational approximation. A good solution should come close to recovering the true topics from which the data was generated (although they might not be in the same order).

Don't be surprised if it takes many iterations before the optimization algorithm converges on a local minimum. (In repeated trials, I found it as many as two thousand iterations to converge to a stationary point of the objective.) This particular problem demonstrates the limitations of limited-memory quasi-Newton approximations to the Hessian; the low storage requirements can come at the cost of slow convergence to the solution.

The C++ interface

One of the nice byproducts of writing a MATLAB interface for L-BFGS-B is that I've ended up with a neat little C++ class that encapsulates the nuts and bolts of executing the solver. A brief description of the Program class can be found in the header file program.h.

Program is an abstract class, which means that its impossible to instantiate a Program object. This is because the class has member functions that aren't yet implemented. These are called pure virtual functions. In order to use the Program class, one needs to define another class that inherits the Program class and that implements the pure virtual functions. In fact, the class MatlabProgram (in matlabprogram.h) is an example of such a class.

The new child class must declare and implement these two functions:

  virtual double computeObjective (int n, double* x);
  virtual void computeGradient (int n, double* x, double* g);

See the header file for a detailed description of these functions. The only remaining detail is calling the Program constructor. After that, it is just a matter of declaring a new object of type MyProgram then calling the function runSolver.

Contents of the tar archive

solver.f
Fortran 77 source for the L-BFGS-B solver routines.

program.h
program.cpp
Header and source file for the C++ interface.

array.h
arrayofmatrices.h
matlabexception.h
matlabmatrix.h
matlabprogram.h
matlabscalar.h
matlabstring.h
arrayofmatrices.cpp
matlabexception.cpp
matlabmatrix.cpp
matlabprogram.cpp
matlabscalar.cpp
matlabstring.cpp
lbfgsb.cpp
Header and source files for the MATLAB interface.

lbfgsb.m
Usage instructions for the MEX file.

examplehs038.m
MATLAB script for solving the H&S test example no. 38.

computeObjectiveHS038.m
computeGradientHS038.m
genericcallback.m
MATLAB functions used by the script examplehs038.m.

exampleldaimages.m
MATLAB script that generates synthetic documents and topics, then computes a mean field variational approximation to the posterior distribution of the latent Dirichlet allocation model, then displays the result.

mflda.m
This function computes a mean field variational approximation to the posterior distribution of the latent Dirichlet allocation model by minimizing the distance between the variational distribution and the true distribution. It uses the L-BFGS-B to minimize the objective function subject to the bound constraints. The objective function also acts as a lower bound on the logarithm of the denominator that appears from the application of Bayes' rule.4

callbackMFLDA.m
computeObjectiveMFLDA.m
computeGradientMFLDA.m
computelnZ.m
computelnZconst.m
computem.m
dirichletrnd.m
gammarnd.m
createsyntheticdata.m
mfldainit.m
Some functions used by exampleldaimages.m and mflda.m.

Footnotes

1 Ciyou Zhu, Richard H. Byrd, Peihuang Lu and Jorge Nocedal (1997). Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization. ACM Transactions on Mathematical Software, Vol. 23, No. 4., pp. 550-560.

1 Another MATLAB interface for the L-BFGS-B routines has been developed by Liam Stewart at the University of Toronto.

3 Willi Hock and Klaus Schittkowski (1981). Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems, Vol. 187, Springer-Verlag.

4 David M. Blei, Andrew Y. Ng, Michael I. Jordan (2003). Latent Dirichlet allocation. Journal of Machine Learning Research, Vol. 3, pp. 993-1022.

5 Tom L. Griffiths and Mark Steyvers (2004). Finding scientific topics. Proceedings of the National Academy of Sciences, Vol. 101, pp. 5228-5235.


March 3, 2007
================================================ FILE: code/gpml/util/lbfgsb/array.h ================================================ #ifndef INCLUDE_ARRAY #define INCLUDE_ARRAY #include "matlabexception.h" #include // Function definitions. // ----------------------------------------------------------------- template void copymemory (const Type* source, Type* dest, int length) { memcpy(dest,source,sizeof(Type)*length); } // Class Array // ----------------------------------------------------------------- // An Array object stores a collection of ordered elements. The key // aspect of Array objects is that they do not necessarily achieve // encapsulation; they do not necessarily retain independent // ownership of their data. That is, the data could be modified // externally by another agent. This behaviour is determined by the // data member "owner". If "owner" is false, the array does not take // care of allocation and deallocation of the data in memory. // // Copy assignment behaves different than usual---it copies the // data, but requires that the destination already have the proper // resources allocated in memory. // // The copy constructor performs a shallow copy. template class Array { public: // This constructor allocates memory for an array of elements of // the specificed length. explicit Array (int length); // This constructor is set to point to an already existing // array. As such, it does not gain ownership of the elements. Array (Type* data, int length); // The copy constructor performs a shallow copy, so that both // arrays share the same copy of the data. Array (const Array& source); // The destructor. ~Array(); // Set all the elements of the array to the specified value. void setvalue (const Type& value); // Copy the elements to from the source array. Both the source and // destination array must have the same length. void inject (const Type* source); void inject (const Array& source); Array& operator= (const Type* source); Array& operator= (const Array& source); // Get the number of elements in the array. int length() const { return n; }; // Elements of an array A can be accessed the subscript operator; // e.g. A[i]. For the sake of efficiency, no checks are made to // ensure that the subscripting is legal. It is up to the user to // ensure that i is non-negative and less than A.length(). Note // that subscripts, as per C++ convention, start at 0. Type& operator[] (int i); const Type& operator[] (int i) const; // Returns true if the two arrays have the same dimensions // (i.e. the same lengths). bool operator== (const Array& a) const { return n == a.n; }; bool operator!= (const Array& a) const { return !(*this == a); }; protected: Type* elems; int n; // The number of elements in the array. bool owner; // Whether the object has ownership of the data. }; // Function definitions for class Array. // ----------------------------------------------------------------- template Array::Array (int length) { n = length; owner = true; elems = 0; elems = new Type[length]; } template Array::Array (Type* data, int length) { n = length; owner = false; elems = data; } template Array::Array (const Array& source) { n = source.n; owner = false; elems = source.elems; } template Array::~Array () { if (owner && elems) delete[] elems; } template void Array::setvalue (const Type& value) { for (int i = 0; i < n; i++) elems[i] = value; } template void Array::inject (const Type* source) { // Check for self-assignment. If there is self-assignment, there // is no need to copy because the objects share the same data! if (elems != source) copymemory(source,elems,n); } template void Array::inject (const Array& source) { if (n != source.n) throw MatlabException("Unable to perform copy; arrays have \ different lengths"); inject(source.elems); } template Array& Array::operator= (const Type* source) { inject(source); return *this; } template Array& Array::operator= (const Array& source) { inject(source); return *this; } template Type& Array::operator[] (int i) { return elems[i]; } template const Type& Array::operator[] (int i) const { return elems[i]; } #endif ================================================ FILE: code/gpml/util/lbfgsb/arrayofmatrices.cpp ================================================ #include "arrayofmatrices.h" #include "matrix.h" // Function definitions for class ArrayOfMatrices. // ----------------------------------------------------------------- ArrayOfMatrices::ArrayOfMatrices (const mxArray* ptr) : Array(getnummatlabmatrices(ptr)) { setvalue(0); if (mxIsCell(ptr)) // Fill in the entries of the array with the entries of the Matlab // cell array. for (int i = 0; i < n; i++) elems[i] = new Matrix(mxGetCell(ptr,i)); else // Fill in the single entry of the array with the Matlab matrix. elems[0] = new Matrix(ptr); } ArrayOfMatrices::ArrayOfMatrices (const mxArray* ptrs[], int numptrs) : Array(numptrs) { setvalue(0); for (int i = 0; i < numptrs; i++) elems[i] = new Matrix(ptrs[i]); } ArrayOfMatrices::ArrayOfMatrices (mxArray* ptrs[], const ArrayOfMatrices& source) : Array(source.length()) { setvalue(0); // Create the new Matlab matrices and copy the data from the // source. for (int i = 0; i < n; i++) { int h = source[i]->height(); int w = source[i]->width(); elems[i] = new Matrix(ptrs[i],h,w); *elems[i] = *source[i]; } } ArrayOfMatrices::ArrayOfMatrices (const ArrayOfMatrices& source) : Array(source) { } ArrayOfMatrices::ArrayOfMatrices (double* data, const ArrayOfMatrices& model) : Array(model.length()) { setvalue(0); // Repeat for each matrix. for (int i = 0; i < n; i++) { const Matrix* A = model[i]; elems[i] = new Matrix(data,A->height(),A->width()); data += A->length(); } } ArrayOfMatrices::~ArrayOfMatrices() { for (int i = 0; i < n; i++) if (elems[i]) delete elems[i]; } int ArrayOfMatrices::numelems() const { int m = 0; // The return value. // Repeat for each matrix. for (int i = 0; i < n; i++) m += elems[i]->length(); return m; } ArrayOfMatrices& ArrayOfMatrices::operator= (const ArrayOfMatrices& source) { // Copy each matrix. for (int i = 0; i < n; i++) *elems[i] = *source.elems[i]; return *this; } int ArrayOfMatrices::getnummatlabmatrices (const mxArray* ptr) { int n; // The return value. if (mxIsCell(ptr)) n = mxGetNumberOfElements(ptr); else if (mxGetNumberOfDimensions(ptr) == 2 && mxIsDouble(ptr)) n = 1; else throw MatlabException("The Matlab array must either be a cell \ array or a double precision matrix"); return n; } ================================================ FILE: code/gpml/util/lbfgsb/arrayofmatrices.h ================================================ #ifndef INCLUDE_ARRAYOFMATRICES #define INCLUDE_ARRAYOFMATRICES #include "array.h" #include "matlabmatrix.h" #include "mex.h" // Class ArrayOfMatrices. // ----------------------------------------------------------------- class ArrayOfMatrices : public Array { public: // This version of the constructor behaves just like its parent. explicit ArrayOfMatrices (int length) : Array(length) { }; // This constructor creates an array of matrices from a Matlab // array. It accepts either a matrix in double precision, or a cell // array with entries that are matrices. explicit ArrayOfMatrices (const mxArray* ptr); // This constructor creates an array of matrices from a collection // of Matlab arrays. The Matlab arrays must be matrices. ArrayOfMatrices (const mxArray* ptrs[], int numptrs); // This constructor creates an array of matrices and the // associated Matlab structures. The Matlab structures are // matrices. The second input argument acts as a template for the // creation of the matrices, but the data from "model" is not // actually copied into the new ArrayOfMatrices object. It is up // to the user to make sure that the array of mxArray pointers has // enough room for the pointers to the Matlab arrays. ArrayOfMatrices (mxArray* ptrs[], const ArrayOfMatrices& model); // This constructor creates an array of matrices using the second // input argument as a model. The input argument "data" contains // the element data. Note that the information is NOT copied from // the model! ArrayOfMatrices (double* data, const ArrayOfMatrices& model); // The copy constructor makes a shallow copy of the source object. ArrayOfMatrices (const ArrayOfMatrices& source); // The destructor. ~ArrayOfMatrices(); // Copy assignment operator that observes the same behaviour as // the Array copy assignment operator. ArrayOfMatrices& operator= (const ArrayOfMatrices& source); // Return the total number of elements in all the matrices. int numelems() const; protected: static int getnummatlabmatrices (const mxArray* ptr); }; #endif ================================================ FILE: code/gpml/util/lbfgsb/lbfgsb.cpp ================================================ #include "mex.h" #include "matrix.h" #include "matlabexception.h" #include "matlabscalar.h" #include "matlabstring.h" #include "matlabmatrix.h" #include "arrayofmatrices.h" #include "program.h" #include "matlabprogram.h" #include #include extern void _main(); // Constants. // ----------------------------------------------------------------- const int minNumInputArgs = 5; // Function definitions. // ----------------------------------------------------------------- void mexFunction (int nlhs, mxArray *plhs[], int nrhs, const mxArray *prhs[]) try { // Check to see if we have the correct number of input and output // arguments. if (nrhs < minNumInputArgs) throw MatlabException("Incorrect number of input arguments"); // Get the starting point for the variables. This is specified in // the first input argument. The variables must be either a single // matrix or a cell array of matrices. int k = 0; // The index of the current input argument. ArrayOfMatrices x0(prhs[k++]); // Create the output, which stores the solution obtained from // running IPOPT. There should be as many output arguments as cell // entries in X. if (nlhs != x0.length()) throw MatlabException("Incorrect number of output arguments"); ArrayOfMatrices x(plhs,x0); // Load the lower and upper bounds on the variables as // ArrayOfMatrices objects. They should have the same structure as // the ArrayOfMatrices object "x". ArrayOfMatrices lb(prhs[k++]); ArrayOfMatrices ub(prhs[k++]); // Check to make sure the bounds make sense. if (lb != x || ub != x) throw MatlabException("Input arguments LB and UB must have the same \ structure as X"); // Get the Matlab callback functions. MatlabString objFunc(prhs[k++]); MatlabString gradFunc(prhs[k++]); // Get the auxiliary data. const mxArray* auxData; const mxArray* ptr = prhs[k++]; if (nrhs > 5) { if (mxIsEmpty(ptr)) auxData = 0; else auxData = ptr; } else auxData = 0; // Get the intermediate callback function. MatlabString* iterFunc; ptr = prhs[k++]; if (nrhs > 6) { if (mxIsEmpty(ptr)) iterFunc = 0; else iterFunc = new MatlabString(ptr); } else iterFunc = 0; // Set the options for the L-BFGS algorithm to their defaults. int maxiter = defaultmaxiter; int m = defaultm; double factr = defaultfactr; double pgtol = defaultpgtol; // Process the remaining input arguments, which set options for // the IPOPT algorithm. while (k < nrhs) { // Get the option label from the Matlab input argument. MatlabString optionLabel(prhs[k++]); if (k < nrhs) { // Get the option value from the Matlab input argument. MatlabScalar optionValue(prhs[k++]); double value = optionValue; if (!strcmp(optionLabel,"maxiter")) maxiter = (int) value; else if (!strcmp(optionLabel,"m")) m = (int) value; else if (!strcmp(optionLabel,"factr")) factr = value / mxGetEps(); else if (!strcmp(optionLabel,"pgtol")) pgtol = value; else { if (iterFunc) delete iterFunc; throw MatlabException("Nonexistent option"); } } } // Create a new instance of the optimization problem. x = x0; MatlabProgram program(x,lb,ub,&objFunc,&gradFunc,iterFunc, (mxArray*) auxData,m,maxiter,factr,pgtol); // Run the L-BFGS-B solver. SolverExitStatus exitStatus = program.runSolver(); if (exitStatus == abnormalTermination) { if (iterFunc) delete iterFunc; throw MatlabException("Solver unable to satisfy convergence \ criteria due to abnormal termination"); } else if (exitStatus == errorOnInput) { if (iterFunc) delete iterFunc; throw MatlabException("Invalid inputs to L-BFGS routine"); } // Free the dynamically allocated memory. if (iterFunc) delete iterFunc; } catch (std::exception& error) { mexErrMsgTxt(error.what()); } ================================================ FILE: code/gpml/util/lbfgsb/license.txt ================================================ Copyright (c) 2009, Peter Carbonetto All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: * Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. * Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution * Neither the name of the University of British Columbia nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. ================================================ FILE: code/gpml/util/lbfgsb/matlabexception.cpp ================================================ #include "matlabexception.h" // Function definitions for class MatlabException. // ----------------------------------------------------------------- MatlabException::MatlabException (const char* message) throw() : exception() { this->message = message; } MatlabException::MatlabException (const MatlabException& source) throw() : exception() { message = source.message; } MatlabException& MatlabException::operator= (const MatlabException& source) { message = source.message; return *this; } ================================================ FILE: code/gpml/util/lbfgsb/matlabexception.h ================================================ #ifndef INCLUDE_MATLABEXCEPTION #define INCLUDE_MATLABEXCEPTION #include // Class MatlabException // ----------------------------------------------------------------- // This class just makes it easier for me to throw exceptions. Its // functionality really has nothing to do with MATLAB. class MatlabException : public std::exception { public: MatlabException (const char* message) throw(); ~MatlabException() throw() { }; // The copy constructor makes a shallow copy. MatlabException (const MatlabException& source) throw(); // The copy assignment operator makes a shallow copy as well. MatlabException& operator= (const MatlabException& source); // Return the message string. virtual const char* what () const throw() { return message; }; private: const char* message; // The error message. }; #endif ================================================ FILE: code/gpml/util/lbfgsb/matlabmatrix.cpp ================================================ #include "matlabmatrix.h" #include "matlabexception.h" // Function definitions. // ----------------------------------------------------------------- double* getMatlabMatrixDouble (const mxArray* ptr) { if (mxGetNumberOfDimensions(ptr) != 2) throw MatlabException("Matlab array must be a matrix"); if (!mxIsDouble(ptr)) throw MatlabException("Matlab array must be of type double"); return mxGetPr(ptr); } double* createMatlabMatrixDouble (mxArray*& ptr, int height, int width) { ptr = mxCreateDoubleMatrix(height,width,mxREAL); return mxGetPr(ptr); } // Function definitions for class Matrix. // ----------------------------------------------------------------- Matrix::Matrix (int height, int width) : Array(height*width) { h = height; w = width; } Matrix::Matrix (double* data, int height, int width) : Array(data,height*width) { h = height; w = width; } Matrix::Matrix (const mxArray* ptr) : Array(getMatlabMatrixDouble(ptr),mxGetNumberOfElements(ptr)) { h = mxGetM(ptr); w = mxGetN(ptr); } Matrix::Matrix (mxArray*& ptr, int height, int width) : Array(createMatlabMatrixDouble(ptr,height,width), height*width) { h = height; w = width; } Matrix::Matrix (const Matrix& source) : Array(source) { h = source.h; w = source.w; } Matrix& Matrix::operator= (const Matrix& source) { inject(source); return *this; } bool Matrix::operator== (const Matrix& X) const { return (h == X.h) && (w == X.w); } double& Matrix::entry (int r, int c) { return elems[h*c + r]; } double Matrix::entry (int r, int c) const { return elems[h*c + r]; } ================================================ FILE: code/gpml/util/lbfgsb/matlabmatrix.h ================================================ #ifndef INCLUDE_MATLABMATRIX #define INCLUDE_MATLABMATRIX #include "array.h" #include "mex.h" // Class Matrix // --------------------------------------------------------------- // A matrix object stores its elements in column-major format, as in // Fortran and Matlab. This means that columns are stored one after // another. For example, the matrix // // 1 2 3 // 4 5 6 // // is stored in memory as // // 1 4 2 5 3 6 // // Like Array objects, Matrix objects are not necessarily // encapsulated, and they exhibit analogous behaviour. class Matrix : public Array { public: // This constructor allocates memory for matrix of the specified // heigth and width. Matrix (int height, int width); // This constructor basically follows the lead of the analagous // constructor for the Array class. Matrix (double* data, int height, int width); // This constructor retrieves a matrix from a Matlab array. This // particular constructor is only defined for Matrix. // Since Matlab handles storage, the object created by this // constructor is not encapsulated. explicit Matrix (const mxArray* ptr); // This constructor creates a new Matlab array as a side effect. // Since Matlab handles storage, the object created by this // constructor is not encapsulated. Matrix (mxArray*& ptr, int height, int width); // The copy constructor makes a shallow copy of the data. Matrix (const Matrix& source); // The destructor. ~Matrix() { }; // Copy assignment operator that observes the same behaviour as // the Array copy assignment operator. Matrix& operator= (const Matrix& source); // Get the height and width of the matrix. int height() const { return h; }; int width () const { return w; }; // Returns true if the two matrices have the same dimensions // (i.e. the same height and width). bool operator== (const Matrix& X) const; bool operator!= (const Matrix& X) const { return !(*this == X); }; // If X is an object of type Matrix, X.entry(r,c) accesses the // entry of the rth row and cth column. double entry (int r, int c) const; double& entry (int r, int c); double operator() (int r, int c) const { return entry(r,c); }; double& operator() (int r, int c) { return entry(r,c); }; protected: int h; // The height of the matrix. int w; // The width of the matrix. }; #endif ================================================ FILE: code/gpml/util/lbfgsb/matlabprogram.cpp ================================================ #include "matlabprogram.h" #include "array.h" // Function definitions. // ----------------------------------------------------------------- // Copy the matrix elements to the destination array. It is assumed // that sufficient memory has been allocated for the destination // array. The copying preserves the column-major ordering of the // matrix elements. void copyElemsFromMatrix (const Matrix& source, double* dest) { Matrix destmatrix(dest,source.height(),source.width()); destmatrix = source; } // Copy the elements from the source array into the matrix. void copyElemsToMatrix (const double* source, Matrix& dest) { dest.inject(source); } // Copy the elements contained in all the matrices to the destination // array. It is assumed that sufficient memory has been allocated for // the destination array. The copying preserves the column-major // ordering of the matrix elements. void copyElemsFromArrayOfMatrices (const ArrayOfMatrices& source, double* dest) { // Repeat for each matrix. for (int i = 0; i < source.length(); i++) { copyElemsFromMatrix(*source[i],dest); dest += source[i]->length(); } } // Copy the elements from the source array into the array of matrices. void copyElemsToArrayOfMatrices (const double* source, ArrayOfMatrices& dest) { // Repeat for each matrix. for (int i = 0; i < dest.length(); i++) { copyElemsToMatrix(source,*dest[i]); source += dest[i]->length(); } } int getBoundType (double& lb, double& ub) { bool haslb = !mxIsInf(lb); bool hasub = !mxIsInf(ub); int btype = haslb + 3*hasub - 2*(haslb && hasub); lb = haslb ? lb : 0; ub = hasub ? ub : 0; return btype; } // Function definitions for class MatlabProgram. // ----------------------------------------------------------------- MatlabProgram::MatlabProgram (ArrayOfMatrices& variables, const ArrayOfMatrices& lowerbounds, const ArrayOfMatrices& upperbounds, const MatlabString* objFunc, const MatlabString* gradFunc, const MatlabString* iterFunc, mxArray* auxData, int m, int maxiter, double factr, double pgtol) : Program(variables.numelems(),0,0,0,0,m,maxiter,factr,pgtol), variables(variables) { x = new double[n]; lb = new double[n]; ub = new double[n]; btype = new int[n]; this->objFunc = objFunc; this->gradFunc = gradFunc; this->iterFunc = iterFunc; // Copy some pieces of information from the input arguments to the // respective program structures. copyElemsFromArrayOfMatrices(variables,x); copyElemsFromArrayOfMatrices(lowerbounds,lb); copyElemsFromArrayOfMatrices(upperbounds,ub); // Set the bound types. for (int i = 0; i < n; i++) btype[i] = getBoundType(lb[i],ub[i]); // Create a Matlab array for the variables. int nv = variables.length(); varInputs = new mxArray*[nv]; varMatlab = new ArrayOfMatrices(varInputs,variables); // Set up the inputs to the objective callback function. numInputsObjFunc = nv + (bool) auxData; inputsObjFunc = new mxArray*[numInputsObjFunc]; copymemory(varInputs,inputsObjFunc,nv); if (auxData) inputsObjFunc[nv] = auxData; // Set up the inputs to the gradient callback function. numInputsGradFunc = numInputsObjFunc; inputsGradFunc = inputsObjFunc; // If necessary, set up the inputs to the iterative callback // function. The first two inputs are the iteration and the function // objective. The next of inputs are the values of the variables // then, optionally, the auxiliary data. if (iterFunc) { numInputsIterFunc = 2 + nv + (bool) auxData; inputsIterFunc = new mxArray*[numInputsIterFunc]; mxArray** ptr = inputsIterFunc; tMatlab = new MatlabScalar(*ptr++,0); fMatlab = new MatlabScalar(*ptr++,0); copymemory(varInputs,ptr,nv); ptr += nv; if (auxData) *ptr = auxData; } } MatlabProgram::~MatlabProgram() { delete[] x; delete[] lb; delete[] ub; delete[] btype; // Deallocate the Matlab arrays. int nv = variables.length(); for (int i = 0; i < nv; i++) mxDestroyArray(varInputs[i]); delete varMatlab; delete[] varInputs; // If necessary, deallocate the additional Matlab arrays that act as // inputs to the iterative callback function. if (iterFunc) { delete fMatlab; delete tMatlab; // delete[] inputsIterFunc; } } double MatlabProgram::computeObjective (int n, double* x) { int nlhs = 1; // The number of outputs from Matlab. mxArray* plhs[nlhs]; // The outputs from the Matlab routine. // Copy the current value of the optimization variables. copyElemsToArrayOfMatrices(x,*varMatlab); // Call the designated Matlab routine for evaluating the objective // function. It takes as input the values of the variables, and // returns a single output, the value of the objective function. if (mexCallMATLAB(nlhs,plhs,numInputsObjFunc,inputsObjFunc,*objFunc)) throw MatlabException("Evaluation of objective function failed in \ call to MATLAB routine"); // Get the result passed back from the Matlab routine. MatlabScalar matlabOutput(plhs[0]); double objective = matlabOutput; // Free the dynamically allocated memory. mxDestroyArray(plhs[0]); return objective; } void MatlabProgram::computeGradient (int n, double* x, double* g) { int nlhs = variables.length(); // The number of outputs from Matlab. mxArray* plhs[nlhs]; // The outputs from the Matlab routine. // Copy the current value of the optimization variables. copyElemsToArrayOfMatrices(x,*varMatlab); // Call the designated Matlab routine for computing the gradient // of the objective function. It takes as input the values of the // variables, and returns as many outputs corresponding to the // partial derivatives of the objective function with respect to // the variables at their curent values. if (mexCallMATLAB(nlhs,plhs,numInputsGradFunc,inputsGradFunc,*gradFunc)) throw MatlabException("Evaluation of objective gradient failed in \ call to MATLAB routine"); // Get the result passed back from the Matlab routine. ArrayOfMatrices matlabOutput((const mxArray**) plhs,nlhs); for (int i = 0; i < variables.length(); i++) if (matlabOutput[i]->length() != variables[i]->length()) throw MatlabException("Invalid gradient passed back from MATLAB \ routine"); copyElemsFromArrayOfMatrices(matlabOutput,g); // Free the dynamically allocated Matlab outputs. for (int i = 0; i < variables.length(); i++) mxDestroyArray(plhs[i]); } void MatlabProgram::iterCallback (int t, double* x, double f) { if (iterFunc) { // Copy the current iteration, the current value of the objective // function, and the current value of the optimization variables. *tMatlab = t; *fMatlab = f; copyElemsToArrayOfMatrices(x,*varMatlab); // Call the Matlab iterative callback routine. Since there are no // outputs, there is no memory to deallocate. if (mexCallMATLAB(0,0,numInputsIterFunc,inputsIterFunc,*iterFunc)) throw MatlabException("Call to MATLAB iterative callback function \ failed"); } } SolverExitStatus MatlabProgram::runSolver() { SolverExitStatus status; // The return value. status = Program::runSolver(); // Copy the solution to the class member "variables". copyElemsToArrayOfMatrices(x,variables); return status; } ================================================ FILE: code/gpml/util/lbfgsb/matlabprogram.h ================================================ #ifndef INCLUDE_MATLABPROGRAM #define INCLUDE_MATLABPROGRAM #include "mex.h" #include "program.h" #include "matlabscalar.h" #include "matlabstring.h" #include "arrayofmatrices.h" // Class MatlabProgram. // ----------------------------------------------------------------- // This is an implementation of the abstract class Program. class MatlabProgram: public Program { public: // On input, "variables" should contain the initial point for the // optimization routine. If no iterative callback function is // specified, "iterFunc" may be set to 0. Also, if no auxiliary data // is needed, it may also be set to 0. MatlabProgram (ArrayOfMatrices& variables, const ArrayOfMatrices& lowerbounds, const ArrayOfMatrices& upperbounds, const MatlabString* objFunc, const MatlabString* gradFunc, const MatlabString* iterFunc, mxArray* auxData, int m = defaultm, int maxiter = defaultmaxiter, double factr = defaultfactr, double pgtol = defaultpgtol); // The destructor. virtual ~MatlabProgram(); // These provide definitions for the pure virtual functions of the // abstract parent class. virtual double computeObjective (int n, double* x); virtual void computeGradient (int n, double* x, double* g); virtual void iterCallback (int t, double* x, double f); // Run the solver. Upon completion, the solution is stored in // "variables". SolverExitStatus runSolver(); protected: ArrayOfMatrices& variables; // Storage for the initial value and // solution. const MatlabString* objFunc; // The objective callback function. const MatlabString* gradFunc; // The gradient callback function. const MatlabString* iterFunc; // The iterative callback function. ArrayOfMatrices* varMatlab; // Inputs to the Matlab callback mxArray** varInputs; // functions representing the // current values of the variables. MatlabScalar* fMatlab; // Input to the Matlab callback // functions representing the // current value of the objective. MatlabScalar* tMatlab; // Input to the Matlab callback // functions representing the // current iteration. int numInputsObjFunc; // The number of inputs passed to the // objective callback function. int numInputsGradFunc; // The number of inputs passed to the // gradient callback function. int numInputsIterFunc; // The number of inputs passed to the // iterative callback function. mxArray** inputsObjFunc; // The inputs passed to the objective // callback function. mxArray** inputsGradFunc; // The inputs passed to the gradient // callback function. mxArray** inputsIterFunc; // The inputs passed to the iterative // callback function. }; #endif ================================================ FILE: code/gpml/util/lbfgsb/matlabscalar.cpp ================================================ #include "matlabscalar.h" #include "matlabexception.h" // Function definitions. // ----------------------------------------------------------------- double& getMatlabScalar (const mxArray* ptr) { if (!mxIsDouble(ptr)) throw MatlabException("Matlab array must be of type double"); if (mxGetNumberOfElements(ptr) != 1) throw MatlabException("The Matlab array must be a scalar"); return *mxGetPr(ptr); } double& createMatlabScalar (mxArray*& ptr) { ptr = mxCreateDoubleScalar(0); return *mxGetPr(ptr); } // Function definitions for class MatlabScalar. // --------------------------------------------------------------- MatlabScalar::MatlabScalar (const mxArray* ptr) : x(getMatlabScalar(ptr)) { } MatlabScalar::MatlabScalar (mxArray*& ptr, double value) : x(createMatlabScalar(ptr)) { x = value; } MatlabScalar::MatlabScalar (MatlabScalar& source) : x(source.x) { } MatlabScalar& MatlabScalar::operator= (double value) { x = value; return *this; } ================================================ FILE: code/gpml/util/lbfgsb/matlabscalar.h ================================================ #ifndef INCLUDE_MATLABSCALAR #define INCLUDE_MATLABSCALAR #include "mex.h" // Class MatlabScalar // ----------------------------------------------------------------- // The main appeal of this class is that one can create a scalar // object that accesses a MATLAB array. // // Note that the copy assignment operator is not valid for this class // because we cannot reassign a reference. class MatlabScalar { public: // This constructor accepts as input a pointer to a Matlab array // which must be a scalar in double precision. explicit MatlabScalar (const mxArray* ptr); // This constructor creates a new Matlab array which is a scalar // in double precision. MatlabScalar (mxArray*& ptr, double value); // The copy constructor. MatlabScalar (MatlabScalar& source); // The destructor. ~MatlabScalar() { }; // Access the value of the scalar. operator const double () const { return x; }; // Assign the value of the scalar. MatlabScalar& operator= (double value); protected: double& x; // The copy assignment operator is kept protected because it is // invalid. MatlabScalar& operator= (const MatlabScalar& source) { return *this; }; }; #endif ================================================ FILE: code/gpml/util/lbfgsb/matlabstring.cpp ================================================ #include "matlabstring.h" #include "matlabexception.h" // Function definitions. // ----------------------------------------------------------------- char* copystring (const char* source) { int n = strlen(source); // The length of the string. char* dest = new char[n+1]; // The return value. strcpy(dest,source); return dest; } // Function definitions for class MatlabString. // ----------------------------------------------------------------- MatlabString::MatlabString (const mxArray* ptr) { s = 0; // Check to make sure the Matlab array is a string. if (!mxIsChar(ptr)) throw MatlabException("Matlab array must be a string (of type CHAR)"); // Get the string passed as a Matlab array. s = mxArrayToString(ptr); if (s == 0) throw MatlabException("Unable to obtain string from Matlab array"); } MatlabString::MatlabString (const MatlabString& source) { s = 0; // Copy the source string. s = copystring(source.s); } MatlabString::~MatlabString() { if (s) mxFree(s); } ================================================ FILE: code/gpml/util/lbfgsb/matlabstring.h ================================================ #ifndef INCLUDE_MATLABSTRING #define INCLUDE_MATLABSTRING #include "mex.h" #include // Function declarations. // ----------------------------------------------------------------- // Copy a C-style string (i.e. a null-terminated character array). char* copystring (const char* source); // Class MatlabString. // ----------------------------------------------------------------- // This class encapsulates read-only access to a MATLAB character // array. class MatlabString { public: // The constructor accepts as input a pointer to a Matlab array, // which must be a valid string (array of type CHAR). explicit MatlabString (const mxArray* ptr); // The copy constructor makes a full copy of the source string. MatlabString (const MatlabString& source); // The destructor. ~MatlabString(); // Conversion operator for null-terminated string. operator const char* () const { return s; }; protected: char* s; // The null-terminated string. // The copy assignment operator is not proper, thus remains // protected. MatlabString& operator= (const MatlabString& source) { return *this; }; }; #endif ================================================ FILE: code/gpml/util/lbfgsb/program.cpp ================================================ #include "program.h" #include // Function definitions. // ----------------------------------------------------------------- // Copy a C-style string (a null-terminated character array) to a // non-C-style string (a simple character array). The length of the // destination character array is given by "ndest". If the source is // shorter than the destination, the destination is padded with blank // characters. void copyCStrToCharArray (const char* source, char* dest, int ndest) { // Get the length of the source C string. int nsource = strlen(source); // Only perform the copy if the source can fit into the destination. if (nsource < ndest) { // Copy the string. strcpy(dest,source); // Fill in the rest of the string with blanks. for (int i = nsource; i < ndest; i++) dest[i] = ' '; } } // Return true if the two strings are the same. The second input // argument must be a C-style string (a null-terminated character // array), but the first input argument need not be one. We only // compare the two strings up to the length of "cstr". bool strIsEqualToCStr (const char* str, const char* cstr) { // Get the length of the C string. int n = strlen(cstr); return !strncmp(str,cstr,n); } // Function definitions for class Program. // ----------------------------------------------------------------- Program::Program (int n, double* x, double* lb, double* ub, int* btype, int m, int maxiter, double factr, double pgtol) { this->n = n; this->x = x; this->lb = lb; this->ub = ub; this->btype = btype; this->m = m; this->maxiter = maxiter; this->factr = factr; this->pgtol = pgtol; this->iprint = defaultprintlevel; this->owner = false; initStructures(); } Program::Program (int n, int m, int maxiter, double factr, double pgtol) { this->n = n; this->x = new double[n]; this->lb = new double[n]; this->ub = new double[n]; this->btype = new int[n]; this->m = m; this->maxiter = maxiter; this->factr = factr; this->pgtol = pgtol; this->iprint = defaultprintlevel; this->owner = true; initStructures(); } Program::~Program() { if (owner) { delete[] x; delete[] lb; delete[] ub; delete[] btype; } delete[] g; delete[] wa; delete[] iwa; } SolverExitStatus Program::runSolver() { SolverExitStatus status = success; // The return value. // Initialize the objective function and gradient to zero. f = 0; for (int i = 0; i < n; i++) g[i] = 0; // This initial call sets up the structures for L-BFGS. callLBFGS("START"); // Repeat until we've reached the maximum number of iterations. int t = 0; while (true) { // Do something according to the "task" from the previous call to // L-BFGS. if (strIsEqualToCStr(task,"FG")) { // Evaluate the objective function and the gradient of the // objective at the current point. f = computeObjective(n,x); computeGradient(n,x,g); } else if (strIsEqualToCStr(task,"NEW_X")) { // Go to the next iteration and call the iterative callback // routine. t++; iterCallback(t,x,f); // If we've reached the maximum number of iterations, terminate // the optimization. if (t == maxiter) { callLBFGS("STOP"); break; } } else if (strIsEqualToCStr(task,"CONV")) break; else if (strIsEqualToCStr(task,"ABNO")) { status = abnormalTermination; break; } else if (strIsEqualToCStr(task,"ERROR")) { status = errorOnInput; break; } // Call L-BFGS again. callLBFGS(); } return status; } void Program::callLBFGS (const char* cmd) { if (cmd) copyCStrToCharArray(cmd,task,60); setulb_(&n,&m,x,lb,ub,btype,&f,g,&factr,&pgtol,wa,iwa,task,&iprint, csave,lsave,isave,dsave); } void Program::initStructures() { f = 0; g = new double[n]; wa = new double[(2*m + 4)*n + 12*m*(m + 1)]; iwa = new int[3*n]; } ================================================ FILE: code/gpml/util/lbfgsb/program.h ================================================ #ifndef INCLUDE_PROGRAM #define INCLUDE_PROGRAM // Type definitions. // ----------------------------------------------------------------- // This defines the possible results of running the L-BFGS-B solver. enum SolverExitStatus { success, // The algorithm has converged to a stationary // point or has reached the maximum number of // iterations. abnormalTermination, // The algorithm has terminated abnormally and // was unable to satisfy the convergence // criteria. errorOnInput // The routine has detected an error in the // input parameters. }; // Constants. // ----------------------------------------------------------------- const int defaultm = 5; const int defaultmaxiter = 100; const double defaultfactr = 1e7; const double defaultpgtol = 1e-5; const int defaultprintlevel = -1; // Function declarations. // ----------------------------------------------------------------- // This is the L-BFGS-B routine implemented in Fortran 77. extern "C" void setulb_ (int* n, int* m, double x[], double l[], double u[], int nbd[], double* f, double g[], double* factr, double* pgtol, double wa[], int iwa[], char task[], int* iprint, char csave[], bool lsave[], int isave[], double dsave[]); // Class Program. // ----------------------------------------------------------------- // This class encapsulates execution of the L-BFGS-B routine for // solving a nonlinear optimization problem with bound constraints // using limited-memory approximations to the Hessian. // // This is an abstract class since some of the class methods have not // been implemented (they are "pure virtual" functions). In order to // use this class, one needs to define a child class and provide // definitions for the pure virtual methods. class Program { public: // The first input argument "n" is the number of variables, and "x" // must be set to the suggested starting point for the optimization // algorithm. See the L-BFGS-B documentation for more information on // the inputs to the constructor. Program (int n, double* x, double* lb, double* ub, int* btype, int m = defaultm, int maxiter = defaultmaxiter, double factr = defaultfactr, double pgtol = defaultpgtol); // This is the same constructor as above, except that the // appropriate amount of memory is dynamically allocated for the // variables, the bounds, and the bound types. The destructor also // makes sure that the memory is properly deallocated, but not so // for the other constructor. Program (int n, int m = defaultm, int maxiter = defaultmaxiter, double factr = defaultfactr, double pgtol = defaultpgtol); // The destructor. virtual ~Program(); // An implementation of this method should return the value of the // objective at the current value of the variables "x", where "n" is // the number of variables. virtual double computeObjective (int n, double* x) = 0; // An implementation of this method should fill in the values of the // gradient "g" and the current point "x". virtual void computeGradient (int n, double* x, double* g) = 0; // The child class may optionally override this method which is // called once per iteration of the L-BFGS-B routine. The input // arguments are the current iteration "t", the current value of the // variables "x" and the current value of the objective "f". virtual void iterCallback (int t, double* x, double f) { }; // Run the solver. Upon completion, the solution is stored in "x". SolverExitStatus runSolver(); protected: // The copy constructor and copy assignment operator are kept // private so that they are not used. Program (const Program& source) { }; Program& operator= (const Program& source) { return *this; }; int n; // The number of variables. double* x; // The current point. double* lb; // The lower bounds. double* ub; // The upper bounds. int* btype; // The bound types. private: // Execute a single step the L-BFGS-B solver routine. void callLBFGS (const char* cmd = 0); // Initialize some of the structures used by the L-BFGS-B routine. void initStructures(); bool owner; // If true, then the blocks of memory pointed to by // x, lb, ub and btype must be deallocated in the // destructor. double f; // The value of the objective. double* g; // The value of the gradient. int iprint; // The print level. int maxiter; // The maximum number of iterations. double pgtol; // Convergence parameter passed to L-BFGS-B. double factr; // Convergence parameter passed to L-BFGS-B. int m; // The number of variable corrections to the // limited-memory approximation to the Hessian. // These are structures used by the L-BFGS-B routine. double* wa; int* iwa; char task[60]; char csave[60]; bool lsave[4]; int isave[44]; double dsave[29]; }; #endif ================================================ FILE: code/gpml/util/lbfgsb/solver.f ================================================ c================ L-BFGS-B (version 2.4) ========================== subroutine setulb(n, m, x, l, u, nbd, f, g, factr, pgtol, wa, iwa, + task, iprint, csave, lsave, isave, dsave) character*60 task, csave logical lsave(4) integer n, m, iprint, + nbd(n), iwa(3*n), isave(44) double precision f, factr, pgtol, x(n), l(n), u(n), g(n), + wa(2*m*n+4*n+11*m*m+8*m), dsave(29) c ************ c c Subroutine setulb c c This subroutine partitions the working arrays wa and iwa, and c then uses the limited memory BFGS method to solve the bound c constrained optimization problem by calling mainlb. c (The direct method will be used in the subspace minimization.) c c n is an integer variable. c On entry n is the dimension of the problem. c On exit n is unchanged. c c m is an integer variable. c On entry m is the maximum number of variable metric corrections c used to define the limited memory matrix. c On exit m is unchanged. c c x is a double precision array of dimension n. c On entry x is an approximation to the solution. c On exit x is the current approximation. c c l is a double precision array of dimension n. c On entry l is the lower bound on x. c On exit l is unchanged. c c u is a double precision array of dimension n. c On entry u is the upper bound on x. c On exit u is unchanged. c c nbd is an integer array of dimension n. c On entry nbd represents the type of bounds imposed on the c variables, and must be specified as follows: c nbd(i)=0 if x(i) is unbounded, c 1 if x(i) has only a lower bound, c 2 if x(i) has both lower and upper bounds, and c 3 if x(i) has only an upper bound. c On exit nbd is unchanged. c c f is a double precision variable. c On first entry f is unspecified. c On final exit f is the value of the function at x. c c g is a double precision array of dimension n. c On first entry g is unspecified. c On final exit g is the value of the gradient at x. c c factr is a double precision variable. c On entry factr >= 0 is specified by the user. The iteration c will stop when c c (f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr*epsmch c c where epsmch is the machine precision, which is automatically c generated by the code. Typical values for factr: 1.d+12 for c low accuracy; 1.d+7 for moderate accuracy; 1.d+1 for extremely c high accuracy. c On exit factr is unchanged. c c pgtol is a double precision variable. c On entry pgtol >= 0 is specified by the user. The iteration c will stop when c c max{|proj g_i | i = 1, ..., n} <= pgtol c c where pg_i is the ith component of the projected gradient. c On exit pgtol is unchanged. c c wa is a double precision working array of length c (2mmax + 4)nmax + 11mmax^2 + 8mmax. c c iwa is an integer working array of length 3nmax. c c task is a working string of characters of length 60 indicating c the current job when entering and quitting this subroutine. c c iprint is an integer variable that must be set by the user. c It controls the frequency and type of output generated: c iprint<0 no output is generated; c iprint=0 print only one line at the last iteration; c 0100 print details of every iteration including x and g; c When iprint > 0, the file iterate.dat will be created to c summarize the iteration. c c csave is a working string of characters of length 60. c c lsave is a logical working array of dimension 4. c On exit with 'task' = NEW_X, the following information is c available: c If lsave(1) = .true. then the initial X has been replaced by c its projection in the feasible set; c If lsave(2) = .true. then the problem is constrained; c If lsave(3) = .true. then each variable has upper and lower c bounds; c c isave is an integer working array of dimension 44. c On exit with 'task' = NEW_X, the following information is c available: c isave(22) = the total number of intervals explored in the c search of Cauchy points; c isave(26) = the total number of skipped BFGS updates before c the current iteration; c isave(30) = the number of current iteration; c isave(31) = the total number of BFGS updates prior the current c iteration; c isave(33) = the number of intervals explored in the search of c Cauchy point in the current iteration; c isave(34) = the total number of function and gradient c evaluations; c isave(36) = the number of function value or gradient c evaluations in the current iteration; c if isave(37) = 0 then the subspace argmin is within the box; c if isave(37) = 1 then the subspace argmin is beyond the box; c isave(38) = the number of free variables in the current c iteration; c isave(39) = the number of active constraints in the current c iteration; c n + 1 - isave(40) = the number of variables leaving the set of c active constraints in the current iteration; c isave(41) = the number of variables entering the set of active c constraints in the current iteration. c c dsave is a double precision working array of dimension 29. c On exit with 'task' = NEW_X, the following information is c available: c dsave(1) = current 'theta' in the BFGS matrix; c dsave(2) = f(x) in the previous iteration; c dsave(3) = factr*epsmch; c dsave(4) = 2-norm of the line search direction vector; c dsave(5) = the machine precision epsmch generated by the code; c dsave(7) = the accumulated time spent on searching for c Cauchy points; c dsave(8) = the accumulated time spent on c subspace minimization; c dsave(9) = the accumulated time spent on line search; c dsave(11) = the slope of the line search function at c the current point of line search; c dsave(12) = the maximum relative step length imposed in c line search; c dsave(13) = the infinity norm of the projected gradient; c dsave(14) = the relative step length in the line search; c dsave(15) = the slope of the line search function at c the starting point of the line search; c dsave(16) = the square of the 2-norm of the line search c direction vector. c c Subprograms called: c c L-BFGS-B Library ... mainlb. c c c References: c c [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited c memory algorithm for bound constrained optimization'', c SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208. c c [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: a c limited memory FORTRAN code for solving bound constrained c optimization problems'', Tech. Report, NAM-11, EECS Department, c Northwestern University, 1994. c c (Postscript files of these papers are available via anonymous c ftp to ece.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.) c c * * * c c NEOS, November 1994. (Latest revision April 1997.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer l1,l2,l3,lws,lr,lz,lt,ld,lwa,lwy,lsy,lss,lwt,lwn,lsnd if (task .eq. 'START') then isave(1) = m*n isave(2) = m**2 isave(3) = 4*m**2 isave(4) = 1 isave(5) = isave(4) + isave(1) isave(6) = isave(5) + isave(1) isave(7) = isave(6) + isave(2) isave(8) = isave(7) + isave(2) isave(9) = isave(8) isave(10) = isave(9) + isave(2) isave(11) = isave(10) + isave(3) isave(12) = isave(11) + isave(3) isave(13) = isave(12) + n isave(14) = isave(13) + n isave(15) = isave(14) + n isave(16) = isave(15) + n endif l1 = isave(1) l2 = isave(2) l3 = isave(3) lws = isave(4) lwy = isave(5) lsy = isave(6) lss = isave(7) lwt = isave(9) lwn = isave(10) lsnd = isave(11) lz = isave(12) lr = isave(13) ld = isave(14) lt = isave(15) lwa = isave(16) call mainlb(n,m,x,l,u,nbd,f,g,factr,pgtol, + wa(lws),wa(lwy),wa(lsy),wa(lss),wa(lwt), + wa(lwn),wa(lsnd),wa(lz),wa(lr),wa(ld),wa(lt), + wa(lwa),iwa(1),iwa(n+1),iwa(2*n+1),task,iprint, + csave,lsave,isave(22),dsave) return end c======================= The end of setulb ============================= subroutine mainlb(n, m, x, l, u, nbd, f, g, factr, pgtol, ws, wy, + sy, ss, wt, wn, snd, z, r, d, t, wa, + index, iwhere, indx2, task, iprint, + csave, lsave, isave, dsave) character*60 task, csave logical lsave(4) integer n, m, iprint, nbd(n), index(n), + iwhere(n), indx2(n), isave(23) double precision f, factr, pgtol, + x(n), l(n), u(n), g(n), z(n), r(n), d(n), t(n), + wa(8*m), ws(n, m), wy(n, m), sy(m, m), ss(m, m), + wt(m, m), wn(2*m, 2*m), snd(2*m, 2*m), dsave(29) c ************ c c Subroutine mainlb c c This subroutine solves bound constrained optimization problems by c using the compact formula of the limited memory BFGS updates. c c n is an integer variable. c On entry n is the number of variables. c On exit n is unchanged. c c m is an integer variable. c On entry m is the maximum number of variable metric c corrections allowed in the limited memory matrix. c On exit m is unchanged. c c x is a double precision array of dimension n. c On entry x is an approximation to the solution. c On exit x is the current approximation. c c l is a double precision array of dimension n. c On entry l is the lower bound of x. c On exit l is unchanged. c c u is a double precision array of dimension n. c On entry u is the upper bound of x. c On exit u is unchanged. c c nbd is an integer array of dimension n. c On entry nbd represents the type of bounds imposed on the c variables, and must be specified as follows: c nbd(i)=0 if x(i) is unbounded, c 1 if x(i) has only a lower bound, c 2 if x(i) has both lower and upper bounds, c 3 if x(i) has only an upper bound. c On exit nbd is unchanged. c c f is a double precision variable. c On first entry f is unspecified. c On final exit f is the value of the function at x. c c g is a double precision array of dimension n. c On first entry g is unspecified. c On final exit g is the value of the gradient at x. c c factr is a double precision variable. c On entry factr >= 0 is specified by the user. The iteration c will stop when c c (f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= factr*epsmch c c where epsmch is the machine precision, which is automatically c generated by the code. c On exit factr is unchanged. c c pgtol is a double precision variable. c On entry pgtol >= 0 is specified by the user. The iteration c will stop when c c max{|proj g_i | i = 1, ..., n} <= pgtol c c where pg_i is the ith component of the projected gradient. c On exit pgtol is unchanged. c c ws, wy, sy, and wt are double precision working arrays used to c store the following information defining the limited memory c BFGS matrix: c ws, of dimension n x m, stores S, the matrix of s-vectors; c wy, of dimension n x m, stores Y, the matrix of y-vectors; c sy, of dimension m x m, stores S'Y; c ss, of dimension m x m, stores S'S; c wt, of dimension m x m, stores the Cholesky factorization c of (theta*S'S+LD^(-1)L'); see eq. c (2.26) in [3]. c c wn is a double precision working array of dimension 2m x 2m c used to store the LEL^T factorization of the indefinite matrix c K = [-D -Y'ZZ'Y/theta L_a'-R_z' ] c [L_a -R_z theta*S'AA'S ] c c where E = [-I 0] c [ 0 I] c c snd is a double precision working array of dimension 2m x 2m c used to store the lower triangular part of c N = [Y' ZZ'Y L_a'+R_z'] c [L_a +R_z S'AA'S ] c c z(n),r(n),d(n),t(n),wa(8*m) are double precision working arrays. c z is used at different times to store the Cauchy point and c the Newton point. c c c index is an integer working array of dimension n. c In subroutine freev, index is used to store the free and fixed c variables at the Generalized Cauchy Point (GCP). c c iwhere is an integer working array of dimension n used to record c the status of the vector x for GCP computation. c iwhere(i)=0 or -3 if x(i) is free and has bounds, c 1 if x(i) is fixed at l(i), and l(i) .ne. u(i) c 2 if x(i) is fixed at u(i), and u(i) .ne. l(i) c 3 if x(i) is always fixed, i.e., u(i)=x(i)=l(i) c -1 if x(i) is always free, i.e., no bounds on it. c c indx2 is an integer working array of dimension n. c Within subroutine cauchy, indx2 corresponds to the array iorder. c In subroutine freev, a list of variables entering and leaving c the free set is stored in indx2, and it is passed on to c subroutine formk with this information. c c task is a working string of characters of length 60 indicating c the current job when entering and leaving this subroutine. c c iprint is an INTEGER variable that must be set by the user. c It controls the frequency and type of output generated: c iprint<0 no output is generated; c iprint=0 print only one line at the last iteration; c 0100 print details of every iteration including x and g; c When iprint > 0, the file iterate.dat will be created to c summarize the iteration. c c csave is a working string of characters of length 60. c c lsave is a logical working array of dimension 4. c c isave is an integer working array of dimension 23. c c dsave is a double precision working array of dimension 29. c c c Subprograms called c c L-BFGS-B Library ... cauchy, subsm, lnsrlb, formk, c c errclb, prn1lb, prn2lb, prn3lb, active, projgr, c c freev, cmprlb, matupd, formt. c c Minpack2 Library ... timer, dpmeps. c c Linpack Library ... dcopy, ddot. c c c References: c c [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited c memory algorithm for bound constrained optimization'', c SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208. c c [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: FORTRAN c Subroutines for Large Scale Bound Constrained Optimization'' c Tech. Report, NAM-11, EECS Department, Northwestern University, c 1994. c c [3] R. Byrd, J. Nocedal and R. Schnabel "Representations of c Quasi-Newton Matrices and their use in Limited Memory Methods'', c Mathematical Programming 63 (1994), no. 4, pp. 129-156. c c (Postscript files of these papers are available via anonymous c ftp to ece.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.) c c * * * c c NEOS, November 1994. (Latest revision April 1997.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ logical prjctd,cnstnd,boxed,updatd,wrk character*3 word integer i,k,nintol,itfile,iback,nskip, + head,col,iter,itail,iupdat, + nint,nfgv,info,ifun, + iword,nfree,nact,ileave,nenter double precision theta,fold,ddot,dr,rr,tol,dpmeps, + xstep,sbgnrm,ddum,dnorm,dtd,epsmch, + cpu1,cpu2,cachyt,sbtime,lnscht,time1,time2, + gd,gdold,stp,stpmx,time double precision one,zero parameter (one=1.0d0,zero=0.0d0) if (task .eq. 'START') then call timer(time1) c Generate the current machine precision. epsmch = dpmeps() fold = 0.0d0 dnorm = 0.0d0 cpu1 = 0.0d0 gd = 0.0d0 sbgnrm = 0.0d0 stp = 0.0d0 stpmx = 0.0d0 gdold = 0.0d0 dtd = 0.0d0 c Initialize counters and scalars when task='START'. c for the limited memory BFGS matrices: col = 0 head = 1 theta = one iupdat = 0 updatd = .false. iback = 0 itail = 0 ifun = 0 iword = 0 nact = 0 ileave = 0 nenter = 0 c for operation counts: iter = 0 nfgv = 0 nint = 0 nintol = 0 nskip = 0 nfree = n c for stopping tolerance: tol = factr*epsmch c for measuring running time: cachyt = 0 sbtime = 0 lnscht = 0 c 'word' records the status of subspace solutions. word = '---' c 'info' records the termination information. info = 0 itfile = 0 if (iprint .ge. 1) then c open a summary file 'iterate.dat' open (8, file = 'iterate.dat', status = 'unknown') itfile = 8 endif c Check the input arguments for errors. call errclb(n,m,factr,l,u,nbd,task,info,k) if (task(1:5) .eq. 'ERROR') then call prn3lb(n,x,f,task,iprint,info,itfile, + iter,nfgv,nintol,nskip,nact,sbgnrm, + zero,nint,word,iback,stp,xstep,k, + cachyt,sbtime,lnscht) return endif call prn1lb(n,m,l,u,x,iprint,itfile,epsmch) c Initialize iwhere & project x onto the feasible set. call active(n,l,u,nbd,x,iwhere,iprint,prjctd,cnstnd,boxed) c The end of the initialization. else c restore local variables. prjctd = lsave(1) cnstnd = lsave(2) boxed = lsave(3) updatd = lsave(4) nintol = isave(1) itfile = isave(3) iback = isave(4) nskip = isave(5) head = isave(6) col = isave(7) itail = isave(8) iter = isave(9) iupdat = isave(10) nint = isave(12) nfgv = isave(13) info = isave(14) ifun = isave(15) iword = isave(16) nfree = isave(17) nact = isave(18) ileave = isave(19) nenter = isave(20) theta = dsave(1) fold = dsave(2) tol = dsave(3) dnorm = dsave(4) epsmch = dsave(5) cpu1 = dsave(6) cachyt = dsave(7) sbtime = dsave(8) lnscht = dsave(9) time1 = dsave(10) gd = dsave(11) stpmx = dsave(12) sbgnrm = dsave(13) stp = dsave(14) gdold = dsave(15) dtd = dsave(16) c After returning from the driver go to the point where execution c is to resume. if (task(1:5) .eq. 'FG_LN') goto 666 if (task(1:5) .eq. 'NEW_X') goto 777 if (task(1:5) .eq. 'FG_ST') goto 111 if (task(1:4) .eq. 'STOP') then if (task(7:9) .eq. 'CPU') then c restore the previous iterate. call dcopy(n,t,1,x,1) call dcopy(n,r,1,g,1) f = fold endif goto 999 endif endif c Compute f0 and g0. task = 'FG_START' c return to the driver to calculate f and g; reenter at 111. goto 1000 111 continue nfgv = 1 c Compute the infinity norm of the (-) projected gradient. call projgr(n,l,u,nbd,x,g,sbgnrm) if (iprint .ge. 1) then write (6,1002) iter,f,sbgnrm write (itfile,1003) iter,nfgv,sbgnrm,f endif if (sbgnrm .le. pgtol) then c terminate the algorithm. task = 'CONVERGENCE: NORM OF PROJECTED GRADIENT <= PGTOL' goto 999 endif c ----------------- the beginning of the loop -------------------------- 222 continue if (iprint .ge. 99) write (6,1001) iter + 1 iword = -1 c if (.not. cnstnd .and. col .gt. 0) then c skip the search for GCP. call dcopy(n,x,1,z,1) wrk = updatd nint = 0 goto 333 endif cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c Compute the Generalized Cauchy Point (GCP). c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc call timer(cpu1) call cauchy(n,x,l,u,nbd,g,indx2,iwhere,t,d,z, + m,wy,ws,sy,wt,theta,col,head, + wa(1),wa(2*m+1),wa(4*m+1),wa(6*m+1),nint, + iprint,sbgnrm,info,epsmch) if (info .ne. 0) then c singular triangular system detected; refresh the lbfgs memory. if(iprint .ge. 1) write (6, 1005) info = 0 col = 0 head = 1 theta = one iupdat = 0 updatd = .false. call timer(cpu2) cachyt = cachyt + cpu2 - cpu1 goto 222 endif call timer(cpu2) cachyt = cachyt + cpu2 - cpu1 nintol = nintol + nint c Count the entering and leaving variables for iter > 0; c find the index set of free and active variables at the GCP. call freev(n,nfree,index,nenter,ileave,indx2, + iwhere,wrk,updatd,cnstnd,iprint,iter) nact = n - nfree 333 continue c If there are no free variables or B=theta*I, then c skip the subspace minimization. if (nfree .eq. 0 .or. col .eq. 0) goto 555 cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c Subspace minimization. c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc call timer(cpu1) c Form the LEL^T factorization of the indefinite c matrix K = [-D -Y'ZZ'Y/theta L_a'-R_z' ] c [L_a -R_z theta*S'AA'S ] c where E = [-I 0] c [ 0 I] if (wrk) call formk(n,nfree,index,nenter,ileave,indx2,iupdat, + updatd,wn,snd,m,ws,wy,sy,theta,col,head,info) if (info .ne. 0) then c nonpositive definiteness in Cholesky factorization; c refresh the lbfgs memory and restart the iteration. if(iprint .ge. 1) write (6, 1006) info = 0 col = 0 head = 1 theta = one iupdat = 0 updatd = .false. call timer(cpu2) sbtime = sbtime + cpu2 - cpu1 goto 222 endif c compute r=-Z'B(xcp-xk)-Z'g (using wa(2m+1)=W'(xcp-x) c from 'cauchy'). call cmprlb(n,m,x,g,ws,wy,sy,wt,z,r,wa,index, + theta,col,head,nfree,cnstnd,info) if (info .ne. 0) goto 444 c call the direct method. call subsm(n,m,nfree,index,l,u,nbd,z,r,ws,wy,theta, + col,head,iword,wa,wn,iprint,info) 444 continue if (info .ne. 0) then c singular triangular system detected; c refresh the lbfgs memory and restart the iteration. if(iprint .ge. 1) write (6, 1005) info = 0 col = 0 head = 1 theta = one iupdat = 0 updatd = .false. call timer(cpu2) sbtime = sbtime + cpu2 - cpu1 goto 222 endif call timer(cpu2) sbtime = sbtime + cpu2 - cpu1 555 continue cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c Line search and optimality tests. c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c Generate the search direction d:=z-x. do 40 i = 1, n d(i) = z(i) - x(i) 40 continue call timer(cpu1) 666 continue call lnsrlb(n,l,u,nbd,x,f,fold,gd,gdold,g,d,r,t,z,stp,dnorm, + dtd,xstep,stpmx,iter,ifun,iback,nfgv,info,task, + boxed,cnstnd,csave,isave(22),dsave(17)) if (info .ne. 0 .or. iback .ge. 20) then c restore the previous iterate. call dcopy(n,t,1,x,1) call dcopy(n,r,1,g,1) f = fold if (col .eq. 0) then c abnormal termination. if (info .eq. 0) then info = -9 c restore the actual number of f and g evaluations etc. nfgv = nfgv - 1 ifun = ifun - 1 iback = iback - 1 endif task = 'ABNORMAL_TERMINATION_IN_LNSRCH' iter = iter + 1 goto 999 else c refresh the lbfgs memory and restart the iteration. if(iprint .ge. 1) write (6, 1008) if (info .eq. 0) nfgv = nfgv - 1 info = 0 col = 0 head = 1 theta = one iupdat = 0 updatd = .false. task = 'RESTART_FROM_LNSRCH' call timer(cpu2) lnscht = lnscht + cpu2 - cpu1 goto 222 endif else if (task(1:5) .eq. 'FG_LN') then c return to the driver for calculating f and g; reenter at 666. goto 1000 else c calculate and print out the quantities related to the new X. call timer(cpu2) lnscht = lnscht + cpu2 - cpu1 iter = iter + 1 c Compute the infinity norm of the projected (-)gradient. call projgr(n,l,u,nbd,x,g,sbgnrm) c Print iteration information. call prn2lb(n,x,f,g,iprint,itfile,iter,nfgv,nact, + sbgnrm,nint,word,iword,iback,stp,xstep) goto 1000 endif 777 continue c Test for termination. if (sbgnrm .le. pgtol) then c terminate the algorithm. task = 'CONVERGENCE: NORM OF PROJECTED GRADIENT <= PGTOL' goto 999 endif ddum = max(abs(fold), abs(f), one) if ((fold - f) .le. tol*ddum) then c terminate the algorithm. task = 'CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH' if (iback .ge. 10) info = -5 c i.e., to issue a warning if iback>10 in the line search. goto 999 endif c Compute d=newx-oldx, r=newg-oldg, rr=y'y and dr=y's. do 42 i = 1, n r(i) = g(i) - r(i) 42 continue rr = ddot(n,r,1,r,1) if (stp .eq. one) then dr = gd - gdold ddum = -gdold else dr = (gd - gdold)*stp call dscal(n,stp,d,1) ddum = -gdold*stp endif if (dr .le. epsmch*ddum) then c skip the L-BFGS update. nskip = nskip + 1 updatd = .false. if (iprint .ge. 1) write (6,1004) dr, ddum goto 888 endif cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc c c Update the L-BFGS matrix. c cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc updatd = .true. iupdat = iupdat + 1 c Update matrices WS and WY and form the middle matrix in B. call matupd(n,m,ws,wy,sy,ss,d,r,itail, + iupdat,col,head,theta,rr,dr,stp,dtd) c Form the upper half of the pds T = theta*SS + L*D^(-1)*L'; c Store T in the upper triangular of the array wt; c Cholesky factorize T to J*J' with c J' stored in the upper triangular of wt. call formt(m,wt,sy,ss,col,theta,info) if (info .ne. 0) then c nonpositive definiteness in Cholesky factorization; c refresh the lbfgs memory and restart the iteration. if(iprint .ge. 1) write (6, 1007) info = 0 col = 0 head = 1 theta = one iupdat = 0 updatd = .false. goto 222 endif c Now the inverse of the middle matrix in B is c [ D^(1/2) O ] [ -D^(1/2) D^(-1/2)*L' ] c [ -L*D^(-1/2) J ] [ 0 J' ] 888 continue c -------------------- the end of the loop ----------------------------- goto 222 999 continue call timer(time2) time = time2 - time1 call prn3lb(n,x,f,task,iprint,info,itfile, + iter,nfgv,nintol,nskip,nact,sbgnrm, + time,nint,word,iback,stp,xstep,k, + cachyt,sbtime,lnscht) 1000 continue c Save local variables. lsave(1) = prjctd lsave(2) = cnstnd lsave(3) = boxed lsave(4) = updatd isave(1) = nintol isave(3) = itfile isave(4) = iback isave(5) = nskip isave(6) = head isave(7) = col isave(8) = itail isave(9) = iter isave(10) = iupdat isave(12) = nint isave(13) = nfgv isave(14) = info isave(15) = ifun isave(16) = iword isave(17) = nfree isave(18) = nact isave(19) = ileave isave(20) = nenter dsave(1) = theta dsave(2) = fold dsave(3) = tol dsave(4) = dnorm dsave(5) = epsmch dsave(6) = cpu1 dsave(7) = cachyt dsave(8) = sbtime dsave(9) = lnscht dsave(10) = time1 dsave(11) = gd dsave(12) = stpmx dsave(13) = sbgnrm dsave(14) = stp dsave(15) = gdold dsave(16) = dtd 1001 format (//,'ITERATION ',i5) 1002 format + (/,'At iterate',i5,4x,'f= ',1p,d12.5,4x,'|proj g|= ',1p,d12.5) 1003 format (2(1x,i4),5x,'-',5x,'-',3x,'-',5x,'-',5x,'-',8x,'-',3x, + 1p,2(1x,d10.3)) 1004 format (' ys=',1p,e10.3,' -gs=',1p,e10.3,' BFGS update SKIPPED') 1005 format (/, +' Singular triangular system detected;',/, +' refresh the lbfgs memory and restart the iteration.') 1006 format (/, +' Nonpositive definiteness in Cholesky factorization in formk;',/, +' refresh the lbfgs memory and restart the iteration.') 1007 format (/, +' Nonpositive definiteness in Cholesky factorization in formt;',/, +' refresh the lbfgs memory and restart the iteration.') 1008 format (/, +' Bad direction in the line search;',/, +' refresh the lbfgs memory and restart the iteration.') return end c======================= The end of mainlb ============================= subroutine active(n, l, u, nbd, x, iwhere, iprint, + prjctd, cnstnd, boxed) logical prjctd, cnstnd, boxed integer n, iprint, nbd(n), iwhere(n) double precision x(n), l(n), u(n) c ************ c c Subroutine active c c This subroutine initializes iwhere and projects the initial x to c the feasible set if necessary. c c iwhere is an integer array of dimension n. c On entry iwhere is unspecified. c On exit iwhere(i)=-1 if x(i) has no bounds c 3 if l(i)=u(i) c 0 otherwise. c In cauchy, iwhere is given finer gradations. c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer nbdd,i double precision zero parameter (zero=0.0d0) c Initialize nbdd, prjctd, cnstnd and boxed. nbdd = 0 prjctd = .false. cnstnd = .false. boxed = .true. c Project the initial x to the easible set if necessary. do 10 i = 1, n if (nbd(i) .gt. 0) then if (nbd(i) .le. 2 .and. x(i) .le. l(i)) then if (x(i) .lt. l(i)) then prjctd = .true. x(i) = l(i) endif nbdd = nbdd + 1 else if (nbd(i) .ge. 2 .and. x(i) .ge. u(i)) then if (x(i) .gt. u(i)) then prjctd = .true. x(i) = u(i) endif nbdd = nbdd + 1 endif endif 10 continue c Initialize iwhere and assign values to cnstnd and boxed. do 20 i = 1, n if (nbd(i) .ne. 2) boxed = .false. if (nbd(i) .eq. 0) then c this variable is always free iwhere(i) = -1 c otherwise set x(i)=mid(x(i), u(i), l(i)). else cnstnd = .true. if (nbd(i) .eq. 2 .and. u(i) - l(i) .le. zero) then c this variable is always fixed iwhere(i) = 3 else iwhere(i) = 0 endif endif 20 continue if (iprint .ge. 0) then if (prjctd) write (6,*) + 'The initial X is infeasible. Restart with its projection.' if (.not. cnstnd) + write (6,*) 'This problem is unconstrained.' endif if (iprint .gt. 0) write (6,1001) nbdd 1001 format (/,'At X0 ',i9,' variables are exactly at the bounds') return end c======================= The end of active ============================= subroutine bmv(m, sy, wt, col, v, p, info) integer m, col, info double precision sy(m, m), wt(m, m), v(2*col), p(2*col) c ************ c c Subroutine bmv c c This subroutine computes the product of the 2m x 2m middle matrix c in the compact L-BFGS formula of B and a 2m vector v; c it returns the product in p. c c m is an integer variable. c On entry m is the maximum number of variable metric corrections c used to define the limited memory matrix. c On exit m is unchanged. c c sy is a double precision array of dimension m x m. c On entry sy specifies the matrix S'Y. c On exit sy is unchanged. c c wt is a double precision array of dimension m x m. c On entry wt specifies the upper triangular matrix J' which is c the Cholesky factor of (thetaS'S+LD^(-1)L'). c On exit wt is unchanged. c c col is an integer variable. c On entry col specifies the number of s-vectors (or y-vectors) c stored in the compact L-BFGS formula. c On exit col is unchanged. c c v is a double precision array of dimension 2col. c On entry v specifies vector v. c On exit v is unchanged. c c p is a double precision array of dimension 2col. c On entry p is unspecified. c On exit p is the product Mv. c c info is an integer variable. c On entry info is unspecified. c On exit info = 0 for normal return, c = nonzero for abnormal return when the system c to be solved by dtrsl is singular. c c Subprograms called: c c Linpack ... dtrsl. c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer i,k,i2 double precision sum if (col .eq. 0) return c PART I: solve [ D^(1/2) O ] [ p1 ] = [ v1 ] c [ -L*D^(-1/2) J ] [ p2 ] [ v2 ]. c solve Jp2=v2+LD^(-1)v1. p(col + 1) = v(col + 1) do 20 i = 2, col i2 = col + i sum = 0.0d0 do 10 k = 1, i - 1 sum = sum + sy(i,k)*v(k)/sy(k,k) 10 continue p(i2) = v(i2) + sum 20 continue c Solve the triangular system call dtrsl(wt,m,col,p(col+1),11,info) if (info .ne. 0) return c solve D^(1/2)p1=v1. cc do 30 i = 1, col cc p(i) = v(i)/sqrt(sy(i,i)) cc30 continue c PART II: solve [ -D^(1/2) D^(-1/2)*L' ] [ p1 ] = [ p1 ] c [ 0 J' ] [ p2 ] [ p2 ]. c solve J^Tp2=p2. call dtrsl(wt,m,col,p(col+1),01,info) if (info .ne. 0) return c compute p1=-D^(-1/2)(p1-D^(-1/2)L'p2) c =-D^(-1/2)p1+D^(-1)L'p2. do 40 i = 1, col cc p(i) = -p(i)/sqrt(sy(i,i)) combined with do 30 loop cc into the next line p(i) = -v(i)/sy(i,i) 40 continue do 60 i = 1, col sum = 0.d0 do 50 k = i + 1, col sum = sum + sy(k,i)*p(col+k)/sy(i,i) 50 continue p(i) = p(i) + sum 60 continue return end c======================== The end of bmv =============================== subroutine cauchy(n, x, l, u, nbd, g, iorder, iwhere, t, d, xcp, + m, wy, ws, sy, wt, theta, col, head, p, c, wbp, + v, nint, iprint, sbgnrm, info, epsmch) integer n, m, head, col, nint, iprint, info, + nbd(n), iorder(n), iwhere(n) double precision theta, epsmch, + x(n), l(n), u(n), g(n), t(n), d(n), xcp(n), + wy(n, col), ws(n, col), sy(m, m), + wt(m, m), p(2*m), c(2*m), wbp(2*m), v(2*m) c ************ c c Subroutine cauchy c c For given x, l, u, g (with sbgnrm > 0), and a limited memory c BFGS matrix B defined in terms of matrices WY, WS, WT, and c scalars head, col, and theta, this subroutine computes the c generalized Cauchy point (GCP), defined as the first local c minimizer of the quadratic c c Q(x + s) = g's + 1/2 s'Bs c c along the projected gradient direction P(x-tg,l,u). c The routine returns the GCP in xcp. c c n is an integer variable. c On entry n is the dimension of the problem. c On exit n is unchanged. c c x is a double precision array of dimension n. c On entry x is the starting point for the GCP computation. c On exit x is unchanged. c c l is a double precision array of dimension n. c On entry l is the lower bound of x. c On exit l is unchanged. c c u is a double precision array of dimension n. c On entry u is the upper bound of x. c On exit u is unchanged. c c nbd is an integer array of dimension n. c On entry nbd represents the type of bounds imposed on the c variables, and must be specified as follows: c nbd(i)=0 if x(i) is unbounded, c 1 if x(i) has only a lower bound, c 2 if x(i) has both lower and upper bounds, and c 3 if x(i) has only an upper bound. c On exit nbd is unchanged. c c g is a double precision array of dimension n. c On entry g is the gradient of f(x). g must be a nonzero vector. c On exit g is unchanged. c c iorder is an integer working array of dimension n. c iorder will be used to store the breakpoints in the piecewise c linear path and free variables encountered. On exit, c iorder(1),...,iorder(nleft) are indices of breakpoints c which have not been encountered; c iorder(nleft+1),...,iorder(nbreak) are indices of c encountered breakpoints; and c iorder(nfree),...,iorder(n) are indices of variables which c have no bound constraits along the search direction. c c iwhere is an integer array of dimension n. c On entry iwhere indicates only the permanently fixed (iwhere=3) c or free (iwhere= -1) components of x. c On exit iwhere records the status of the current x variables. c iwhere(i)=-3 if x(i) is free and has bounds, but is not moved c 0 if x(i) is free and has bounds, and is moved c 1 if x(i) is fixed at l(i), and l(i) .ne. u(i) c 2 if x(i) is fixed at u(i), and u(i) .ne. l(i) c 3 if x(i) is always fixed, i.e., u(i)=x(i)=l(i) c -1 if x(i) is always free, i.e., it has no bounds. c c t is a double precision working array of dimension n. c t will be used to store the break points. c c d is a double precision array of dimension n used to store c the Cauchy direction P(x-tg)-x. c c xcp is a double precision array of dimension n used to return the c GCP on exit. c c m is an integer variable. c On entry m is the maximum number of variable metric corrections c used to define the limited memory matrix. c On exit m is unchanged. c c ws, wy, sy, and wt are double precision arrays. c On entry they store information that defines the c limited memory BFGS matrix: c ws(n,m) stores S, a set of s-vectors; c wy(n,m) stores Y, a set of y-vectors; c sy(m,m) stores S'Y; c wt(m,m) stores the c Cholesky factorization of (theta*S'S+LD^(-1)L'). c On exit these arrays are unchanged. c c theta is a double precision variable. c On entry theta is the scaling factor specifying B_0 = theta I. c On exit theta is unchanged. c c col is an integer variable. c On entry col is the actual number of variable metric c corrections stored so far. c On exit col is unchanged. c c head is an integer variable. c On entry head is the location of the first s-vector c (or y-vector) in S (or Y). c On exit col is unchanged. c c p is a double precision working array of dimension 2m. c p will be used to store the vector p = W^(T)d. c c c is a double precision working array of dimension 2m. c c will be used to store the vector c = W^(T)(xcp-x). c c wbp is a double precision working array of dimension 2m. c wbp will be used to store the row of W corresponding c to a breakpoint. c c v is a double precision working array of dimension 2m. c c nint is an integer variable. c On exit nint records the number of quadratic segments explored c in searching for the GCP. c c iprint is an INTEGER variable that must be set by the user. c It controls the frequency and type of output generated: c iprint<0 no output is generated; c iprint=0 print only one line at the last iteration; c 0100 print details of every iteration including x and g; c When iprint > 0, the file iterate.dat will be created to c summarize the iteration. c c sbgnrm is a double precision variable. c On entry sbgnrm is the norm of the projected gradient at x. c On exit sbgnrm is unchanged. c c info is an integer variable. c On entry info is 0. c On exit info = 0 for normal return, c = nonzero for abnormal return when the the system c used in routine bmv is singular. c c Subprograms called: c c L-BFGS-B Library ... hpsolb, bmv. c c Linpack ... dscal dcopy, daxpy. c c c References: c c [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited c memory algorithm for bound constrained optimization'', c SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208. c c [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: FORTRAN c Subroutines for Large Scale Bound Constrained Optimization'' c Tech. Report, NAM-11, EECS Department, Northwestern University, c 1994. c c (Postscript files of these papers are available via anonymous c ftp to ece.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.) c c * * * c c NEOS, November 1994. (Latest revision April 1997.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ logical xlower,xupper,bnded integer i,j,col2,nfree,nbreak,pointr, + ibp,nleft,ibkmin,iter double precision f1,f2,dt,dtm,tsum,dibp,zibp,dibp2,bkmin, + tu,tl,wmc,wmp,wmw,ddot,tj,tj0,neggi,sbgnrm, + f2_org double precision one,zero parameter (one=1.0d0,zero=0.0d0) c Check the status of the variables, reset iwhere(i) if necessary; c compute the Cauchy direction d and the breakpoints t; initialize c the derivative f1 and the vector p = W'd (for theta = 1). if (sbgnrm .le. zero) then if (iprint .ge. 0) write (6,*) 'Subgnorm = 0. GCP = X.' call dcopy(n,x,1,xcp,1) return endif bnded = .true. nfree = n + 1 nbreak = 0 ibkmin = 0 bkmin = zero col2 = 2*col f1 = zero if (iprint .ge. 99) write (6,3010) c We set p to zero and build it up as we determine d. do 20 i = 1, col2 p(i) = zero 20 continue c In the following loop we determine for each variable its bound c status and its breakpoint, and update p accordingly. c Smallest breakpoint is identified. do 50 i = 1, n neggi = -g(i) if (iwhere(i) .ne. 3 .and. iwhere(i) .ne. -1) then c if x(i) is not a constant and has bounds, c compute the difference between x(i) and its bounds. if (nbd(i) .le. 2) tl = x(i) - l(i) if (nbd(i) .ge. 2) tu = u(i) - x(i) c If a variable is close enough to a bound c we treat it as at bound. xlower = nbd(i) .le. 2 .and. tl .le. zero xupper = nbd(i) .ge. 2 .and. tu .le. zero c reset iwhere(i). iwhere(i) = 0 if (xlower) then if (neggi .le. zero) iwhere(i) = 1 else if (xupper) then if (neggi .ge. zero) iwhere(i) = 2 else if (abs(neggi) .le. zero) iwhere(i) = -3 endif endif pointr = head if (iwhere(i) .ne. 0 .and. iwhere(i) .ne. -1) then d(i) = zero else d(i) = neggi f1 = f1 - neggi*neggi c calculate p := p - W'e_i* (g_i). do 40 j = 1, col p(j) = p(j) + wy(i,pointr)* neggi p(col + j) = p(col + j) + ws(i,pointr)*neggi pointr = mod(pointr,m) + 1 40 continue if (nbd(i) .le. 2 .and. nbd(i) .ne. 0 + .and. neggi .lt. zero) then c x(i) + d(i) is bounded; compute t(i). nbreak = nbreak + 1 iorder(nbreak) = i t(nbreak) = tl/(-neggi) if (nbreak .eq. 1 .or. t(nbreak) .lt. bkmin) then bkmin = t(nbreak) ibkmin = nbreak endif else if (nbd(i) .ge. 2 .and. neggi .gt. zero) then c x(i) + d(i) is bounded; compute t(i). nbreak = nbreak + 1 iorder(nbreak) = i t(nbreak) = tu/neggi if (nbreak .eq. 1 .or. t(nbreak) .lt. bkmin) then bkmin = t(nbreak) ibkmin = nbreak endif else c x(i) + d(i) is not bounded. nfree = nfree - 1 iorder(nfree) = i if (abs(neggi) .gt. zero) bnded = .false. endif endif 50 continue c The indices of the nonzero components of d are now stored c in iorder(1),...,iorder(nbreak) and iorder(nfree),...,iorder(n). c The smallest of the nbreak breakpoints is in t(ibkmin)=bkmin. if (theta .ne. one) then c complete the initialization of p for theta not= one. call dscal(col,theta,p(col+1),1) endif c Initialize GCP xcp = x. call dcopy(n,x,1,xcp,1) if (nbreak .eq. 0 .and. nfree .eq. n + 1) then c is a zero vector, return with the initial xcp as GCP. if (iprint .gt. 100) write (6,1010) (xcp(i), i = 1, n) return endif c Initialize c = W'(xcp - x) = 0. do 60 j = 1, col2 c(j) = zero 60 continue c Initialize derivative f2. f2 = -theta*f1 f2_org = f2 if (col .gt. 0) then call bmv(m,sy,wt,col,p,v,info) if (info .ne. 0) return f2 = f2 - ddot(col2,v,1,p,1) endif dtm = -f1/f2 tsum = zero nint = 1 if (iprint .ge. 99) + write (6,*) 'There are ',nbreak,' breakpoints ' c If there are no breakpoints, locate the GCP and return. if (nbreak .eq. 0) goto 888 nleft = nbreak iter = 1 tj = zero c------------------- the beginning of the loop ------------------------- 777 continue c Find the next smallest breakpoint; c compute dt = t(nleft) - t(nleft + 1). tj0 = tj if (iter .eq. 1) then c Since we already have the smallest breakpoint we need not do c heapsort yet. Often only one breakpoint is used and the c cost of heapsort is avoided. tj = bkmin ibp = iorder(ibkmin) else if (iter .eq. 2) then c Replace the already used smallest breakpoint with the c breakpoint numbered nbreak > nlast, before heapsort call. if (ibkmin .ne. nbreak) then t(ibkmin) = t(nbreak) iorder(ibkmin) = iorder(nbreak) endif c Update heap structure of breakpoints c (if iter=2, initialize heap). endif call hpsolb(nleft,t,iorder,iter-2) tj = t(nleft) ibp = iorder(nleft) endif dt = tj - tj0 if (dt .ne. zero .and. iprint .ge. 100) then write (6,4011) nint,f1,f2 write (6,5010) dt write (6,6010) dtm endif c If a minimizer is within this interval, c locate the GCP and return. if (dtm .lt. dt) goto 888 c Otherwise fix one variable and c reset the corresponding component of d to zero. tsum = tsum + dt nleft = nleft - 1 iter = iter + 1 dibp = d(ibp) d(ibp) = zero if (dibp .gt. zero) then zibp = u(ibp) - x(ibp) xcp(ibp) = u(ibp) iwhere(ibp) = 2 else zibp = l(ibp) - x(ibp) xcp(ibp) = l(ibp) iwhere(ibp) = 1 endif if (iprint .ge. 100) write (6,*) 'Variable ',ibp,' is fixed.' if (nleft .eq. 0 .and. nbreak .eq. n) then c all n variables are fixed, c return with xcp as GCP. dtm = dt goto 999 endif c Update the derivative information. nint = nint + 1 dibp2 = dibp**2 c Update f1 and f2. c temporarily set f1 and f2 for col=0. f1 = f1 + dt*f2 + dibp2 - theta*dibp*zibp f2 = f2 - theta*dibp2 if (col .gt. 0) then c update c = c + dt*p. call daxpy(col2,dt,p,1,c,1) c choose wbp, c the row of W corresponding to the breakpoint encountered. pointr = head do 70 j = 1,col wbp(j) = wy(ibp,pointr) wbp(col + j) = theta*ws(ibp,pointr) pointr = mod(pointr,m) + 1 70 continue c compute (wbp)Mc, (wbp)Mp, and (wbp)M(wbp)'. call bmv(m,sy,wt,col,wbp,v,info) if (info .ne. 0) return wmc = ddot(col2,c,1,v,1) wmp = ddot(col2,p,1,v,1) wmw = ddot(col2,wbp,1,v,1) c update p = p - dibp*wbp. call daxpy(col2,-dibp,wbp,1,p,1) c complete updating f1 and f2 while col > 0. f1 = f1 + dibp*wmc f2 = f2 + 2.0d0*dibp*wmp - dibp2*wmw endif f2 = max(epsmch*f2_org,f2) if (nleft .gt. 0) then dtm = -f1/f2 goto 777 c to repeat the loop for unsearched intervals. else if(bnded) then f1 = zero f2 = zero dtm = zero else dtm = -f1/f2 endif c------------------- the end of the loop ------------------------------- 888 continue if (iprint .ge. 99) then write (6,*) write (6,*) 'GCP found in this segment' write (6,4010) nint,f1,f2 write (6,6010) dtm endif if (dtm .le. zero) dtm = zero tsum = tsum + dtm c Move free variables (i.e., the ones w/o breakpoints) and c the variables whose breakpoints haven't been reached. call daxpy(n,tsum,d,1,xcp,1) 999 continue c Update c = c + dtm*p = W'(x^c - x) c which will be used in computing r = Z'(B(x^c - x) + g). if (col .gt. 0) call daxpy(col2,dtm,p,1,c,1) if (iprint .gt. 100) write (6,1010) (xcp(i),i = 1,n) if (iprint .ge. 99) write (6,2010) 1010 format ('Cauchy X = ',/,(4x,1p,6(1x,d11.4))) 2010 format (/,'---------------- exit CAUCHY----------------------',/) 3010 format (/,'---------------- CAUCHY entered-------------------') 4010 format ('Piece ',i3,' --f1, f2 at start point ',1p,2(1x,d11.4)) 4011 format (/,'Piece ',i3,' --f1, f2 at start point ', + 1p,2(1x,d11.4)) 5010 format ('Distance to the next break point = ',1p,d11.4) 6010 format ('Distance to the stationary point = ',1p,d11.4) return end c====================== The end of cauchy ============================== subroutine cmprlb(n, m, x, g, ws, wy, sy, wt, z, r, wa, index, + theta, col, head, nfree, cnstnd, info) logical cnstnd integer n, m, col, head, nfree, info, index(n) double precision theta, + x(n), g(n), z(n), r(n), wa(4*m), + ws(n, m), wy(n, m), sy(m, m), wt(m, m) c ************ c c Subroutine cmprlb c c This subroutine computes r=-Z'B(xcp-xk)-Z'g by using c wa(2m+1)=W'(xcp-x) from subroutine cauchy. c c Subprograms called: c c L-BFGS-B Library ... bmv. c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer i,j,k,pointr double precision a1,a2 if (.not. cnstnd .and. col .gt. 0) then do 26 i = 1, n r(i) = -g(i) 26 continue else do 30 i = 1, nfree k = index(i) r(i) = -theta*(z(k) - x(k)) - g(k) 30 continue call bmv(m,sy,wt,col,wa(2*m+1),wa(1),info) if (info .ne. 0) then info = -8 return endif pointr = head do 34 j = 1, col a1 = wa(j) a2 = theta*wa(col + j) do 32 i = 1, nfree k = index(i) r(i) = r(i) + wy(k,pointr)*a1 + ws(k,pointr)*a2 32 continue pointr = mod(pointr,m) + 1 34 continue endif return end c======================= The end of cmprlb ============================= subroutine errclb(n, m, factr, l, u, nbd, task, info, k) character*60 task integer n, m, info, k, nbd(n) double precision factr, l(n), u(n) c ************ c c Subroutine errclb c c This subroutine checks the validity of the input data. c c c * * * c c NEOS, November 1994. (Latest revision April 1997.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer i double precision zero parameter (zero=0.0d0) c Check the input arguments for errors. if (n .le. 0) task = 'ERROR: N .LE. 0' if (m .le. 0) task = 'ERROR: M .LE. 0' if (factr .lt. zero) task = 'ERROR: FACTR .LT. 0' c Check the validity of the arrays nbd(i), u(i), and l(i). do 10 i = 1, n if (nbd(i) .lt. 0 .or. nbd(i) .gt. 3) then c return task = 'ERROR: INVALID NBD' info = -6 k = i endif if (nbd(i) .eq. 2) then if (l(i) .gt. u(i)) then c return task = 'ERROR: NO FEASIBLE SOLUTION' info = -7 k = i endif endif 10 continue return end c======================= The end of errclb ============================= subroutine formk(n, nsub, ind, nenter, ileave, indx2, iupdat, + updatd, wn, wn1, m, ws, wy, sy, theta, col, + head, info) integer n, nsub, m, col, head, nenter, ileave, iupdat, + info, ind(n), indx2(n) double precision theta, wn(2*m, 2*m), wn1(2*m, 2*m), + ws(n, m), wy(n, m), sy(m, m) logical updatd c ************ c c Subroutine formk c c This subroutine forms the LEL^T factorization of the indefinite c c matrix K = [-D -Y'ZZ'Y/theta L_a'-R_z' ] c [L_a -R_z theta*S'AA'S ] c where E = [-I 0] c [ 0 I] c The matrix K can be shown to be equal to the matrix M^[-1]N c occurring in section 5.1 of [1], as well as to the matrix c Mbar^[-1] Nbar in section 5.3. c c n is an integer variable. c On entry n is the dimension of the problem. c On exit n is unchanged. c c nsub is an integer variable c On entry nsub is the number of subspace variables in free set. c On exit nsub is not changed. c c ind is an integer array of dimension nsub. c On entry ind specifies the indices of subspace variables. c On exit ind is unchanged. c c nenter is an integer variable. c On entry nenter is the number of variables entering the c free set. c On exit nenter is unchanged. c c ileave is an integer variable. c On entry indx2(ileave),...,indx2(n) are the variables leaving c the free set. c On exit ileave is unchanged. c c indx2 is an integer array of dimension n. c On entry indx2(1),...,indx2(nenter) are the variables entering c the free set, while indx2(ileave),...,indx2(n) are the c variables leaving the free set. c On exit indx2 is unchanged. c c iupdat is an integer variable. c On entry iupdat is the total number of BFGS updates made so far. c On exit iupdat is unchanged. c c updatd is a logical variable. c On entry 'updatd' is true if the L-BFGS matrix is updatd. c On exit 'updatd' is unchanged. c c wn is a double precision array of dimension 2m x 2m. c On entry wn is unspecified. c On exit the upper triangle of wn stores the LEL^T factorization c of the 2*col x 2*col indefinite matrix c [-D -Y'ZZ'Y/theta L_a'-R_z' ] c [L_a -R_z theta*S'AA'S ] c c wn1 is a double precision array of dimension 2m x 2m. c On entry wn1 stores the lower triangular part of c [Y' ZZ'Y L_a'+R_z'] c [L_a+R_z S'AA'S ] c in the previous iteration. c On exit wn1 stores the corresponding updated matrices. c The purpose of wn1 is just to store these inner products c so they can be easily updated and inserted into wn. c c m is an integer variable. c On entry m is the maximum number of variable metric corrections c used to define the limited memory matrix. c On exit m is unchanged. c c ws, wy, sy, and wtyy are double precision arrays; c theta is a double precision variable; c col is an integer variable; c head is an integer variable. c On entry they store the information defining the c limited memory BFGS matrix: c ws(n,m) stores S, a set of s-vectors; c wy(n,m) stores Y, a set of y-vectors; c sy(m,m) stores S'Y; c wtyy(m,m) stores the Cholesky factorization c of (theta*S'S+LD^(-1)L') c theta is the scaling factor specifying B_0 = theta I; c col is the number of variable metric corrections stored; c head is the location of the 1st s- (or y-) vector in S (or Y). c On exit they are unchanged. c c info is an integer variable. c On entry info is unspecified. c On exit info = 0 for normal return; c = -1 when the 1st Cholesky factorization failed; c = -2 when the 2st Cholesky factorization failed. c c Subprograms called: c c Linpack ... dcopy, dpofa, dtrsl. c c c References: c [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited c memory algorithm for bound constrained optimization'', c SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208. c c [2] C. Zhu, R.H. Byrd, P. Lu, J. Nocedal, ``L-BFGS-B: a c limited memory FORTRAN code for solving bound constrained c optimization problems'', Tech. Report, NAM-11, EECS Department, c Northwestern University, 1994. c c (Postscript files of these papers are available via anonymous c ftp to ece.nwu.edu in the directory pub/lbfgs/lbfgs_bcm.) c c * * * c c NEOS, November 1994. (Latest revision April 1997.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer m2,ipntr,jpntr,iy,is,jy,js,is1,js1,k1,i,k, + col2,pbegin,pend,dbegin,dend,upcl double precision ddot,temp1,temp2,temp3,temp4 double precision zero parameter (zero=0.0d0) c Form the lower triangular part of c WN1 = [Y' ZZ'Y L_a'+R_z'] c [L_a+R_z S'AA'S ] c where L_a is the strictly lower triangular part of S'AA'Y c R_z is the upper triangular part of S'ZZ'Y. if (updatd) then if (iupdat .gt. m) then c shift old part of WN1. do 10 jy = 1, m - 1 js = m + jy call dcopy(m-jy,wn1(jy+1,jy+1),1,wn1(jy,jy),1) call dcopy(m-jy,wn1(js+1,js+1),1,wn1(js,js),1) call dcopy(m-1,wn1(m+2,jy+1),1,wn1(m+1,jy),1) 10 continue endif c put new rows in blocks (1,1), (2,1) and (2,2). pbegin = 1 pend = nsub dbegin = nsub + 1 dend = n iy = col is = m + col ipntr = head + col - 1 if (ipntr .gt. m) ipntr = ipntr - m jpntr = head do 20 jy = 1, col js = m + jy temp1 = zero temp2 = zero temp3 = zero c compute element jy of row 'col' of Y'ZZ'Y do 15 k = pbegin, pend k1 = ind(k) temp1 = temp1 + wy(k1,ipntr)*wy(k1,jpntr) 15 continue c compute elements jy of row 'col' of L_a and S'AA'S do 16 k = dbegin, dend k1 = ind(k) temp2 = temp2 + ws(k1,ipntr)*ws(k1,jpntr) temp3 = temp3 + ws(k1,ipntr)*wy(k1,jpntr) 16 continue wn1(iy,jy) = temp1 wn1(is,js) = temp2 wn1(is,jy) = temp3 jpntr = mod(jpntr,m) + 1 20 continue c put new column in block (2,1). jy = col jpntr = head + col - 1 if (jpntr .gt. m) jpntr = jpntr - m ipntr = head do 30 i = 1, col is = m + i temp3 = zero c compute element i of column 'col' of R_z do 25 k = pbegin, pend k1 = ind(k) temp3 = temp3 + ws(k1,ipntr)*wy(k1,jpntr) 25 continue ipntr = mod(ipntr,m) + 1 wn1(is,jy) = temp3 30 continue upcl = col - 1 else upcl = col endif c modify the old parts in blocks (1,1) and (2,2) due to changes c in the set of free variables. ipntr = head do 45 iy = 1, upcl is = m + iy jpntr = head do 40 jy = 1, iy js = m + jy temp1 = zero temp2 = zero temp3 = zero temp4 = zero do 35 k = 1, nenter k1 = indx2(k) temp1 = temp1 + wy(k1,ipntr)*wy(k1,jpntr) temp2 = temp2 + ws(k1,ipntr)*ws(k1,jpntr) 35 continue do 36 k = ileave, n k1 = indx2(k) temp3 = temp3 + wy(k1,ipntr)*wy(k1,jpntr) temp4 = temp4 + ws(k1,ipntr)*ws(k1,jpntr) 36 continue wn1(iy,jy) = wn1(iy,jy) + temp1 - temp3 wn1(is,js) = wn1(is,js) - temp2 + temp4 jpntr = mod(jpntr,m) + 1 40 continue ipntr = mod(ipntr,m) + 1 45 continue c modify the old parts in block (2,1). ipntr = head do 60 is = m + 1, m + upcl jpntr = head do 55 jy = 1, upcl temp1 = zero temp3 = zero do 50 k = 1, nenter k1 = indx2(k) temp1 = temp1 + ws(k1,ipntr)*wy(k1,jpntr) 50 continue do 51 k = ileave, n k1 = indx2(k) temp3 = temp3 + ws(k1,ipntr)*wy(k1,jpntr) 51 continue if (is .le. jy + m) then wn1(is,jy) = wn1(is,jy) + temp1 - temp3 else wn1(is,jy) = wn1(is,jy) - temp1 + temp3 endif jpntr = mod(jpntr,m) + 1 55 continue ipntr = mod(ipntr,m) + 1 60 continue c Form the upper triangle of WN = [D+Y' ZZ'Y/theta -L_a'+R_z' ] c [-L_a +R_z S'AA'S*theta] m2 = 2*m do 70 iy = 1, col is = col + iy is1 = m + iy do 65 jy = 1, iy js = col + jy js1 = m + jy wn(jy,iy) = wn1(iy,jy)/theta wn(js,is) = wn1(is1,js1)*theta 65 continue do 66 jy = 1, iy - 1 wn(jy,is) = -wn1(is1,jy) 66 continue do 67 jy = iy, col wn(jy,is) = wn1(is1,jy) 67 continue wn(iy,iy) = wn(iy,iy) + sy(iy,iy) 70 continue c Form the upper triangle of c WN= [ LL' L^-1(-L_a'+R_z')] c [(-L_a +R_z)L'^-1 S'AA'S*theta ] c first Cholesky factor (1,1) block of wn to get LL' c with L' stored in the upper triangle of wn. call dpofa(wn,m2,col,info) if (info .ne. 0) then info = -1 return endif c then form L^-1(-L_a'+R_z') in the (1,2) block. col2 = 2*col do 71 js = col+1 ,col2 call dtrsl(wn,m2,col,wn(1,js),11,info) 71 continue c Form S'AA'S*theta + (L^-1(-L_a'+R_z'))'L^-1(-L_a'+R_z') in the c upper triangle of (2,2) block of wn. do 72 is = col+1, col2 do 74 js = is, col2 wn(is,js) = wn(is,js) + ddot(col,wn(1,is),1,wn(1,js),1) 74 continue 72 continue c Cholesky factorization of (2,2) block of wn. call dpofa(wn(col+1,col+1),m2,col,info) if (info .ne. 0) then info = -2 return endif return end c======================= The end of formk ============================== subroutine formt(m, wt, sy, ss, col, theta, info) integer m, col, info double precision theta, wt(m, m), sy(m, m), ss(m, m) c ************ c c Subroutine formt c c This subroutine forms the upper half of the pos. def. and symm. c T = theta*SS + L*D^(-1)*L', stores T in the upper triangle c of the array wt, and performs the Cholesky factorization of T c to produce J*J', with J' stored in the upper triangle of wt. c c Subprograms called: c c Linpack ... dpofa. c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer i,j,k,k1 double precision ddum double precision zero parameter (zero=0.0d0) c Form the upper half of T = theta*SS + L*D^(-1)*L', c store T in the upper triangle of the array wt. do 52 j = 1, col wt(1,j) = theta*ss(1,j) 52 continue do 55 i = 2, col do 54 j = i, col k1 = min(i,j) - 1 ddum = zero do 53 k = 1, k1 ddum = ddum + sy(i,k)*sy(j,k)/sy(k,k) 53 continue wt(i,j) = ddum + theta*ss(i,j) 54 continue 55 continue c Cholesky factorize T to J*J' with c J' stored in the upper triangle of wt. call dpofa(wt,m,col,info) if (info .ne. 0) then info = -3 endif return end c======================= The end of formt ============================== subroutine freev(n, nfree, index, nenter, ileave, indx2, + iwhere, wrk, updatd, cnstnd, iprint, iter) integer n, nfree, nenter, ileave, iprint, iter, + index(n), indx2(n), iwhere(n) logical wrk, updatd, cnstnd c ************ c c Subroutine freev c c This subroutine counts the entering and leaving variables when c iter > 0, and finds the index set of free and active variables c at the GCP. c c cnstnd is a logical variable indicating whether bounds are present c c index is an integer array of dimension n c for i=1,...,nfree, index(i) are the indices of free variables c for i=nfree+1,...,n, index(i) are the indices of bound variables c On entry after the first iteration, index gives c the free variables at the previous iteration. c On exit it gives the free variables based on the determination c in cauchy using the array iwhere. c c indx2 is an integer array of dimension n c On entry indx2 is unspecified. c On exit with iter>0, indx2 indicates which variables c have changed status since the previous iteration. c For i= 1,...,nenter, indx2(i) have changed from bound to free. c For i= ileave+1,...,n, indx2(i) have changed from free to bound. c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer iact,i,k nenter = 0 ileave = n + 1 if (iter .gt. 0 .and. cnstnd) then c count the entering and leaving variables. do 20 i = 1, nfree k = index(i) if (iwhere(k) .gt. 0) then ileave = ileave - 1 indx2(ileave) = k if (iprint .ge. 100) write (6,*) + 'Variable ',k,' leaves the set of free variables' endif 20 continue do 22 i = 1 + nfree, n k = index(i) if (iwhere(k) .le. 0) then nenter = nenter + 1 indx2(nenter) = k if (iprint .ge. 100) write (6,*) + 'Variable ',k,' enters the set of free variables' endif 22 continue if (iprint .ge. 99) write (6,*) + n+1-ileave,' variables leave; ',nenter,' variables enter' endif wrk = (ileave .lt. n+1) .or. (nenter .gt. 0) .or. updatd c Find the index set of free and active variables at the GCP. nfree = 0 iact = n + 1 do 24 i = 1, n if (iwhere(i) .le. 0) then nfree = nfree + 1 index(nfree) = i else iact = iact - 1 index(iact) = i endif 24 continue if (iprint .ge. 99) write (6,*) + nfree,' variables are free at GCP ',iter + 1 return end c======================= The end of freev ============================== subroutine hpsolb(n, t, iorder, iheap) integer iheap, n, iorder(n) double precision t(n) c ************ c c Subroutine hpsolb c c This subroutine sorts out the least element of t, and puts the c remaining elements of t in a heap. c c n is an integer variable. c On entry n is the dimension of the arrays t and iorder. c On exit n is unchanged. c c t is a double precision array of dimension n. c On entry t stores the elements to be sorted, c On exit t(n) stores the least elements of t, and t(1) to t(n-1) c stores the remaining elements in the form of a heap. c c iorder is an integer array of dimension n. c On entry iorder(i) is the index of t(i). c On exit iorder(i) is still the index of t(i), but iorder may be c permuted in accordance with t. c c iheap is an integer variable specifying the task. c On entry iheap should be set as follows: c iheap .eq. 0 if t(1) to t(n) is not in the form of a heap, c iheap .ne. 0 if otherwise. c On exit iheap is unchanged. c c c References: c Algorithm 232 of CACM (J. W. J. Williams): HEAPSORT. c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c ************ integer i,j,k,indxin,indxou double precision ddum,out if (iheap .eq. 0) then c Rearrange the elements t(1) to t(n) to form a heap. do 20 k = 2, n ddum = t(k) indxin = iorder(k) c Add ddum to the heap. i = k 10 continue if (i.gt.1) then j = i/2 if (ddum .lt. t(j)) then t(i) = t(j) iorder(i) = iorder(j) i = j goto 10 endif endif t(i) = ddum iorder(i) = indxin 20 continue endif c Assign to 'out' the value of t(1), the least member of the heap, c and rearrange the remaining members to form a heap as c elements 1 to n-1 of t. if (n .gt. 1) then i = 1 out = t(1) indxou = iorder(1) ddum = t(n) indxin = iorder(n) c Restore the heap 30 continue j = i+i if (j .le. n-1) then if (t(j+1) .lt. t(j)) j = j+1 if (t(j) .lt. ddum ) then t(i) = t(j) iorder(i) = iorder(j) i = j goto 30 endif endif t(i) = ddum iorder(i) = indxin c Put the least member in t(n). t(n) = out iorder(n) = indxou endif return end c====================== The end of hpsolb ============================== subroutine lnsrlb(n, l, u, nbd, x, f, fold, gd, gdold, g, d, r, t, + z, stp, dnorm, dtd, xstep, stpmx, iter, ifun, + iback, nfgv, info, task, boxed, cnstnd, csave, + isave, dsave) character*60 task, csave logical boxed, cnstnd integer n, iter, ifun, iback, nfgv, info, + nbd(n), isave(2) double precision f, fold, gd, gdold, stp, dnorm, dtd, xstep, + stpmx, x(n), l(n), u(n), g(n), d(n), r(n), t(n), + z(n), dsave(13) c ********** c c Subroutine lnsrlb c c This subroutine calls subroutine dcsrch from the Minpack2 library c to perform the line search. Subroutine dscrch is safeguarded so c that all trial points lie within the feasible region. c c Subprograms called: c c Minpack2 Library ... dcsrch. c c Linpack ... dtrsl, ddot. c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ********** integer i double precision ddot,a1,a2 double precision one,zero,big parameter (one=1.0d0,zero=0.0d0,big=1.0d+10) double precision ftol,gtol,xtol parameter (ftol=1.0d-3,gtol=0.9d0,xtol=0.1d0) if (task(1:5) .eq. 'FG_LN') goto 556 dtd = ddot(n,d,1,d,1) dnorm = sqrt(dtd) c Determine the maximum step length. stpmx = big if (cnstnd) then if (iter .eq. 0) then stpmx = one else do 43 i = 1, n a1 = d(i) if (nbd(i) .ne. 0) then if (a1 .lt. zero .and. nbd(i) .le. 2) then a2 = l(i) - x(i) if (a2 .ge. zero) then stpmx = zero else if (a1*stpmx .lt. a2) then stpmx = a2/a1 endif else if (a1 .gt. zero .and. nbd(i) .ge. 2) then a2 = u(i) - x(i) if (a2 .le. zero) then stpmx = zero else if (a1*stpmx .gt. a2) then stpmx = a2/a1 endif endif endif 43 continue endif endif if (iter .eq. 0 .and. .not. boxed) then stp = min(one/dnorm, stpmx) else stp = one endif call dcopy(n,x,1,t,1) call dcopy(n,g,1,r,1) fold = f ifun = 0 iback = 0 csave = 'START' 556 continue gd = ddot(n,g,1,d,1) if (ifun .eq. 0) then gdold=gd if (gd .ge. zero) then c the directional derivative >=0. c Line search is impossible. info = -4 return endif endif call dcsrch(f,gd,stp,ftol,gtol,xtol,zero,stpmx,csave,isave,dsave) xstep = stp*dnorm if (csave(1:4) .ne. 'CONV' .and. csave(1:4) .ne. 'WARN') then task = 'FG_LNSRCH' ifun = ifun + 1 nfgv = nfgv + 1 iback = ifun - 1 if (stp .eq. one) then call dcopy(n,z,1,x,1) else do 41 i = 1, n x(i) = stp*d(i) + t(i) 41 continue endif else task = 'NEW_X' endif return end c======================= The end of lnsrlb ============================= subroutine matupd(n, m, ws, wy, sy, ss, d, r, itail, + iupdat, col, head, theta, rr, dr, stp, dtd) integer n, m, itail, iupdat, col, head double precision theta, rr, dr, stp, dtd, d(n), r(n), + ws(n, m), wy(n, m), sy(m, m), ss(m, m) c ************ c c Subroutine matupd c c This subroutine updates matrices WS and WY, and forms the c middle matrix in B. c c Subprograms called: c c Linpack ... dcopy, ddot. c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer j,pointr double precision ddot double precision one parameter (one=1.0d0) c Set pointers for matrices WS and WY. if (iupdat .le. m) then col = iupdat itail = mod(head+iupdat-2,m) + 1 else itail = mod(itail,m) + 1 head = mod(head,m) + 1 endif c Update matrices WS and WY. call dcopy(n,d,1,ws(1,itail),1) call dcopy(n,r,1,wy(1,itail),1) c Set theta=yy/ys. theta = rr/dr c Form the middle matrix in B. c update the upper triangle of SS, c and the lower triangle of SY: if (iupdat .gt. m) then c move old information do 50 j = 1, col - 1 call dcopy(j,ss(2,j+1),1,ss(1,j),1) call dcopy(col-j,sy(j+1,j+1),1,sy(j,j),1) 50 continue endif c add new information: the last row of SY c and the last column of SS: pointr = head do 51 j = 1, col - 1 sy(col,j) = ddot(n,d,1,wy(1,pointr),1) ss(j,col) = ddot(n,ws(1,pointr),1,d,1) pointr = mod(pointr,m) + 1 51 continue if (stp .eq. one) then ss(col,col) = dtd else ss(col,col) = stp*stp*dtd endif sy(col,col) = dr return end c======================= The end of matupd ============================= subroutine prn1lb(n, m, l, u, x, iprint, itfile, epsmch) integer n, m, iprint, itfile double precision epsmch, x(n), l(n), u(n) c ************ c c Subroutine prn1lb c c This subroutine prints the input data, initial point, upper and c lower bounds of each variable, machine precision, as well as c the headings of the output. c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer i if (iprint .ge. 0) then write (6,7001) epsmch write (6,*) 'N = ',n,' M = ',m if (iprint .ge. 1) then write (itfile,2001) epsmch write (itfile,*)'N = ',n,' M = ',m write (itfile,9001) if (iprint .gt. 100) then write (6,1004) 'L =',(l(i),i = 1,n) write (6,1004) 'X0 =',(x(i),i = 1,n) write (6,1004) 'U =',(u(i),i = 1,n) endif endif endif 1004 format (/,a4, 1p, 6(1x,d11.4),/,(4x,1p,6(1x,d11.4))) 2001 format ('RUNNING THE L-BFGS-B CODE',/,/, + 'it = iteration number',/, + 'nf = number of function evaluations',/, + 'nint = number of segments explored during the Cauchy search',/, + 'nact = number of active bounds at the generalized Cauchy point' + ,/, + 'sub = manner in which the subspace minimization terminated:' + ,/,' con = converged, bnd = a bound was reached',/, + 'itls = number of iterations performed in the line search',/, + 'stepl = step length used',/, + 'tstep = norm of the displacement (total step)',/, + 'projg = norm of the projected gradient',/, + 'f = function value',/,/, + ' * * *',/,/, + 'Machine precision =',1p,d10.3) 7001 format ('RUNNING THE L-BFGS-B CODE',/,/, + ' * * *',/,/, + 'Machine precision =',1p,d10.3) 9001 format (/,3x,'it',3x,'nf',2x,'nint',2x,'nact',2x,'sub',2x,'itls', + 2x,'stepl',4x,'tstep',5x,'projg',8x,'f') return end c======================= The end of prn1lb ============================= subroutine prn2lb(n, x, f, g, iprint, itfile, iter, nfgv, nact, + sbgnrm, nint, word, iword, iback, stp, xstep) character*3 word integer n, iprint, itfile, iter, nfgv, nact, nint, + iword, iback double precision f, sbgnrm, stp, xstep, x(n), g(n) c ************ c c Subroutine prn2lb c c This subroutine prints out new information after a successful c line search. c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer i,imod c 'word' records the status of subspace solutions. if (iword .eq. 0) then c the subspace minimization converged. word = 'con' else if (iword .eq. 1) then c the subspace minimization stopped at a bound. word = 'bnd' else if (iword .eq. 5) then c the truncated Newton step has been used. word = 'TNT' else word = '---' endif if (iprint .ge. 99) then write (6,*) 'LINE SEARCH',iback,' times; norm of step = ',xstep write (6,2001) iter,f,sbgnrm if (iprint .gt. 100) then write (6,1004) 'X =',(x(i), i = 1, n) write (6,1004) 'G =',(g(i), i = 1, n) endif else if (iprint .gt. 0) then imod = mod(iter,iprint) if (imod .eq. 0) write (6,2001) iter,f,sbgnrm endif if (iprint .ge. 1) write (itfile,3001) + iter,nfgv,nint,nact,word,iback,stp,xstep,sbgnrm,f 1004 format (/,a4, 1p, 6(1x,d11.4),/,(4x,1p,6(1x,d11.4))) 2001 format + (/,'At iterate',i5,4x,'f= ',1p,d12.5,4x,'|proj g|= ',1p,d12.5) 3001 format(2(1x,i4),2(1x,i5),2x,a3,1x,i4,1p,2(2x,d7.1),1p,2(1x,d10.3)) return end c======================= The end of prn2lb ============================= subroutine prn3lb(n, x, f, task, iprint, info, itfile, + iter, nfgv, nintol, nskip, nact, sbgnrm, + time, nint, word, iback, stp, xstep, k, + cachyt, sbtime, lnscht) character*60 task character*3 word integer n, iprint, info, itfile, iter, nfgv, nintol, + nskip, nact, nint, iback, k double precision f, sbgnrm, time, stp, xstep, cachyt, sbtime, + lnscht, x(n) c ************ c c Subroutine prn3lb c c This subroutine prints out information when either a built-in c convergence test is satisfied or when an error message is c generated. c c c * * * c c NEOS, November 1994. (Latest revision April 1997.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer i if (task(1:5) .eq. 'ERROR') goto 999 if (iprint .ge. 0) then write (6,3003) write (6,3004) write(6,3005) n,iter,nfgv,nintol,nskip,nact,sbgnrm,f if (iprint .ge. 100) then write (6,1004) 'X =',(x(i),i = 1,n) endif if (iprint .ge. 1) write (6,*) ' F =',f endif 999 continue if (iprint .ge. 0) then write (6,3009) task if (info .ne. 0) then if (info .eq. -1) write (6,9011) if (info .eq. -2) write (6,9012) if (info .eq. -3) write (6,9013) if (info .eq. -4) write (6,9014) if (info .eq. -5) write (6,9015) if (info .eq. -6) write (6,*)' Input nbd(',k,') is invalid.' if (info .eq. -7) + write (6,*)' l(',k,') > u(',k,'). No feasible solution.' if (info .eq. -8) write (6,9018) if (info .eq. -9) write (6,9019) endif if (iprint .ge. 1) write (6,3007) cachyt,sbtime,lnscht write (6,3008) time if (iprint .ge. 1) then if (info .eq. -4 .or. info .eq. -9) then write (itfile,3002) + iter,nfgv,nint,nact,word,iback,stp,xstep endif write (itfile,3009) task if (info .ne. 0) then if (info .eq. -1) write (itfile,9011) if (info .eq. -2) write (itfile,9012) if (info .eq. -3) write (itfile,9013) if (info .eq. -4) write (itfile,9014) if (info .eq. -5) write (itfile,9015) if (info .eq. -8) write (itfile,9018) if (info .eq. -9) write (itfile,9019) endif write (itfile,3008) time endif endif 1004 format (/,a4, 1p, 6(1x,d11.4),/,(4x,1p,6(1x,d11.4))) 3002 format(2(1x,i4),2(1x,i5),2x,a3,1x,i4,1p,2(2x,d7.1),6x,'-',10x,'-') 3003 format (/, + ' * * *',/,/, + 'Tit = total number of iterations',/, + 'Tnf = total number of function evaluations',/, + 'Tnint = total number of segments explored during', + ' Cauchy searches',/, + 'Skip = number of BFGS updates skipped',/, + 'Nact = number of active bounds at final generalized', + ' Cauchy point',/, + 'Projg = norm of the final projected gradient',/, + 'F = final function value',/,/, + ' * * *') 3004 format (/,3x,'N',3x,'Tit',2x,'Tnf',2x,'Tnint',2x, + 'Skip',2x,'Nact',5x,'Projg',8x,'F') 3005 format (i5,2(1x,i4),(1x,i6),(2x,i4),(1x,i5),1p,2(2x,d10.3)) 3007 format (/,' Cauchy time',1p,e10.3,' seconds.',/ + ' Subspace minimization time',1p,e10.3,' seconds.',/ + ' Line search time',1p,e10.3,' seconds.') 3008 format (/,' Total User time',1p,e10.3,' seconds.',/) 3009 format (/,a60) 9011 format (/, +' Matrix in 1st Cholesky factorization in formk is not Pos. Def.') 9012 format (/, +' Matrix in 2st Cholesky factorization in formk is not Pos. Def.') 9013 format (/, +' Matrix in the Cholesky factorization in formt is not Pos. Def.') 9014 format (/, +' Derivative >= 0, backtracking line search impossible.',/, +' Previous x, f and g restored.',/, +' Possible causes: 1 error in function or gradient evaluation;',/, +' 2 rounding errors dominate computation.') 9015 format (/, +' Warning: more than 10 function and gradient',/, +' evaluations in the last line search. Termination',/, +' may possibly be caused by a bad search direction.') 9018 format (/,' The triangular system is singular.') 9019 format (/, +' Line search cannot locate an adequate point after 20 function',/ +,' and gradient evaluations. Previous x, f and g restored.',/, +' Possible causes: 1 error in function or gradient evaluation;',/, +' 2 rounding error dominate computation.') return end c======================= The end of prn3lb ============================= subroutine projgr(n, l, u, nbd, x, g, sbgnrm) integer n, nbd(n) double precision sbgnrm, x(n), l(n), u(n), g(n) c ************ c c Subroutine projgr c c This subroutine computes the infinity norm of the projected c gradient. c c c * * * c c NEOS, November 1994. (Latest revision April 1997.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ integer i double precision gi double precision zero parameter (zero=0.0d0) sbgnrm = zero do 15 i = 1, n gi = g(i) if (nbd(i) .ne. 0) then if (gi .lt. zero) then if (nbd(i) .ge. 2) gi = max((x(i)-u(i)),gi) else if (nbd(i) .le. 2) gi = min((x(i)-l(i)),gi) endif endif sbgnrm = max(sbgnrm,abs(gi)) 15 continue return end c======================= The end of projgr ============================= subroutine subsm(n, m, nsub, ind, l, u, nbd, x, d, ws, wy, theta, + col, head, iword, wv, wn, iprint, info) integer n, m, nsub, col, head, iword, iprint, info, + ind(nsub), nbd(n) double precision theta, + l(n), u(n), x(n), d(n), + ws(n, m), wy(n, m), + wv(2*m), wn(2*m, 2*m) c ************ c c Subroutine subsm c c Given xcp, l, u, r, an index set that specifies c the active set at xcp, and an l-BFGS matrix B c (in terms of WY, WS, SY, WT, head, col, and theta), c this subroutine computes an approximate solution c of the subspace problem c c (P) min Q(x) = r'(x-xcp) + 1/2 (x-xcp)' B (x-xcp) c c subject to l<=x<=u c x_i=xcp_i for all i in A(xcp) c c along the subspace unconstrained Newton direction c c d = -(Z'BZ)^(-1) r. c c The formula for the Newton direction, given the L-BFGS matrix c and the Sherman-Morrison formula, is c c d = (1/theta)r + (1/theta*2) Z'WK^(-1)W'Z r. c c where c K = [-D -Y'ZZ'Y/theta L_a'-R_z' ] c [L_a -R_z theta*S'AA'S ] c c Note that this procedure for computing d differs c from that described in [1]. One can show that the matrix K is c equal to the matrix M^[-1]N in that paper. c c n is an integer variable. c On entry n is the dimension of the problem. c On exit n is unchanged. c c m is an integer variable. c On entry m is the maximum number of variable metric corrections c used to define the limited memory matrix. c On exit m is unchanged. c c nsub is an integer variable. c On entry nsub is the number of free variables. c On exit nsub is unchanged. c c ind is an integer array of dimension nsub. c On entry ind specifies the coordinate indices of free variables. c On exit ind is unchanged. c c l is a double precision array of dimension n. c On entry l is the lower bound of x. c On exit l is unchanged. c c u is a double precision array of dimension n. c On entry u is the upper bound of x. c On exit u is unchanged. c c nbd is a integer array of dimension n. c On entry nbd represents the type of bounds imposed on the c variables, and must be specified as follows: c nbd(i)=0 if x(i) is unbounded, c 1 if x(i) has only a lower bound, c 2 if x(i) has both lower and upper bounds, and c 3 if x(i) has only an upper bound. c On exit nbd is unchanged. c c x is a double precision array of dimension n. c On entry x specifies the Cauchy point xcp. c On exit x(i) is the minimizer of Q over the subspace of c free variables. c c d is a double precision array of dimension n. c On entry d is the reduced gradient of Q at xcp. c On exit d is the Newton direction of Q. c c ws and wy are double precision arrays; c theta is a double precision variable; c col is an integer variable; c head is an integer variable. c On entry they store the information defining the c limited memory BFGS matrix: c ws(n,m) stores S, a set of s-vectors; c wy(n,m) stores Y, a set of y-vectors; c theta is the scaling factor specifying B_0 = theta I; c col is the number of variable metric corrections stored; c head is the location of the 1st s- (or y-) vector in S (or Y). c On exit they are unchanged. c c iword is an integer variable. c On entry iword is unspecified. c On exit iword specifies the status of the subspace solution. c iword = 0 if the solution is in the box, c 1 if some bound is encountered. c c wv is a double precision working array of dimension 2m. c c wn is a double precision array of dimension 2m x 2m. c On entry the upper triangle of wn stores the LEL^T factorization c of the indefinite matrix c c K = [-D -Y'ZZ'Y/theta L_a'-R_z' ] c [L_a -R_z theta*S'AA'S ] c where E = [-I 0] c [ 0 I] c On exit wn is unchanged. c c iprint is an INTEGER variable that must be set by the user. c It controls the frequency and type of output generated: c iprint<0 no output is generated; c iprint=0 print only one line at the last iteration; c 0100 print details of every iteration including x and g; c When iprint > 0, the file iterate.dat will be created to c summarize the iteration. c c info is an integer variable. c On entry info is unspecified. c On exit info = 0 for normal return, c = nonzero for abnormal return c when the matrix K is ill-conditioned. c c Subprograms called: c c Linpack dtrsl. c c c References: c c [1] R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, ``A limited c memory algorithm for bound constrained optimization'', c SIAM J. Scientific Computing 16 (1995), no. 5, pp. 1190--1208. c c c c * * * c c NEOS, November 1994. (Latest revision June 1996.) c Optimization Technology Center. c Argonne National Laboratory and Northwestern University. c Written by c Ciyou Zhu c in collaboration with R.H. Byrd, P. Lu-Chen and J. Nocedal. c c c ************ logical temp1_updated integer pointr,m2,col2,ibd,jy,js,i,j,k double precision alpha,dk,temp1,temp2 double precision one,zero parameter (one=1.0d0,zero=0.0d0) if (nsub .le. 0) return if (iprint .ge. 99) write (6,1001) c Compute wv = W'Zd. pointr = head do 20 i = 1, col temp1 = zero temp2 = zero do 10 j = 1, nsub k = ind(j) temp1 = temp1 + wy(k,pointr)*d(j) temp2 = temp2 + ws(k,pointr)*d(j) 10 continue wv(i) = temp1 wv(col + i) = theta*temp2 pointr = mod(pointr,m) + 1 20 continue c Compute wv:=K^(-1)wv. m2 = 2*m col2 = 2*col call dtrsl(wn,m2,col2,wv,11,info) if (info .ne. 0) return do 25 i = 1, col wv(i) = -wv(i) 25 continue call dtrsl(wn,m2,col2,wv,01,info) if (info .ne. 0) return c Compute d = (1/theta)d + (1/theta**2)Z'W wv. pointr = head do 40 jy = 1, col js = col + jy do 30 i = 1, nsub k = ind(i) d(i) = d(i) + wy(k,pointr)*wv(jy)/theta + + ws(k,pointr)*wv(js) 30 continue pointr = mod(pointr,m) + 1 40 continue do 50 i = 1, nsub d(i) = d(i)/theta 50 continue c Backtrack to the feasible region. alpha = one temp1 = alpha do 60 i = 1, nsub k = ind(i) dk = d(i) if (nbd(k) .ne. 0) then temp1_updated = .false. if (dk .lt. zero .and. nbd(k) .le. 2) then temp2 = l(k) - x(k) if (temp2 .ge. zero) then temp1 = zero temp1_updated = .true. else if (dk*alpha .lt. temp2) then temp1 = temp2/dk temp1_updated = .true. endif else if (dk .gt. zero .and. nbd(k) .ge. 2) then temp2 = u(k) - x(k) if (temp2 .le. zero) then temp1 = zero temp1_updated = .true. else if (dk*alpha .gt. temp2) then temp1 = temp2/dk temp1_updated = .true. endif endif cc logical variable temp1_updated added to eliminate unexpected cc trigger of the if statement due to possible difference between cc hardware precision and double precision. if (temp1_updated .and. temp1 .lt. alpha) then alpha = temp1 ibd = i endif endif 60 continue if (alpha .lt. one) then dk = d(ibd) k = ind(ibd) if (dk .gt. zero) then x(k) = u(k) d(ibd) = zero else if (dk .lt. zero) then x(k) = l(k) d(ibd) = zero endif endif do 70 i = 1, nsub k = ind(i) x(k) = x(k) + alpha*d(i) 70 continue if (iprint .ge. 99) then if (alpha .lt. one) then write (6,1002) alpha else write (6,*) 'SM solution inside the box' end if if (iprint .gt.100) write (6,1003) (x(i),i=1,n) endif if (alpha .lt. one) then iword = 1 else iword = 0 endif if (iprint .ge. 99) write (6,1004) 1001 format (/,'----------------SUBSM entered-----------------',/) 1002 format ( 'ALPHA = ',f7.5,' backtrack to the BOX') 1003 format ('Subspace solution X = ',/,(4x,1p,6(1x,d11.4))) 1004 format (/,'----------------exit SUBSM --------------------',/) return end c====================== The end of subsm =============================== subroutine dcsrch(f,g,stp,ftol,gtol,xtol,stpmin,stpmax, + task,isave,dsave) character*(*) task integer isave(2) double precision f,g,stp,ftol,gtol,xtol,stpmin,stpmax double precision dsave(13) c ********** c c Subroutine dcsrch c c This subroutine finds a step that satisfies a sufficient c decrease condition and a curvature condition. c c Each call of the subroutine updates an interval with c endpoints stx and sty. The interval is initially chosen c so that it contains a minimizer of the modified function c c psi(stp) = f(stp) - f(0) - ftol*stp*f'(0). c c If psi(stp) <= 0 and f'(stp) >= 0 for some step, then the c interval is chosen so that it contains a minimizer of f. c c The algorithm is designed to find a step that satisfies c the sufficient decrease condition c c f(stp) <= f(0) + ftol*stp*f'(0), c c and the curvature condition c c abs(f'(stp)) <= gtol*abs(f'(0)). c c If ftol is less than gtol and if, for example, the function c is bounded below, then there is always a step which satisfies c both conditions. c c If no step can be found that satisfies both conditions, then c the algorithm stops with a warning. In this case stp only c satisfies the sufficient decrease condition. c c A typical invocation of dcsrch has the following outline: c c task = 'START' c 10 continue c call dcsrch( ... ) c if (task .eq. 'FG') then c Evaluate the function and the gradient at stp c goto 10 c end if c c NOTE: The user must no alter work arrays between calls. c c The subroutine statement is c c subroutine dcsrch(f,g,stp,ftol,gtol,xtol,stpmin,stpmax, c task,isave,dsave) c where c c f is a double precision variable. c On initial entry f is the value of the function at 0. c On subsequent entries f is the value of the c function at stp. c On exit f is the value of the function at stp. c c g is a double precision variable. c On initial entry g is the derivative of the function at 0. c On subsequent entries g is the derivative of the c function at stp. c On exit g is the derivative of the function at stp. c c stp is a double precision variable. c On entry stp is the current estimate of a satisfactory c step. On initial entry, a positive initial estimate c must be provided. c On exit stp is the current estimate of a satisfactory step c if task = 'FG'. If task = 'CONV' then stp satisfies c the sufficient decrease and curvature condition. c c ftol is a double precision variable. c On entry ftol specifies a nonnegative tolerance for the c sufficient decrease condition. c On exit ftol is unchanged. c c gtol is a double precision variable. c On entry gtol specifies a nonnegative tolerance for the c curvature condition. c On exit gtol is unchanged. c c xtol is a double precision variable. c On entry xtol specifies a nonnegative relative tolerance c for an acceptable step. The subroutine exits with a c warning if the relative difference between sty and stx c is less than xtol. c On exit xtol is unchanged. c c stpmin is a double precision variable. c On entry stpmin is a nonnegative lower bound for the step. c On exit stpmin is unchanged. c c stpmax is a double precision variable. c On entry stpmax is a nonnegative upper bound for the step. c On exit stpmax is unchanged. c c task is a character variable of length at least 60. c On initial entry task must be set to 'START'. c On exit task indicates the required action: c c If task(1:2) = 'FG' then evaluate the function and c derivative at stp and call dcsrch again. c c If task(1:4) = 'CONV' then the search is successful. c c If task(1:4) = 'WARN' then the subroutine is not able c to satisfy the convergence conditions. The exit value of c stp contains the best point found during the search. c c If task(1:5) = 'ERROR' then there is an error in the c input arguments. c c On exit with convergence, a warning or an error, the c variable task contains additional information. c c isave is an integer work array of dimension 2. c c dsave is a double precision work array of dimension 13. c c Subprograms called c c MINPACK-2 ... dcstep c c MINPACK-1 Project. June 1983. c Argonne National Laboratory. c Jorge J. More' and David J. Thuente. c c MINPACK-2 Project. October 1993. c Argonne National Laboratory and University of Minnesota. c Brett M. Averick, Richard G. Carter, and Jorge J. More'. c c ********** double precision zero,p5,p66 parameter(zero=0.0d0,p5=0.5d0,p66=0.66d0) double precision xtrapl,xtrapu parameter(xtrapl=1.1d0,xtrapu=4.0d0) logical brackt integer stage double precision finit,ftest,fm,fx,fxm,fy,fym,ginit,gtest, + gm,gx,gxm,gy,gym,stx,sty,stmin,stmax,width,width1 c Initialization block. if (task(1:5) .eq. 'START') then c Check the input arguments for errors. if (stp .lt. stpmin) task = 'ERROR: STP .LT. STPMIN' if (stp .gt. stpmax) task = 'ERROR: STP .GT. STPMAX' if (g .ge. zero) task = 'ERROR: INITIAL G .GE. ZERO' if (ftol .lt. zero) task = 'ERROR: FTOL .LT. ZERO' if (gtol .lt. zero) task = 'ERROR: GTOL .LT. ZERO' if (xtol .lt. zero) task = 'ERROR: XTOL .LT. ZERO' if (stpmin .lt. zero) task = 'ERROR: STPMIN .LT. ZERO' if (stpmax .lt. stpmin) task = 'ERROR: STPMAX .LT. STPMIN' c Exit if there are errors on input. if (task(1:5) .eq. 'ERROR') return c Initialize local variables. brackt = .false. stage = 1 finit = f ginit = g gtest = ftol*ginit width = stpmax - stpmin width1 = width/p5 c The variables stx, fx, gx contain the values of the step, c function, and derivative at the best step. c The variables sty, fy, gy contain the value of the step, c function, and derivative at sty. c The variables stp, f, g contain the values of the step, c function, and derivative at stp. stx = zero fx = finit gx = ginit sty = zero fy = finit gy = ginit stmin = zero stmax = stp + xtrapu*stp task = 'FG' goto 1000 else c Restore local variables. if (isave(1) .eq. 1) then brackt = .true. else brackt = .false. endif stage = isave(2) ginit = dsave(1) gtest = dsave(2) gx = dsave(3) gy = dsave(4) finit = dsave(5) fx = dsave(6) fy = dsave(7) stx = dsave(8) sty = dsave(9) stmin = dsave(10) stmax = dsave(11) width = dsave(12) width1 = dsave(13) endif c If psi(stp) <= 0 and f'(stp) >= 0 for some step, then the c algorithm enters the second stage. ftest = finit + stp*gtest if (stage .eq. 1 .and. f .le. ftest .and. g .ge. zero) + stage = 2 c Test for warnings. if (brackt .and. (stp .le. stmin .or. stp .ge. stmax)) + task = 'WARNING: ROUNDING ERRORS PREVENT PROGRESS' if (brackt .and. stmax - stmin .le. xtol*stmax) + task = 'WARNING: XTOL TEST SATISFIED' if (stp .eq. stpmax .and. f .le. ftest .and. g .le. gtest) + task = 'WARNING: STP = STPMAX' if (stp .eq. stpmin .and. (f .gt. ftest .or. g .ge. gtest)) + task = 'WARNING: STP = STPMIN' cc New warning statement added to eliminate the unexpected case cc of stp=stx due to possible difference between hardware precision cc and double precision. if (stp .eq. stx) + task = 'WARNING: STP = STX' c Test for convergence. if (f .le. ftest .and. abs(g) .le. gtol*(-ginit)) + task = 'CONVERGENCE' c Test for termination. if (task(1:4) .eq. 'WARN' .or. task(1:4) .eq. 'CONV') goto 1000 c A modified function is used to predict the step during the c first stage if a lower function value has been obtained but c the decrease is not sufficient. if (stage .eq. 1 .and. f .le. fx .and. f .gt. ftest) then c Define the modified function and derivative values. fm = f - stp*gtest fxm = fx - stx*gtest fym = fy - sty*gtest gm = g - gtest gxm = gx - gtest gym = gy - gtest c Call dcstep to update stx, sty, and to compute the new step. call dcstep(stx,fxm,gxm,sty,fym,gym,stp,fm,gm, + brackt,stmin,stmax) c Reset the function and derivative values for f. fx = fxm + stx*gtest fy = fym + sty*gtest gx = gxm + gtest gy = gym + gtest else c Call dcstep to update stx, sty, and to compute the new step. call dcstep(stx,fx,gx,sty,fy,gy,stp,f,g, + brackt,stmin,stmax) endif c Decide if a bisection step is needed. if (brackt) then if (abs(sty-stx) .ge. p66*width1) stp = stx + p5*(sty - stx) width1 = width width = abs(sty-stx) endif c Set the minimum and maximum steps allowed for stp. if (brackt) then stmin = min(stx,sty) stmax = max(stx,sty) else stmin = stp + xtrapl*(stp - stx) stmax = stp + xtrapu*(stp - stx) endif c Force the step to be within the bounds stpmax and stpmin. stp = max(stp,stpmin) stp = min(stp,stpmax) c If further progress is not possible, let stp be the best c point obtained during the search. if (brackt .and. (stp .le. stmin .or. stp .ge. stmax) + .or. (brackt .and. stmax-stmin .le. xtol*stmax)) stp = stx c Obtain another function and derivative. task = 'FG' 1000 continue c Save local variables. if (brackt) then isave(1) = 1 else isave(1) = 0 endif isave(2) = stage dsave(1) = ginit dsave(2) = gtest dsave(3) = gx dsave(4) = gy dsave(5) = finit dsave(6) = fx dsave(7) = fy dsave(8) = stx dsave(9) = sty dsave(10) = stmin dsave(11) = stmax dsave(12) = width dsave(13) = width1 end c====================== The end of dcsrch ============================== subroutine dcstep(stx,fx,dx,sty,fy,dy,stp,fp,dp,brackt, + stpmin,stpmax) logical brackt double precision stx,fx,dx,sty,fy,dy,stp,fp,dp,stpmin,stpmax c ********** c c Subroutine dcstep c c This subroutine computes a safeguarded step for a search c procedure and updates an interval that contains a step that c satisfies a sufficient decrease and a curvature condition. c c The parameter stx contains the step with the least function c value. If brackt is set to .true. then a minimizer has c been bracketed in an interval with endpoints stx and sty. c The parameter stp contains the current step. c The subroutine assumes that if brackt is set to .true. then c c min(stx,sty) < stp < max(stx,sty), c c and that the derivative at stx is negative in the direction c of the step. c c The subroutine statement is c c subroutine dcstep(stx,fx,dx,sty,fy,dy,stp,fp,dp,brackt, c stpmin,stpmax) c c where c c stx is a double precision variable. c On entry stx is the best step obtained so far and is an c endpoint of the interval that contains the minimizer. c On exit stx is the updated best step. c c fx is a double precision variable. c On entry fx is the function at stx. c On exit fx is the function at stx. c c dx is a double precision variable. c On entry dx is the derivative of the function at c stx. The derivative must be negative in the direction of c the step, that is, dx and stp - stx must have opposite c signs. c On exit dx is the derivative of the function at stx. c c sty is a double precision variable. c On entry sty is the second endpoint of the interval that c contains the minimizer. c On exit sty is the updated endpoint of the interval that c contains the minimizer. c c fy is a double precision variable. c On entry fy is the function at sty. c On exit fy is the function at sty. c c dy is a double precision variable. c On entry dy is the derivative of the function at sty. c On exit dy is the derivative of the function at the exit sty. c c stp is a double precision variable. c On entry stp is the current step. If brackt is set to .true. c then on input stp must be between stx and sty. c On exit stp is a new trial step. c c fp is a double precision variable. c On entry fp is the function at stp c On exit fp is unchanged. c c dp is a double precision variable. c On entry dp is the the derivative of the function at stp. c On exit dp is unchanged. c c brackt is an logical variable. c On entry brackt specifies if a minimizer has been bracketed. c Initially brackt must be set to .false. c On exit brackt specifies if a minimizer has been bracketed. c When a minimizer is bracketed brackt is set to .true. c c stpmin is a double precision variable. c On entry stpmin is a lower bound for the step. c On exit stpmin is unchanged. c c stpmax is a double precision variable. c On entry stpmax is an upper bound for the step. c On exit stpmax is unchanged. c c MINPACK-1 Project. June 1983 c Argonne National Laboratory. c Jorge J. More' and David J. Thuente. c c MINPACK-2 Project. October 1993. c Argonne National Laboratory and University of Minnesota. c Brett M. Averick and Jorge J. More'. c c ********** double precision zero,p66,two,three parameter(zero=0.0d0,p66=0.66d0,two=2.0d0,three=3.0d0) double precision gamma,p,q,r,s,sgnd,stpc,stpf,stpq,theta sgnd = dp*(dx/abs(dx)) c First case: A higher function value. The minimum is bracketed. c If the cubic step is closer to stx than the quadratic step, the c cubic step is taken, otherwise the average of the cubic and c quadratic steps is taken. if (fp .gt. fx) then theta = three*(fx - fp)/(stp - stx) + dx + dp s = max(abs(theta),abs(dx),abs(dp)) gamma = s*sqrt((theta/s)**2 - (dx/s)*(dp/s)) if (stp .lt. stx) gamma = -gamma p = (gamma - dx) + theta q = ((gamma - dx) + gamma) + dp r = p/q stpc = stx + r*(stp - stx) stpq = stx + ((dx/((fx - fp)/(stp - stx) + dx))/two)* + (stp - stx) if (abs(stpc-stx) .lt. abs(stpq-stx)) then stpf = stpc else stpf = stpc + (stpq - stpc)/two endif brackt = .true. c Second case: A lower function value and derivatives of opposite c sign. The minimum is bracketed. If the cubic step is farther from c stp than the secant step, the cubic step is taken, otherwise the c secant step is taken. else if (sgnd .lt. zero) then theta = three*(fx - fp)/(stp - stx) + dx + dp s = max(abs(theta),abs(dx),abs(dp)) gamma = s*sqrt((theta/s)**2 - (dx/s)*(dp/s)) if (stp .gt. stx) gamma = -gamma p = (gamma - dp) + theta q = ((gamma - dp) + gamma) + dx r = p/q stpc = stp + r*(stx - stp) stpq = stp + (dp/(dp - dx))*(stx - stp) if (abs(stpc-stp) .gt. abs(stpq-stp)) then stpf = stpc else stpf = stpq endif brackt = .true. c Third case: A lower function value, derivatives of the same sign, c and the magnitude of the derivative decreases. else if (abs(dp) .lt. abs(dx)) then c The cubic step is computed only if the cubic tends to infinity c in the direction of the step or if the minimum of the cubic c is beyond stp. Otherwise the cubic step is defined to be the c secant step. theta = three*(fx - fp)/(stp - stx) + dx + dp s = max(abs(theta),abs(dx),abs(dp)) c The case gamma = 0 only arises if the cubic does not tend c to infinity in the direction of the step. gamma = s*sqrt(max(zero,(theta/s)**2-(dx/s)*(dp/s))) if (stp .gt. stx) gamma = -gamma p = (gamma - dp) + theta q = (gamma + (dx - dp)) + gamma r = p/q if (r .lt. zero .and. gamma .ne. zero) then stpc = stp + r*(stx - stp) else if (stp .gt. stx) then stpc = stpmax else stpc = stpmin endif stpq = stp + (dp/(dp - dx))*(stx - stp) if (brackt) then c A minimizer has been bracketed. If the cubic step is c closer to stp than the secant step, the cubic step is c taken, otherwise the secant step is taken. if (abs(stpc-stp) .lt. abs(stpq-stp)) then stpf = stpc else stpf = stpq endif if (stp .gt. stx) then stpf = min(stp+p66*(sty-stp),stpf) else stpf = max(stp+p66*(sty-stp),stpf) endif else c A minimizer has not been bracketed. If the cubic step is c farther from stp than the secant step, the cubic step is c taken, otherwise the secant step is taken. if (abs(stpc-stp) .gt. abs(stpq-stp)) then stpf = stpc else stpf = stpq endif stpf = min(stpmax,stpf) stpf = max(stpmin,stpf) endif c Fourth case: A lower function value, derivatives of the c same sign, and the magnitude of the derivative does not c decrease. If the minimum is not bracketed, the step is either c stpmin or stpmax, otherwise the cubic step is taken. else if (brackt) then theta = three*(fp - fy)/(sty - stp) + dy + dp s = max(abs(theta),abs(dy),abs(dp)) gamma = s*sqrt((theta/s)**2 - (dy/s)*(dp/s)) if (stp .gt. sty) gamma = -gamma p = (gamma - dp) + theta q = ((gamma - dp) + gamma) + dy r = p/q stpc = stp + r*(sty - stp) stpf = stpc else if (stp .gt. stx) then stpf = stpmax else stpf = stpmin endif endif c Update the interval which contains a minimizer. if (fp .gt. fx) then sty = stp fy = fp dy = dp else if (sgnd .lt. zero) then sty = stx fy = fx dy = dx endif stx = stp fx = fp dx = dp endif c Compute the new step. stp = stpf end c====================== The end of dcstep ============================== subroutine timer(ttime) double precision ttime c ********* c c Subroutine timer c c This subroutine is used to determine user time. In a typical c application, the user time for a code segment requires calls c to subroutine timer to determine the initial and final time. c c The subroutine statement is c c subroutine timer(ttime) c c where c c ttime is an output variable which specifies the user time. c c Argonne National Laboratory and University of Minnesota. c MINPACK-2 Project. c c Modified October 1990 by Brett M. Averick. c c ********** real temp real tarray(2) real etime c The first element of the array tarray specifies user time temp = etime(tarray) ttime = dble(tarray(1)) return end c====================== The end of timer =============================== double precision function dnrm2(n,x,incx) integer n,incx double precision x(n) c ********** c c Function dnrm2 c c Given a vector x of length n, this function calculates the c Euclidean norm of x with stride incx. c c The function statement is c c double precision function dnrm2(n,x,incx) c c where c c n is a positive integer input variable. c c x is an input array of length n. c c incx is a positive integer variable that specifies the c stride of the vector. c c Subprograms called c c FORTRAN-supplied ... abs, max, sqrt c c MINPACK-2 Project. February 1991. c Argonne National Laboratory. c Brett M. Averick. c c ********** integer i double precision scale dnrm2 = 0.0d0 scale = 0.0d0 do 10 i = 1, n, incx scale = max(scale, abs(x(i))) 10 continue if (scale .eq. 0.0d0) return do 20 i = 1, n, incx dnrm2 = dnrm2 + (x(i)/scale)**2 20 continue dnrm2 = scale*sqrt(dnrm2) return end c====================== The end of dnrm2 =============================== double precision function dpmeps() c ********** c c Subroutine dpeps c c This subroutine computes the machine precision parameter c dpmeps as the smallest floating point number such that c 1 + dpmeps differs from 1. c c This subroutine is based on the subroutine machar described in c c W. J. Cody, c MACHAR: A subroutine to dynamically determine machine parameters, c ACM Trans. Math. Soft., 14, 1988, pages 303-311. c c The subroutine statement is: c c subroutine dpeps(dpmeps) c c where c c dpmeps is a double precision variable. c On entry dpmeps need not be specified. c On exit dpmeps is the machine precision. c c MINPACK-2 Project. February 1991. c Argonne National Laboratory and University of Minnesota. c Brett M. Averick. c c ******* integer i,ibeta,irnd,it,itemp,negep double precision a,b,beta,betain,betah,temp,tempa,temp1, + zero,one,two data zero,one,two /0.0d0,1.0d0,2.0d0/ c determine ibeta, beta ala malcolm. a = one b = one 10 continue a = a + a temp = a + one temp1 = temp - a if (temp1 - one .eq. zero) go to 10 20 continue b = b + b temp = a + b itemp = int(temp - a) if (itemp .eq. 0) go to 20 ibeta = itemp beta = dble(ibeta) c determine it, irnd. it = 0 b = one 30 continue it = it + 1 b = b * beta temp = b + one temp1 = temp - b if (temp1 - one .eq. zero) go to 30 irnd = 0 betah = beta/two temp = a + betah if (temp - a .ne. zero) irnd = 1 tempa = a + beta temp = tempa + betah if ((irnd .eq. 0) .and. (temp - tempa .ne. zero)) irnd = 2 c determine dpmeps. negep = it + 3 betain = one/beta a = one do 40 i = 1, negep a = a*betain 40 continue 50 continue temp = one + a if (temp - one .ne. zero) go to 60 a = a*beta go to 50 60 continue dpmeps = a if ((ibeta .eq. 2) .or. (irnd .eq. 0)) go to 70 a = (a*(one + a))/two temp = one + a if (temp - one .ne. zero) dpmeps = a 70 return end c====================== The end of dpmeps ============================== subroutine daxpy(n,da,dx,incx,dy,incy) c c constant times a vector plus a vector. c uses unrolled loops for increments equal to one. c jack dongarra, linpack, 3/11/78. c double precision dx(*),dy(*),da integer i,incx,incy,ix,iy,m,mp1,n c if(n.le.0)return if (da .eq. 0.0d0) return if(incx.eq.1.and.incy.eq.1)go to 20 c c code for unequal increments or equal increments c not equal to 1 c ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do 10 i = 1,n dy(iy) = dy(iy) + da*dx(ix) ix = ix + incx iy = iy + incy 10 continue return c c code for both increments equal to 1 c c c clean-up loop c 20 m = mod(n,4) if( m .eq. 0 ) go to 40 do 30 i = 1,m dy(i) = dy(i) + da*dx(i) 30 continue if( n .lt. 4 ) return 40 mp1 = m + 1 do 50 i = mp1,n,4 dy(i) = dy(i) + da*dx(i) dy(i + 1) = dy(i + 1) + da*dx(i + 1) dy(i + 2) = dy(i + 2) + da*dx(i + 2) dy(i + 3) = dy(i + 3) + da*dx(i + 3) 50 continue return end c====================== The end of daxpy =============================== subroutine dcopy(n,dx,incx,dy,incy) c c copies a vector, x, to a vector, y. c uses unrolled loops for increments equal to one. c jack dongarra, linpack, 3/11/78. c double precision dx(*),dy(*) integer i,incx,incy,ix,iy,m,mp1,n c if(n.le.0)return if(incx.eq.1.and.incy.eq.1)go to 20 c c code for unequal increments or equal increments c not equal to 1 c ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do 10 i = 1,n dy(iy) = dx(ix) ix = ix + incx iy = iy + incy 10 continue return c c code for both increments equal to 1 c c c clean-up loop c 20 m = mod(n,7) if( m .eq. 0 ) go to 40 do 30 i = 1,m dy(i) = dx(i) 30 continue if( n .lt. 7 ) return 40 mp1 = m + 1 do 50 i = mp1,n,7 dy(i) = dx(i) dy(i + 1) = dx(i + 1) dy(i + 2) = dx(i + 2) dy(i + 3) = dx(i + 3) dy(i + 4) = dx(i + 4) dy(i + 5) = dx(i + 5) dy(i + 6) = dx(i + 6) 50 continue return end c====================== The end of dcopy =============================== double precision function ddot(n,dx,incx,dy,incy) c c forms the dot product of two vectors. c uses unrolled loops for increments equal to one. c jack dongarra, linpack, 3/11/78. c double precision dx(*),dy(*),dtemp integer i,incx,incy,ix,iy,m,mp1,n c ddot = 0.0d0 dtemp = 0.0d0 if(n.le.0)return if(incx.eq.1.and.incy.eq.1)go to 20 c c code for unequal increments or equal increments c not equal to 1 c ix = 1 iy = 1 if(incx.lt.0)ix = (-n+1)*incx + 1 if(incy.lt.0)iy = (-n+1)*incy + 1 do 10 i = 1,n dtemp = dtemp + dx(ix)*dy(iy) ix = ix + incx iy = iy + incy 10 continue ddot = dtemp return c c code for both increments equal to 1 c c c clean-up loop c 20 m = mod(n,5) if( m .eq. 0 ) go to 40 do 30 i = 1,m dtemp = dtemp + dx(i)*dy(i) 30 continue if( n .lt. 5 ) go to 60 40 mp1 = m + 1 do 50 i = mp1,n,5 dtemp = dtemp + dx(i)*dy(i) + dx(i + 1)*dy(i + 1) + * dx(i + 2)*dy(i + 2) + dx(i + 3)*dy(i + 3) + dx(i + 4)*dy(i + 4) 50 continue 60 ddot = dtemp return end c====================== The end of ddot ================================ subroutine dpofa(a,lda,n,info) integer lda,n,info double precision a(lda,*) c c dpofa factors a double precision symmetric positive definite c matrix. c c dpofa is usually called by dpoco, but it can be called c directly with a saving in time if rcond is not needed. c (time for dpoco) = (1 + 18/n)*(time for dpofa) . c c on entry c c a double precision(lda, n) c the symmetric matrix to be factored. only the c diagonal and upper triangle are used. c c lda integer c the leading dimension of the array a . c c n integer c the order of the matrix a . c c on return c c a an upper triangular matrix r so that a = trans(r)*r c where trans(r) is the transpose. c the strict lower triangle is unaltered. c if info .ne. 0 , the factorization is not complete. c c info integer c = 0 for normal return. c = k signals an error condition. the leading minor c of order k is not positive definite. c c linpack. this version dated 08/14/78 . c cleve moler, university of new mexico, argonne national lab. c c subroutines and functions c c blas ddot c fortran sqrt c c internal variables c double precision ddot,t double precision s integer j,jm1,k c begin block with ...exits to 40 c c do 30 j = 1, n info = j s = 0.0d0 jm1 = j - 1 if (jm1 .lt. 1) go to 20 do 10 k = 1, jm1 t = a(k,j) - ddot(k-1,a(1,k),1,a(1,j),1) t = t/a(k,k) a(k,j) = t s = s + t*t 10 continue 20 continue s = a(j,j) - s c ......exit if (s .le. 0.0d0) go to 40 a(j,j) = sqrt(s) 30 continue info = 0 40 continue return end c====================== The end of dpofa =============================== subroutine dscal(n,da,dx,incx) c c scales a vector by a constant. c uses unrolled loops for increment equal to one. c jack dongarra, linpack, 3/11/78. c modified 3/93 to return if incx .le. 0. c double precision da,dx(*) integer i,incx,m,mp1,n,nincx c if( n.le.0 .or. incx.le.0 )return if(incx.eq.1)go to 20 c c code for increment not equal to 1 c nincx = n*incx do 10 i = 1,nincx,incx dx(i) = da*dx(i) 10 continue return c c code for increment equal to 1 c c c clean-up loop c 20 m = mod(n,5) if( m .eq. 0 ) go to 40 do 30 i = 1,m dx(i) = da*dx(i) 30 continue if( n .lt. 5 ) return 40 mp1 = m + 1 do 50 i = mp1,n,5 dx(i) = da*dx(i) dx(i + 1) = da*dx(i + 1) dx(i + 2) = da*dx(i + 2) dx(i + 3) = da*dx(i + 3) dx(i + 4) = da*dx(i + 4) 50 continue return end c====================== The end of dscal =============================== subroutine dtrsl(t,ldt,n,b,job,info) integer ldt,n,job,info double precision t(ldt,*),b(*) c c c dtrsl solves systems of the form c c t * x = b c or c trans(t) * x = b c c where t is a triangular matrix of order n. here trans(t) c denotes the transpose of the matrix t. c c on entry c c t double precision(ldt,n) c t contains the matrix of the system. the zero c elements of the matrix are not referenced, and c the corresponding elements of the array can be c used to store other information. c c ldt integer c ldt is the leading dimension of the array t. c c n integer c n is the order of the system. c c b double precision(n). c b contains the right hand side of the system. c c job integer c job specifies what kind of system is to be solved. c if job is c c 00 solve t*x=b, t lower triangular, c 01 solve t*x=b, t upper triangular, c 10 solve trans(t)*x=b, t lower triangular, c 11 solve trans(t)*x=b, t upper triangular. c c on return c c b b contains the solution, if info .eq. 0. c otherwise b is unaltered. c c info integer c info contains zero if the system is nonsingular. c otherwise info contains the index of c the first zero diagonal element of t. c c linpack. this version dated 08/14/78 . c g. w. stewart, university of maryland, argonne national lab. c c subroutines and functions c c blas daxpy,ddot c fortran mod c c internal variables c double precision ddot,temp integer case,j,jj c c begin block permitting ...exits to 150 c c check for zero diagonal elements. c do 10 info = 1, n c ......exit if (t(info,info) .eq. 0.0d0) go to 150 10 continue info = 0 c c determine the task and go to it. c case = 1 if (mod(job,10) .ne. 0) case = 2 if (mod(job,100)/10 .ne. 0) case = case + 2 go to (20,50,80,110), case c c solve t*x=b for t lower triangular c 20 continue b(1) = b(1)/t(1,1) if (n .lt. 2) go to 40 do 30 j = 2, n temp = -b(j-1) call daxpy(n-j+1,temp,t(j,j-1),1,b(j),1) b(j) = b(j)/t(j,j) 30 continue 40 continue go to 140 c c solve t*x=b for t upper triangular. c 50 continue b(n) = b(n)/t(n,n) if (n .lt. 2) go to 70 do 60 jj = 2, n j = n - jj + 1 temp = -b(j+1) call daxpy(j,temp,t(1,j+1),1,b(1),1) b(j) = b(j)/t(j,j) 60 continue 70 continue go to 140 c c solve trans(t)*x=b for t lower triangular. c 80 continue b(n) = b(n)/t(n,n) if (n .lt. 2) go to 100 do 90 jj = 2, n j = n - jj + 1 b(j) = b(j) - ddot(jj-1,t(j+1,j),1,b(j+1),1) b(j) = b(j)/t(j,j) 90 continue 100 continue go to 140 c c solve trans(t)*x=b for t upper triangular. c 110 continue b(1) = b(1)/t(1,1) if (n .lt. 2) go to 130 do 120 j = 2, n b(j) = b(j) - ddot(j-1,t(1,j),1,b(1),1) b(j) = b(j)/t(j,j) 120 continue 130 continue 140 continue 150 continue return end c====================== The end of dtrsl =============================== ================================================ FILE: code/gpml/util/lbfgsb.m ================================================ % LBFGSB Call the nonlinear bound-constrained solver that uses % limited-memory BFGS quasi-Newton updates. % % The basic function call is % % LBFGSB(x0,lb,ub,objfunc,gradfunc) % % The first input argument x0 is either a matrix or a cell array of % matrices. It declares the starting point for the solver. % % The second and third input arguments lb and ub must be of the exact same % structure as x0. They declare the lower and upper bounds on the % variables, respectively. Set an entry to Inf to declare no bound. % % The next two input arguments must be the names of MATLAB routines % (M-files): % % objfunc Calculates the objective function at the current % point. The routine must accept as many inputs as cell % entry in x0 (or one input if x0 is a matrix). The % output must be a scalar representing the objective % evaluated at the current point. % % gradfunc Computes the gradient of the objective at the current % point. The input is the same as objfunc, but it must % return as many outputs as there are inputs, and each % of the outputs have the same matrix structure as its % corresponding input. % % Optionally, one may choose to pass additional auxiliary data to the % MATLAB callback routines listed above through the function call % LBFGSB(...,auxdata). If auxdata is the empty matrix, no extra % information is passed. It is important to observe that the auxiliary % data MAY NOT CHANGE through the course of the L-BFGS optimization! The % auxiliary data keep the same values as they possessed in the initial % call. If you need variables that change over time, you may want to % consider global variables (type HELP GLOBAL). % % LBFGSB(...,auxdata,iterfunc) specifies an additional callback routine % which is called once per algorithm iteration. The callback routine must % take the form ITERFUNC(T,F,X1,X2,...,AUXDATA). T is the current % iteration of the algorithm. F is the current value of the objective. The % inputs X1, X2,... are exactly the same as the ones passed to objfunc and % gradfunc. Finally, extra information may be passed through the input % AUXDATA. No outputs are expected from iterfunc. If iterfunc is the empty % string, no routine is called. % % These inputs may be followed by parameter/value pairs to modify the % default algorithm options. The algorithm option 'maxiter' sets the % maximum number of iterations. The rest of the options 'm', 'factr' and % 'pgtol' are described in the L-BFGS-B documentation. Note that in this % implementation we divide factr by the machine precision, so that the % convergence condition is satisfied when it is less than factr, not less % than factr times the machine precision. % % Interface has been written by Peter Carbonetto. % Compilation instructions by Hannes Nickisch, 2010-11-24 function x = lbfgsb(x,varargin) fprintf('\n\n\n') fprintf('##########################################################################\n') fprintf('In order to use the L-BFGS minimiser you have to compile Peter Carbonetto''s\n') fprintf('"Matlab interface for L-BFGS-B" [1]. The challenge here is the Fortran 77 code.\n') fprintf('We provide a Makefile suitable for Linux 32/64 bit and Mac whenever you have g77\n') fprintf('properly installed.Under Ubuntu, you can achieve this by installing the package\n') fprintf('fort77, for example.\n\n') fprintf('Compilation is done by editing lbfgsb/Makefile. In any case, you need to\n') fprintf('provide MATLAB_HOME which can be found by the commands ''locate matlab'' or \n') fprintf('''find / -name "matlab"''. You can choose between two compilation modes:\n') fprintf('a) using the mex utility by Matlab [default]\n') fprintf(' provide MEX, then type ''make mex''\n\n') fprintf('b) without mex utility by Matlab\n') fprintf(' provide MEX_SUFFIX and MATLAB_LIB, then type ''make nomex''\n\n') fprintf('In Ubuntu 10.04 LTS, the libg2c library needed for both a) and b) is not\n') fprintf('included per default. If ''ls /usr/lib/libg2c.*'' does not list anything\n') fprintf('this is the case on your machine. You then whant to install the packages\n') fprintf('gcc-3.4-base and libg2c0 e.g. from\n') fprintf('http://packages.ubuntu.com/hardy/gcc-3.4-base and\n') fprintf('http://packages.ubuntu.com/hardy/libg2c0.\n') fprintf('After installation, you have to create a symbolic link by ''cd /usr/lib'' and\n') fprintf('''ln -s libg2c.so.0 libg2c.so''.\n\n') fprintf('[1] http://www.cs.ubc.ca/~pcarbo/lbfgsb-for-matlab.html\n') fprintf('[2] http://www.mathworks.com/support/compilers/R2010a\n') fprintf('##########################################################################\n') fprintf('\n\n\n') error ================================================ FILE: code/gpml/util/make.m ================================================ % For compilation under Linux/Octave you need to install the package % octavex.x-headers. % Under Windows/Octave, compilation works out of the box. % % There is a big variety of platforms running Matlab around i.e. % 'SUN4','SOL2','HP700','SGI','SGI64','IBM_RS','ALPHA', % 'LNX86','GLNX86','GLNXA64', % 'MAC', 'MACI' % the following script is only tested for the most common ones (Mac,Linux,Win) % but should work with minor modifications on your architecture. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch 2010-12-16. OCTAVE = exist('OCTAVE_VERSION') ~= 0; % check if we run Matlab or Octave %% 0) change directory me = mfilename; % what is my filename mydir = which(me); mydir = mydir(1:end-2-numel(me)); % where am I located cd(mydir) if OCTAVE %% compile solve_chol mkoctfile --mex solve_chol.c delete solve_chol.o else % MATLAB %% compile solve_chol if ispc % Windows libext = 'lib'; % remove the trailing underscore after the Lapack function "dpotrs" % we do that in a clumsy way since there is no sed/awk under Windows fid = fopen('solve_chol.c','r'); fid_win = fopen('solve_chol2XXXX.c','w'); while 1 tline = fgetl(fid); if ~ischar(tline), break, end id = strfind(tline,'dpotrs'); if numel(id) tline = [tline(1:id-1),'dpotrs',tline(id+7:end)]; end fprintf(fid_win,[tline,'\n']); end fclose(fid_win); fclose(fid); if numel(strfind(computer,'64')) % 64 bit ospath = 'extern/lib/win64/microsoft'; else % 32 bit ospath = 'extern/lib/win32/lcc'; end % compile comp_cmd = 'mex -O solve_chol2XXXX.c -output solve_chol'; eval([comp_cmd,' ''',matlabroot,'/',ospath,'/libmwlapack.',libext,'''']) delete solve_chol2XXXX* else % Linux, MAC, Solaris mex -O -lmwlapack solve_chol.c end end ================================================ FILE: code/gpml/util/minimize.m ================================================ % minimize.m - minimize a smooth differentiable multivariate function using % LBFGS (Limited memory LBFGS) or CG (Conjugate Gradients) % Usage: [X, fX, i] = minimize(X, F, p, other, ... ) % where % X is an initial guess (any type: vector, matrix, cell array, struct) % F is the objective function (function pointer or name) % p parameters - if p is a number, it corresponds to p.length below % p.length allowed 1) # linesearches or 2) if -ve minus # func evals % p.method minimization method, 'LBFGS' (default) or 'CG' % p.verbosity 0 quiet, 1 line, 2 line + warnings (default), 3 graphical % p.mem number of directions used in LBFGS (default 4) % other, ... other parameters, passed to the function F % X returned minimizer % fX vector of function values showing minimization progress % i final number of linesearches or function evaluations % The function F must take the following syntax [f, df] = F(X, other, ...) % where f is the function value and df its partial derivatives. The types of X % and df must be identical (vector, matrix, cell array, struct, etc). % % Copyright (C) 1996 - 2011 by Carl Edward Rasmussen, 2011-05-20. % Permission is hereby granted, free of charge, to any person OBTAINING A COPY % OF THIS SOFTWARE AND ASSOCIATED DOCUMENTATION FILES (THE "SOFTWARE"), TO DEAL % IN THE SOFTWARE WITHOUT RESTRICTION, INCLUDING WITHOUT LIMITATION THE RIGHTS % to use, copy, modify, merge, publish, distribute, sublicense, and/or sell % copies of the Software, and to permit persons to whom the Software is % furnished to do so, subject to the following conditions: % % The above copyright notice and this permission notice shall be included in % all copies or substantial portions of the Software. % % THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR % IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, % FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE % AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER % LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, % OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE % SOFTWARE. function [X, fX, i] = minimize(X, F, p, varargin) if isnumeric(p), p = struct('length', p); end % convert p to struct if p.length > 0, p.S = 'linesearch #'; else p.S = 'function evaluation #'; end; if ~isfield(p,'method'), p.method = @LBFGS; end % set default method if ~isfield(p,'verbosity'), p.verbosity = 2; end % default 1 line text output if ~isfield(p,'MFEPLS'), p.MFEPLS = 20; end % Max Func Evals Per Line Search if ~isfield(p,'MSR'), p.MSR = 100; end % Max Slope Ratio default x = unwrap(X); % convert initial guess to vector f(F, X, varargin{:}); % set up the function f [fx, dfx] = f(x); % initial function value and derivatives if p.verbosity, printf('Initial Function Value %4.6e\r', fx); end if p.verbosity > 2, clf; subplot(211); hold on; xlabel(p.S); ylabel('function value'); plot(p.length < 0, fx, '+'); drawnow; end [x, fX, i] = feval(p.method, x, fx, dfx, p); % minimize using direction method X = rewrap(X, x); % convert answer to original representation if p.verbosity, printf('\n'); end function [x, fx, i] = CG(x0, fx0, dfx0, p); i = p.length < 0; % initialize resource counter r = -dfx0; s = -r'*r; b = -1/s; bs = -1; ok = 0; fx = fx0; % steepest descent while i < abs(p.length) b = b*bs/min(b*s,bs/p.MSR); % suitable initial step size using slope ratio [x, b, fx0, dfx, i] = lineSearch(x0, fx0, dfx0, r, s, b, i, p); if i < 0 % if line search failed i = -i; if ok, ok = 0; r = -dfx; else break; end % try steepest or stop else ok = 1; bs = b*s; % record step times slope (for slope ratio method) r = (dfx'*(dfx-dfx0))/(dfx0'*dfx0)*r - dfx; % Polack-Ribiere CG end s = r'*dfx; if s >= 0, r = -dfx; s = r'*dfx; ok = 0; end % slope must be -ve x0 = x; dfx0 = dfx; fx = [fx; fx0]; % replace old values with new ones end function [x, fx, i] = LBFGS(x0, fx0, dfx0, p); n = length(x0); k = 0; ok = 0; fx = fx0; bs = -1/p.MSR; if isfield(p, 'mem'), m = p.mem; else m = min(4, n); end t = zeros(n, m); y = zeros(n, m); % allocate memory i = p.length < 0; % initialize resource counter while i < abs(p.length) q = dfx0; for j = rem(k-1:-1:max(0,k-m),m)+1 a(j) = t(:,j)'*q/rho(j); q = q-a(j)*y(:,j); end if k == 0, r = -q/(q'*q); else r = -t(:,j)'*y(:,j)/(y(:,j)'*y(:,j))*q; end for j = rem(max(0,k-m):k-1,m)+1 r = r-t(:,j)*(a(j)+y(:,j)'*r/rho(j)); end s = r'*dfx0; if s >= 0, r = -dfx0; s = r'*dfx0; k = 0; ok = 0; end b = bs/min(bs,s/p.MSR); % suitable initial step size (usually 1) [x, b, fx0, dfx, i] = lineSearch(x0, fx0, dfx0, r, s, b, i, p); if i < 0 % if line search failed i = -i; if ok, ok = 0; k = 0; else break; end % try steepest or stop else j = rem(k,m)+1; t(:,j) = x-x0; y(:,j) = dfx-dfx0; rho(j) = t(:,j)'*y(:,j); ok = 1; k = k+1; bs = b*s; end x0 = x; dfx0 = dfx; fx = [fx; fx0]; % replace and add values end function [x, a, fx, df, i] = lineSearch(x0, f0, df0, d, s, a, i, p) INT = 0.1; EXT = 5.0; % interpolate and extrapolation limits SIG = 0.1; RHO = SIG/2; % Wolfe-Powell condition constants, 0 2 A = [-a a]/5; nd = norm(d); subplot(212); hold off; plot(0, f0, 'k+'); hold on; plot(nd*A, f0+s*A, 'k-'); xlabel('distance in line search direction'); ylabel('function value'); end while 1 % keep extrapolating as long as necessary ok = 0; while ~ok & j < LIMIT try % try, catch and bisect to safeguard extrapolation evaluation [p3.f p3.df] = f(x0+p3.x*d); j = j+1; p3.s = p3.df'*d; ok = 1; if isnan(p3.f+p3.s) | isinf(p3.f+p3.s) error('Objective function returned Inf or NaN',''); end; catch if p.verbosity > 1, printf('\n'); warning(lasterr); end % warn or silence p3.x = (p1.x+p3.x)/2; ok = 0; p3.f = NaN; % bisect, and retry end end if p.verbosity > 2 plot(nd*p3.x, p3.f, 'b+'); plot(nd*(p3.x+A), p3.f+p3.s*A, 'b-'); drawnow end if p3.s > -maxS | p3.f > f0+p3.x*RHO*s | j >= LIMIT, break; end % done? maxS = max(-p3.s*SIG, maxS); % modify max slope if function gets steeper p0 = p1; p1 = p3; % move points back one unit a = p1.x-p0.x; b = minCubic(a, p1.f-p0.f, p0.s, p1.s); % cubic extrapolation if ~isreal(b) || isnan(b) || isinf(b) || b < a || b > a*EXT % if b is bad b = a*EXT; % then extrapolate maximum amount end p3.x = p0.x+max(b, p1.x+INT*a); % move away from current, off-set end while 1 % keep interpolating as long as necessary if p1.f > p3.f, p2 = p3; else p2 = p1; end % make p2 the best so far if abs(p2.s) < maxS & p2.f < f0+a*RHO*s | j >= LIMIT, break; end % done? if p2.s*SIG < -maxS & p2.f < f0+a*RHO*s, maxS = -p2.s*SIG; end; % modify maxS a = p3.x-p1.x; b = minCubic(a, p3.f-p1.f, p1.s, p3.s); % cubic interpolation if ~isreal(b) || isnan(b) || isinf(b) || b < 0 || b > a % if b is bad b = a/2; % then bisect end p2.x = p1.x+min(max(b, INT*a), (1-INT)*a); % move away from current, off-set [p2.f p2.df] = f(x0+p2.x*d); j = j+1; p2.s = p2.df'*d; if p.verbosity > 2 plot(nd*p2.x, p2.f, 'r+'); plot(nd*(p2.x+A), p2.f+p2.s*A, 'r'); drawnow end if p2.s > 0 | p2.f > f0+p2.x*RHO*s, p3 = p2; else p1 = p2; end % new bracket end x = x0+p2.x*d; fx = p2.f; df = p2.df; a = p2.x; % return the value found if p.length < 0, i = i+j; else i = i+1; end % count func evals or line searches if p.verbosity, printf('%s %6i; value %4.6e\r', p.S, i, fx); end if abs(p2.s) > maxS || p2.f > f0+a*RHO*s, i = -i; end % indicate faliure if p.verbosity > 2 if i>0, plot(norm(d)*p2.x, fx, 'go'); end subplot(211); plot(abs(i), fx, '+'); drawnow; end function x = minCubic(x, df, s0, s1) % return minimizer of approximating cubic A = -6*df+3*(s0+s1)*x; B = 3*df-(2*s0+s1)*x; x = -s0*x*x/(B+sqrt(B*B-A*s0*x)); function [fx, dfx] = f(varargin); persistent F p; if nargout == 0 p = varargin; if ischar(p{1}), F = str2func(p{1}); else F = p{1}; end else [fx, dfx] = F(rewrap(p{2}, varargin{1}), p{3:end}); dfx = unwrap(dfx); end function v = unwrap(s) % extract num elements of s (any type) into v (vector) v = []; if isnumeric(s) v = s(:); % numeric values are recast to column vector elseif isstruct(s) v = unwrap(struct2cell(orderfields(s))); % alphabetize, conv to cell, recurse elseif iscell(s) % cell array elements are for i = 1:numel(s), v = [v; unwrap(s{i})]; end % handled sequentially end % other types are ignored function [s v] = rewrap(s, v) % map elements of v (vector) onto s (any type) if isnumeric(s) if numel(v) < numel(s) error('The vector for conversion contains too few elements') end s = reshape(v(1:numel(s)), size(s)); % numeric values are reshaped v = v(numel(s)+1:end); % remaining arguments passed on elseif isstruct(s) [s p] = orderfields(s); p(p) = 1:numel(p); % alphabetize, store ordering [t v] = rewrap(struct2cell(s), v); % convert to cell, recurse s = orderfields(cell2struct(t,fieldnames(s),1),p); % conv to struct, reorder elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially [s{i} v] = rewrap(s{i}, v); end end % other types are not processed function printf(varargin); fprintf(varargin{:}); if exist('fflush','builtin') fflush(stdout); end ================================================ FILE: code/gpml/util/minimize_bfgs.m ================================================ % minimize.m - minimize a smooth differentiable multivariate function using % LBFGS (Limited memory LBFGS) or CG (Conjugate Gradients) % Usage: [X, fX, i] = minimize(X, F, p, other, ... ) % where % X is an initial guess (any type: vector, matrix, cell array, struct) % F is the objective function (function pointer or name) % p parameters - if p is a number, it corresponds to p.length below % p.length allowed 1) # linesearches or 2) if -ve minus # func evals % p.method minimization method, 'BFGS', 'LBFGS' or 'CG' % p.verbosity 0 quiet, 1 line, 2 line + warnings (default), 3 graphical % p.mem number of directions used in LBFGS (default 100) % other, ... other parameters, passed to the function F % X returned minimizer % fX vector of function values showing minimization progress % i final number of linesearches or function evaluations % The function F must take the following syntax [f, df] = F(X, other, ...) % where f is the function value and df its partial derivatives. The types of X % and df must be identical (vector, matrix, cell array, struct, etc). % % Copyright (C) 1996 - 2011 by Carl Edward Rasmussen, 2011-10-13. % Permission is hereby granted, free of charge, to any person OBTAINING A COPY % OF THIS SOFTWARE AND ASSOCIATED DOCUMENTATION FILES (THE "SOFTWARE"), TO DEAL % IN THE SOFTWARE WITHOUT RESTRICTION, INCLUDING WITHOUT LIMITATION THE RIGHTS % to use, copy, modify, merge, publish, distribute, sublicense, and/or sell % copies of the Software, and to permit persons to whom the Software is % furnished to do so, subject to the following conditions: % % The above copyright notice and this permission notice shall be included in % all copies or substantial portions of the Software. % % THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR % IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, % FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE % AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER % LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, % OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE % SOFTWARE. function [X, fX, i] = minimize(X, F, p, varargin) if isnumeric(p), p = struct('length', p); end % convert p to struct if p.length > 0, p.S = 'linesearch #'; else p.S = 'function evaluation #'; end; x = unwrap(X); % convert initial guess to vector if ~isfield(p,'method'), if length(x) > 1000, p.method = @LBFGS; else p.method = @BFGS; end; end % set default method if ~isfield(p,'verbosity'), p.verbosity = 2; end % default 1 line text output if ~isfield(p,'MFEPLS'), p.MFEPLS = 10; end % Max Func Evals Per Line Search if ~isfield(p,'MSR'), p.MSR = 100; end % Max Slope Ratio default f(F, X, varargin{:}); % set up the function f [fx, dfx] = f(x); % initial function value and derivatives if p.verbosity, printf('Initial Function Value %4.6e\r', fx); end if p.verbosity > 2, clf; subplot(211); hold on; xlabel(p.S); ylabel('function value'); plot(p.length < 0, fx, '+'); drawnow; end [x, fX, i] = feval(p.method, x, fx, dfx, p); % minimize using direction method X = rewrap(X, x); % convert answer to original representation if p.verbosity, printf('\n'); end function [x, fx, i] = CG(x0, fx0, dfx0, p); if ~isfield(p, 'SIG'), p.SIG = 0.1; end % default for line search quality i = p.length < 0; ok = 0; % initialize resource counter r = -dfx0; s = -r'*r; b = -1/(s-1); bs = -1; fx = fx0; % steepest descent while i < abs(p.length) b = b*bs/min(b*s,bs/p.MSR); % suitable initial step size using slope ratio [x, b, fx0, dfx, i] = lineSearch(x0, fx0, dfx0, r, s, b, i, p); if i < 0 % if line search failed i = -i; if ok, ok = 0; r = -dfx; else break; end % try steepest or stop else ok = 1; bs = b*s; % record step times slope (for slope ratio method) r = (dfx'*(dfx-dfx0))/(dfx0'*dfx0)*r - dfx; % Polack-Ribiere CG end s = r'*dfx; if s >= 0, r = -dfx; s = r'*dfx; ok = 0; end % slope must be -ve x0 = x; dfx0 = dfx; fx = [fx; fx0]; % replace old values with new ones end function [x, fx, i] = BFGS(x0, fx0, dfx0, p); if ~isfield(p, 'SIG'), p.SIG = 0.5; end % default for line search quality i = p.length < 0; ok = 0; % initialize resource counter x = x0; fx = fx0; r = -dfx0; s = -r'*r; b = -1/(s-1); H = eye(length(x0)); while i < abs(p.length) [x, b, fx0, dfx, i] = lineSearch(x0, fx0, dfx0, r, s, b, i, p); if i < 0 i = -i; if ok, ok = 0; else break; end; % try steepest or stop else ok = 1; t = x - x0; y = dfx - dfx0; ty = t'*y; Hy = H*y; H = H + (ty+y'*Hy)/ty^2*t*t' - 1/ty*Hy*t' - 1/ty*t*Hy'; % BFGS update end r = -H*dfx; s = r'*dfx; x0 = x; dfx0 = dfx; fx = [fx; fx0]; end function [x, fx, i] = LBFGS(x0, fx0, dfx0, p); if ~isfield(p, 'SIG'), p.SIG = 0.5; end % default for line search quality n = length(x0); k = 0; ok = 0; x = x0; fx = fx0; bs = -1/p.MSR; if isfield(p, 'mem'), m = p.mem; else m = min(100, n); end % set memory size a = zeros(1, m); t = zeros(n, m); y = zeros(n, m); % allocate memory i = p.length < 0; % initialize resource counter while i < abs(p.length) q = dfx0; for j = rem(k-1:-1:max(0,k-m),m)+1 a(j) = t(:,j)'*q/rho(j); q = q-a(j)*y(:,j); end if k == 0, r = -q/(q'*q); else r = -t(:,j)'*y(:,j)/(y(:,j)'*y(:,j))*q; end for j = rem(max(0,k-m):k-1,m)+1 r = r-t(:,j)*(a(j)+y(:,j)'*r/rho(j)); end s = r'*dfx0; if s >= 0, r = -dfx0; s = r'*dfx0; k = 0; ok = 0; end b = bs/min(bs,s/p.MSR); % suitable initial step size (usually 1) if isnan(r) | isinf(r) % if nonsense direction i = -i; % try steepest or stop else [x, b, fx0, dfx, i] = lineSearch(x0, fx0, dfx0, r, s, b, i, p); end if i < 0 % if line search failed i = -i; if ok, ok = 0; k = 0; else break; end % try steepest or stop else j = rem(k,m)+1; t(:,j) = x-x0; y(:,j) = dfx-dfx0; rho(j) = t(:,j)'*y(:,j); ok = 1; k = k+1; bs = b*s; end x0 = x; dfx0 = dfx; fx = [fx; fx0]; % replace and add values end function [x, a, fx, df, i] = lineSearch(x0, f0, df0, d, s, a, i, p) if p.length < 0, LIMIT = min(p.MFEPLS, -i-p.length); else LIMIT = p.MFEPLS; end p0.x = 0.0; p0.f = f0; p0.df = df0; p0.s = s; p1 = p0; % init p0 and p1 j = 0; p3.x = a; wp(p0, p.SIG, 0); % set step & Wolfe-Powell conditions if p.verbosity > 2 A = [-a a]/5; nd = norm(d); subplot(212); hold off; plot(0, f0, 'k+'); hold on; plot(nd*A, f0+s*A, 'k-'); xlabel('distance in line search direction'); ylabel('function value'); end while 1 % keep extrapolating as long as necessary ok = 0; while ~ok & j < LIMIT try % try, catch and bisect to safeguard extrapolation evaluation j = j+1; [p3.f p3.df] = f(x0+p3.x*d); p3.s = p3.df'*d; ok = 1; if isnan(p3.f+p3.s) | isinf(p3.f+p3.s) error('Objective function returned Inf or NaN',''); end; catch if p.verbosity > 1, printf('\n'); warning(lasterr); end % warn or silence p3.x = (p1.x+p3.x)/2; ok = 0; p3.f = NaN; % bisect, and retry end end if p.verbosity > 2 plot(nd*p3.x, p3.f, 'b+'); plot(nd*(p3.x+A), p3.f+p3.s*A, 'b-'); drawnow end if wp(p3) | j >= LIMIT, break; end % done? p0 = p1; p1 = p3; % move points back one unit p3.x = p0.x + minCubic(p1.x-p0.x, p1.f-p0.f, p0.s, p1.s, 1); % cubic extrap end while 1 % keep interpolating as long as necessary if p1.f > p3.f, p2 = p3; else p2 = p1; end % make p2 the best so far if wp(p2) > 1 | j >= LIMIT, break; end % done? p2.x = p1.x + minCubic(p3.x-p1.x, p3.f-p1.f, p1.s, p3.s, 0); % cubic interp j = j+1; [p2.f p2.df] = f(x0+p2.x*d); p2.s = p2.df'*d; if p.verbosity > 2 plot(nd*p2.x, p2.f, 'r+'); plot(nd*(p2.x+A), p2.f+p2.s*A, 'r'); drawnow end if wp(p2) > -1 & p2.s > 0 | wp(p2) < -1, p3 = p2; else p1 = p2; end % bracket end x = x0+p2.x*d; fx = p2.f; df = p2.df; a = p2.x; % return the value found if p.length < 0, i = i+j; else i = i+1; end % count func evals or line searches if p.verbosity, printf('%s %6i; value %4.6e\r', p.S, i, fx); end if wp(p2) < 2, i = -i; end % indicate faliure if p.verbosity > 2 if i>0, plot(norm(d)*p2.x, fx, 'go'); end subplot(211); plot(abs(i), fx, '+'); drawnow; end function z = minCubic(x, df, s0, s1, extr) % minimizer of approximating cubic INT = 0.1; EXT = 5.0; % interpolate and extrapolation limits A = -6*df+3*(s0+s1)*x; B = 3*df-(2*s0+s1)*x; if B<0, z = s0*x/(s0-s1); else z = -s0*x*x/(B+sqrt(B*B-A*s0*x)); end if extr % are we extrapolating? if ~isreal(z) | ~isfinite(z) | z < x | z > x*EXT, z = EXT*x; end % fix bad z z = max(z, (1+INT)*x); % extrapolate by at least INT else % else, we are interpolating if ~isreal(z) | ~isfinite(z) | z < 0 | z > x, z = x/2; end; % fix bad z z = min(max(z, INT*x), (1-INT)*x); % at least INT away from the boundaries end function y = wp(p, SIG, RHO) persistent a b c sig rho; if nargin == 3 % if three arguments, then set up the Wolfe-Powell conditions a = RHO*p.s; b = p.f; c = -SIG*p.s; sig = SIG; rho = RHO; y= 0; else if p.f > a*p.x+b % function value too large? if a > 0, y = -1; else y = -2; end else if p.s < -c, y = 0; elseif p.s > c, y = 1; else y = 2; end % if sig*abs(p.s) > c, a = rho*p.s; b = p.f-a*p.x; c = sig*abs(p.s); end end end function [fx, dfx] = f(varargin); persistent F p; if nargout == 0 p = varargin; if ischar(p{1}), F = str2func(p{1}); else F = p{1}; end else [fx, dfx] = F(rewrap(p{2}, varargin{1}), p{3:end}); dfx = unwrap(dfx); end function v = unwrap(s) % extract num elements of s (any type) into v (vector) v = []; if isnumeric(s) v = s(:); % numeric values are recast to column vector elseif isstruct(s) v = unwrap(struct2cell(orderfields(s))); % alphabetize, conv to cell, recurse elseif iscell(s) % cell array elements are for i = 1:numel(s), v = [v; unwrap(s{i})]; end % handled sequentially end % other types are ignored function [s v] = rewrap(s, v) % map elements of v (vector) onto s (any type) if isnumeric(s) if numel(v) < numel(s) error('The vector for conversion contains too few elements') end s = reshape(v(1:numel(s)), size(s)); % numeric values are reshaped v = v(numel(s)+1:end); % remaining arguments passed on elseif isstruct(s) [s p] = orderfields(s); p(p) = 1:numel(p); % alphabetize, store ordering [t v] = rewrap(struct2cell(s), v); % convert to cell, recurse s = orderfields(cell2struct(t,fieldnames(s),1),p); % conv to struct, reorder elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially [s{i} v] = rewrap(s{i}, v); end end % other types are not processed function printf(varargin); fprintf(varargin{:}); if exist('fflush','builtin') fflush(stdout); end ================================================ FILE: code/gpml/util/minimize_lbfgsb.m ================================================ function [X, fX, i] = minimize_lbfgsb(X, f, length, varargin) % Minimize a differentiable multivariate function using quasi Newton. % % Usage: [X, fX, i] = minimize_lbfgsb(X, f, length, P1, P2, P3, ... ) % % X initial guess; may be of any type, including struct and cell array % f the name or pointer to the function to be minimized. The function % f must return two arguments, the value of the function, and it's % partial derivatives wrt the elements of X. The partial derivative % must have the same type as X. % length length of the run; if it is positive, it gives the maximum number of % line searches, if negative its absolute gives the maximum allowed % number of function evaluations. Optionally, length can have a second % component, which will indicate the reduction in function value to be % expected in the first line-search (defaults to 1.0). % P1, P2 ... parameters are passed to the function f. % % X the returned solution % fX vector of function values indicating progress made % i number of iterations (line searches or function evaluations, % depending on the sign of "length") used at termination. % % The function returns when either its length is up, or if no further progress % can be made (ie, we are at a (local) minimum, or so close that due to % numerical problems, we cannot get any closer). NOTE: If the function % terminates within a few iterations, it could be an indication that the % function values and derivatives are not consistent (ie, there may be a bug in % the implementation of your "f" function). % % Copyright (C) 2010 by Hannes Nickisch, 2010-02-05 % global variables serve as communication interface between calls global minimize_lbfgsb_iteration_number global minimize_lbfgsb_objective global minimize_lbfgsb_gradient global minimize_lbfgsb_X % init global variables minimize_lbfgsb_iteration_number = 0; minimize_lbfgsb_objective = Inf; minimize_lbfgsb_gradient = 0*unwrap(X); minimize_lbfgsb_X = X; X0 = X; lb = -Inf*ones(size(unwrap(X0))); ub = Inf*ones(size(unwrap(X0))); maxiter = abs(length); % max number of iterations % no callback routine used so far % m is the number of saved vectors used to estimate the Hessian % factr is the precision 1e-12 X = lbfgsb( unwrap(X0), lb, ub, 'minimize_lbfgsb_objfun', ... 'minimize_lbfgsb_gradfun', ... {f,X0,varargin{:}}, [], ... 'maxiter',maxiter, 'm',4, 'factr',1e-12, 'pgtol',1e-5); i = minimize_lbfgsb_iteration_number; fX = minimize_lbfgsb_objective; X = rewrap(X0,X); % clear global variables clear minimize_lbfgsb_iteration_number clear minimize_lbfgsb_objective clear minimize_lbfgsb_gradient clear minimize_lbfgsb_X % Extract the numerical values from "s" into the column vector "v". The % variable "s" can be of any type, including struct and cell array. % Non-numerical elements are ignored. See also the reverse rewrap.m. function v = unwrap(s) v = []; if isnumeric(s) v = s(:); % numeric values are recast to column vector elseif isstruct(s) v = unwrap(struct2cell(orderfields(s)));% alphabetize, conv to cell, recurse elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially v = [v; unwrap(s{i})]; end end % other types are ignored % Map the numerical elements in the vector "v" onto the variables "s" which can % be of any type. The number of numerical elements must match; on exit "v" % should be empty. Non-numerical entries are just copied. See also unwrap.m. function [s v] = rewrap(s, v) if isnumeric(s) if numel(v) < numel(s) error('The vector for conversion contains too few elements') end s = reshape(v(1:numel(s)), size(s)); % numeric values are reshaped v = v(numel(s)+1:end); % remaining arguments passed on elseif isstruct(s) [s p] = orderfields(s); p(p) = 1:numel(p); % alphabetize, store ordering [t v] = rewrap(struct2cell(s), v); % convert to cell, recurse s = orderfields(cell2struct(t,fieldnames(s),1),p); % conv to struct, reorder elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially [s{i} v] = rewrap(s{i}, v); end end % other types are not processed ================================================ FILE: code/gpml/util/minimize_lbfgsb_gradfun.m ================================================ function G = minimize_lbfgsb_gradfun(X,varargin) % extract input arguments varargin = varargin{1}; strctX = varargin{2}; f = varargin{1}; % global variables serve as communication interface between calls global minimize_lbfgsb_iteration_number global minimize_lbfgsb_objective global minimize_lbfgsb_gradient global minimize_lbfgsb_X if norm(unwrap(X)-unwrap(minimize_lbfgsb_X))>1e-10 [y,G] = feval(f,rewrap(strctX,X),varargin{3:end}); else y = minimize_lbfgsb_objective(minimize_lbfgsb_iteration_number); G = minimize_lbfgsb_gradient; end % memorise gradient and position minimize_lbfgsb_gradient = G; minimize_lbfgsb_X = X; G = unwrap(G); % Extract the numerical values from "s" into the column vector "v". The % variable "s" can be of any type, including struct and cell array. % Non-numerical elements are ignored. See also the reverse rewrap.m. function v = unwrap(s) v = []; if isnumeric(s) v = s(:); % numeric values are recast to column vector elseif isstruct(s) v = unwrap(struct2cell(orderfields(s)));% alphabetize, conv to cell, recurse elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially v = [v; unwrap(s{i})]; end end % other types are ignored % Map the numerical elements in the vector "v" onto the variables "s" which can % be of any type. The number of numerical elements must match; on exit "v" % should be empty. Non-numerical entries are just copied. See also unwrap.m. function [s v] = rewrap(s, v) if isnumeric(s) if numel(v) < numel(s) error('The vector for conversion contains too few elements') end s = reshape(v(1:numel(s)), size(s)); % numeric values are reshaped v = v(numel(s)+1:end); % remaining arguments passed on elseif isstruct(s) [s p] = orderfields(s); p(p) = 1:numel(p); % alphabetize, store ordering [t v] = rewrap(struct2cell(s), v); % convert to cell, recurse s = orderfields(cell2struct(t,fieldnames(s),1),p); % conv to struct, reorder elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially [s{i} v] = rewrap(s{i}, v); end end % other types are not processed ================================================ FILE: code/gpml/util/minimize_lbfgsb_objfun.m ================================================ function y = minimize_lbfgsb_objfun(X,varargin) % extract input arguments varargin = varargin{1}; strctX = varargin{2}; f = varargin{1}; % global variables serve as communication interface between calls global minimize_lbfgsb_iteration_number global minimize_lbfgsb_objective global minimize_lbfgsb_gradient global minimize_lbfgsb_X recompute = 0; if minimize_lbfgsb_iteration_number==0 recompute = 1; else if norm(unwrap(X)-unwrap(minimize_lbfgsb_X))>1e-10 recompute = 1; end end if recompute [y,G] = feval(f,rewrap(strctX,X),varargin{3:end}); else y = minimize_lbfgsb_objective(minimize_lbfgsb_iteration_number); G = minimize_lbfgsb_gradient; end % increase global counter, memorise objective function, gradient and position minimize_lbfgsb_iteration_number = minimize_lbfgsb_iteration_number+1; minimize_lbfgsb_objective(minimize_lbfgsb_iteration_number,1) = y; minimize_lbfgsb_gradient = G; minimize_lbfgsb_X = X; % Extract the numerical values from "s" into the column vector "v". The % variable "s" can be of any type, including struct and cell array. % Non-numerical elements are ignored. See also the reverse rewrap.m. function v = unwrap(s) v = []; if isnumeric(s) v = s(:); % numeric values are recast to column vector elseif isstruct(s) v = unwrap(struct2cell(orderfields(s)));% alphabetize, conv to cell, recurse elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially v = [v; unwrap(s{i})]; end end % other types are ignored % Map the numerical elements in the vector "v" onto the variables "s" which can % be of any type. The number of numerical elements must match; on exit "v" % should be empty. Non-numerical entries are just copied. See also unwrap.m. function [s v] = rewrap(s, v) if isnumeric(s) if numel(v) < numel(s) error('The vector for conversion contains too few elements') end s = reshape(v(1:numel(s)), size(s)); % numeric values are reshaped v = v(numel(s)+1:end); % remaining arguments passed on elseif isstruct(s) [s p] = orderfields(s); p(p) = 1:numel(p); % alphabetize, store ordering [t v] = rewrap(struct2cell(s), v); % convert to cell, recurse s = orderfields(cell2struct(t,fieldnames(s),1),p); % conv to struct, reorder elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially [s{i} v] = rewrap(s{i}, v); end end % other types are not processed ================================================ FILE: code/gpml/util/minimize_may18.m ================================================ % minimize.m - minimize a smooth differentiable multivariate function using % LBFGS (Limited memory LBFGS) or CG (Conjugate Gradients) % Usage: [X, fX, i] = minimize(X, F, p, other, ... ) % where % X is an initial guess (any type: vector, matrix, cell array, struct) % F is the objective function (function pointer or name) % p parameters - if p is a number, it corresponds to p.length below % p.length allowed 1) # linesearches or 2) if -ve minus # func evals % p.method minimization method, 'LBFGS' (default) or 'CG' % p.verbosity 0 quiet, 1 line, 2 line + warnings (default), 3 graphical % p.mem number of directions used in LBFGS (default 4) % other, ... other parameters, passed to the function F % X returned minimizer % fX vector of function values showing minimization progress % i final number of linesearches or function evaluations % The function F must take the following syntax [f, df] = F(X, other, ...) % where f is the function value and df its partial derivatives. The types of X % and df must be identical (vector, matrix, cell array, struct, etc). function [X, fX, i] = minimize(X, F, p, varargin) if isnumeric(p), p = struct('length', p); end % convert p to struct if p.length > 0, p.S = 'linesearch #'; else p.S = 'function evaluation #'; end; if ~isfield(p,'method'), p.method = @LBFGS; end % set default method if ~isfield(p,'verbosity'), p.verbosity = 2; end % default 1 line text output if ~isfield(p,'MFEPLS'), p.MFEPLS = 20; end % Max Func Evals Per Line Search if ~isfield(p,'MSR'), p.MSR = 100; end % Max Slope Ratio default x = unwrap(X); % convert initial guess to vector f(F, X, varargin{:}); % set up the function f [fx, dfx] = f(x); % initial function value and derivatives if p.verbosity, printf('Initial Function Value %4.6e\r', fx); end if p.verbosity > 2, clf; subplot(211); hold on; xlabel(p.S); ylabel('function value'); plot(p.length < 0, fx, '+'); drawnow; end [x, fX, i] = feval(p.method, x, fx, dfx, p); % minimize using direction method X = rewrap(X, x); % convert answer to original representation if p.verbosity, printf('\n'); end function [x, fx, i] = CG(x0, fx0, dfx0, p); i = p.length < 0; % initialize resource counter r = -dfx0; s = -r'*r; b = -1/s; bs = -1; ok = 0; fx = fx0; % steepest descent while i < abs(p.length) [x, b, fx0, dfx, i] = lineSearch(x0, fx0, dfx0, r, s, b*bs/min(b*s,bs/p.MSR), i, p); if i < 0 % if line search failed i = -i; if ok, ok = 0; r = -dfx; else break; end % try steepest or stop else ok = 1; bs = b*s; % record step times slope (for slope ratio method) r = (dfx'*(dfx-dfx0))/(dfx0'*dfx0)*r - dfx; % Polack-Ribiere CG end s = r'*dfx; if s >= 0, r = -dfx; s = r'*dfx; ok = 0; end % slope must be -ve x0 = x; dfx0 = dfx; fx = [fx; fx0]; % replace old values with new ones end function [x, fx, i] = LBFGS(x0, fx0, dfx0, p); n = length(x0); k = 0; ok = 0; fx = fx0; bs = -dfx0'*dfx0/p.MSR; if isfield(p, 'mem'), m = p.mem; else m = min(4, n); end t = zeros(n, m); y = zeros(n, m); % allocate memory i = p.length < 0; % initialize resource counter while i < abs(p.length) q = dfx0; for j = rem(k-1:-1:max(0,k-m),m)+1 a(j) = t(:,j)'*q/rho(j); q = q-a(j)*y(:,j); end if k == 0, r = -q/(q'*q); else r = -t(:,j)'*y(:,j)/(y(:,j)'*y(:,j))*q; end for j = rem(max(0,k-m):k-1,m)+1 r = r-t(:,j)*(a(j)+y(:,j)'*r/rho(j)); end s = r'*dfx0; if s >= 0, r = -dfx0; s = r'*dfx0; k = 0; ok = 0; end [x, b, fx0, dfx, i] = lineSearch(x0, fx0, dfx0, r, s, bs/min(bs,s/p.MSR), i, p); if i < 0 % if line search failed i = -i; if ok, ok = 0; k = 0; else break; end % try steepest or stop else j = rem(k,m)+1; t(:,j) = x-x0; y(:,j) = dfx-dfx0; rho(j) = t(:,j)'*y(:,j); ok = 1; k = k+1; bs = b*s; end x0 = x; dfx0 = dfx; fx = [fx; fx0]; % replace and add values end function [x, a, fx, df, i] = lineSearch(x0, f0, df0, d, s, a, i, p) INT = 0.1; EXT = 5.0; % interpolate and extrapolation limits SIG = 0.1; RHO = SIG/2; % Wolfe-Powell condition constants, 0 2 A = [-a a]/5; nd = norm(d); subplot(212); hold off; plot(0, f0, 'k+'); hold on; plot(nd*A, f0+s*A, 'k-'); xlabel('distance in line search direction'); ylabel('function value'); end while 1 % keep extrapolating as long as necessary ok = 0; while ~ok & j < LIMIT try % try, catch and bisect to safeguard extrapolation evaluation [p3.f p3.df] = f(x0+p3.x*d); j = j+1; p3.s = p3.df'*d; ok = 1; if isnan(p3.f+p3.s) | isinf(p3.f+p3.s) error('Objective function returned Inf or NaN',''); end; catch if p.verbosity > 1, printf('\n'); warning(lasterr); end % warn or silence p3.x = (p1.x+p3.x)/2; ok = 0; p3.f = NaN; % bisect, and retry end end if p.verbosity > 2 plot(nd*p3.x, p3.f, 'b+'); plot(nd*(p3.x+A), p3.f+p3.s*A, 'b-'); drawnow end if p3.s > -maxS | p3.f > f0+p3.x*RHO*s | j >= LIMIT, break; end % done? p0 = p1; p1 = p3; % move points back one unit a = p1.x-p0.x; b = minCubic(a, p1.f-p0.f, p0.s, p1.s); % cubic extrapolation if ~isreal(b) || isnan(b) || isinf(b) || b < a || b > a*EXT % if b is bad b = a*EXT; % then extrapolate maximum amount end p3.x = p0.x+max(b, p1.x+INT*a); % move away from current, off-set end while 1 % keep interpolating as long as necessary if p1.f > p3.f, p2 = p3; else p2 = p1; end % make p2 the best so far if abs(p2.s) < maxS & p2.f < f0+a*RHO*s | j >= LIMIT, break; end % done? a = p3.x-p1.x; b = minCubic(a, p3.f-p1.f, p1.s, p3.s); % cubic interpolation if ~isreal(b) || isnan(b) || isinf(b) || b < 0 || b > a % if b is bad b = a/2; % then bisect end p2.x = p1.x+min(max(b, INT*a), (1-INT)*a); % move away from current, off-set [p2.f p2.df] = f(x0+p2.x*d); j = j+1; p2.s = p2.df'*d; if p.verbosity > 2 plot(nd*p2.x, p2.f, 'r+'); plot(nd*(p2.x+A), p2.f+p2.s*A, 'r'); drawnow end if p2.s > 0 | p2.f > f0+p2.x*RHO*s, p3 = p2; else p1 = p2; end % new bracket end x = x0+p2.x*d; fx = p2.f; df = p2.df; a = p2.x; % return the value found if p.length < 0, i = i+j; else i = i+1; end % count func evals or line searches if p.verbosity, printf('%s %6i; value %4.6e\r', p.S, i, fx); end if abs(p2.s) > maxS || p2.f > f0+a*RHO*s, i = -i; end % indicate faliure if p.verbosity > 2 if i>0, plot(norm(d)*p2.x, fx, 'go'); end subplot(211); plot(abs(i), fx, '+'); drawnow; end function x = minCubic(dx, df, s0, s1) A = -6*df+3*(s0+s1)*dx; B = 3*df-(2*s0+s1)*dx; x = -s0*dx*dx/(B+sqrt(B*B-A*s0*dx)); % num. error possible, ok function [fx, dfx] = f(varargin); persistent F p; if nargout == 0 p = varargin; if ischar(p{1}), F = str2func(p{1}); else F = p{1}; end else [fx, dfx] = F(rewrap(p{2}, varargin{1}), p{3:end}); dfx = unwrap(dfx); end % Extract the numerical values from "s" into the column vector "v". The % variable "s" can be of any type, including struct and cell array. % Non-numerical elements are ignored. See also the reverse rewrap.m. function v = unwrap(s) v = []; if isnumeric(s) v = s(:); % numeric values are recast to column vector elseif isstruct(s) v = unwrap(struct2cell(orderfields(s))); % alphabetize, conv to cell, recurse elseif iscell(s) % cell array elements are for i = 1:numel(s), v = [v; unwrap(s{i})]; end % handled sequentially end % other types are ignored % Map the numerical elements in the vector "v" onto the variables "s" which can % be of any type. The number of numerical elements must match; on exit "v" % should be empty. Non-numerical entries are just copied. See also unwrap.m. function [s v] = rewrap(s, v) if isnumeric(s) if numel(v) < numel(s) error('The vector for conversion contains too few elements') end s = reshape(v(1:numel(s)), size(s)); % numeric values are reshaped v = v(numel(s)+1:end); % remaining arguments passed on elseif isstruct(s) [s p] = orderfields(s); p(p) = 1:numel(p); % alphabetize, store ordering [t v] = rewrap(struct2cell(s), v); % convert to cell, recurse s = orderfields(cell2struct(t,fieldnames(s),1),p); % conv to struct, reorder elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially [s{i} v] = rewrap(s{i}, v); end end % other types are not processed function printf(varargin); fprintf(varargin{:}); if exist('fflush','builtin') fflush(stdout); end ================================================ FILE: code/gpml/util/minimize_may21.m ================================================ % minimize.m - minimize a smooth differentiable multivariate function using % LBFGS (Limited memory LBFGS) or CG (Conjugate Gradients) % Usage: [X, fX, i] = minimize(X, F, p, other, ... ) % where % X is an initial guess (any type: vector, matrix, cell array, struct) % F is the objective function (function pointer or name) % p parameters - if p is a number, it corresponds to p.length below % p.length allowed 1) # linesearches or 2) if -ve minus # func evals % p.method minimization method, 'LBFGS' (default) or 'CG' % p.verbosity 0 quiet, 1 line, 2 line + warnings (default), 3 graphical % p.mem number of directions used in LBFGS (default 4) % other, ... other parameters, passed to the function F % X returned minimizer % fX vector of function values showing minimization progress % i final number of linesearches or function evaluations % The function F must take the following syntax [f, df] = F(X, other, ...) % where f is the function value and df its partial derivatives. The types of X % and df must be identical (vector, matrix, cell array, struct, etc). % % Copyright (C) 1996 - 2011 by Carl Edward Rasmussen, 2011-05-20. % Permission is hereby granted, free of charge, to any person OBTAINING A COPY % OF THIS SOFTWARE AND ASSOCIATED DOCUMENTATION FILES (THE "SOFTWARE"), TO DEAL % IN THE SOFTWARE WITHOUT RESTRICTION, INCLUDING WITHOUT LIMITATION THE RIGHTS % to use, copy, modify, merge, publish, distribute, sublicense, and/or sell % copies of the Software, and to permit persons to whom the Software is % furnished to do so, subject to the following conditions: % % The above copyright notice and this permission notice shall be included in % all copies or substantial portions of the Software. % % THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR % IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, % FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE % AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER % LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, % OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE % SOFTWARE. function [X, fX, i] = minimize(X, F, p, varargin) if isnumeric(p), p = struct('length', p); end % convert p to struct if p.length > 0, p.S = 'linesearch #'; else p.S = 'function evaluation #'; end; if ~isfield(p,'method'), p.method = @LBFGS; end % set default method if ~isfield(p,'verbosity'), p.verbosity = 2; end % default 1 line text output if ~isfield(p,'MFEPLS'), p.MFEPLS = 20; end % Max Func Evals Per Line Search if ~isfield(p,'MSR'), p.MSR = 100; end % Max Slope Ratio default x = unwrap(X); % convert initial guess to vector f(F, X, varargin{:}); % set up the function f [fx, dfx] = f(x); % initial function value and derivatives if p.verbosity, printf('Initial Function Value %4.6e\r', fx); end if p.verbosity > 2, clf; subplot(211); hold on; xlabel(p.S); ylabel('function value'); plot(p.length < 0, fx, '+'); drawnow; end [x, fX, i] = feval(p.method, x, fx, dfx, p); % minimize using direction method X = rewrap(X, x); % convert answer to original representation if p.verbosity, printf('\n'); end function [x, fx, i] = CG(x0, fx0, dfx0, p); i = p.length < 0; % initialize resource counter r = -dfx0; s = -r'*r; b = -1/s; bs = -1; ok = 0; fx = fx0; % steepest descent while i < abs(p.length) b = b*bs/min(b*s,bs/p.MSR); % suitable initial step size using slope ratio [x, b, fx0, dfx, i] = lineSearch(x0, fx0, dfx0, r, s, b, i, p); if i < 0 % if line search failed i = -i; if ok, ok = 0; r = -dfx; else break; end % try steepest or stop else ok = 1; bs = b*s; % record step times slope (for slope ratio method) r = (dfx'*(dfx-dfx0))/(dfx0'*dfx0)*r - dfx; % Polack-Ribiere CG end s = r'*dfx; if s >= 0, r = -dfx; s = r'*dfx; ok = 0; end % slope must be -ve x0 = x; dfx0 = dfx; fx = [fx; fx0]; % replace old values with new ones end function [x, fx, i] = LBFGS(x0, fx0, dfx0, p); n = length(x0); k = 0; ok = 0; fx = fx0; bs = -1/p.MSR; if isfield(p, 'mem'), m = p.mem; else m = min(4, n); end t = zeros(n, m); y = zeros(n, m); % allocate memory i = p.length < 0; % initialize resource counter while i < abs(p.length) q = dfx0; for j = rem(k-1:-1:max(0,k-m),m)+1 a(j) = t(:,j)'*q/rho(j); q = q-a(j)*y(:,j); end if k == 0, r = -q/(q'*q); else r = -t(:,j)'*y(:,j)/(y(:,j)'*y(:,j))*q; end for j = rem(max(0,k-m):k-1,m)+1 r = r-t(:,j)*(a(j)+y(:,j)'*r/rho(j)); end s = r'*dfx0; if s >= 0, r = -dfx0; s = r'*dfx0; k = 0; ok = 0; end b = bs/min(bs,s/p.MSR); % suitable initial step size (usually 1) [x, b, fx0, dfx, i] = lineSearch(x0, fx0, dfx0, r, s, b, i, p); if i < 0 % if line search failed i = -i; if ok, ok = 0; k = 0; else break; end % try steepest or stop else j = rem(k,m)+1; t(:,j) = x-x0; y(:,j) = dfx-dfx0; rho(j) = t(:,j)'*y(:,j); ok = 1; k = k+1; bs = b*s; end x0 = x; dfx0 = dfx; fx = [fx; fx0]; % replace and add values end function [x, a, fx, df, i] = lineSearch(x0, f0, df0, d, s, a, i, p) INT = 0.1; EXT = 5.0; % interpolate and extrapolation limits SIG = 0.1; RHO = SIG/2; % Wolfe-Powell condition constants, 0 2 A = [-a a]/5; nd = norm(d); subplot(212); hold off; plot(0, f0, 'k+'); hold on; plot(nd*A, f0+s*A, 'k-'); xlabel('distance in line search direction'); ylabel('function value'); end while 1 % keep extrapolating as long as necessary ok = 0; while ~ok & j < LIMIT try % try, catch and bisect to safeguard extrapolation evaluation [p3.f p3.df] = f(x0+p3.x*d); j = j+1; p3.s = p3.df'*d; ok = 1; if isnan(p3.f+p3.s) | isinf(p3.f+p3.s) error('Objective function returned Inf or NaN',''); end; catch if p.verbosity > 1, printf('\n'); warning(lasterr); end % warn or silence p3.x = (p1.x+p3.x)/2; ok = 0; p3.f = NaN; % bisect, and retry end end if p.verbosity > 2 plot(nd*p3.x, p3.f, 'b+'); plot(nd*(p3.x+A), p3.f+p3.s*A, 'b-'); drawnow end if p3.s > -maxS | p3.f > f0+p3.x*RHO*s | j >= LIMIT, break; end % done? maxS = max(-p3.s*SIG, maxS); % modify max slope if function gets steeper p0 = p1; p1 = p3; % move points back one unit a = p1.x-p0.x; b = minCubic(a, p1.f-p0.f, p0.s, p1.s); % cubic extrapolation if ~isreal(b) || isnan(b) || isinf(b) || b < a || b > a*EXT % if b is bad b = a*EXT; % then extrapolate maximum amount end p3.x = p0.x+max(b, p1.x+INT*a); % move away from current, off-set end while 1 % keep interpolating as long as necessary if p1.f > p3.f, p2 = p3; else p2 = p1; end % make p2 the best so far if abs(p2.s) < maxS & p2.f < f0+a*RHO*s | j >= LIMIT, break; end % done? if p2.s*SIG < -maxS & p2.f < f0+a*RHO*s, maxS = -p2.s*SIG; end; % modify maxS a = p3.x-p1.x; b = minCubic(a, p3.f-p1.f, p1.s, p3.s); % cubic interpolation if ~isreal(b) || isnan(b) || isinf(b) || b < 0 || b > a % if b is bad b = a/2; % then bisect end p2.x = p1.x+min(max(b, INT*a), (1-INT)*a); % move away from current, off-set [p2.f p2.df] = f(x0+p2.x*d); j = j+1; p2.s = p2.df'*d; if p.verbosity > 2 plot(nd*p2.x, p2.f, 'r+'); plot(nd*(p2.x+A), p2.f+p2.s*A, 'r'); drawnow end if p2.s > 0 | p2.f > f0+p2.x*RHO*s, p3 = p2; else p1 = p2; end % new bracket end x = x0+p2.x*d; fx = p2.f; df = p2.df; a = p2.x; % return the value found if p.length < 0, i = i+j; else i = i+1; end % count func evals or line searches if p.verbosity, printf('%s %6i; value %4.6e\r', p.S, i, fx); end if abs(p2.s) > maxS || p2.f > f0+a*RHO*s, i = -i; end % indicate faliure if p.verbosity > 2 if i>0, plot(norm(d)*p2.x, fx, 'go'); end subplot(211); plot(abs(i), fx, '+'); drawnow; end function x = minCubic(x, df, s0, s1) % return minimizer of approximating cubic A = -6*df+3*(s0+s1)*x; B = 3*df-(2*s0+s1)*x; x = -s0*x*x/(B+sqrt(B*B-A*s0*x)); function [fx, dfx] = f(varargin); persistent F p; if nargout == 0 p = varargin; if ischar(p{1}), F = str2func(p{1}); else F = p{1}; end else [fx, dfx] = F(rewrap(p{2}, varargin{1}), p{3:end}); dfx = unwrap(dfx); end function v = unwrap(s) % extract num elements of s (any type) into v (vector) v = []; if isnumeric(s) v = s(:); % numeric values are recast to column vector elseif isstruct(s) v = unwrap(struct2cell(orderfields(s))); % alphabetize, conv to cell, recurse elseif iscell(s) % cell array elements are for i = 1:numel(s), v = [v; unwrap(s{i})]; end % handled sequentially end % other types are ignored function [s v] = rewrap(s, v) % map elements of v (vector) onto s (any type) if isnumeric(s) if numel(v) < numel(s) error('The vector for conversion contains too few elements') end s = reshape(v(1:numel(s)), size(s)); % numeric values are reshaped v = v(numel(s)+1:end); % remaining arguments passed on elseif isstruct(s) [s p] = orderfields(s); p(p) = 1:numel(p); % alphabetize, store ordering [t v] = rewrap(struct2cell(s), v); % convert to cell, recurse s = orderfields(cell2struct(t,fieldnames(s),1),p); % conv to struct, reorder elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially [s{i} v] = rewrap(s{i}, v); end end % other types are not processed function printf(varargin); fprintf(varargin{:}); if exist('fflush','builtin') fflush(stdout); end ================================================ FILE: code/gpml/util/minimize_pre_oct26.m ================================================ % minimize.m - minimize a smooth differentiable multivariate function using % LBFGS (Limited memory LBFGS) or CG (Conjugate Gradients) % Usage: [X, fX, i] = minimize(X, F, p, other, ... ) % where % X is an initial guess (any type: vector, matrix, cell array, struct) % F is the objective function (function pointer or name) % p parameters - if p is a number, it corresponds to p.length below % p.length allowed 1) # linesearches or 2) if -ve minus # func evals % p.method minimization method, 'LBFGS' (default) or 'CG' % p.verbosity 0 quiet, 1 line, 2 line + warnings (default), 3 graphical % p.mem number of directions used in LBFGS (default 4) % other, ... other parameters, passed to the function F % X returned minimizer % fX vector of function values showing minimization progress % i final number of linesearches or function evaluations % The function F must take the following syntax [f, df] = F(X, other, ...) % where f is the function value and df its partial derivatives. The types of X % and df must be identical (vector, matrix, cell array, struct, etc). % % Copyright (C) 1996 - 2011 by Carl Edward Rasmussen, 2011-05-20. % Permission is hereby granted, free of charge, to any person OBTAINING A COPY % OF THIS SOFTWARE AND ASSOCIATED DOCUMENTATION FILES (THE "SOFTWARE"), TO DEAL % IN THE SOFTWARE WITHOUT RESTRICTION, INCLUDING WITHOUT LIMITATION THE RIGHTS % to use, copy, modify, merge, publish, distribute, sublicense, and/or sell % copies of the Software, and to permit persons to whom the Software is % furnished to do so, subject to the following conditions: % % The above copyright notice and this permission notice shall be included in % all copies or substantial portions of the Software. % % THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR % IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, % FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE % AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER % LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, % OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE % SOFTWARE. function [X, fX, i] = minimize(X, F, p, varargin) if isnumeric(p), p = struct('length', p); end % convert p to struct if p.length > 0, p.S = 'linesearch #'; else p.S = 'function evaluation #'; end; if ~isfield(p,'method'), p.method = @LBFGS; end % set default method if ~isfield(p,'verbosity'), p.verbosity = 2; end % default 1 line text output if ~isfield(p,'MFEPLS'), p.MFEPLS = 20; end % Max Func Evals Per Line Search if ~isfield(p,'MSR'), p.MSR = 100; end % Max Slope Ratio default x = unwrap(X); % convert initial guess to vector f(F, X, varargin{:}); % set up the function f [fx, dfx] = f(x); % initial function value and derivatives if p.verbosity, printf('Initial Function Value %4.6e\r', fx); end if p.verbosity > 2, clf; subplot(211); hold on; xlabel(p.S); ylabel('function value'); plot(p.length < 0, fx, '+'); drawnow; end [x, fX, i] = feval(p.method, x, fx, dfx, p); % minimize using direction method X = rewrap(X, x); % convert answer to original representation if p.verbosity, printf('\n'); end function [x, fx, i] = CG(x0, fx0, dfx0, p); i = p.length < 0; % initialize resource counter r = -dfx0; s = -r'*r; b = -1/s; bs = -1; ok = 0; fx = fx0; % steepest descent while i < abs(p.length) b = b*bs/min(b*s,bs/p.MSR); % suitable initial step size using slope ratio [x, b, fx0, dfx, i] = lineSearch(x0, fx0, dfx0, r, s, b, i, p); if i < 0 % if line search failed i = -i; if ok, ok = 0; r = -dfx; else break; end % try steepest or stop else ok = 1; bs = b*s; % record step times slope (for slope ratio method) r = (dfx'*(dfx-dfx0))/(dfx0'*dfx0)*r - dfx; % Polack-Ribiere CG end s = r'*dfx; if s >= 0, r = -dfx; s = r'*dfx; ok = 0; end % slope must be -ve x0 = x; dfx0 = dfx; fx = [fx; fx0]; % replace old values with new ones end function [x, fx, i] = LBFGS(x0, fx0, dfx0, p); n = length(x0); k = 0; ok = 0; fx = fx0; bs = -1/p.MSR; if isfield(p, 'mem'), m = p.mem; else m = min(4, n); end t = zeros(n, m); y = zeros(n, m); % allocate memory i = p.length < 0; % initialize resource counter while i < abs(p.length) q = dfx0; for j = rem(k-1:-1:max(0,k-m),m)+1 a(j) = t(:,j)'*q/rho(j); q = q-a(j)*y(:,j); end if k == 0, r = -q/(q'*q); else r = -t(:,j)'*y(:,j)/(y(:,j)'*y(:,j))*q; end for j = rem(max(0,k-m):k-1,m)+1 r = r-t(:,j)*(a(j)+y(:,j)'*r/rho(j)); end s = r'*dfx0; if s >= 0, r = -dfx0; s = r'*dfx0; k = 0; ok = 0; end b = bs/min(bs,s/p.MSR); % suitable initial step size (usually 1) [x, b, fx0, dfx, i] = lineSearch(x0, fx0, dfx0, r, s, b, i, p); if i < 0 % if line search failed i = -i; if ok, ok = 0; k = 0; else break; end % try steepest or stop else j = rem(k,m)+1; t(:,j) = x-x0; y(:,j) = dfx-dfx0; rho(j) = t(:,j)'*y(:,j); ok = 1; k = k+1; bs = b*s; end x0 = x; dfx0 = dfx; fx = [fx; fx0]; % replace and add values end function [x, a, fx, df, i] = lineSearch(x0, f0, df0, d, s, a, i, p) INT = 0.1; EXT = 5.0; % interpolate and extrapolation limits SIG = 0.1; RHO = SIG/2; % Wolfe-Powell condition constants, 0 2 A = [-a a]/5; nd = norm(d); subplot(212); hold off; plot(0, f0, 'k+'); hold on; plot(nd*A, f0+s*A, 'k-'); xlabel('distance in line search direction'); ylabel('function value'); end while 1 % keep extrapolating as long as necessary ok = 0; while ~ok & j < LIMIT try % try, catch and bisect to safeguard extrapolation evaluation [p3.f p3.df] = f(x0+p3.x*d); j = j+1; p3.s = p3.df'*d; ok = 1; if isnan(p3.f+p3.s) | isinf(p3.f+p3.s) error('Objective function returned Inf or NaN',''); end; catch if p.verbosity > 1, printf('\n'); warning(lasterr); end % warn or silence p3.x = (p1.x+p3.x)/2; ok = 0; p3.f = NaN; % bisect, and retry end end if p.verbosity > 2 plot(nd*p3.x, p3.f, 'b+'); plot(nd*(p3.x+A), p3.f+p3.s*A, 'b-'); drawnow end if p3.s > -maxS | p3.f > f0+p3.x*RHO*s | j >= LIMIT, break; end % done? maxS = max(-p3.s*SIG, maxS); % modify max slope if function gets steeper p0 = p1; p1 = p3; % move points back one unit a = p1.x-p0.x; b = minCubic(a, p1.f-p0.f, p0.s, p1.s); % cubic extrapolation if ~isreal(b) || isnan(b) || isinf(b) || b < a || b > a*EXT % if b is bad b = a*EXT; % then extrapolate maximum amount end p3.x = p0.x+max(b, p1.x+INT*a); % move away from current, off-set end while 1 % keep interpolating as long as necessary if p1.f > p3.f, p2 = p3; else p2 = p1; end % make p2 the best so far if abs(p2.s) < maxS & p2.f < f0+a*RHO*s | j >= LIMIT, break; end % done? if p2.s*SIG < -maxS & p2.f < f0+a*RHO*s, maxS = -p2.s*SIG; end; % modify maxS a = p3.x-p1.x; b = minCubic(a, p3.f-p1.f, p1.s, p3.s); % cubic interpolation if ~isreal(b) || isnan(b) || isinf(b) || b < 0 || b > a % if b is bad b = a/2; % then bisect end p2.x = p1.x+min(max(b, INT*a), (1-INT)*a); % move away from current, off-set [p2.f p2.df] = f(x0+p2.x*d); j = j+1; p2.s = p2.df'*d; if p.verbosity > 2 plot(nd*p2.x, p2.f, 'r+'); plot(nd*(p2.x+A), p2.f+p2.s*A, 'r'); drawnow end if p2.s > 0 | p2.f > f0+p2.x*RHO*s, p3 = p2; else p1 = p2; end % new bracket end x = x0+p2.x*d; fx = p2.f; df = p2.df; a = p2.x; % return the value found if p.length < 0, i = i+j; else i = i+1; end % count func evals or line searches if p.verbosity, printf('%s %6i; value %4.6e\r', p.S, i, fx); end if abs(p2.s) > maxS || p2.f > f0+a*RHO*s, i = -i; end % indicate faliure if p.verbosity > 2 if i>0, plot(norm(d)*p2.x, fx, 'go'); end subplot(211); plot(abs(i), fx, '+'); drawnow; end function x = minCubic(x, df, s0, s1) % return minimizer of approximating cubic A = -6*df+3*(s0+s1)*x; B = 3*df-(2*s0+s1)*x; x = -s0*x*x/(B+sqrt(B*B-A*s0*x)); function [fx, dfx] = f(varargin); persistent F p; if nargout == 0 p = varargin; if ischar(p{1}), F = str2func(p{1}); else F = p{1}; end else [fx, dfx] = F(rewrap(p{2}, varargin{1}), p{3:end}); dfx = unwrap(dfx); end function v = unwrap(s) % extract num elements of s (any type) into v (vector) v = []; if isnumeric(s) v = s(:); % numeric values are recast to column vector elseif isstruct(s) v = unwrap(struct2cell(orderfields(s))); % alphabetize, conv to cell, recurse elseif iscell(s) % cell array elements are for i = 1:numel(s), v = [v; unwrap(s{i})]; end % handled sequentially end % other types are ignored function [s v] = rewrap(s, v) % map elements of v (vector) onto s (any type) if isnumeric(s) if numel(v) < numel(s) error('The vector for conversion contains too few elements') end s = reshape(v(1:numel(s)), size(s)); % numeric values are reshaped v = v(numel(s)+1:end); % remaining arguments passed on elseif isstruct(s) [s p] = orderfields(s); p(p) = 1:numel(p); % alphabetize, store ordering [t v] = rewrap(struct2cell(s), v); % convert to cell, recurse s = orderfields(cell2struct(t,fieldnames(s),1),p); % conv to struct, reorder elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially [s{i} v] = rewrap(s{i}, v); end end % other types are not processed function printf(varargin); fprintf(varargin{:}); if exist('fflush','builtin') fflush(stdout); end ================================================ FILE: code/gpml/util/old_minimize.m ================================================ function [X, fX, i] = minimize(X, f, length, varargin) % Minimize a differentiable multivariate function using conjugate gradients. % % Usage: [X, fX, i] = minimize(X, f, length, P1, P2, P3, ... ) % % X initial guess; may be of any type, including struct and cell array % f the name or pointer to the function to be minimized. The function % f must return two arguments, the value of the function, and it's % partial derivatives wrt the elements of X. The partial derivative % must have the same type as X. % length length of the run; if it is positive, it gives the maximum number of % line searches, if negative its absolute gives the maximum allowed % number of function evaluations. Optionally, length can have a second % component, which will indicate the reduction in function value to be % expected in the first line-search (defaults to 1.0). % P1, P2, ... parameters are passed to the function f. % % X the returned solution % fX vector of function values indicating progress made % i number of iterations (line searches or function evaluations, % depending on the sign of "length") used at termination. % % The function returns when either its length is up, or if no further progress % can be made (ie, we are at a (local) minimum, or so close that due to % numerical problems, we cannot get any closer). NOTE: If the function % terminates within a few iterations, it could be an indication that the % function values and derivatives are not consistent (ie, there may be a bug in % the implementation of your "f" function). % % The Polack-Ribiere flavour of conjugate gradients is used to compute search % directions, and a line search using quadratic and cubic polynomial % approximations and the Wolfe-Powell stopping criteria is used together with % the slope ratio method for guessing initial step sizes. Additionally a bunch % of checks are made to make sure that exploration is taking place and that % extrapolation will not be unboundedly large. % % See also: checkgrad % % Copyright (C) 2001 - 2010 by Carl Edward Rasmussen, 2010-01-03 INT = 0.1; % don't reevaluate within 0.1 of the limit of the current bracket EXT = 3.0; % extrapolate maximum 3 times the current step-size MAX = 20; % max 20 function evaluations per line search RATIO = 10; % maximum allowed slope ratio SIG = 0.1; RHO = SIG/2; % SIG and RHO are the constants controlling the Wolfe- % Powell conditions. SIG is the maximum allowed absolute ratio between % previous and new slopes (derivatives in the search direction), thus setting % SIG to low (positive) values forces higher precision in the line-searches. % RHO is the minimum allowed fraction of the expected (from the slope at the % initial point in the linesearch). Constants must satisfy 0 < RHO < SIG < 1. % Tuning of SIG (depending on the nature of the function to be optimized) may % speed up the minimization; it is probably not worth playing much with RHO. % The code falls naturally into 3 parts, after the initial line search is % started in the direction of steepest descent. 1) we first enter a while loop % which uses point 1 (p1) and (p2) to compute an extrapolation (p3), until we % have extrapolated far enough (Wolfe-Powell conditions). 2) if necessary, we % enter the second loop which takes p2, p3 and p4 chooses the subinterval % containing a (local) minimum, and interpolates it, unil an acceptable point % is found (Wolfe-Powell conditions). Note, that points are always maintained % in order p0 <= p1 <= p2 < p3 < p4. 3) compute a new search direction using % conjugate gradients (Polack-Ribiere flavour), or revert to steepest if there % was a problem in the previous line-search. Return the best value so far, if % two consecutive line-searches fail, or whenever we run out of function % evaluations or line-searches. During extrapolation, the "f" function may fail % either with an error or returning Nan or Inf, and minimize should handle this % gracefully. if max(size(length)) == 2, red=length(2); length=length(1); else red=1; end if length>0, S='Linesearch'; else S='Function evaluation'; end i = 0; % zero the run length counter ls_failed = 0; % no previous line search has failed [f0 df0] = feval(f, X, varargin{:}); % get function value and gradient Z = X; X = unwrap(X); df0 = unwrap(df0); fprintf('%s %6i; Value %4.6e\r', S, i, f0); if exist('fflush','builtin') fflush(stdout); end fX = f0; i = i + (length<0); % count epochs?! s = -df0; d0 = -s'*s; % initial search direction (steepest) and slope x3 = red/(1-d0); % initial step is red/(|s|+1) while i < abs(length) % while not finished i = i + (length>0); % count iterations?! X0 = X; F0 = f0; dF0 = df0; % make a copy of current values if length>0, M = MAX; else M = min(MAX, -length-i); end while 1 % keep extrapolating as long as necessary x2 = 0; f2 = f0; d2 = d0; f3 = f0; df3 = df0; success = 0; while ~success && M > 0 try M = M - 1; i = i + (length<0); % count epochs?! [f3 df3] = feval(f, rewrap(Z,X+x3*s), varargin{:}); df3 = unwrap(df3); if isnan(f3) || isinf(f3) || any(isnan(df3)+isinf(df3)), error(''), end success = 1; catch % catch any error which occured in f x3 = (x2+x3)/2; % bisect and try again end end if f3 < F0, X0 = X+x3*s; F0 = f3; dF0 = df3; end % keep best values d3 = df3'*s; % new slope if d3 > SIG*d0 || f3 > f0+x3*RHO*d0 || M == 0 % are we done extrapolating? break end x1 = x2; f1 = f2; d1 = d2; % move point 2 to point 1 x2 = x3; f2 = f3; d2 = d3; % move point 3 to point 2 A = 6*(f1-f2)+3*(d2+d1)*(x2-x1); % make cubic extrapolation B = 3*(f2-f1)-(2*d1+d2)*(x2-x1); x3 = x1-d1*(x2-x1)^2/(B+sqrt(B*B-A*d1*(x2-x1))); % num. error possible, ok! if ~isreal(x3) || isnan(x3) || isinf(x3) || x3 < 0 % num prob | wrong sign? x3 = x2*EXT; % extrapolate maximum amount elseif x3 > x2*EXT % new point beyond extrapolation limit? x3 = x2*EXT; % extrapolate maximum amount elseif x3 < x2+INT*(x2-x1) % new point too close to previous point? x3 = x2+INT*(x2-x1); end end % end extrapolation while (abs(d3) > -SIG*d0 || f3 > f0+x3*RHO*d0) && M > 0 % keep interpolating if d3 > 0 || f3 > f0+x3*RHO*d0 % choose subinterval x4 = x3; f4 = f3; d4 = d3; % move point 3 to point 4 else x2 = x3; f2 = f3; d2 = d3; % move point 3 to point 2 end if f4 > f0 x3 = x2-(0.5*d2*(x4-x2)^2)/(f4-f2-d2*(x4-x2)); % quadratic interpolation else A = 6*(f2-f4)/(x4-x2)+3*(d4+d2); % cubic interpolation B = 3*(f4-f2)-(2*d2+d4)*(x4-x2); x3 = x2+(sqrt(B*B-A*d2*(x4-x2)^2)-B)/A; % num. error possible, ok! end if isnan(x3) || isinf(x3) x3 = (x2+x4)/2; % if we had a numerical problem then bisect end x3 = max(min(x3, x4-INT*(x4-x2)),x2+INT*(x4-x2)); % don't accept too close [f3 df3] = feval(f, rewrap(Z,X+x3*s), varargin{:}); df3 = unwrap(df3); if f3 < F0, X0 = X+x3*s; F0 = f3; dF0 = df3; end % keep best values M = M - 1; i = i + (length<0); % count epochs?! d3 = df3'*s; % new slope end % end interpolation if abs(d3) < -SIG*d0 && f3 < f0+x3*RHO*d0 % if line search succeeded X = X+x3*s; f0 = f3; fX = [fX' f0]'; % update variables fprintf('%s %6i; Value %4.6e\r', S, i, f0); if exist('fflush','builtin') fflush(stdout); end s = (df3'*df3-df0'*df3)/(df0'*df0)*s - df3; % Polack-Ribiere CG direction df0 = df3; % swap derivatives d3 = d0; d0 = df0'*s; if d0 > 0 % new slope must be negative s = -df0; d0 = -s'*s; % otherwise use steepest direction end x3 = x3 * min(RATIO, d3/(d0-realmin)); % slope ratio but max RATIO ls_failed = 0; % this line search did not fail else X = X0; f0 = F0; df0 = dF0; % restore best point so far if ls_failed || i > abs(length) % line search failed twice in a row break; % or we ran out of time, so we give up end s = -df0; d0 = -s'*s; % try steepest x3 = 1/(1-d0); ls_failed = 1; % this line search failed end end X = rewrap(Z,X); fprintf('\n'); if exist('fflush','builtin') fflush(stdout); end % Extract the numerical values from "s" into the column vector "v". The % variable "s" can be of any type, including struct and cell array. % Non-numerical elements are ignored. See also the reverse rewrap.m. function v = unwrap(s) v = []; if isnumeric(s) v = s(:); % numeric values are recast to column vector elseif isstruct(s) v = unwrap(struct2cell(orderfields(s))); % alphabetize, conv to cell, recurse elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially v = [v; unwrap(s{i})]; end end % other types are ignored % Map the numerical elements in the vector "v" onto the variables "s" which can % be of any type. The number of numerical elements must match; on exit "v" % should be empty. Non-numerical entries are just copied. See also unwrap.m. function [s v] = rewrap(s, v) if isnumeric(s) if numel(v) < numel(s) error('The vector for conversion contains too few elements') end s = reshape(v(1:numel(s)), size(s)); % numeric values are reshaped v = v(numel(s)+1:end); % remaining arguments passed on elseif isstruct(s) [s p] = orderfields(s); p(p) = 1:numel(p); % alphabetize, store ordering [t v] = rewrap(struct2cell(s), v); % convert to cell, recurse s = orderfields(cell2struct(t,fieldnames(s),1),p); % conv to struct, reorder elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially [s{i} v] = rewrap(s{i}, v); end end % other types are not processed ================================================ FILE: code/gpml/util/rewrap.m ================================================ % Map the numerical elements in the vector "v" onto the variables "s" which can % be of any type. The number of numerical elements must match; on exit "v" % should be empty. Non-numerical entries are just copied. See also unwrap.m. function [s v] = rewrap(s, v) if isnumeric(s) if numel(v) < numel(s) error('The vector for conversion contains too few elements') end s = reshape(v(1:numel(s)), size(s)); % numeric values are reshaped v = v(numel(s)+1:end); % remaining arguments passed on elseif isstruct(s) [s p] = orderfields(s); p(p) = 1:numel(p); % alphabetize, store ordering [t v] = rewrap(struct2cell(s), v); % convert to cell, recurse s = orderfields(cell2struct(t,fieldnames(s),1),p); % conv to struct, reorder elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially [s{i} v] = rewrap(s{i}, v); end end % other types are not processed ================================================ FILE: code/gpml/util/solve_chol.c ================================================ /* solve_chol - solve a linear system A*X = B using the cholesky factorization of A (where A is square, symmetric and positive definite. Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch 2010-12-16. */ #include "mex.h" #include #include extern int dpotrs_(char *, long *, long *, double *, long *, double *, long *, long *); void mexFunction(int nlhs, mxArray *plhs[], int nrhs, const mxArray *prhs[]) { double *C; long n, m, q; if (nrhs != 2 || nlhs > 1) /* check the input */ mexErrMsgTxt("Usage: X = solve_chol(R, B)"); n = mxGetN(prhs[0]); if (n != mxGetM(prhs[0])) mexErrMsgTxt("Error: First argument matrix must be square"); if (n != mxGetM(prhs[1])) mexErrMsgTxt("Error: First and second argument matrices must have same number of rows"); m = mxGetN(prhs[1]); plhs[0] = mxCreateDoubleMatrix(n, m, mxREAL); /* allocate space for output */ C = mxGetPr(plhs[0]); if (n==0) return; /* if argument was empty matrix, do no more */ memcpy( C, mxGetPr(prhs[1]), n*m*sizeof(double) );/* copy argument matrix */ dpotrs_("U", &n, &m, mxGetPr(prhs[0]), &n, C, &n, &q); /* solve system */ if (q > 0) mexErrMsgTxt("Error: illegal input to solve_chol"); } ================================================ FILE: code/gpml/util/solve_chol.m ================================================ % solve_chol - solve linear equations from the Cholesky factorization. % Solve A*X = B for X, where A is square, symmetric, positive definite. The % input to the function is R the Cholesky decomposition of A and the matrix B. % Example: X = solve_chol(chol(A),B); % % NOTE: The program code is written in the C language for efficiency and is % contained in the file solve_chol.c, and should be compiled using matlabs mex % facility. However, this file also contains a (less efficient) matlab % implementation, supplied only as a help to people unfamiliar with mex. If % the C code has been properly compiled and is avaiable, it automatically % takes precendence over the matlab code in this file. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch 2010-09-18. function X = solve_chol(L, B) if nargin ~= 2 || nargout > 1 error('Wrong number of arguments.'); end if size(L,1) ~= size(L,2) || size(L,1) ~= size(B,1) error('Wrong sizes of matrix arguments.'); end X = L\(L'\B); ================================================ FILE: code/gpml/util/sq_dist.m ================================================ % sq_dist - a function to compute a matrix of all pairwise squared distances % between two sets of vectors, stored in the columns of the two matrices, a % (of size D by n) and b (of size D by m). If only a single argument is given % or the second matrix is empty, the missing matrix is taken to be identical % to the first. % % Usage: C = sq_dist(a, b) % or: C = sq_dist(a) or equiv.: C = sq_dist(a, []) % % Where a is of size Dxn, b is of size Dxm (or empty), C is of size nxm. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-12-13. function C = sq_dist(a, b) if nargin<1 || nargin>3 || nargout>1, error('Wrong number of arguments.'); end bsx = exist('bsxfun','builtin'); % since Matlab R2007a 7.4.0 and Octave 3.0 if ~bsx, bsx = exist('bsxfun'); end % bsxfun is not yes "builtin" in Octave [D, n] = size(a); % Computation of a^2 - 2*a*b + b^2 is less stable than (a-b)^2 because numerical % precision can be lost when both a and b have very large absolute value and the % same sign. For that reason, we subtract the mean from the data beforehand to % stabilise the computations. This is OK because the squared error is % independent of the mean. if nargin==1 % subtract mean mu = mean(a,2); if bsx a = bsxfun(@minus,a,mu); else a = a - repmat(mu,1,size(a,2)); end b = a; m = n; else [d, m] = size(b); if d ~= D, error('Error: column lengths must agree.'); end mu = (m/(n+m))*mean(b,2) + (n/(n+m))*mean(a,2); if bsx a = bsxfun(@minus,a,mu); b = bsxfun(@minus,b,mu); else a = a - repmat(mu,1,n); b = b - repmat(mu,1,m); end end if bsx % compute squared distances C = bsxfun(@plus,sum(a.*a,1)',bsxfun(@minus,sum(b.*b,1),2*a'*b)); else C = repmat(sum(a.*a,1)',1,m) + repmat(sum(b.*b,1),n,1) - 2*a'*b; end C = max(C,0); % numerical noise can cause C to negative i.e. C > -1e-14 ================================================ FILE: code/gpml/util/unwrap.m ================================================ % Extract the numerical values from "s" into the column vector "v". The % variable "s" can be of any type, including struct and cell array. % Non-numerical elements are ignored. See also the reverse rewrap.m. function v = unwrap(s) v = []; if isnumeric(s) v = s(:); % numeric values are recast to column vector elseif isstruct(s) v = unwrap(struct2cell(orderfields(s))); % alphabetize, conv to cell, recurse elseif iscell(s) for i = 1:numel(s) % cell array elements are handled sequentially v = [v; unwrap(s{i})]; end end % other types are ignored ================================================ FILE: code/plot_additive_decomp.m ================================================ function plot_additive_decomp( X, y, kernel_components, kernel_params, log_noise, show_samples, savefigs, fileprefix ) % % Decomposes an additive GP model into component parts % % X,y: The original data % kernel_components: a cell array of covariance functions, to be added % together. This code can figure out which dimension % each kernel applies to. % kernel_params: a cell array of kernel hyperparameters. % % David Duvenaud % March 2014 addpath(genpath( 'utils' )); addpath(genpath( 'gpml' )); if nargin < 7; savefigs = false; end if nargin < 6; show_samples = true; end nstar = 300; [N,D] = size(X); num_components = length(kernel_components); assert(num_components == length(kernel_params)); noise_var = exp(2*log_noise); % First, compute and plot the entire model. % concatenate all additive components complete_cov = { 'covSum', kernel_components }; complete_hypers = vertcat( kernel_params{:} ); noise_cov = eye(length(y)).*noise_var; complete_sigma = feval(complete_cov{:}, complete_hypers, X) + noise_cov; %complete_sigmastar = feval(complete_cov{:}, complete_hypers, X, xrange); %complete_sigmastarstart = feval(complete_cov{:}, complete_hypers, xrange); % First, plot the data %complete_mean = complete_sigmastar' / complete_sigma * y; %complete_var = complete_sigmastarstart - complete_sigmastar' / complete_sigma * complete_sigmastar; %figure(1); clf; %filename = sptrintf( '%s-complete', fileprefix ); %nice_oned_plot( X, y, xstar, complete_mean, complete_var, true, filename, savefigs) % Next, show the posterior of each component, one at a time. for i = 1:num_components cur_cov = kernel_components{i}; cur_hyp = kernel_params{i}; % Figure out which dimension this kernel applies to. assert( strcmp( cur_cov{1}, 'covMask' ) ); d = cur_cov{2}{1}; left_extend = 0.1; right_extend = 0.1; x_left = min(X(:,d)) - (max(X(:,d)) - min(X(:,d)))*left_extend; x_right = max(X(:,d)) + (max(X(:,d)) - min(X(:,d)))*right_extend; xrange = linspace(x_left, x_right, nstar)'; xstar = NaN(nstar,D); xstar(:, d) = xrange; % Compute Gram matrices of just this component. component_sigma = feval(cur_cov{:}, cur_hyp, X); component_sigma_star = feval(cur_cov{:}, cur_hyp, xstar, X); component_sigma_starstar = feval(cur_cov{:}, cur_hyp, xstar); % Compute posterior of just this component. component_mean = component_sigma_star / complete_sigma * y; component_var = component_sigma_starstar - component_sigma_star / complete_sigma * component_sigma_star'; data_mean = component_sigma / complete_sigma * y; %data_mean = y - (complete_sigma - component_sigma) / complete_sigma * y; % Plot posterior mean and variance. figure(i); clf; filename = sprintf( '%s-component-%d', fileprefix, d ); [h_data_orig, h_gp_post, h_sample] = nice_oned_plot( X(:,d), y, data_mean, xrange, component_mean, component_var, show_samples, savefigs, filename); end % Plot legend if savefigs set(h_data_orig,'Visible','off'); set(h_sample,'Visible','off'); set(h_gp_post,'Visible','off'); axis off h_legend = legend([h_data_orig, h_gp_post, h_sample], ... 'Data', 'Posterior density', 'Posterior samples', 'Location', 'Best'); set(h_legend,'FontSize',12); set(h_legend,'FontName','Times'); save2pdf([filename '-legend.pdf'], gcf, 600, true ); end complete_mean = (complete_sigma - noise_cov) / complete_sigma * y; rs = 1 - var(complete_mean - y)/ var(y) end function [h_data_orig, h_gp_post, h_sample] = nice_oned_plot( X, y, y_adjusted, xstar, mean, full_variance, show_samples, savefigs, filename) % Makes a nice plot of a GP posterior. % % David Duvenaud % March 2014 variance = diag(full_variance); % How figure looks. num_quantiles = 10; num_rand_samples = 10; dpi = 600; if nargin < 7; savefigs = false; end quantiles = linspace(0,0.5,num_quantiles+1); quantiles = quantiles(2:end); for s = quantiles edges = [mean + norminv(s, 0, 1).*sqrt(variance); ... flipdim(mean - norminv(s, 0, 1).*sqrt(variance),1)]; h_gp_post = fill([xstar; flipdim(xstar,1)], edges, color_spectrum(2*s), ... 'EdgeColor', 'none'); hold on; end h_data_orig = plot( X, y, 'kx', 'Linewidth', 1.5, 'Markersize', 10, 'Color', colorbrew(2)); hold on; %h_data_orig = plot( X, y, 'kx', 'Linewidth', 1.5, 'Markersize', 10); hold on; if show_samples % Plot posterior samples seed=0; % fixing the seed of the random generators randn('state',seed); rand('state',seed); for n_sample = 1:num_rand_samples L = chol(full_variance + eye(length(xstar)).*max(full_variance(:)).*0.0001); sample = mean + L'*randn(length(xstar),1); %sample = mvnrnd( mean, full_variance + eye(length(xstar)).*max(full_variance(:)).*0.0001, 1); h_sample = plot( xstar, sample, '-', 'Color', colorbrew(n_sample), 'Linewidth', 1); hold on; % end end show_mean = false; if show_mean h_data_adjust = plot( X, y_adjusted, 'k+', 'Linewidth', 1.5, 'Markersize', 10, 'Color', colorbrew(3)); hold on; end %ylim( ylimits); xlim( [xstar(1), xstar(end)]); set(gca,'Layer','top') % Show the axes again %legend_handle = legend( [hc1 h2 ], { 'GP Posterior Uncertainty', 'Data'}, 'Location', 'SouthEast'); %set_thesis_fonts( gca, legend_handle ); set( gca, 'XTick', [] ); set( gca, 'yTick', [] ); set( gca, 'XTickLabel', '' ); set( gca, 'yTickLabel', '' ); %xlabel( '$x$' ); %ylabel( '$f(x)$\qquad' ); set(get(gca,'XLabel'),'Rotation',0,'Interpreter','latex', 'Fontsize', 16); set(get(gca,'YLabel'),'Rotation',0,'Interpreter','latex', 'Fontsize', 16); set(gcf, 'color', 'white'); % Add axes, legend, make the plot look nice, and save it. tightfig(); set_fig_units_cm(10,6); if savefigs %filename = sprintf('%s/%s-%d', introfigsdir, 'fuzzy', N ); %myaa('publish'); %savepng(gcf, filename); save2pdf([filename '.pdf'], gcf, dpi, true ); end end function col = color_spectrum(p) no_col = [1 1 1]; full_col = [ 1 0 0 ]; col = (1 - p)*no_col + p*full_col; end ================================================ FILE: code/plot_additive_decomp_cov.m ================================================ function plot_additive_decomp_cov( X, y, kernel_components, kernel_params, log_noise, savefigs, file_prefix, column_names ) % % Decomposes an additive GP model into component parts % % X,y: The original data % kernel_components: a cell array of covariance functions, to be added % together. This code can figure out which dimension % each kernel applies to. % kernel_params: a cell array of kernel hyperparameters. % % David Duvenaud % March 2014 addpath(genpath( 'utils' )); addpath(genpath( 'gpml' )); if nargin < 7; savefigs = false; end if nargin < 6; show_samples = true; end nstar = 400; dpi = 300; [N,D] = size(X); num_components = length(kernel_components); assert(num_components == length(kernel_params)); noise_var = exp(2*log_noise); % First, compute and plot the entire model. % concatenate all additive components complete_cov = { 'covSum', kernel_components }; complete_hypers = unwrap( kernel_params ); noise_cov = eye(length(y)).*noise_var; complete_sigma = feval(complete_cov{:}, complete_hypers, X) + noise_cov; %complete_sigmastar = feval(complete_cov{:}, complete_hypers, X, xrange); %complete_sigmastarstart = feval(complete_cov{:}, complete_hypers, xrange); % First, plot the data %complete_mean = complete_sigmastar' / complete_sigma * y; %complete_var = complete_sigmastarstart - complete_sigmastar' / complete_sigma * complete_sigmastar; % Detect categorical dimensions. categorical = zeros(1,D); for d = 1:D num_unique = length(unique(X(:,d))); if num_unique < 10 categorical(d) = true; fprintf('Dimension %d is categorical with %d different values\n', ... d, num_unique); end end figure; clf; %figure(1); clf; %filename = sptrintf( '%s-complete', fileprefix ); %nice_oned_plot( X, y, xstar, complete_mean, complete_var, true, filename, savefigs) left_extend = 0.05; right_extend = 0.05; cmax = -Inf; cmin = Inf; % Next, show the posterior of each component, one at a time. for i = 1:num_components for j = 1:num_components cur_cov1 = kernel_components{i}; cur_hyp1 = kernel_params{i}; cur_cov2 = kernel_components{j}; cur_hyp2 = kernel_params{j}; % Figure out which dimension this kernel applies to. assert( strcmp( cur_cov1{1}, 'covMask' ) ); d1 = cur_cov1{2}{1}; assert( strcmp( cur_cov2{1}, 'covMask' ) ); d2 = cur_cov2{2}{1}; % Compute ranges of test sets. x1_left = min(X(:,d1)) - (max(X(:,d1)) - min(X(:,d1)))*left_extend; x1_right = max(X(:,d1)) + (max(X(:,d1)) - min(X(:,d1)))*right_extend; x2_left = min(X(:,d2)) - (max(X(:,d2)) - min(X(:,d2)))*left_extend; x2_right = max(X(:,d2)) + (max(X(:,d2)) - min(X(:,d2)))*right_extend; x1range = linspace(x1_left, x1_right, nstar)'; x1star = NaN(nstar,D); x1star(:, d1) = x1range; x2range = linspace(x2_left, x2_right, nstar)'; x2star = NaN(nstar,D); x2star(:, d2) = x2range; % Compute posterior covariance between these two components. component1_sigma_star = feval(cur_cov1{:}, cur_hyp1, x1star, X); component1_sigma_starstar = feval(cur_cov1{:}, cur_hyp1, x1star); component2_sigma_star = feval(cur_cov2{:}, cur_hyp2, x2star, X); component2_sigma_starstar = feval(cur_cov2{:}, cur_hyp2, x2star); covar = -component1_sigma_star / complete_sigma * component2_sigma_star'; var1 = -component1_sigma_star / complete_sigma * component1_sigma_star'; var2 = -component2_sigma_star / complete_sigma * component2_sigma_star'; % Diagonals have an extra term. if i == j covar = covar + component1_sigma_starstar; % is equal to component2_sigma_starstar assert(all(component1_sigma_starstar(:) == component2_sigma_starstar(:))); var1 = var1 + component1_sigma_starstar; var2 = var2 + component2_sigma_starstar; end correlation = covar ./ sqrt(bsxfun(@times, diag(var1), diag(var2)')); assert(all(abs(correlation(:)) <= 1)); % Plot posterior mean and variance. if savefigs clf; imagesc(correlation); caxis([-0.5 1]); % Make the plot look nice. set( gca, 'XTick', [] ); set( gca, 'yTick', [] ); set( gca, 'XTickLabel', '' ); set( gca, 'yTickLabel', '' ); set(gcf, 'color', 'white'); tightfig(); set_fig_units_cm(4,4); title1 = strrep(column_names{i}, ' ', '-'); title2 = strrep(column_names{j}, ' ', '-'); filename = sprintf('%s-%s', title1, title2); save2pdf([file_prefix, '-', filename '.pdf'], gcf, dpi, true ); filenames{i}{j} = filename; else subplot(num_components, num_components, j + (i - 1) * num_components); imagesc(correlation); caxis([-0.5 1]); end % Keep track of max and min for unified colobar. cmax = max([cmax, max(correlation(:))]); cmin = min([cmin, min(correlation(:))]); end end % Print colorbar figure; caxis([-0.5 1]); hc = colorbar('vert'); axis off; set(hc, 'FontName', 'Times New Roman', 'Fontsize', 12); %tightfig(); set_fig_units_cm(10,4); save2pdf([file_prefix, '-colorbar.pdf'], gcf, dpi, true ); if savefigs smalltext = ''; if num_components > 7; smalltext = '\small'; end % Print a latex table. fprintf('\n\\renewcommand{\\tabcolsep}{1mm}'); fprintf('\n\\def\\incpic#1{\\includegraphics[width=%1.3f\\columnwidth]{%s-#1}}', ... 1/(num_components + 1), file_prefix); %fprintf('\n\\begin{tabular}{p{2mm}*{%d}{p{%1.3f\\columnwidth}}}\n & ', ... % num_components, 1/(num_components + 1)); fprintf('\n\\begin{tabular}{p{2mm}*{%d}{c}}\n & ', ... num_components); for i = 1:num_components % print header if i == 1 for j = 1:num_components fprintf('%s{%s}', smalltext, column_names{j}); if j < num_components; fprintf(' & '); else fprintf(' \\\\ \n '); end end end fprintf('\\rotatebox{90}{%s{%s}} & ', smalltext, column_names{i}); for j = 1:num_components fprintf('\\incpic{%s}', filenames{i}{j}); if j < num_components; fprintf(' & '); else fprintf(' \\\\ \n '); end end end fprintf('\\end{tabular}\n'); end end ================================================ FILE: code/plot_oned_gp.m ================================================ function plot_oned_gp( savefigs ) % % A demo showing what happens if you condition on GPs. % % David Duvenaud % March 2014 addpath(genpath( 'utils' )); if nargin < 1; savefigs = false; end % How to save figure. introfigsdir = '../figures/intro'; dpi = 600; % How figure looks. num_quantiles = 10; Nstar = 200; ylimits = [-1 1.1]; show_samples = true; sample_ix_subset = false; num_rand_samples = 10; % Specify GP model. sigma = .02; length_scale = 20; output_variance = 0.14; % Make up some data x = [ 20 80 35 ]'; y = [ -10 55 40 ]' ./100; max_N = length(x); % Condition on one datapoint at a time. for N = 0:max_N % Fill in gram matrix K = NaN(N,N); for j = 1:N for k = 1:N K(j,k) = kernel( x(j), x(k), length_scale, output_variance ); end end % Compute inverse covariance Kn = K + sigma^2 .* diag(ones(N, 1)); % Compute covariance with test points. xstar = linspace(-10,110, Nstar); Kstar = NaN(Nstar, N); kfull = NaN(Nstar, Nstar); for j = 1:Nstar for k = 1:N Kstar(j,k) = kernel( xstar(j), x(k), length_scale, output_variance); end kstarstar = kernel( xstar(j), xstar(j), length_scale, output_variance); for k = 1:Nstar kfull(j,k) = kernel( xstar(j), xstar(k), length_scale, output_variance); end end % Compute posterior mean and variance. f = (Kstar / Kn) * y(1:N); variance = kstarstar - diag((Kstar / Kn) * Kstar'); full_variance = kfull - (Kstar / Kn) * Kstar'; % Plot posterior mean and variance. figure(N+1); clf; quantiles = linspace(0,0.5,num_quantiles+1);%0:0.05:0.5; quantiles = quantiles(2:end); for s = quantiles edges = [f+norminv(s, 0, 1).*sqrt(variance); ... flipdim(f-norminv(s, 0, 1).*sqrt(variance),1)]; hc1 = fill([xstar'; flipdim(xstar',1)], edges, color_spectrum(2*s), 'EdgeColor', 'none'); hold on; end if show_samples % Plot posterior samples seed=0; % fixing the seed of the random generators randn('state',seed); rand('state',seed); for n_sample = 1:num_rand_samples if sample_ix_subset cur_ixs = sort(randperm(Nstar, 2)); cur_range = cur_ixs(1):cur_ixs(2); else cur_range = 1:Nstar; end sample = mvnrnd( f, full_variance, 1); hs = plot( xstar(cur_range), sample(cur_range), '-', 'Color', colorbrew(n_sample), 'Linewidth', 1); hold on; % end end h2 = plot( x(1:N), y(1:N), 'k+', 'Linewidth', 1.5, 'Markersize', 10); hold on; ylim( ylimits); xlim( [xstar(1), xstar(end)]); set(gca,'Layer','top') % Show the axes again %legend_handle = legend( [hc1 h2 ], { 'GP Posterior Uncertainty', 'Data'}, 'Location', 'SouthEast'); %set_thesis_fonts( gca, legend_handle ); set( gca, 'XTick', [] ); set( gca, 'yTick', [] ); set( gca, 'XTickLabel', '' ); set( gca, 'yTickLabel', '' ); %xlabel( '$x$' ); %ylabel( '$f(x)$\qquad' ); set(get(gca,'XLabel'),'Rotation',0,'Interpreter','latex', 'Fontsize', 16); set(get(gca,'YLabel'),'Rotation',0,'Interpreter','latex', 'Fontsize', 16); set(gcf, 'color', 'white'); % Add axes, legend, make the plot look nice, and save it. tightfig(); set_fig_units_cm(10,6); if savefigs filename = sprintf('%s/%s-%d', introfigsdir, 'fuzzy', N ); %myaa('publish'); %savepng(gcf, filename); save2pdf(filename, gcf, dpi, true ); end end end function col = color_spectrum(p) no_col = [1 1 1]; full_col = [ 1 0 0 ]; col = (1 - p)*no_col + p*full_col; end function d = kernel(x, y, length_scale, output_variance) d = output_variance * exp( - 0.5 * ( (( x - y ) ./ length_scale) .^ 2 ) ); ... %+ 3.*output_variance * exp( - 0.5 * ( ( ( x - y ) ./ (length_scale * 10) ).^ 2 ) ) ... %+ 0.01 * output_variance * exp( - 0.5 * ( ( ( x - y ) ./ (length_scale ./ 10) ).^ 2 ) ); end ================================================ FILE: code/plot_oned_gplvm.m ================================================ % A simple demo to help me understand variational methods. % % David Duvenaud % March 2013 % fix the seed of the random number generators. %seed=0; randn('state',seed); rand('state',seed); % Specify settings. function gplvm_demo(seed) if nargin < 1; seed=0; end % fixing the seed of the random generators randn('state',seed); rand('state',seed); n = 1000; nxbins = 20; nybins = 20; nstar = 300; histprop = 0.25; sidemargin = 0.08; intermargin = 0.04; % Generate data. x_prior = @(x) normpdf( x, 0, 1 ); true_x = randn(n,1); x_lim = [min(true_x) max(true_x)]; n_sample_points = 200; sample_points = linspace( x_lim(1), x_lim(2), n_sample_points); % Generate a random function. numerical_noise = 1e-5; se_length_scale = 2.5; se_outout_var = 2; se_kernel = @(x,y) se_outout_var*exp( - 0.5 * ( ( x - y ) .^ 2 ) ./ se_length_scale^2 ); K = bsxfun(se_kernel, sample_points', sample_points ) + eye(n_sample_points).*numerical_noise; % Evaluate prior. %Kstar = bsxfun(se_kernel, xstar', xrange ) + eye(n_xstar).*numerical_noise; % Evaluate prior. y = mvnrnd( zeros(size(sample_points)), K, 1)'; weights = K \ y; % Now compute kernel function weights. true_f = @(x)(bsxfun(se_kernel, x', sample_points' )' * weights); % Construct posterior function. % Condition on samples. %f = (Kstar / Kn) * y(1:N); %true_f = @(x) 4 - x.^2; y = true_f(true_x); %x_lim = [-2 2]; %y_lim = [0 4]; y_lim = [min(y) max(y)]; xrange = linspace(x_lim(1), x_lim(2), nstar); figure(100); clf; h=axes('position',[histprop+sidemargin+intermargin, histprop+sidemargin+intermargin, 1-histprop-sidemargin-2*intermargin, 1-histprop-1*sidemargin-2*intermargin]); % [left bottom width height] % where left and bottom define the distance from the lower-left corner of the % container. % Plot data plot(true_x, y_lim(1).*ones(size(true_x)), 'k*' ); hold on; plot(x_lim(1).*ones(size(y)), y, 'k*' ); gray = [0.9 0.9 0.9]; % Plot connecting lines. for i = 1:n; line([true_x(i), true_x(i)], [y_lim(1), true_f(true_x(i))], 'Color', gray ); line([x_lim(1), true_x(i)], [true_f(true_x(i)), true_f(true_x(i))], 'Color', gray ); end % Plot true warping function. plot(xrange, true_f(xrange'), '-', 'Linewidth', 2); hold on; %plot(true_x, true_f(true_x), '-', 'Linewidth', 2); hold on; xlim(x_lim); ylim(y_lim); set( gca, 'yTick', [] ); set( gca, 'yTickLabel', '' ); set( gca, 'XTick', [] ); set( gca, 'XTickLabel', '' ); % Plot x dist h=axes('position',[histprop+sidemargin+intermargin, sidemargin, 1-histprop-sidemargin-2*intermargin, histprop]); [xcounts, xbins] = hist(true_x, nxbins); xprob = xcounts ./ sum(xcounts); bar(xbins, xprob); hold on; xb = get(gca,'child'); set(xb,'FaceColor', colorbrew(2)); plot(xrange, x_prior(xrange)./max(x_prior(xrange))*max(xprob), 'r--'); xlim(x_lim); set( gca, 'yTick', [] ); set( gca, 'yTickLabel', '' ); xlabel('Latent x coordinate'); % Plot y dist h=axes('position',[sidemargin histprop+sidemargin+intermargin, histprop, 1-histprop-sidemargin-2*intermargin]); [ycounts, ybins] = hist(y, nybins); yprob = ycounts ./ sum(ycounts); h = barh(ybins, yprob); hold on; set(get(h,'Parent'),'xdir','r') yb = get(gca,'child'); set(yb,'FaceColor', colorbrew(3)); ylim(y_lim); set( gca, 'XTick', [] ); set( gca, 'XTickLabel', '' ); ylabel('observed y coordinate'); % Make plot prettier. set(gcf, 'color', 'white'); set(gca, 'TickDir', 'out'); set_fig_units_cm( 14, 8 ); save2pdf([ 'figures/gplvm_1d_draw_', num2str(seed), '.pdf'], gcf, 600, true); ================================================ FILE: code/plot_structure_examples.m ================================================ function plot_structure_examples(seed) % A script to make a series of plots demonstrating how structure can be % reflected in a kernel. % % This version updated to be more amenable to tiny plots. % % David Duvenaud % Jan 2013 if nargin < 1; seed = 1; end randn('state',seed); rand('state',seed); savefigs = true; figpath = '../figures/grammar/structure_examples/'; addpath(genpath('utils')) % Make up some data %X = [ -2 -1 0 1 2 ]' .* 2; %y = [ -1 1 1 -0.8 1.1 ]'; X = [ 1 2 3 ]' .* 2; y = [ 1 2 4 ]'; y = y - mean(y); N = length(X); n_samples = 2; n_xstar = 201; xrange = linspace(-10, 10, n_xstar)'; post_xrange = linspace(-3, 10, n_xstar)'; x0 = 0; numerical_noise = 1e-5; model_noise = 2; se_length_scale = 2.5; se_outout_var = 2; se_kernel = @(x,y) se_outout_var*exp( - 0.5 * ( ( x - y ) .^ 2 ) ./ se_length_scale^2 ); lin_output_var = 0.5; lin_kernel = @(x,y) lin_output_var*( (x + 1) .* (y + 1) ); longse_length_scale = 20; longse_output_var = 10; longse_kernel = @(x,y) longse_output_var*exp( - 0.5 * ( ( x - y ) .^ 2 ) ./ longse_length_scale^2 ); shortse_length_scale = 0.5; shortse_output_var = 0.5; shortse_kernel = @(x,y) shortse_output_var*exp( - 0.5 * ( ( x - y ) .^ 2 ) ./ shortse_length_scale^2 ); medse_length_scale = 10; medse_output_var = 5; medse_kernel = @(x,y) medse_output_var*exp( - 0.5 * ( ( x - y ) .^ 2 ) ./ medse_length_scale^2 ); per_length_scale = 1; per_period = 4; per_output_var = 1.1; per_kernel = @(x,y) per_output_var*exp( - 2 * ( sin(pi*( x - y )./per_period) .^ 2 ) ./ per_length_scale^2 ); cos_period = 4; cos_output_var = 1.1; cos_kernel = @(x,y) cos_output_var*cos( 2 * pi * ( x - y )./cos_period); wn_output_var = 1.1; wn_kernel = @(x,y) wn_output_var*double( x == y ); rq_length_scale = 2.5; rq_outout_var = 2; rq_alpha = 1.1; rq_kernel = @(x,y) rq_outout_var*( 1 + ( x - y ) .^ 2 ./ (2*rq_alpha*rq_length_scale^2 )).^-rq_alpha; c_kernel = @(x, y) ones(size(x*y)); se_plus_lin = @(x,y) se_kernel(x, y) + lin_kernel(x, y); se_plus_per = @(x,y) se_kernel(x, y) + per_kernel(x, y); se_times_lin = @(x,y) se_kernel(x, y) .* lin_kernel(x, y); se_times_per = @(x,y) se_kernel(x, y) .* per_kernel(x, y); lin_times_per = @(x,y) lin_kernel(x, y) .* per_kernel(x, y); lin_plus_per = @(x,y) lin_kernel(x, y) + per_kernel(x, y); lin_times_lin = @(x,y) lin_kernel(x, y) .* lin_kernel(x, y); longse_times_per = @(x,y) longse_kernel(x, y) .* per_kernel(x, y); longse_plus_per = @(x,y) longse_kernel(x, y) + per_kernel(x, y); longse_times_lin = @(x,y) longse_kernel(x, y) .* lin_kernel(x, y); longse_plus_se = @(x,y) longse_kernel(x, y) + se_kernel(x, y); shortse_plus_medse = @(x,y) shortse_kernel(x, y) + medse_kernel(x, y); % kernel_names = {'se_kernel', 'lin_kernel', 'per_kernel', 'longse_kernel', ... % 'se_plus_lin', 'se_plus_per', 'se_times_lin', 'se_times_per', ... % 'lin_times_per', 'lin_plus_per', 'lin_times_lin', ... % 'longse_times_per', 'longse_plus_per', 'longse_times_lin', ... % 'rq_kernel','c_kernel'}; kernel_names = {'shortse_plus_medse'}; % Automatically build kernel names from function names. for i = 1:numel(kernel_names) kernels{i} = eval(kernel_names{i}); end %color_ixs = [ 1, 2, 3, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 4 ]; color_ixs = repmat(10, 1, numel(kernels)); if 1 % plot the kernels. for k = 1:numel(kernels) figure(k); clf; cur_kernel = kernels{k}; kvals = bsxfun(cur_kernel, xrange, x0 ); kernel_plot( xrange, kvals, color_ixs(k) ); if savefigs save2pdf([ figpath, kernel_names{k}, '.pdf'], gcf, 600, true); end pause(0.01); drawnow; end end % Plot draws from the kernels. for k = 1:numel(kernels) figure(10 + k); clf; K = bsxfun(kernels{k}, xrange', xrange ) + eye(n_xstar).*numerical_noise; % Evaluate prior. %L = chol(K); samples = mvnrnd( zeros(size(xrange)), K, n_samples)'; samples = samples + repmat((1:n_samples) * 5 - 10, n_xstar, 1); samples_plot( xrange, samples, [1:n_samples] ); if savefigs save2pdf([ figpath, kernel_names{k} '_draws_s', int2str(seed), '.pdf'], gcf, 600, true); end pause(0.01); drawnow; end if 0 % Plot posterior dists of the kernels. for k = 1:numel(kernels) figure(20 + k); clf; cur_kernel = kernels{k}; K = bsxfun(cur_kernel, X, X' ) + eye(N).*model_noise; % Evaluate prior. weights = K \ y; posterior = @(x)(bsxfun(cur_kernel, post_xrange, X') * weights); % Construct posterior function. posterior_variance = @(x)diag(bsxfun(cur_kernel, post_xrange', post_xrange) - (bsxfun(cur_kernel, X', xrange) / K * bsxfun(cur_kernel, xrange', X))); posterior_plot( post_xrange, posterior(post_xrange), posterior_variance(post_xrange), X, y ); if savefigs save2pdf([ figpath, kernel_names{k} '_post.pdf'], gcf, 600, true); end pause(0.01); drawnow; end end end function kernel_plot( xrange, vals, color_ix ) % Figure settings. lw = 2; fontsize = 10; plot(xrange, vals, 'Color', colorbrew(color_ix), 'LineWidth', lw); hold on; if all( vals >= 0 ); lowlim = 0; else lowlim = min(vals) * 1.05; end % Make plot prettier. xlim([min(xrange), max(xrange)]); ylim([min(lowlim,0), max(vals) * 1.05]); set( gca, 'XTick', [ 0 ] ); set( gca, 'yTick', [ 0 ] ); set( gca, 'XTickLabel', '' ); %set( gca, 'yTickLabel', '' ); % xlabel( '$x - x''$', 'Fontsize', fontsize ); %xlabel(' '); %ylabel( '$k(x, 0)$', 'Fontsize', fontsize ); set(get(gca,'XLabel'),'Rotation',0,'Interpreter','latex', 'Fontsize', fontsize); set(get(gca,'YLabel'),'Rotation',90,'Interpreter','latex', 'Fontsize', fontsize); %set(gca, 'TickDir', 'out') set(gca, 'Box', 'off'); set(gcf, 'color', 'white'); set(gca, 'YGrid', 'off'); set_fig_units_cm( 4,3 ); end function samples_plot( xrange, samples, color_ix ) % Figure settings. lw = 2; fontsize = 10; for i = 1:size(samples, 2); plot(xrange, samples(:,i), 'Color', colorbrew(color_ix(i)), 'LineWidth', lw); hold on; end % Make plot prettier. xlim([min(xrange), max(xrange)]); set( gca, 'XTick', [ 0 ] ); %set( gca, 'yTick', [ 0 ] ); set( gca, 'yTick', [] ); set( gca, 'XTickLabel', '' ); set( gca, 'yTickLabel', '' ); %xlabel( '$x$', 'Fontsize', fontsize ); %ylabel( '$f(x)$', 'Fontsize', fontsize ); set(get(gca,'XLabel'),'Rotation',0,'Interpreter','latex', 'Fontsize', fontsize); set(get(gca,'YLabel'),'Rotation',90,'Interpreter','latex', 'Fontsize', fontsize); %set(gca, 'TickDir', 'out') set(gca, 'Box', 'off'); set(gcf, 'color', 'white'); set(gca, 'YGrid', 'off'); set_fig_units_cm( 4,3 ); end function posterior_plot( xrange, f, v, X, y ) % Figure settings. lw = 2; fontsize = 10; opacity = 1; fake_opacity = 0.1; jbfill( xrange', ... f' + 2.*sqrt(v)', ... f' - 2.*sqrt(v)', ... colorbrew(2).^fake_opacity, 'none', 1, opacity); hold on; plot( xrange, f, '-', 'Linewidth', lw, 'Color', colorbrew(2)); hold on; % Plot data. plot( X, y, 'kx', 'Linewidth', lw, 'MarkerSize', 6); hold on; % Make plot prettier. width = abs(min(xrange) - max(xrange)); % Widen xlim slightly. xlim([min(xrange) - width/50, max(xrange) + width/50]); %set( gca, 'XTick', [ -1 0 1 ] ); %set( gca, 'yTick', [ 0 1 ] ); set( gca, 'XTickLabel', '' ); %set( gca, 'yTickLabel', '' ); % xlabel( '$x$', 'Fontsize', fontsize ); ylabel( '$f(x)$', 'Fontsize', fontsize ); set(get(gca,'XLabel'),'Rotation',0,'Interpreter','latex', 'Fontsize', fontsize); set(get(gca,'YLabel'),'Rotation',90,'Interpreter','latex', 'Fontsize', fontsize); %set(gca, 'TickDir', 'out') set(gca, 'Box', 'off'); set(gcf, 'color', 'white'); set(gca, 'YGrid', 'off'); ylim([-5, 5]) set_fig_units_cm( 4,3 ); end ================================================ FILE: code/plot_symmetry_samples.m ================================================ function plot_symmetry_samples % Simple demo of drawing from a 2-dimensional GP % % David Duvenaud % March 2014 % -=-=-=-=-=-=-=--=-=-=-=-=-=-=-=-=- close all N_1d = 50; % Fineness of grid in each dimension. range = linspace( -5, 5, N_1d); % Choose a set of x locations. %plot_2d_gp_and_kernel( range, @gnn_kernel ); %plot_2d_gp_and_kernel( range, @add_kernel ); %plot_2d_gp_and_kernel( range, @fa_kernel ); %plot_2d_gp_and_kernel( range, @se_kernel ); %plot_2d_gp_and_kernel( range, @se_1d_kernel ); %plot_2d_gp_and_kernel( range, @se_1d_kernel_plus_2d ); %plot_2d_gp_and_kernel( range, @symm_xy_kernel_naive ); plot_2d_gp_and_kernel( range, @symm_xy_kernel_prod ); %plot_2d_gp_and_kernel( range, @symm_xy_kernel_proj ); %plot_2d_gp_and_kernel( range, @spline_kernel ); end function plot_2d_gp_and_kernel( range, kernel ) figdir = '../figures/symmetries/'; titlestring = kernel(); % Call with no arguments to get a description of the kernel. filename = [figdir titlestring] % Set the random seed. randn('state', 0); rand('twister', 0); [x1, x2] = meshgrid( range); x = [x1(:) x2(:)]'; N = size(x, 2); mu = zeros(N, 1); % Set the mean of the GP to zero everywhere. figure; % Plot kernel. %subplot(1,2,1); sigma = kernel( x, [ 0; 0 ] ); surf(x1, x2, reshape(sigma, sqrt(numel(sigma)), sqrt(numel(sigma)))); nice_figure_save([filename, '-kernel']) %title('kernel'); % Plot a draw from a GP with that kernel. %subplot(1,2,2); sigma = kernel( x, x ); sigma = 0.5*(sigma + sigma'); sigma = sigma + eye(length(sigma)) .* 1e-6; f = mvnrnd( mu, sigma ); % Draw a sample from a multivariate Gaussian. surf(x1, x2, reshape(f, sqrt(numel(f)), sqrt(numel(f)))); % Plot the drawn value at each location. %title('sample'); nice_figure_save([filename, '-sample']) %mtit(titlestring); %set(gcf,'units','centimeters') %set(gcf,'Position',[1 1 40 15]) %savepng(gcf, titlestring ); end function nice_figure_save(filename) % axis off set(gcf, 'color', 'white'); set( gca, 'xTickLabel', '' ); set( gca, 'yTickLabel', '' ); set( gca, 'zTickLabel', '' ); set(gca, 'TickDir', 'in') tightfig set_fig_units_cm(12,12); myaa('publish'); savepng(gcf, filename); %filename_eps = ['../figures/additive/3d-kernel/', filename, '.eps'] %filename_pdf = ['../figures/additive/3d-kernel/', filename, '.pdf'] %print -depsc2 filename_eps %eps2pdf( filename_eps, filename_pdf, true); end % Stationary covariance functions % =================================== function sigma = gnn_kernel(x, y) if nargin == 0 sigma = 'gaussian-nn'; return; end [dx, nx] = size(x); [dy, ny] = size(y); n = min(nx, ny); sigma = exp( -0.15.*sq_dist(x, zeros(dx, n))) ... .* exp( -0.15.*sq_dist(y, zeros(dy, n)))' ... % Note the transpose! .* exp( -0.5.*sq_dist(x, y)); end function sigma = symm_xy_kernel_naive(x, y) if nargin == 0 sigma = 'symmetric-xy-naive'; return; end xs = [x(2, :); x(1, :)]; sigma = 0.5.*exp( -0.5.*sq_dist(x, y)) + 0.5.*exp( -0.5.*sq_dist(xs, y)); end function sigma = symm_xy_kernel_prod(x, y) if nargin == 0 sigma = 'symmetric-xy-prod'; return; end xs = [x(2, :); x(1, :)]; sigma = 0.5.*exp( -0.5.*sq_dist(x, y)) .* exp( -0.5.*sq_dist(xs, y)); end function sigma = symm_xy_kernel_proj(x, y) if nargin == 0 sigma = 'symmetric-xy-projection'; return; end projx = [(x(1,:) + x(2,:))./2; abs(x(1,:) - x(2,:))./2]; projy = [(y(1,:) + y(2,:))./2; abs(y(1,:) - y(2,:))./2]; sigma = exp( -0.5.*sq_dist(projx, projy)); end function sigma = se_kernel(x, y) if nargin == 0 sigma = 'squared-exp'; return; end sigma = exp( -0.5.*sq_dist(x, y)); end function sigma = se_1d_kernel(x, y) if nargin == 0 sigma = '1d-squared-exp'; return; end A = [ 0.5 -0.7 ]; sigma = exp( -0.5.*sq_dist(A*x, A*y)); end function sigma = se_1d_kernel_plus_2d(x, y) if nargin == 0 sigma = '1d-squared-exp-plus-2d'; return; end A = [ .7 -.9 ]; sigma = exp( -0.5.*(sq_dist(A*x, A*y)) + 0.1.*exp( -0.5.*sq_dist(x, y))); end function sigma = fa_kernel(x, y) if nargin == 0 sigma = 'factor-analysis'; return; end A = [ .7 -.9 ]; sigma = exp( -0.5.*(sq_dist(A*x, A*y) + 0.05.*sq_dist(x, y))); end function sigma = add_kernel(x, y) if nargin == 0 sigma = 'factor-analysis-plus-se'; return; end sigma = 0.4.*exp( -0.5.*( sq_dist(x(1,:), y(1,:)))) + 0.4.*exp( -0.5.*(sq_dist(x(2,:), y(2,:)))) + 0.1.*exp( -0.5.*(sq_dist(x, y))); end function sigma = spline_kernel(x, y) if nargin == 0 sigma = 'exp-y-spline'; return; end sigma = exp( -sqrt(sq_dist(x(1,:), y(1,:))/100) -sqrt(sq_dist(x(2,:), y(2,:))/100)); end ================================================ FILE: code/utils/colorbrew.m ================================================ function c = colorbrew( i, flag, N ) % % Nice colors taken from % http://colorbrewer2.org/ % % David Duvenaud % March 2012 if nargin < 2; flag = 'qual'; end switch flag case 'qual' c_array(1, :) = [ 228, 26, 28 ]; % red c_array(2, :) = [ 55, 126, 184 ]; % blue c_array(3, :) = [ 77, 175, 74 ]; % green c_array(4, :) = [ 152, 78, 163 ]; % purple c_array(5, :) = [ 255, 127, 0 ]; % orange c_array(6, :) = [ 255, 255, 51 ]; % yellow c_array(7, :) = [ 166, 86, 40 ]; % brown c_array(8, :) = [ 247, 129, 191 ]; % pink c_array(9, :) = [ 153, 153, 153]; % grey c_array(10, :) = [ 0, 0, 0]; % black % Wrap around to the end. c = c_array( mod(i - 1, 10) + 1, : ) ./ 255; case 'seq' c_array_d = [254, 196, 79; 254, 153, 41; 236, 112, 20; 204, 76, 2; 153, 52, 4; 102, 37, 6]; d = size(c_array_d, 1); c_array = interp1((1:d)', c_array_d, linspace(1, d, N)'); c = c_array( i, : ) ./ 255; case 'div' c_array_d = [158, 1, 66; 213, 62, 79; 244, 109, 67; 253, 174, 97; 171, 221, 164; 102, 194, 165; 50, 136, 189; 94, 79, 162]; d = size(c_array_d, 1); c_array = interp1((1:d)', c_array_d, linspace(1, d, N)'); c = c_array( i, : ) ./ 255; end end ================================================ FILE: code/utils/coord_to_color.m ================================================ function colors = coord_to_color(x, scale) % Takes a set of 2d points and turns them into colors % % David Duvenaud % March 2014 if nargin < 2; scale = 5; end corner_colors = colorbrew([5 2 3 4 1]); [n,d] = size(x); assert(d == 2); % Rescale points from 0 to 1 along each dimension. %x(:,1) = x(:,1) - min(x(:,1)); %x(:,2) = x(:,2) - min(x(:,2)); %xmax = max(x(:,1)); %ymax = max(x(:,2)); %x(:, 1) = x(:, 1) ./ xmax; %x(:, 2) = x(:, 2) ./ ymax; %xmin = 0.1; %xmax = 0.9; %ymin = 0.1; %ymax = 0.9; theta = (0:(2*pi/4):2*pi) + pi/4; theta = theta(1:end-1); coords = [[sin(theta) 0]; [cos(theta) 0]]; %coords = [[xmin ymin]; [xmin ymax]; [xmax ymin]; [xmax ymax]]; weights = se_kernel( x' .*scale, coords.*scale); % Take each point, and its color will be a linear combination of the 4 % original colors, depending on their location. weight_sums = sum(weights, 2); weights = weights ./ repmat( weight_sums, 1, length(coords)); colors(:,1) = weights * corner_colors(:, 1); colors(:,2) = weights * corner_colors(:, 2); colors(:,3) = weights * corner_colors(:, 3); %{ xcolors = bsxfun( @times, x(:,1), corner_colors(1,:)) ... + bsxfun( @times, 1 - x(:,1), corner_colors(2,:)); ycolors = bsxfun( @times, x(:,2), corner_colors(3,:)) ... + bsxfun( @times, 1 - x(:,2), corner_colors(4,:)); xcolors = (tanh((xcolors - 0.5) .* 2) + 1) ./ 2; ycolors = (tanh((ycolors - 0.5) .* 2) + 1) ./ 2; colors = (xcolors + ycolors)./2; %colors = colors .^ 2; %colors = (tanh((colors - 0.5) .* 5) + 1) ./ 2; %} end function sigma = se_kernel(x, y) if nargin == 0 sigma = 'Normal SE covariance.'; return; end sigma = exp( -0.5.*sq_dist(x, y)); end ================================================ FILE: code/utils/coord_to_image.m ================================================ function colors = coord_to_image(x) % Takes a set of 2d points and turns them into colors % % David Duvenaud % March 2014 x = mod(x,1); arrow = [... [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]; ... [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]; ... [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]; ... [ 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 ]; ... [ 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 ]; ... [ 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 ]; ... [ 0 0 0 0 1 1 0 1 1 0 1 1 0 0 0 0 ]; ... [ 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 ]; ... [ 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0 0 ]; ... [ 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 ]; ... [ 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 ]; ... [ 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 ]; ... [ 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 ]; ... [ 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 ]; ... [ 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 ]; ... [ 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 ]; ... [ 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 ]; ... [ 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 ]; ... [ 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 ]; ... ]; % add some padding arrow = [zeros( size(arrow, 1), 4), arrow, zeros( size(arrow, 1), 4)]; arrow = [zeros( 4, size(arrow, 2)); arrow; zeros( 4, size(arrow, 2))]; [n,d] = size(x); assert(d == 2); grid_resolution = 14; %coords = [[sin(theta) 0]; [cos(theta) 0]]; xrange1 = linspace( 0, 1, size(arrow,1)); xrange2 = linspace( 0, 1, size(arrow,2)); [x1, x2] = ndgrid( xrange1, xrange2 ); % Make a grid coords = [ x1(:), x2(:) ]; N = size(coords, 1); corner_colors = repmat(colorbrew(1), N, 1); corner_colors(logical(arrow(:)), :) = repmat(colorbrew(2), sum(arrow(:)), 1); %coords = [[xmin ymin]; [xmin ymax]; [xmax ymin]; [xmax ymax]]; weights = se_kernel( x', coords'); % Take each point, and its color will be a linear combination of the 4 % original colors, depending on their location. weight_sums = sum(weights, 2); weights = weights ./ repmat( weight_sums, 1, length(coords)); colors(:,1) = weights * corner_colors(:, 1); colors(:,2) = weights * corner_colors(:, 2); colors(:,3) = weights * corner_colors(:, 3); end function sigma = se_kernel(x, y) if nargin == 0 sigma = 'Normal SE covariance.'; return; end sigma = exp( -0.5.*sq_dist(x, y).*1000); end ================================================ FILE: code/utils/draw_warped_density.m ================================================ % Plot the density in the observed space. % % David and Tomo function draw_warped_density( X, Y, log_hypers, ... circle_size, circle_alpha,N_points) figure(5432100); clf; [N,Q] = size(X); [N,D] = size(Y); if nargin < 4 circle_size = 0.05; circle_alpha = 0.2; N_points = 1000; end latent_draws = mvnrnd( zeros(1, Q), eye(Q), N_points); % Compute Gram gram matrix. hyp(1) = -log_hypers.gamma/2; hyp(2) = log_hypers.alpha/2; K = covSEiso(hyp, X) + eye(N)*max(exp(log_hypers.betainv), 1e-3); % Add a little noise. % Compute conditional posterior. crosscov = covSEiso(hyp, latent_draws, X); post_mean = crosscov*(K\Y); prior_var = covSEiso(hyp, latent_draws, 'diag'); post_var = prior_var - sum(bsxfun(@times, crosscov/K, crosscov), 2); samples = post_mean + randn(N_points, D) .* repmat(post_var, 1, D); plot_little_circles(samples(:,1), samples(:,2), ... circle_size, colorbrew(3), circle_alpha ); hold on; % Draw the original data and current assigments % markers = {'x', 'o', 's', 'd', '.', '>', '<', '^'}; plot(Y(:, 1), Y(:, 2), 'o', 'MarkerEdgeColor', 'k'); hold on; title('Observed space'); end ================================================ FILE: code/utils/logdet.m ================================================ function ld = logdet(K) % returns the log-determinant of posdef matrix K. ld = 2*sum(log(diag(chol(K)))); end ================================================ FILE: code/utils/logmvnpdf.m ================================================ function logp = logmvnpdf(x,mu,Sigma) % Log of multivariate normal pdf. % % David Duvenaud % January 2012. dim = length(mu); logdetcov = logdet(Sigma); a = bsxfun(@minus, x, mu); logp = (-dim/2)*log(2*pi) + (-.5)*logdetcov +... (-.5.*sum(bsxfun( @times, a / Sigma, a), 2)); % Evaluate for multiple inputs. end function ld = logdet(K) % returns the log-determinant of posdef matrix K. % This is probably horribly slow. ld = NaN; try ld = 2*sum(log(diag(chol(K)))); catch e e; end end ================================================ FILE: code/utils/logsumexp.m ================================================ function s = logsumexp(x, dim) % Returns log(sum(exp(x),dim)) while avoiding numerical underflow. % Default is dim = 1 (columns). % Written by Mo Chen (mochen@ie.cuhk.edu.hk). March 2009. if nargin == 1, % Determine which dimension sum will use dim = find(size(x)~=1,1); if isempty(dim), dim = 1; end end % subtract the largest in each column y = max(x,[],dim); x = bsxfun(@minus,x,y); s = y + log(sum(exp(x),dim)); i = find(~isfinite(y)); if ~isempty(i) s(i) = y(i); end ================================================ FILE: code/utils/logsumexp2.m ================================================ function s = logsumexp(a, dim) % Returns log(sum(exp(a),dim)) while avoiding numerical underflow. % Default is dim = 1 (rows) or dim=2 for a row vector % logsumexp(a, 2) will sum across columns instead of rows % Written by Tom Minka, modified by Kevin Murphy if nargin < 2 dim = 1; if ndims(a) <= 2 & size(a,1)==1 dim = 2; end end % subtract the largest in each column [y, i] = max(a,[],dim); dims = ones(1,ndims(a)); dims(dim) = size(a,dim); a = a - repmat(y, dims); s = y + log(sum(exp(a),dim)); %i = find(~finite(y)); %if ~isempty(i) % s(i) = y(i); %end ================================================ FILE: code/utils/myaa.m ================================================ function [varargout] = myaa(varargin) %MYAA Render figure with anti-aliasing. % MYAA % Anti-aliased rendering of the current figure. This makes graphics look % a lot better than in a standard matlab figure, which is useful for % publishing results on the web or to better see the fine details in a % complex and cluttered plot. Some simple keyboard commands allow % the user to set the rendering quality interactively, zoom in/out and % re-render when needed. % % Usage: % myaa: Renders an anti-aliased version of the current figure. % % myaa(K): Sets the supersampling factor to K. Using a % higher K yields better rendering but takes longer time. If K is % omitted, it defaults to 4. It may be useful to run e.g. myaa(2) to % make a low-quality rendering as a first try, because it is a lot % faster than the default. % % myaa([K D]): Sets supersampling factor to K but downsamples the % image by a factor of D. If D is larger than K, the rendered image % will be smaller than the original. If D is smaller than K, the % rendering will be bigger. % % myaa('publish'): An experimental parameter, useful for publishing % matlab programs (see example 3). Beware, it kills the current figure. % % Interactivity: % The anti-aliased figure can be updated with the following keyboard % commands: % % Re-render image (to reflect changes in the figure) % + Zoom in (decrease downsampling factor) % - Zoom out (increase downsampling factor) % 1 ... 9 Change supersampling and downsampling factor to ... % q Quit, i.e. close the anti-aliased figure % % Myaa can also be called with up to 3 parameters. % FIG = MYAA(K,AAMETHOD,FIGMODE) % Parameters and output: % K Subsampling factor. If a vector is specified, [K D], then % the second element will describe the downsampling factor. % Default is K = 4 and D = 4. % AAMETHOD Downsampling method. Normally this is chosen automatically. % 'standard': convolution based filtering and downsampling % 'imresize': downsampling using the imresize command from % the image toolbox. % 'noshrink': used internally % FIGMODE Display mode % 'figure': the normal mode, a new figure is created % 'update': internally used for interactive sessions % 'publish': used internally % FIG A handle to the new anti-aliased figure % % Example 1: % spharm2; % myaa; % % Press '1', '2' or '4' and try '+' and '-' % % Press 'r' or to update the anti-aliased rendering, e.g. % % after rotating the 3-D object in the original figure. % % Example 2: % line(randn(2500,2)',randn(2500,2)','color','black','linewidth',0.01) % myaa(8); % % Example 3: % xpklein; % myaa(2,'standard'); % % Example 3: % Put the following in test.m % %% My test publish % % Testing to publish some anti-aliased images % % % spharm2; % Produce some nice graphics % myaa('publish'); % Render an anti-aliased version % % Then run: % publish test.m; % showdemo test; % % % BUGS: % Dotted and dashed lines in plots are not rendered correctly. This is % probably due to a bug in Matlab and it will hopefully be fixed in a % future version. % The OpenGL renderer does not always manage to render an image large % enough. Try the zbuffer renderer if you have problems or decrease the % K factor. You can set the current renderer to zbuffer by running e.g. % set(gcf,'renderer','zbuffer'). % % See also PUBLISH, PRINT % % Version 1.1, 2008-08-21 % Version 1.0, 2008-08-05 % % Author: Anders Brun % anders@cb.uu.se % %% Force drawing of graphics drawnow; %% Find out about the current DPI... screen_DPI = get(0,'ScreenPixelsPerInch'); %% Determine the best choice of convolver. % If IPPL is available, imfilter is much faster. Otherwise it does not % matter too much. try if ippl() myconv = @imfilter; else myconv = @conv2; end catch myconv = @conv2; end %% Set default options and interpret arguments if isempty(varargin) self.K = [4 4]; try imfilter(zeros(2,2),zeros(2,2)); self.aamethod = 'imresize'; catch self.aamethod = 'standard'; end self.figmode = 'figure'; elseif strcmp(varargin{1},'publish') self.K = [4 4]; self.aamethod = 'noshrink'; self.figmode = 'publish'; elseif strcmp(varargin{1},'mypublish') self.K = [8 4]; self.aamethod = 'imresize'; self.figmode = 'publish'; self.filename = varargin{2}; % Save image directly to file. elseif strcmp(varargin{1},'update') self = get(gcf,'UserData'); figure(self.source_fig); drawnow; self.figmode = 'update'; elseif strcmp(varargin{1},'lazyupdate') self = get(gcf,'UserData'); self.figmode = 'lazyupdate'; elseif length(varargin) == 1 self.K = varargin{1}; if length(self.K) == 1 self.K = [self.K self.K]; end if self.K(1) > 16 error('To avoid excessive use of memory, K has been limited to max 16. Change the code to fix this on your own risk.'); end try imfilter(zeros(2,2),zeros(2,2)); self.aamethod = 'imresize'; catch self.aamethod = 'standard'; end self.figmode = 'figure'; elseif length(varargin) == 2 self.K = varargin{1}; self.aamethod = varargin{2}; self.figmode = 'figure'; elseif length(varargin) == 3 self.K = varargin{1}; self.aamethod = varargin{2}; self.figmode = varargin{3}; if strcmp(self.figmode,'publish') && ~strcmp(varargin{2},'noshrink') printf('\nThe AAMETHOD was not set to ''noshrink'': Fixed.\n\n'); self.aamethod = 'noshrink'; end else error('Wrong syntax, run: help myaa'); end if length(self.K) == 1 self.K = [self.K self.K]; end %% Capture current figure in high resolution if ~strcmp(self.figmode,'lazyupdate'); tempfile = 'myaa_temp_screendump.png'; self.source_fig = gcf; current_paperpositionmode = get(self.source_fig,'PaperPositionMode'); current_inverthardcopy = get(self.source_fig,'InvertHardcopy'); set(self.source_fig,'PaperPositionMode','auto'); set(self.source_fig,'InvertHardcopy','off'); print(self.source_fig,['-r',num2str(screen_DPI*self.K(1))], '-dpng', tempfile); set(self.source_fig,'InvertHardcopy',current_inverthardcopy); set(self.source_fig,'PaperPositionMode',current_paperpositionmode); self.raw_hires = imread(tempfile); delete(tempfile); end %% Start filtering to remove aliasing w = warning; warning off; if strcmp(self.aamethod,'standard') || strcmp(self.aamethod,'noshrink') % Subsample hires figure image with standard anti-aliasing using a % butterworth filter kk = lpfilter(self.K(2)*3,self.K(2)*0.9,2); mm = myconv(ones(size(self.raw_hires(:,:,1))),kk,'same'); a1 = max(min(myconv(single(self.raw_hires(:,:,1))/(256),kk,'same'),1),0)./mm; a2 = max(min(myconv(single(self.raw_hires(:,:,2))/(256),kk,'same'),1),0)./mm; a3 = max(min(myconv(single(self.raw_hires(:,:,3))/(256),kk,'same'),1),0)./mm; if strcmp(self.aamethod,'standard') if abs(1-self.K(2)) > 0.001 raw_lowres = double(cat(3,a1(2:self.K(2):end,2:self.K(2):end),a2(2:self.K(2):end,2:self.K(2):end),a3(2:self.K(2):end,2:self.K(2):end))); else raw_lowres = self.raw_hires; end else raw_lowres = double(cat(3,a1,a2,a3)); end elseif strcmp(self.aamethod,'imresize') % This is probably the fastest method available at this moment... raw_lowres = single(imresize(self.raw_hires,1/self.K(2),'bilinear'))/256; end warning(w); %% Place the anti-aliased image in some image on the screen ... if strcmp(self.figmode,'figure'); % Create a new figure at the same place as the previous % The content of this new image is just a bitmap... oldpos = get(gcf,'Position'); self.myaa_figure = figure; fig = self.myaa_figure; set(fig,'Menubar','none'); set(fig,'Resize','off'); sz = size(raw_lowres); set(fig,'Units','pixels'); pos = [oldpos(1:2) sz(2:-1:1)]; set(fig,'Position',pos); ax = axes; image(raw_lowres); set(ax,'Units','pixels'); set(ax,'Position',[1 1 sz(2) sz(1)]); axis off; elseif strcmp(self.figmode,'publish'); % Create a new figure at the same place as the previous % The content of this new image is just a bitmap... self.myaa_figure = figure; fig = self.myaa_figure; current_units = get(self.source_fig,'Units'); set(self.source_fig,'Units','pixels'); pos = get(self.source_fig,'Position'); set(self.source_fig,'Units',current_units); set(fig,'Position',[pos(1) pos(2) pos(3) pos(4)]); ax = axes; % Fix by DD: % Clip the color data to about out-of-bounds errors raw_lowres(raw_lowres>1) = 1; raw_lowres(raw_lowres<0) = 0; image(raw_lowres); set(ax,'Units','normalized'); set(ax,'Position',[0 0 1 1]); axis off; close(self.source_fig); if strcmp(varargin{1},'mypublish') imwrite(raw_lowres, [self.filename, '.png']); end elseif strcmp(self.figmode,'update'); fig = self.myaa_figure; figure(fig); clf; set(fig,'Menubar','none'); set(fig,'Resize','off'); sz = size(raw_lowres); set(fig,'Units','pixels'); pos = get(fig,'Position'); pos(3:4) = sz(2:-1:1); set(fig,'Position',pos); ax = axes; image(raw_lowres); set(ax,'Units','pixels'); set(ax,'Position',[1 1 sz(2) sz(1)]); axis off; elseif strcmp(self.figmode,'lazyupdate'); clf; fig = self.myaa_figure; sz = size(raw_lowres); pos = get(fig,'Position'); pos(3:4) = sz(2:-1:1); set(fig,'Position',pos); ax = axes; image(raw_lowres); set(ax,'Units','pixels'); set(ax,'Position',[1 1 sz(2) sz(1)]); axis off; end %% Store current state set(gcf,'userdata',self); set(gcf,'KeyPressFcn',@keypress); set(gcf,'Interruptible','off'); %% Avoid unnecessary console output if nargout == 1 varargout(1) = {fig}; end %% A simple lowpass filter kernel (Butterworth). % sz is the size of the filter % subsmp is the downsampling factor to be used later % n is the degree of the butterworth filter function kk = lpfilter(sz, subsmp, n) sz = 2*floor(sz/2)+1; % make sure the size of the filter is odd cut_frequency = 0.5 / subsmp; range = (-(sz-1)/2:(sz-1)/2)/(sz-1); [ii,jj] = ndgrid(range,range); rr = sqrt(ii.^2+jj.^2); kk = ifftshift(1./(1+(rr./cut_frequency).^(2*n))); kk = fftshift(real(ifft2(kk))); kk = kk./sum(kk(:)); function keypress(src,evnt) if isempty(evnt.Character) return end recognized = 0; self = get(gcf,'userdata'); if evnt.Character == '+' self.K(2) = max(self.K(2).*0.5^(1/2),1); recognized = 1; set(gcf,'userdata',self); myaa('lazyupdate'); elseif evnt.Character == '-' self.K(2) = min(self.K(2).*2^(1/2),16); recognized = 1; set(gcf,'userdata',self); myaa('lazyupdate'); elseif evnt.Character == ' ' || evnt.Character == 'r' || evnt.Character == 'R' set(gcf,'userdata',self); myaa('update'); elseif evnt.Character == 'q' close(gcf); elseif find('123456789' == evnt.Character) self.K = [str2double(evnt.Character) str2double(evnt.Character)]; set(gcf,'userdata',self); myaa('update'); end ================================================ FILE: code/utils/parseArgs.m ================================================ function ArgStruct=parseArgs(args,ArgStruct,varargin) % Helper function for parsing varargin. % % % ArgStruct=parseArgs(varargin,ArgStruct[,FlagtypeParams[,Aliases]]) % % * ArgStruct is the structure full of named arguments with default values. % * Flagtype params is params that don't require a value. (the value will be set to 1 if it is present) % * Aliases can be used to map one argument-name to several argstruct fields % % % example usage: % -------------- % function parseargtest(varargin) % % %define the acceptable named arguments and assign default values % Args=struct('Holdaxis',0, ... % 'SpacingVertical',0.05,'SpacingHorizontal',0.05, ... % 'PaddingLeft',0,'PaddingRight',0,'PaddingTop',0,'PaddingBottom',0, ... % 'MarginLeft',.1,'MarginRight',.1,'MarginTop',.1,'MarginBottom',.1, ... % 'rows',[],'cols',[]); % % %The capital letters define abrreviations. % % Eg. parseargtest('spacingvertical',0) is equivalent to parseargtest('sv',0) % % Args=parseArgs(varargin,Args, ... % fill the arg-struct with values entered by the user % {'Holdaxis'}, ... %this argument has no value (flag-type) % {'Spacing' {'sh','sv'}; 'Padding' {'pl','pr','pt','pb'}; 'Margin' {'ml','mr','mt','mb'}}); % % disp(Args) % % % % % % Aslak Grinsted 2003 Aliases={}; FlagTypeParams=''; if (length(varargin)>0) FlagTypeParams=strvcat(varargin{1}); if length(varargin)>1 Aliases=varargin{2}; end end %---------------Get "numeric" arguments NumArgCount=1; while (NumArgCount<=size(args,2))&(~ischar(args{NumArgCount})) NumArgCount=NumArgCount+1; end NumArgCount=NumArgCount-1; if (NumArgCount>0) ArgStruct.NumericArguments={args{1:NumArgCount}}; else ArgStruct.NumericArguments={}; end %--------------Make an accepted fieldname matrix (case insensitive) Fnames=fieldnames(ArgStruct); for i=1:length(Fnames) name=lower(Fnames{i,1}); Fnames{i,2}=name; %col2=lower AbbrevIdx=find(Fnames{i,1}~=name); Fnames{i,3}=[name(AbbrevIdx) ' ']; %col3=abreviation letters (those that are uppercase in the ArgStruct) e.g. SpacingHoriz->sh %the space prevents strvcat from removing empty lines Fnames{i,4}=isempty(strmatch(Fnames{i,2},FlagTypeParams)); %Does this parameter have a value? (e.g. not flagtype) end FnamesFull=strvcat(Fnames{:,2}); FnamesAbbr=strvcat(Fnames{:,3}); if length(Aliases)>0 for i=1:length(Aliases) name=lower(Aliases{i,1}); FieldIdx=strmatch(name,FnamesAbbr,'exact'); %try abbreviations (must be exact) if isempty(FieldIdx) FieldIdx=strmatch(name,FnamesFull); %&??????? exact or not? end Aliases{i,2}=FieldIdx; AbbrevIdx=find(Aliases{i,1}~=name); Aliases{i,3}=[name(AbbrevIdx) ' ']; %the space prevents strvcat from removing empty lines Aliases{i,1}=name; %dont need the name in uppercase anymore for aliases end %Append aliases to the end of FnamesFull and FnamesAbbr FnamesFull=strvcat(FnamesFull,strvcat(Aliases{:,1})); FnamesAbbr=strvcat(FnamesAbbr,strvcat(Aliases{:,3})); end %--------------get parameters-------------------- l=NumArgCount+1; while (l<=length(args)) a=args{l}; if ischar(a) paramHasValue=1; % assume that the parameter has is of type 'param',value a=lower(a); FieldIdx=strmatch(a,FnamesAbbr,'exact'); %try abbreviations (must be exact) if isempty(FieldIdx) FieldIdx=strmatch(a,FnamesFull); end if (length(FieldIdx)>1) %shortest fieldname should win [mx,mxi]=max(sum(FnamesFull(FieldIdx,:)==' ',2)); FieldIdx=FieldIdx(mxi); end if FieldIdx>length(Fnames) %then it's an alias type. FieldIdx=Aliases{FieldIdx-length(Fnames),2}; end if isempty(FieldIdx) error(['Unknown named parameter: ' a]) end for curField=FieldIdx' %if it is an alias it could be more than one. if (Fnames{curField,4}) val=args{l+1}; else val=1; %parameter is of flag type and is set (1=true).... end ArgStruct.(Fnames{curField,1})=val; end l=l+1+Fnames{FieldIdx(1),4}; %if a wildcard matches more than one else error(['Expected a named parameter: ' num2str(a)]) end end ================================================ FILE: code/utils/plot_little_circles.m ================================================ function p = plot_little_circles(x, y, circle, col, alpha, fast) %AG_PLOT_LITTLE_CIRCLES Plot circular circles of relative size circle % Returns handles to all patches plotted if nargin < 6 fast = false; end [n, d] = size(x); assert( d == 1); assert( size(y, 2) == 1); if size(col, 2) == 1 col = repmat( col, n, 1); end % aspect is width / height %fPos = get(gcf, 'Position'); % need width, height in data values xl = xlim(); yl = ylim(); w = circle*(xl(2)-xl(1));%/fPos(3); h = circle*(yl(2)-yl(1));%/fPos(4); theta = 0:(2*pi/5):2*pi; mx = w*sin(theta); my = h*cos(theta); if fast %for k = 1:length(x) % p(k) = plot(x(k), y(k), '.', 'Color', col(k,:)); %end set(0,'DefaultAxesColorOrder',jet(length(x))) p = plot(x, y, '.'); else for k = 1:length(x) p(k) = patch(x(k)+mx, y(k)+my, col(k,:), 'FaceColor', col(k,:), 'FaceAlpha', alpha, 'EdgeColor', 'none'); end end end ================================================ FILE: code/utils/plot_little_circles_3d.m ================================================ function p = plot_little_circles_3d(x, y, z, circle, col) %AG_PLOT_LITTLE_CIRCLES Plot circular circles of relative size circle % Returns handles to all patches plotted if nargin < 6 fast = false; end [n, d] = size(x); assert( d == 1); assert( size(y, 2) == 1); if size(col, 1) == 1 col = repmat( col, n, 1); end % aspect is width / height %fPos = get(gcf, 'Position'); % need width, height in data values xl = xlim(); yl = ylim(); w = circle*(xl(2)-xl(1));%/fPos(3); h = circle*(yl(2)-yl(1));%/fPos(4); theta = 0:(2*pi/5):2*pi; mx = w*sin(theta); my = h*cos(theta); %if fast for k = 1:length(x) p(k) = plot3(x(k), y(k), z(k), '.', 'Color', col(k,:)); hold on; end %else % for k = 1:length(x) % p(k) = patch(x(k)+mx, y(k)+my, col(k,:), 'FaceColor', col(k,:), 'FaceAlpha', alpha, 'EdgeColor', 'none'); % end %end end ================================================ FILE: code/utils/save2pdf.m ================================================ %SAVE2PDF Saves a figure as a properly cropped pdf % % save2pdf(pdfFileName,handle,dpi) % % - pdfFileName: Destination to write the pdf to. % - handle: (optional) Handle of the figure to write to a pdf. If % omitted, the current figure is used. Note that handles % are typically the figure number. % - dpi: (optional) Integer value of dots per inch (DPI). Sets % resolution of output pdf. Note that 150 dpi is the Matlab % default and this function's default, but 600 dpi is typical for % production-quality. % % Saves figure as a pdf with margins cropped to match the figure size. % (c) Gabe Hoffmann, gabe.hoffmann@gmail.com % Written 8/30/2007 % Revised 9/22/2007 % Revised 1/14/2007 function save2pdf(pdfFileName,handle,dpi, crop) % Verify correct number of arguments %error(nargchk(0,3,nargin)); % If no handle is provided, use the current figure as default if nargin<1 [fileName,pathName] = uiputfile('*.pdf','Save to PDF file:'); if fileName == 0; return; end pdfFileName = [pathName,fileName]; end if nargin<2 handle = gcf; end if nargin<3 dpi = 150; end if nargin<4 crop = false; end % Backup previous settings prePaperType = get(handle,'PaperType'); prePaperUnits = get(handle,'PaperUnits'); preUnits = get(handle,'Units'); prePaperPosition = get(handle,'PaperPosition'); prePaperSize = get(handle,'PaperSize'); % Make changing paper type possible set(handle,'PaperType',''); % Set units to all be the same set(handle,'PaperUnits','inches'); set(handle,'Units','inches'); % Set the page size and position to match the figure's dimensions paperPosition = get(handle,'PaperPosition'); position = get(handle,'Position'); set(handle,'PaperPosition',[0,0,position(3:4)]); set(handle,'PaperSize',position(3:4)); % Save the pdf (this is the same method used by "saveas") print(handle,'-dpdf',pdfFileName,sprintf('-r%d',dpi)) % Restore the previous settings set(handle,'PaperType',prePaperType); set(handle,'PaperUnits',prePaperUnits); set(handle,'Units',preUnits); set(handle,'PaperPosition',prePaperPosition); set(handle,'PaperSize',prePaperSize); if crop system(sprintf('pdfcrop %s %s', pdfFileName, pdfFileName)); end ================================================ FILE: code/utils/saveeps.m ================================================ function saveeps( name ) handle = gcf; % Make changing paper type possible set(handle,'PaperType',''); % Set units to all be the same set(handle,'PaperUnits','inches'); set(handle,'Units','inches'); % Set the page size and position to match the figure's dimensions paperPosition = get(handle,'PaperPosition'); position = get(handle,'Position'); set(handle,'PaperPosition',[0,0,position(3:4)]); set(handle,'PaperSize',position(3:4)); print( gcf, sprintf('-r%d', ceil(72*(4/3))), '-deps', [ name '.eps'] ); end ================================================ FILE: code/utils/savepng.m ================================================ function savepng( handle, name ) % Make changing paper type possible set(handle,'PaperType',''); % Set units to all be the same set(handle,'PaperUnits','inches'); set(handle,'Units','inches'); % Set the page size and position to match the figure's dimensions paperPosition = get(handle,'PaperPosition'); position = get(handle,'Position'); set(handle,'PaperPosition',[0,0,position(3:4)]); set(handle,'PaperSize',position(3:4)); print( gcf, sprintf('-r%d', ceil(72*(4/3))), '-dpng', [ name '.png'] ); end ================================================ FILE: code/utils/set_fig_units_cm.m ================================================ function set_fig_units_cm( width, height ) set(gcf,'units','centimeters'); pos = get(gcf,'position'); set(gcf,'position',[pos(1:2),width,height]); ================================================ FILE: code/utils/sq_dist.m ================================================ % sq_dist - a function to compute a matrix of all pairwise squared distances % between two sets of vectors, stored in the columns of the two matrices, a % (of size D by n) and b (of size D by m). If only a single argument is given % or the second matrix is empty, the missing matrix is taken to be identical % to the first. % % Usage: C = sq_dist(a, b) % or: C = sq_dist(a) or equiv.: C = sq_dist(a, []) % % Where a is of size Dxn, b is of size Dxm (or empty), C is of size nxm. % % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-12-13. function C = sq_dist(a, b) if nargin<1 || nargin>3 || nargout>1, error('Wrong number of arguments.'); end bsx = exist('bsxfun','builtin'); % since Matlab R2007a 7.4.0 and Octave 3.0 if ~bsx, bsx = exist('bsxfun'); end % bsxfun is not yes "builtin" in Octave [D, n] = size(a); % Computation of a^2 - 2*a*b + b^2 is less stable than (a-b)^2 because numerical % precision can be lost when both a and b have very large absolute value and the % same sign. For that reason, we subtract the mean from the data beforehand to % stabilise the computations. This is OK because the squared error is % independent of the mean. if nargin==1 % subtract mean mu = mean(a,2); if bsx a = bsxfun(@minus,a,mu); else a = a - repmat(mu,1,size(a,2)); end b = a; m = n; else [d, m] = size(b); if d ~= D, error('Error: column lengths must agree.'); end mu = (m/(n+m))*mean(b,2) + (n/(n+m))*mean(a,2); if bsx a = bsxfun(@minus,a,mu); b = bsxfun(@minus,b,mu); else a = a - repmat(mu,1,n); b = b - repmat(mu,1,m); end end if bsx % compute squared distances C = bsxfun(@plus,sum(a.*a,1)',bsxfun(@minus,sum(b.*b,1),2*a'*b)); else C = repmat(sum(a.*a,1)',1,m) + repmat(sum(b.*b,1),n,1) - 2*a'*b; end C = max(C,0); % numerical noise can cause C to negative i.e. C > -1e-14 ================================================ FILE: code/utils/subaxis.m ================================================ function h=subaxis(varargin) %SUBAXIS Create axes in tiled positions. (just like subplot) % Usage: % h=subaxis(rows,cols,cellno[,settings]) % h=subaxis(rows,cols,cellx,celly[,settings]) % h=subaxis(rows,cols,cellx,celly,spanx,spany[,settings]) % % SETTINGS: Spacing,SpacingHoriz,SpacingVert % Padding,PaddingRight,PaddingLeft,PaddingTop,PaddingBottom % Margin,MarginRight,MarginLeft,MarginTop,MarginBottom % Holdaxis % % all units are relative (e.g from 0 to 1) % % Abbreviations of parameters can be used.. (Eg MR instead of MarginRight) % (holdaxis means that it wont delete any axes below.) % % % Example: % % >> subaxis(2,1,1,'SpacingVert',0,'MR',0); % >> imagesc(magic(3)) % >> subaxis(2,'p',.02); % >> imagesc(magic(4)) % % 2001 / Aslak Grinsted (Feel free to modify this code.) f=gcf; Args=[]; UserDataArgsOK=0; Args=get(f,'UserData'); if isstruct(Args) UserDataArgsOK=isfield(Args,'SpacingHorizontal')&isfield(Args,'Holdaxis')&isfield(Args,'rows')&isfield(Args,'cols'); end OKToStoreArgs=isempty(Args)|UserDataArgsOK; if isempty(Args)&(~UserDataArgsOK) Args=struct('Holdaxis',0, ... 'SpacingVertical',0.05,'SpacingHorizontal',0.05, ... 'PaddingLeft',0,'PaddingRight',0,'PaddingTop',0,'PaddingBottom',0, ... 'MarginLeft',.1,'MarginRight',.1,'MarginTop',.1,'MarginBottom',.1, ... 'rows',[],'cols',[]); end Args=parseArgs(varargin,Args,{'Holdaxis'},{'Spacing' {'sh','sv'}; 'Padding' {'pl','pr','pt','pb'}; 'Margin' {'ml','mr','mt','mb'}}); if (length(Args.NumericArguments)>1) Args.rows=Args.NumericArguments{1}; Args.cols=Args.NumericArguments{2}; %remove these 2 numerical arguments Args.NumericArguments={Args.NumericArguments{3:end}}; end if OKToStoreArgs set(f,'UserData',Args); end switch length(Args.NumericArguments) case 0 return % no arguments but rows/cols.... case 1 x1=mod((Args.NumericArguments{1}-1),Args.cols)+1; x2=x1; y1=floor((Args.NumericArguments{1}-1)/Args.cols)+1; y2=y1; case 2 x1=Args.NumericArguments{1};x2=x1; y1=Args.NumericArguments{2};y2=y1; case 4 x1=Args.NumericArguments{1};x2=x1+Args.NumericArguments{3}-1; y1=Args.NumericArguments{2};y2=y1+Args.NumericArguments{4}-1; otherwise error('subaxis argument error') end cellwidth=((1-Args.MarginLeft-Args.MarginRight)-(Args.cols-1)*Args.SpacingHorizontal)/Args.cols; cellheight=((1-Args.MarginTop-Args.MarginBottom)-(Args.rows-1)*Args.SpacingVertical)/Args.rows; xpos1=Args.MarginLeft+Args.PaddingLeft+cellwidth*(x1-1)+Args.SpacingHorizontal*(x1-1); xpos2=Args.MarginLeft-Args.PaddingRight+cellwidth*x2+Args.SpacingHorizontal*(x2-1); ypos1=Args.MarginTop+Args.PaddingTop+cellheight*(y1-1)+Args.SpacingVertical*(y1-1); ypos2=Args.MarginTop-Args.PaddingBottom+cellheight*y2+Args.SpacingVertical*(y2-1); if Args.Holdaxis h=axes('position',[xpos1 1-ypos2 xpos2-xpos1 ypos2-ypos1]); else h=subplot('position',[xpos1 1-ypos2 xpos2-xpos1 ypos2-ypos1]); end set(h,'box','on'); %h=axes('position',[x1 1-y2 x2-x1 y2-y1]); set(h,'units',get(gcf,'defaultaxesunits')); set(h,'tag','subaxis'); if (nargout==0) clear h; end; ================================================ FILE: code/utils/test_coord_to_color.m ================================================ % test coord to color n = 1000; x = randn( n, 2); colors = coord_to_color(x); %close all; figure; for i = 1:n plot(x(i,1), x(i,2), '.', 'color', colors(i,:)); hold on; end ================================================ FILE: code/utils/test_coord_to_image.m ================================================ % test coord to color grid_resolution = 100; xrange = linspace( 0, 2,grid_resolution); [x1, x2] = ndgrid( xrange, xrange ); % Make a grid x = [ x1(:), x2(:) ]; colors = coord_to_image(x); imagesc(reshape(colors, [grid_resolution, grid_resolution, 3])); ================================================ FILE: code/utils/tightfig.m ================================================ function hfig = tightfig(hfig) % tightfig: Alters a figure so that it has the minimum size necessary to % enclose all axes in the figure without excess space around them. % % Note that tightfig will expand the figure to completely encompass all % axes if necessary. If any 3D axes are present which have been zoomed, % tightfig will produce an error, as these cannot easily be dealt with. % % hfig - handle to figure, if not supplied, the current figure will be used % instead. if nargin == 0 hfig = gcf; end % There can be an issue with tightfig when the user has been modifying % the contnts manually, the code below is an attempt to resolve this, % but it has not yet been satisfactorily fixed % origwindowstyle = get(hfig, 'WindowStyle'); set(hfig, 'WindowStyle', 'normal'); % 1 point is 0.3528 mm for future use % get all the axes handles note this will also fetch legends and % colorbars as well hax = findall(hfig, 'type', 'axes'); % get the original axes units, so we can change and reset these again % later origaxunits = get(hax, 'Units'); % change the axes units to cm set(hax, 'Units', 'centimeters'); % get various position parameters of the axes if numel(hax) > 1 % fsize = cell2mat(get(hax, 'FontSize')); ti = cell2mat(get(hax,'TightInset')); pos = cell2mat(get(hax, 'Position')); else % fsize = get(hax, 'FontSize'); ti = get(hax,'TightInset'); pos = get(hax, 'Position'); end % ensure very tiny border so outer box always appears ti(ti < 0.1) = 0.15; % we will check if any 3d axes are zoomed, to do this we will check if % they are not being viewed in any of the 2d directions views2d = [0,90; 0,0; 90,0]; for i = 1:numel(hax) set(hax(i), 'LooseInset', ti(i,:)); % set(hax(i), 'LooseInset', [0,0,0,0]); % get the current viewing angle of the axes [az,el] = view(hax(i)); % determine if the axes are zoomed iszoomed = strcmp(get(hax(i), 'CameraViewAngleMode'), 'manual'); % test if we are viewing in 2d mode or a 3d view is2d = all(bsxfun(@eq, [az,el], views2d), 2); if iszoomed && ~any(is2d) error('TIGHTFIG:haszoomed3d', 'Cannot make figures containing zoomed 3D axes tight.') end end % we will move all the axes down and to the left by the amount % necessary to just show the bottom and leftmost axes and labels etc. moveleft = min(pos(:,1) - ti(:,1)); movedown = min(pos(:,2) - ti(:,2)); % we will also alter the height and width of the figure to just % encompass the topmost and rightmost axes and lables figwidth = max(pos(:,1) + pos(:,3) + ti(:,3) - moveleft); figheight = max(pos(:,2) + pos(:,4) + ti(:,4) - movedown); % move all the axes for i = 1:numel(hax) set(hax(i), 'Position', [pos(i,1:2) - [moveleft,movedown], pos(i,3:4)]); end origfigunits = get(hfig, 'Units'); set(hfig, 'Units', 'centimeters'); % change the size of the figure figpos = get(hfig, 'Position'); set(hfig, 'Position', [figpos(1), figpos(2), figwidth, figheight]); % change the size of the paper set(hfig, 'PaperUnits','centimeters'); set(hfig, 'PaperSize', [figwidth, figheight]); set(hfig, 'PaperPositionMode', 'manual'); set(hfig, 'PaperPosition',[0 0 figwidth figheight]); % reset to original units for axes and figure if ~iscell(origaxunits) origaxunits = {origaxunits}; end for i = 1:numel(hax) set(hax(i), 'Units', origaxunits{i}); end set(hfig, 'Units', origfigunits); % set(hfig, 'WindowStyle', origwindowstyle); end ================================================ FILE: common/CUEDbiblio.bst ================================================ % BibTeX standard bibliography style `plain' % version 0.99a for BibTeX versions 0.99a or later, LaTeX version 2.09. % Copyright (C) 1985, all rights reserved. % Copying of this file is authorized only if either % (1) you make absolutely no changes to your copy, including name, or % (2) if you do make changes, you name it something other than % btxbst.doc, plain.bst, unsrt.bst, alpha.bst, and abbrv.bst. % This restriction helps ensure that all standard styles are identical. % The file btxbst.doc has the documentation for this style. ENTRY { address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year } {} { label } INTEGERS { output.state before.all mid.sentence after.sentence after.block } FUNCTION {init.state.consts} { #0 'before.all := #1 'mid.sentence := #2 'after.sentence := #3 'after.block := } STRINGS { s t } FUNCTION {output.nonnull} { 's := output.state mid.sentence = { ", " * write$ } { output.state after.block = { add.period$ write$ newline$ "\newblock " write$ } { output.state before.all = 'write$ { add.period$ " " * write$ } if$ } if$ mid.sentence 'output.state := } if$ s } FUNCTION {output} { duplicate$ empty$ 'pop$ 'output.nonnull if$ } FUNCTION {output.check} { 't := duplicate$ empty$ { pop$ "empty " t * " in " * cite$ * warning$ } 'output.nonnull if$ } FUNCTION {output.bibitem} { newline$ "\bibitem{" write$ cite$ write$ "}" write$ newline$ "" before.all 'output.state := } FUNCTION {fin.entry} { add.period$ write$ newline$ } FUNCTION {new.block} { output.state before.all = 'skip$ { after.block 'output.state := } if$ } FUNCTION {new.sentence} { output.state after.block = 'skip$ { output.state before.all = 'skip$ { after.sentence 'output.state := } if$ } if$ } FUNCTION {not} { { #0 } { #1 } if$ } FUNCTION {and} { 'skip$ { pop$ #0 } if$ } FUNCTION {or} { { pop$ #1 } 'skip$ if$ } FUNCTION {new.block.checka} { empty$ 'skip$ 'new.block if$ } FUNCTION {new.block.checkb} { empty$ swap$ empty$ and 'skip$ 'new.block if$ } FUNCTION {new.sentence.checka} { empty$ 'skip$ 'new.sentence if$ } FUNCTION {new.sentence.checkb} { empty$ swap$ empty$ and 'skip$ 'new.sentence if$ } FUNCTION {field.or.null} { duplicate$ empty$ { pop$ "" } 'skip$ if$ } FUNCTION {emphasize} { duplicate$ empty$ { pop$ "" } { "{\em " swap$ * "}" * } if$ } FUNCTION {scapify} { duplicate$ empty$ { pop$ "" } { "{\sc " swap$ * "}" * } if$ } FUNCTION {boldify} { duplicate$ empty$ { pop$ "" } { "{\bf " swap$ * "}" * } if$ } INTEGERS { nameptr namesleft numnames } FUNCTION {format.names} { 's := #1 'nameptr := s num.names$ 'numnames := numnames 'namesleft := { namesleft #0 > } { s nameptr "{ff~}{vv~}{ll}{, jj}" format.name$ 't := nameptr #1 > { namesleft #1 > { ", " * t * } { numnames #2 > { "," * } 'skip$ if$ t "others" = { " et~al." * } { " and " * t * } if$ } if$ } 't if$ nameptr #1 + 'nameptr := namesleft #1 - 'namesleft := } while$ } FUNCTION {format.authors} { author empty$ { "" } { author format.names scapify } if$ } FUNCTION {format.editors} { editor empty$ { "" } { editor format.names scapify editor num.names$ #1 > { ", editors" * } { ", editor" * } if$ } if$ } FUNCTION {format.title} { title empty$ { "" } { title "t" change.case$ } if$ } FUNCTION {n.dashify} { 't := "" { t empty$ not } { t #1 #1 substring$ "-" = { t #1 #2 substring$ "--" = not { "--" * t #2 global.max$ substring$ 't := } { { t #1 #1 substring$ "-" = } { "-" * t #2 global.max$ substring$ 't := } while$ } if$ } { t #1 #1 substring$ * t #2 global.max$ substring$ 't := } if$ } while$ } FUNCTION {format.date} { year empty$ { month empty$ { "" } { "there's a month but no year in " cite$ * warning$ month } if$ } { month empty$ 'year { month " " * year * } if$ } if$ } FUNCTION {format.btitle} { title emphasize } FUNCTION {tie.or.space.connect} { duplicate$ text.length$ #3 < { "~" } { " " } if$ swap$ * * } FUNCTION {either.or.check} { empty$ 'pop$ { "can't use both " swap$ * " fields in " * cite$ * warning$ } if$ } FUNCTION {format.bvolume} { volume empty$ { "" } { "" volume boldify tie.or.space.connect series empty$ 'skip$ { " of " * series emphasize * } if$ "volume and number" number either.or.check } if$ } FUNCTION {format.number.series} { volume empty$ { number empty$ { series field.or.null } { output.state mid.sentence = { "number" } { "Number" } if$ number tie.or.space.connect series empty$ { "there's a number but no series in " cite$ * warning$ } { " in " * series * } if$ } if$ } { "" } if$ } FUNCTION {format.edition} { edition empty$ { "" } { output.state mid.sentence = { edition "l" change.case$ " edition" * } { edition "t" change.case$ " edition" * } if$ } if$ } INTEGERS { multiresult } FUNCTION {multi.page.check} { 't := #0 'multiresult := { multiresult not t empty$ not and } { t #1 #1 substring$ duplicate$ "-" = swap$ duplicate$ "," = swap$ "+" = or or { #1 'multiresult := } { t #2 global.max$ substring$ 't := } if$ } while$ multiresult } FUNCTION {format.pages} { pages empty$ { "" } { pages multi.page.check { "pages" pages n.dashify tie.or.space.connect } { "page" pages tie.or.space.connect } if$ } if$ } FUNCTION {format.vol.num.pages} { volume field.or.null boldify number empty$ 'skip$ { "[" number * "]" * * volume empty$ { "there's a number but no volume in " cite$ * warning$ } 'skip$ if$ } if$ pages empty$ 'skip$ { duplicate$ empty$ { pop$ format.pages } { ":" * pages n.dashify * } if$ } if$ } FUNCTION {format.chapter.pages} { chapter empty$ 'format.pages { type empty$ { "chapter" } { type "l" change.case$ } if$ chapter tie.or.space.connect pages empty$ 'skip$ { ", " * format.pages * } if$ } if$ } FUNCTION {format.in.ed.booktitle} { booktitle empty$ { "" } { editor empty$ { "In " booktitle emphasize * } { "In " format.editors * ", " * booktitle emphasize * } if$ } if$ } FUNCTION {empty.misc.check} { author empty$ title empty$ howpublished empty$ month empty$ year empty$ note empty$ and and and and and key empty$ not and { "all relevant fields are empty in " cite$ * warning$ } 'skip$ if$ } FUNCTION {format.thesis.type} { type empty$ 'skip$ { pop$ type "t" change.case$ } if$ } FUNCTION {format.tr.number} { type empty$ { "Technical Report" } 'type if$ number empty$ { "t" change.case$ } { number tie.or.space.connect } if$ } FUNCTION {format.article.crossref} { key empty$ { journal empty$ { "need key or journal for " cite$ * " to crossref " * crossref * warning$ "" } { "In {\em " journal * "\/}" * } if$ } { "In " key * } if$ " \cite{" * crossref * "}" * } FUNCTION {format.crossref.editor} { editor #1 "{vv~}{ll}" format.name$ editor num.names$ duplicate$ #2 > { pop$ " et~al." * } { #2 < 'skip$ { editor #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" = { " et~al." * } { " and " * editor #2 "{vv~}{ll}" format.name$ * } if$ } if$ } if$ } FUNCTION {format.book.crossref} { volume empty$ { "empty volume in " cite$ * "'s crossref of " * crossref * warning$ "In " } { "Volume" volume tie.or.space.connect " of " * } if$ editor empty$ editor field.or.null author field.or.null = or { key empty$ { series empty$ { "need editor, key, or series for " cite$ * " to crossref " * crossref * warning$ "" * } { "{\em " * series * "\/}" * } if$ } { key * } if$ } { format.crossref.editor * } if$ " \cite{" * crossref * "}" * } FUNCTION {format.incoll.inproc.crossref} { editor empty$ editor field.or.null author field.or.null = or { key empty$ { booktitle empty$ { "need editor, key, or booktitle for " cite$ * " to crossref " * crossref * warning$ "" } { "In {\em " booktitle * "\/}" * } if$ } { "In " key * } if$ } { "In " format.crossref.editor * } if$ " \cite{" * crossref * "}" * } FUNCTION {article} { output.bibitem format.authors "author" output.check new.block format.title "title" output.check new.block crossref missing$ { journal emphasize "journal" output.check format.vol.num.pages output format.date "year" output.check } { format.article.crossref output.nonnull format.pages output } if$ new.block note output fin.entry } FUNCTION {book} { output.bibitem author empty$ { format.editors "author and editor" output.check } { format.authors output.nonnull crossref missing$ { "author and editor" editor either.or.check } 'skip$ if$ } if$ new.block format.btitle "title" output.check crossref missing$ { format.bvolume output new.block format.number.series output new.sentence publisher "publisher" output.check address output } { new.block format.book.crossref output.nonnull } if$ format.edition output format.date "year" output.check new.block note output fin.entry } FUNCTION {booklet} { output.bibitem format.authors output new.block format.title "title" output.check howpublished address new.block.checkb howpublished output address output format.date output new.block note output fin.entry } FUNCTION {inbook} { output.bibitem author empty$ { format.editors "author and editor" output.check } { format.authors output.nonnull crossref missing$ { "author and editor" editor either.or.check } 'skip$ if$ } if$ new.block format.btitle "title" output.check crossref missing$ { format.bvolume output format.chapter.pages "chapter and pages" output.check new.block format.number.series output new.sentence publisher "publisher" output.check address output } { format.chapter.pages "chapter and pages" output.check new.block format.book.crossref output.nonnull } if$ format.edition output format.date "year" output.check new.block note output fin.entry } FUNCTION {incollection} { output.bibitem format.authors "author" output.check new.block format.title "title" output.check new.block crossref missing$ { format.in.ed.booktitle "booktitle" output.check format.bvolume output format.number.series output format.chapter.pages output new.sentence publisher "publisher" output.check address output format.edition output format.date "year" output.check } { format.incoll.inproc.crossref output.nonnull format.chapter.pages output } if$ new.block note output fin.entry } FUNCTION {inproceedings} { output.bibitem format.authors "author" output.check new.block format.title "title" output.check new.block crossref missing$ { format.in.ed.booktitle "booktitle" output.check format.bvolume output format.number.series output format.pages output address empty$ { organization publisher new.sentence.checkb organization output publisher output format.date "year" output.check } { address output.nonnull format.date "year" output.check new.sentence organization output publisher output } if$ } { format.incoll.inproc.crossref output.nonnull format.pages output } if$ new.block note output fin.entry } FUNCTION {conference} { inproceedings } FUNCTION {manual} { output.bibitem author empty$ { organization empty$ 'skip$ { organization output.nonnull address output } if$ } { format.authors output.nonnull } if$ new.block format.btitle "title" output.check author empty$ { organization empty$ { address new.block.checka address output } 'skip$ if$ } { organization address new.block.checkb organization output address output } if$ format.edition output format.date output new.block note output fin.entry } FUNCTION {mastersthesis} { output.bibitem format.authors "author" output.check new.block format.btitle "title" output.check new.block "Master's thesis" format.thesis.type output.nonnull school "school" output.check address output format.date "year" output.check new.block note output fin.entry } FUNCTION {misc} { output.bibitem format.authors output title howpublished new.block.checkb format.title output howpublished new.block.checka howpublished output format.date output new.block note output fin.entry empty.misc.check } FUNCTION {phdthesis} { output.bibitem format.authors "author" output.check new.block format.btitle "title" output.check new.block "PhD thesis" format.thesis.type output.nonnull school "school" output.check address output format.date "year" output.check new.block note output fin.entry } FUNCTION {dphilthesis} { output.bibitem format.authors "author" output.check new.block format.btitle "title" output.check new.block "DPhil thesis" format.thesis.type output.nonnull school "school" output.check address output format.date "year" output.check new.block note output fin.entry } FUNCTION {proceedings} { output.bibitem editor empty$ { organization output } { format.editors output.nonnull } if$ new.block format.btitle "title" output.check format.bvolume output format.number.series output address empty$ { editor empty$ { publisher new.sentence.checka } { organization publisher new.sentence.checkb organization output } if$ publisher output format.date "year" output.check } { address output.nonnull format.date "year" output.check new.sentence editor empty$ 'skip$ { organization output } if$ publisher output } if$ new.block note output fin.entry } FUNCTION {techreport} { output.bibitem format.authors "author" output.check new.block format.title "title" output.check new.block format.tr.number output.nonnull institution "institution" output.check address output format.date "year" output.check new.block note output fin.entry } FUNCTION {unpublished} { output.bibitem format.authors "author" output.check new.block format.title "title" output.check new.block note "note" output.check format.date output fin.entry } FUNCTION {default.type} { misc } MACRO {jan} {"January"} MACRO {feb} {"February"} MACRO {mar} {"March"} MACRO {apr} {"April"} MACRO {may} {"May"} MACRO {jun} {"June"} MACRO {jul} {"July"} MACRO {aug} {"August"} MACRO {sep} {"September"} MACRO {oct} {"October"} MACRO {nov} {"November"} MACRO {dec} {"December"} MACRO {acmcs} {"ACM Computing Surveys"} MACRO {acta} {"Acta Informatica"} MACRO {cacm} {"Communications of the ACM"} MACRO {ibmjrd} {"IBM Journal of Research and Development"} MACRO {ibmsj} {"IBM Systems Journal"} MACRO {ieeese} {"IEEE Transactions on Software Engineering"} MACRO {ieeetc} {"IEEE Transactions on Computers"} MACRO {ieeetcad} {"IEEE Transactions on Computer-Aided Design of Integrated Circuits"} MACRO {ipl} {"Information Processing Letters"} MACRO {jacm} {"Journal of the ACM"} MACRO {jcss} {"Journal of Computer and System Sciences"} MACRO {scp} {"Science of Computer Programming"} MACRO {sicomp} {"SIAM Journal on Computing"} MACRO {tocs} {"ACM Transactions on Computer Systems"} MACRO {tods} {"ACM Transactions on Database Systems"} MACRO {tog} {"ACM Transactions on Graphics"} MACRO {toms} {"ACM Transactions on Mathematical Software"} MACRO {toois} {"ACM Transactions on Office Information Systems"} MACRO {toplas} {"ACM Transactions on Programming Languages and Systems"} MACRO {tcs} {"Theoretical Computer Science"} READ FUNCTION {sortify} { purify$ "l" change.case$ } INTEGERS { len } FUNCTION {chop.word} { 's := 'len := s #1 len substring$ = { s len #1 + global.max$ substring$ } 's if$ } FUNCTION {sort.format.names} { 's := #1 'nameptr := "" s num.names$ 'numnames := numnames 'namesleft := { namesleft #0 > } { nameptr #1 > { " " * } 'skip$ if$ s nameptr "{vv{ } }{ll{ }}{ ff{ }}{ jj{ }}" format.name$ 't := nameptr numnames = t "others" = and { "et al" * } { t sortify * } if$ nameptr #1 + 'nameptr := namesleft #1 - 'namesleft := } while$ } FUNCTION {sort.format.title} { 't := "A " #2 "An " #3 "The " #4 t chop.word chop.word chop.word sortify #1 global.max$ substring$ } FUNCTION {author.sort} { author empty$ { key empty$ { "to sort, need author or key in " cite$ * warning$ "" } { key sortify } if$ } { author sort.format.names } if$ } FUNCTION {author.editor.sort} { author empty$ { editor empty$ { key empty$ { "to sort, need author, editor, or key in " cite$ * warning$ "" } { key sortify } if$ } { editor sort.format.names } if$ } { author sort.format.names } if$ } FUNCTION {author.organization.sort} { author empty$ { organization empty$ { key empty$ { "to sort, need author, organization, or key in " cite$ * warning$ "" } { key sortify } if$ } { "The " #4 organization chop.word sortify } if$ } { author sort.format.names } if$ } FUNCTION {editor.organization.sort} { editor empty$ { organization empty$ { key empty$ { "to sort, need editor, organization, or key in " cite$ * warning$ "" } { key sortify } if$ } { "The " #4 organization chop.word sortify } if$ } { editor sort.format.names } if$ } FUNCTION {presort} { type$ "book" = type$ "inbook" = or 'author.editor.sort { type$ "proceedings" = 'editor.organization.sort { type$ "manual" = 'author.organization.sort 'author.sort if$ } if$ } if$ " " * year field.or.null sortify * " " * title field.or.null sort.format.title * #1 entry.max$ substring$ 'sort.key$ := } ITERATE {presort} SORT STRINGS { longest.label } INTEGERS { number.label longest.label.width } FUNCTION {initialize.longest.label} { "" 'longest.label := #1 'number.label := #0 'longest.label.width := } FUNCTION {longest.label.pass} { number.label int.to.str$ 'label := number.label #1 + 'number.label := label width$ longest.label.width > { label 'longest.label := label width$ 'longest.label.width := } 'skip$ if$ } EXECUTE {initialize.longest.label} ITERATE {longest.label.pass} FUNCTION {begin.bib} { preamble$ empty$ 'skip$ { preamble$ write$ newline$ } if$ "\begin{thebibliography}{" longest.label * "}" * write$ newline$ } EXECUTE {begin.bib} EXECUTE {init.state.consts} ITERATE {call.type$} FUNCTION {end.bib} { newline$ "\end{thebibliography}" write$ newline$ } EXECUTE {end.bib} ================================================ FILE: common/PhDThesisPSnPDF.cls ================================================ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% Class ``PhD Thesis PSnPDF'' %% %% %% %% A PhD thesis LaTeX template for Cambridge University Engineering Department%% %% %% %% Version: v1.0 %% %% Authors: Krishna Kumar %% %% Date: 2013/11/16 (inception) %% %% License: MIT License (c) 2013 Krishna Kumar %% %% GitHub Repo: https://github.com/kks32/phd-thesis-template/ %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % ************************** Class Identification ****************************** \NeedsTeXFormat{LaTeX2e} \ProvidesClass{common/PhDThesisPSnPDF}[2013/12/10 version 1.0 by Krishna Kumar] \typeout{https://github.com/kks32/phd-thesis-template/} % ****************************************************************************** % **************************** Class Definition ******************************** % ****************************************************************************** % *********************** Define a Print/Online Version ************************ \newif\if@print\@printfalse \DeclareOption{print}{\@printtrue} % ****************************** Define index ********************************** \newif\ifPHD@index\PHD@indexfalse \DeclareOption{index}{\PHD@indextrue} % ******************************* Font Option ********************************** \newif\ifsetFont\setFontfalse % Font is not set \newif\ifPHD@times\PHD@timesfalse % Times with Math Support \DeclareOption{times}{\PHD@timestrue} \newif\ifPHD@fourier\PHD@fourierfalse % Fourier with Math Support \DeclareOption{fourier}{\PHD@fouriertrue} \newif\ifPHD@customfont\PHD@customfontfalse % Custom Font with Math Support \DeclareOption{customfont}{\PHD@customfonttrue} % ******************************* Bibliography ********************************* \newif\ifsetBib\setBibfalse % Custom Bibliography = true/false \newif\ifsetBiBLaTeX\setBiBLaTeXfalse % BiBLaTeX = True / False \newif\ifPHD@biblatex\PHD@biblatexfalse % BiBLaTeX \DeclareOption{biblatex}{\PHD@biblatextrue} \newif\ifPHD@authoryear\PHD@authoryearfalse % Author-Year citation \DeclareOption{authoryear}{\PHD@authoryeartrue} \newif\ifPHD@numbered\PHD@numberedfalse % Numbered citiation \DeclareOption{numbered}{\PHD@numberedtrue} \newif\ifPHD@custombib\PHD@custombibfalse % Custom Bibliography \DeclareOption{custombib}{\PHD@custombibtrue} % ************************* Header / Footer Styling **************************** \newif\ifPHD@pageStyleI\PHD@pageStyleIfalse % Set Page StyleI \DeclareOption{PageStyleI}{\PHD@pageStyleItrue} \newif\ifPHD@pageStyleII\PHD@pageStyleIIfalse % Set Page StyleII \DeclareOption{PageStyleII}{\PHD@pageStyleIItrue} % ***************************** Custom Margins ******************************** \newif\ifsetMargin\setMarginfalse % Margins are not set \newif\ifPHD@custommargin\PHD@custommarginfalse % Custom margin \DeclareOption{custommargin}{\PHD@custommargintrue} % **************************** Separate Abstract ****************************** \newif \ifdefineAbstract\defineAbstractfalse %To enable Separate abstract \newif\ifPHD@abstract\PHD@abstractfalse % Enable Separate Abstract \DeclareOption{abstract}{ \PHD@abstracttrue \ClassWarning{PhDThesisPSnPDF}{You have chosen an option that generates only the Title page and an abstract with PhD title and author name, if this was intentional, ignore this warning. Congratulations on submitting your thesis!! If not, please remove the option `abstract' from the document class and recompile. Good luck with your writing!} \PassOptionsToClass{oneside}{book} } % ****************** Chapter Mode - To print only selected chapters ************ \newif \ifdefineChapter\defineChapterfalse %To enable Separate abstract \newif\ifPHD@chapter\PHD@chapterfalse % Enable Separate Abstract \DeclareOption{chapter}{ \PHD@chaptertrue \ClassWarning{PhDThesisPSnPDF}{You have chosen an option that generates only selected chapters with references, if this was intentional, ignore this warning. If not, please remove the option `chapter' from the document class and recompile. Good luck with your writing!} } \ProcessOptions\relax% % *************************** Pre-defined Options ****************************** % Font Size \newcommand\PHD@ptsize{12pt} %Set Default Size as 12 \DeclareOption{10pt}{ \ClassWarning{PhDThesisPSnPDF}{The Cambridge University PhD thesis guidelines recommend using a minimum font size of 11pt (12pt is preferred) and 10pt for footnotes.} \renewcommand\PHD@ptsize{10pt} } \DeclareOption{11pt}{\renewcommand\PHD@ptsize{11pt}}% \DeclareOption{12pt}{\renewcommand\PHD@ptsize{12pt}}% \PassOptionsToClass{\PHD@ptsize}{book}% % Page Size \newcommand\PHD@papersize{a4paper} % Set Default as a4paper \DeclareOption{a4paper}{\renewcommand\PHD@papersize{a4paper}} \DeclareOption{a5paper}{\renewcommand\PHD@papersize{a5paper}} \DeclareOption{letterpaper}{ \ClassWarning{PhDThesisPSnPDF}{The Cambridge University Engineering Deparment PhD thesis guidelines recommend using A4 or A5paper} \renewcommand\PHD@papersize{letterpaper} } \PassOptionsToClass{\PHD@papersize}{book}% % Column layout \DeclareOption{oneside}{\PassOptionsToClass{\CurrentOption}{book}}% \DeclareOption{twoside}{\PassOptionsToClass{\CurrentOption}{book}}% % Draft Mode \DeclareOption{draft}{\PassOptionsToClass{\CurrentOption}{book}}% % Generates Warning for unknown options \DeclareOption*{ \ClassWarning{PhDThesisPSnPDF}{Unknown or non-standard option '\CurrentOption'. I'll see if I can load it from the book class. If you get a warning unused global option(s): `\CurrentOption` then the option is not supported!} \PassOptionsToClass{\CurrentOption}{book} } % Determine whether to run pdftex or dvips \ProcessOptions\relax% \newif\ifsetDVI\setDVIfalse \ifx\pdfoutput\undefined % we are not running PDFLaTeX \setDVItrue \LoadClass[dvips,fleqn,openright]{book}% \else % we are running PDFLaTeX \ifnum \pdfoutput>0 %PDF-Output \setDVIfalse \LoadClass[pdftex,fleqn,openright]{book}% \else %DVI-output \setDVItrue \LoadClass[fleqn,openright]{book}% \fi \fi %* ***************************** Print / Online ******************************** % Defines a print / online version to define page-layout and hyperrefering \ifsetDVI \special{papersize=\the\paperwidth,\the\paperheight} \RequirePackage[dvips,unicode=true]{hyperref} \else \RequirePackage[unicode=true]{hyperref} \pdfpagewidth=\the\paperwidth \pdfpageheight=\the\paperheight \fi \RequirePackage[usenames, dvipsnames]{xcolor} % Stuff added by David Duvenaud \definecolor{mydarkblue}{rgb}{0.00,0.08,0.45} \if@print % For Print version \hypersetup{ plainpages=false, pdfstartview=FitV, pdftoolbar=true, pdfmenubar=true, bookmarksopen=true, bookmarksnumbered=true, breaklinks=true, linktocpage, colorlinks=true, linkcolor=black, urlcolor=black, citecolor=black, anchorcolor=black } \ifPHD@custommargin \setMarginfalse \else \ifsetDVI % Odd and Even side Margin for binding and set viewmode for PDF \RequirePackage[dvips,paper=\PHD@papersize,hmarginratio=1:1,vmarginratio=1:1,scale=0.75,bindingoffset=0mm]{geometry} \else \RequirePackage[pdftex,paper=\PHD@papersize,hmarginratio=1:1,vmarginratio=1:1,scale=0.75,bindingoffset=0mm]{geometry} \fi \setMargintrue \fi \if@twoside \hypersetup{pdfpagelayout=OneColumn} \else \hypersetup{pdfpagelayout=OneColumn} \fi \else % For PDF Online version \hypersetup{ plainpages=false, pdfstartview=FitV, pdftoolbar=true, pdfmenubar=true, bookmarksopen=true, bookmarksnumbered=true, breaklinks=true, linktocpage, colorlinks=true, % linkcolor=black, % urlcolor=black, % citecolor=black, anchorcolor=black, linkcolor=mydarkblue, urlcolor=mydarkblue, citecolor=mydarkblue, % anchorcolor=green } \ifPHD@custommargin \setMarginfalse \else % No Margin staggering on Odd and Even side \ifsetDVI \RequirePackage[dvips,paper=\PHD@papersize,hmarginratio=1:1,vmarginratio=1:1,scale=0.75]{geometry} \else \RequirePackage[pdftex,paper=\PHD@papersize,hmarginratio=1:1,vmarginratio=1:1,scale=0.75]{geometry} \fi \setMargintrue \fi \hypersetup{pdfpagelayout=OneColumn} \fi % ****************************************************************************** % ******************************** Packages ************************************ % ****************************************************************************** % ************************** Layout and Formatting ***************************** \def\pdfshellescape{1} \RequirePackage{lscape} % Supports Landscape Layout \RequirePackage{setspace} % Define line spacing in paragraph \RequirePackage{calc} % To calculate vertical spacing % ************************* Conditional Statements ***************************** \RequirePackage{ifthen} % Used in LaTeX Class files for conditional statements \RequirePackage{ifpdf} % Check for pdfLaTeX % *********************** Table of Contents & Appendices *********************** % add Bibliography, List of figures and tables to contents \RequirePackage{tocbibind} % Add appendices \RequirePackage[title,titletoc]{appendix} % *************************** Graphics and Figures ***************************** %\ifpdf % Convert eps figures to pdf % \RequirePackage{epstopdf} \RequirePackage[pdftex]{graphicx} % \DeclareGraphicsExtensions{.png, .jpg, .pdf} % \pdfcompresslevel=9 \graphicspath{{misc/}} %\else % \RequirePackage{graphicx} % \DeclareGraphicsExtensions{.eps, .ps} % \graphicspath{{crests/Vector/}{crests/}} %\fi % ************************ URL Package and Definition ************************** \RequirePackage{url} % Redefining urlstyle to use smaller fontsize in References with URLs \newcommand{\url@leostyle}{% \@ifundefined{selectfont}{\renewcommand{\UrlFont}{\sf}} {\renewcommand{\UrlFont}{\small\ttfamily}}} \urlstyle{leo} % ******************************* Bibliography ********************************* \ifPHD@authoryear \ifPHD@biblatex \RequirePackage[backend=biber, style=authoryear, citestyle=alphabetic, sorting=nty, natbib=true]{biblatex} \setBiBLaTeXtrue \else \RequirePackage[round, sort, numbers, authoryear]{natbib} %author year \fi \setBibtrue \else \ifPHD@numbered \ifPHD@biblatex \RequirePackage[backend=biber, style=numeric-comp, citestyle=numeric, sorting=none, natbib=true]{biblatex} \setBiBLaTeXtrue \else \RequirePackage[numbers,sort&compress]{natbib} % numbered citation \fi \setBibtrue \else \ifPHD@custombib \setBibfalse \ifPHD@biblatex \setBiBLaTeXtrue \fi \else \ifPHD@biblatex \RequirePackage[backend=biber, style=numeric-comp, citestyle=numeric, sorting=none, natbib=true]{biblatex} \setBiBLaTeXtrue \else \RequirePackage[numbers,sort&compress]{natbib} % Default - numbered \fi \setBibtrue \ClassWarning{PhDThesisPSnPDF}{No bibliography style was specified. Default numbered style is used. If you would like to use a different style, use `authoryear' or `numbered' in the options in documentclass or use `custombib` and define the natbib package or biblatex package with required style in the Preamble.tex file} \fi \fi \fi % *********************** To copy ligatures and Fonts ************************** \RequirePackage{textcomp} % Font Selection \ifPHD@times %\RequirePackage{mathptmx} % times roman, including math (where possible) \setFonttrue \else \ifPHD@fourier \RequirePackage{fourier} % Fourier \setFonttrue \else \ifPHD@customfont \setFontfalse \else \ClassWarning{PhDThesisPSnPDf}{Using default font Latin Modern. If you would like to use other pre-defined fonts use `times' (The Cambridge University PhD thesis guidelines recommend using Times font) or `fourier' or load a custom font in the preamble.tex file by specifying `customfont' in the class options} \RequirePackage{lmodern} \setFonttrue \fi \fi \fi \RequirePackage[utf8]{inputenc} \RequirePackage[T1]{fontenc} \input{glyphtounicode} \pdfglyphtounicode{f_f}{FB00} \pdfglyphtounicode{f_i}{FB01} \pdfglyphtounicode{f_l}{FB02} \pdfglyphtounicode{f_f_i}{FB03} \pdfglyphtounicode{f_f_l}{FB04} \pdfgentounicode=1 % ****************************************************************************** % **************************** Pre-defined Settings **************************** % ****************************************************************************** % *************************** Setting PDF Meta-Data **************************** \ifpdf \AtBeginDocument{ \hypersetup{ pdftitle = {\@title}, %pdftitle = {DRAFT} pdfauthor = {\@author}, pdfsubject={\@subject}, pdfkeywords={\@keywords} } } \fi % ************************** TOC and Hide Sections ***************************** \newcommand{\nocontentsline}[3]{} \newcommand{\tochide}[2]{ \bgroup\let \addcontentsline=\nocontentsline#1{#2} \egroup} % Removes pagenumber appearing from TOC \addtocontents{toc}{\protect\thispagestyle{empty}} % ***************************** Header Formatting ****************************** % Custom Header with Chapter Number, Page Number and Section Numbering \RequirePackage{fancyhdr} % Define custom header % Set Fancy Header Command is defined to Load FancyHdr after Geometry is defined \newcommand{\setFancyHdr}{ \pagestyle{fancy} \ifPHD@pageStyleI % Style 1: Sets Page Number at the Top and Chapter/Section Name on LE/RO \renewcommand{\chaptermark}[1]{\markboth{##1}{}} \renewcommand{\sectionmark}[1]{\markright{\thesection\ ##1\ }} \fancyhf{} \fancyhead[RO]{\nouppercase \rightmark\hspace{0.25em} | \hspace{0.25em} \bfseries{\thepage}} \fancyhead[LE]{ {\bfseries\thepage} \hspace{0.25em} | \hspace{0.25em} \nouppercase \leftmark} \else \ifPHD@pageStyleII % Style 2: Sets Page Number at the Bottom with Chapter/Section Name on LO/RE \renewcommand{\chaptermark}[1]{\markboth{##1}{}} \renewcommand{\sectionmark}[1]{\markright{\thesection\ ##1}} \fancyhf{} \fancyhead[RO]{\bfseries\nouppercase \rightmark} \fancyhead[LE]{\bfseries \nouppercase \leftmark} \fancyfoot[C]{\thepage} \else % Default Style: Sets Page Number at the Top (LE/RO) with Chapter/Section Name % on LO/RE and an empty footer \renewcommand{\chaptermark}[1]{\markboth {##1}{}} \renewcommand{\sectionmark}[1]{\markright{\thesection\ ##1}} \fancyhf{} \fancyhead[LO]{\nouppercase \rightmark} \fancyhead[LE,RO]{\bfseries\thepage} \fancyhead[RE]{\nouppercase \leftmark} \fi \fi } \setlength{\headheight}{14.5pt} %\renewcommand{\headrulewidth}{0.5pt} %\renewcommand{\footrulewidth}{0pt} \fancypagestyle{plain}{ \fancyhead{} \renewcommand{\headrulewidth}{0pt} } % If Margin has been set (default margin print/online version) \ifsetMargin \setFancyHdr % Apply fancy header settings otherwise apply it in preamble \fi % **************** Clear Header Style on the Last Empty Odd pages ************** \renewcommand{\cleardoublepage}{\clearpage\if@twoside \ifodd\c@page\else% \if@print \hbox{}% \thispagestyle{empty} % Empty header styles \newpage% \if@twocolumn\hbox{}\newpage\fi \fi \fi\fi} % ******************************** Roman Pages ********************************* % The romanpages environment set the page numbering to lowercase roman one % for the contents and figures lists. It also resets % page-numbering for the remainder of the dissertation (arabic, starting at 1). \newenvironment{romanpages}{ \setcounter{page}{1} \renewcommand{\thepage}{\roman{page}}} {\newpage\renewcommand{\thepage}{\arabic{page}}} % ******************************* Appendices ********************************** % Appending the appendix name to the letter in TOC % Especially when backmatter is enabled %\renewcommand\backmatter{ % \def\chaptermark##1{\markboth{% % \ifnum \c@secnumdepth > \m@ne \@chapapp\ \thechapter: \fi ##1}{% % \ifnum \c@secnumdepth > \m@ne \@chapapp\ \thechapter: \fi ##1}}% % \def\sectionmark##1{\relax}} % ****************************************************************************** % **************************** Macro Definitions ******************************* % ****************************************************************************** % These macros are used to declare arguments needed for the % construction of the title page and other preamble. % The year and term the degree will be officially conferred \newcommand{\@degreedate}{} \newcommand{\degreedate}[1]{\renewcommand{\@degreedate}{#1}} % The full (unabbreviated) name of the degree \newcommand{\@degree}{} \newcommand{\degree}[1]{\renewcommand{\@degree}{#1}} % The name of your department(eg. Engineering, Maths, Physics) \newcommand{\@dept}{} \newcommand{\dept}[1]{\renewcommand{\@dept}{#1}} % The name of your college (eg. King's) \newcommand{\@college}{} \newcommand{\college}[1]{\renewcommand{\@college}{#1}} % The name of your University \newcommand{\@university}{} \newcommand{\university}[1]{\renewcommand{\@university}{#1}} % Defining the crest \newcommand{\@crest}{} \newcommand{\crest}[1]{\renewcommand{\@crest}{#1}} % Submission Text \newcommand{\submissiontext}{This dissertation is submitted for the degree of } % keywords (These keywords will appear in the PDF meta-information % called `pdfkeywords`.) \newcommand{\@keywords}{} \newcommand{\keywords}[1]{\renewcommand{\@keywords}{#1}} % subjectline (This subject will appear in the PDF meta-information % called `pdfsubject`.) \newcommand{\@subject}{} \newcommand{\subject}[1]{\renewcommand{\@subject}{#1}} % These macros define an environment for front matter that is always % single column even in a double-column document. \newenvironment{alwayssingle}{% \@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn \else\newpage\fi} {\if@restonecol\twocolumn\else\newpage\fi} % Set single column even under two column layout \newcommand{\setsinglecolumn}{ \if@twocolumn \onecolumn \else \fi } % ****************************************************************************** % *************************** Front Matter Layout ****************************** % ****************************************************************************** % ******************************** Title Page ********************************** \renewcommand{\maketitle}{ % To compute the free vertical space in Title page \computeTitlePageSpacing \begin{singlespace} \begin{center} % Title {\Huge \bfseries{\@title} \par} \vspace{.25\PHD@titlepagespacing} % Crest {\@crest \par} \vspace{.2\PHD@titlepagespacing} % Author {\Large \bfseries{\@author} \par} \vspace*{1ex} % Department and University {\large \@dept \par} \vspace*{1ex} {\large \@university \par} \vspace{.2\PHD@titlepagespacing} % Submission Text {\large \submissiontext \par} \vspace*{1ex} {\large \it {\@degree} \par} \end{center} \vfill \large \begin{minipage}{0.49\textwidth} \flushleft\hspace*{\oddsidemargin}\@college \end{minipage} \begin{minipage}{0.49\textwidth} \flushright \@degreedate \end{minipage} \end{singlespace} } % ********************************* Dedication ********************************* % The dedication environment makes sure the dedication gets its % own page, centered \newenvironment{dedication} { \cleardoublepage \setsinglecolumn \vspace*{0.2\textheight} \thispagestyle{empty} \centering } % ******************************* Declaration ********************************** % The declaration environment puts a large, bold, centered % "Declaration" label at the top of the page. \newenvironment{declaration}{ \cleardoublepage \setsinglecolumn \chapter*{\centering \Large Declaration} \thispagestyle{empty} }{ \flushright \@author{}\\ \@degreedate{} \vfill } % ****************************** Acknowledgements ******************************** % The acknowledgements environment puts a large, bold, centered % "Acknowledgements" label at the top of the page. \newenvironment{acknowledgements}{ \cleardoublepage \setsinglecolumn \chapter*{\centering \Large Acknowledgements} \thispagestyle{empty} } % ******************************* Nomenclature ********************************* \usepackage{nomencl} \makenomenclature \renewcommand{\nomgroup}[1]{% \ifthenelse{\equal{#1}{A}}{\item[\textbf{Roman Symbols}]}{% \ifthenelse{\equal{#1}{G}}{\item[\textbf{Greek Symbols}]}{% \ifthenelse{\equal{#1}{Z}}{\item[\textbf{Acronyms / Abbreviations}]}{% \ifthenelse{\equal{#1}{R}}{\item[\textbf{Superscripts}]}{% \ifthenelse{\equal{#1}{S}}{\item[\textbf{Subscripts}]}{% \ifthenelse{\equal{#1}{X}}{\item[\textbf{Other Symbols}]} {} }% matches mathematical symbols > X }% matches Subscripts > S }% matches Superscripts > R }% matches Abbreviations > Z }% matches Greek Symbols > G }% matches Roman Symbols > A % To add nomenclature in the header \renewcommand{\nompreamble}{\markboth{\nomname}{\nomname}} % Add nomenclature to contents and print out nomenclature \newcommand{\printnomencl}[1][]{ \ifthenelse{\equal {#1}{}} {\printnomenclature} {\printnomenclature[#1]} \addcontentsline{toc}{chapter}{\nomname} } % ***************************** Create the index ******************************* \ifPHD@index \RequirePackage{makeidx} \makeindex \newcommand{\printthesisindex}{ \cleardoublepage \phantomsection \printindex} \else \newcommand{\printthesisindex}{} \fi % ***************************** Chapter Mode *********************************** % The chapter mode allows user to only print particular chapters with references % All other options are disabled by default % To include only specific chapters without TOC, LOF, Title and Front Matter % To send it to supervisior for changes \ifPHD@chapter \defineChaptertrue % Disable the table of contents, figures, tables, index and nomenclature \renewcommand{\maketitle}{} \renewcommand{\tableofcontents}{} \renewcommand{\listoffigures}{} \renewcommand{\listoftables}{} \renewcommand{\printnomencl}{} \renewcommand{\printthesisindex}{} \else \defineChapterfalse \fi % ******************************** Abstract ************************************ % The abstract environment puts a large, bold, centered "Abstract" label at % the top of the page. Defines both abstract and separate abstract environment % To include only the Title and the abstract pages for submission to BoGS \ifPHD@abstract \defineAbstracttrue % Disable the table of contents, figures, tables, index and nomenclature \renewcommand{\tableofcontents}{} \renewcommand{\listoffigures}{} \renewcommand{\listoftables}{} \renewcommand{\printnomencl}{} \renewcommand{\printthesisindex}{} \renewcommand{\bibname}{} \renewcommand{\bibliography}[1]{\thispagestyle{empty}} \else \defineAbstractfalse \fi \newenvironment{abstract} { \ifPHD@abstract % Separate abstract as per Student Registry guidelines \thispagestyle{empty} \setsinglecolumn \begin{center} { \Large {\bfseries {\@title}} \par} {{\large \vspace*{1em} \@author} \par} \end{center} \else % Normal abstract in the thesis \cleardoublepage \setsinglecolumn \chapter*{\centering \Large Abstract} \thispagestyle{empty} \fi } % ******************************** Line Spacing ******************************** % Set spacing as 1.5 line spacing for the PhD Thesis \onehalfspace % ******************** To compute empty space in title page ******************** % Boxes below are used to space differt contents on the title page \newcommand{\computeTitlePageSpacing}{ % Title Box \newsavebox{\PHD@Title} \begin{lrbox}{\PHD@Title} \begin{minipage}[c]{0.98\textwidth} \centering \Huge \bfseries{\@title} \end{minipage} \end{lrbox} % University Crest Box \newsavebox{\PHD@crest} \begin{lrbox}{\PHD@crest} \@crest \end{lrbox} % Author Box \newsavebox{\PHD@author} \begin{lrbox}{\PHD@author} \begin{minipage}[c]{\textwidth} \centering \Large \bfseries{\@author} \end{minipage} \end{lrbox} % Department Box \newsavebox{\PHD@dept} \begin{lrbox}{\PHD@dept} \begin{minipage}[c]{\textwidth} \centering {\large \@dept \par} \vspace*{1ex} {\large \@university \par} \end{minipage} \end{lrbox} % Submission Box \newsavebox{\PHD@submission} \begin{lrbox}{\PHD@submission} \begin{minipage}[c]{\textwidth} \begin{center} \large \submissiontext \par \vspace*{1ex} \large \it {\@degree} \par \end{center} \end{minipage} \end{lrbox} % College and Date Box \newsavebox{\PHD@collegedate} \begin{lrbox}{\PHD@collegedate} \begin{minipage}[c]{\textwidth} \large \begin{minipage}{0.45\textwidth} \flushleft\@college \end{minipage} \begin{minipage}{0.45\textwidth} \flushright \@degreedate \end{minipage} \end{minipage} \end{lrbox} % Now to compute the free vertical space \newlength{\PHD@titlepagespacing} \setlength{\PHD@titlepagespacing}{ \textheight % - \totalheightof{\usebox{\PHD@Title}} - \totalheightof{\usebox{\PHD@crest}} - \totalheightof{\usebox{\PHD@author}} - \totalheightof{\usebox{\PHD@dept}} - \totalheightof{\usebox{\PHD@submission}} - \totalheightof{\usebox{\PHD@collegedate}} } } ================================================ FILE: common/commenting.tex ================================================ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% EDITING HELPER FUNCTIONS %%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% NA: needs attention (rough writing whose correctness needs to be verified) %% TBD: instructions for how to fix a gap ("Describe the propagation by ...") %% PROBLEM: bug or missing crucial bit %% use \fXXX versions of these macros to put additional explanation into a footnote. %% The idea is that we don't want to interrupt the flow of the paper or make it %% impossible to read because there are a bunch of comments. %% NA's (and TBDs, those less crucially) should be written so %% that they flow with the text. \definecolor{WowColor}{rgb}{.75,0,.75} \definecolor{SubtleColor}{rgb}{0,0,.50} % inline \newcommand{\NA}[1]{\textcolor{SubtleColor}{ {\tiny \bf ($\star$)} #1}} \newcommand{\LATER}[1]{\textcolor{SubtleColor}{ {\tiny \bf ($\dagger$)} #1}} \newcommand{\TBD}[1]{\textcolor{SubtleColor}{ {\tiny \bf (!)} #1}} \newcommand{\PROBLEM}[1]{\textcolor{WowColor}{ {\bf (!!)} {\bf #1}}} % as margin notes \newcounter{margincounter} 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\pdfglyphtounicode{Fourroman}{2163} \pdfglyphtounicode{Fsmall}{F766} \pdfglyphtounicode{G}{0047} \pdfglyphtounicode{GBsquare}{3387} \pdfglyphtounicode{Gacute}{01F4} \pdfglyphtounicode{Gamma}{0393} \pdfglyphtounicode{Gammaafrican}{0194} \pdfglyphtounicode{Gangiacoptic}{03EA} \pdfglyphtounicode{Gbreve}{011E} \pdfglyphtounicode{Gcaron}{01E6} \pdfglyphtounicode{Gcedilla}{0122} \pdfglyphtounicode{Gcircle}{24BC} \pdfglyphtounicode{Gcircumflex}{011C} \pdfglyphtounicode{Gcommaaccent}{0122} \pdfglyphtounicode{Gdot}{0120} \pdfglyphtounicode{Gdotaccent}{0120} \pdfglyphtounicode{Gecyrillic}{0413} \pdfglyphtounicode{Ghadarmenian}{0542} \pdfglyphtounicode{Ghemiddlehookcyrillic}{0494} \pdfglyphtounicode{Ghestrokecyrillic}{0492} \pdfglyphtounicode{Gheupturncyrillic}{0490} \pdfglyphtounicode{Ghook}{0193} \pdfglyphtounicode{Gimarmenian}{0533} \pdfglyphtounicode{Gjecyrillic}{0403} \pdfglyphtounicode{Gmacron}{1E20} \pdfglyphtounicode{Gmonospace}{FF27} \pdfglyphtounicode{Grave}{F6CE} \pdfglyphtounicode{Gravesmall}{F760} \pdfglyphtounicode{Gsmall}{F767} \pdfglyphtounicode{Gsmallhook}{029B} \pdfglyphtounicode{Gstroke}{01E4} \pdfglyphtounicode{H}{0048} \pdfglyphtounicode{H18533}{25CF} \pdfglyphtounicode{H18543}{25AA} \pdfglyphtounicode{H18551}{25AB} \pdfglyphtounicode{H22073}{25A1} \pdfglyphtounicode{HPsquare}{33CB} \pdfglyphtounicode{Haabkhasiancyrillic}{04A8} \pdfglyphtounicode{Hadescendercyrillic}{04B2} \pdfglyphtounicode{Hardsigncyrillic}{042A} \pdfglyphtounicode{Hbar}{0126} \pdfglyphtounicode{Hbrevebelow}{1E2A} \pdfglyphtounicode{Hcedilla}{1E28} \pdfglyphtounicode{Hcircle}{24BD} \pdfglyphtounicode{Hcircumflex}{0124} \pdfglyphtounicode{Hdieresis}{1E26} \pdfglyphtounicode{Hdotaccent}{1E22} \pdfglyphtounicode{Hdotbelow}{1E24} \pdfglyphtounicode{Hmonospace}{FF28} \pdfglyphtounicode{Hoarmenian}{0540} \pdfglyphtounicode{Horicoptic}{03E8} \pdfglyphtounicode{Hsmall}{F768} \pdfglyphtounicode{Hungarumlaut}{F6CF} \pdfglyphtounicode{Hungarumlautsmall}{F6F8} \pdfglyphtounicode{Hzsquare}{3390} \pdfglyphtounicode{I}{0049} \pdfglyphtounicode{IAcyrillic}{042F} \pdfglyphtounicode{IJ}{0132} \pdfglyphtounicode{IUcyrillic}{042E} \pdfglyphtounicode{Iacute}{00CD} \pdfglyphtounicode{Iacutesmall}{F7ED} \pdfglyphtounicode{Ibreve}{012C} \pdfglyphtounicode{Icaron}{01CF} \pdfglyphtounicode{Icircle}{24BE} \pdfglyphtounicode{Icircumflex}{00CE} \pdfglyphtounicode{Icircumflexsmall}{F7EE} \pdfglyphtounicode{Icyrillic}{0406} \pdfglyphtounicode{Idblgrave}{0208} \pdfglyphtounicode{Idieresis}{00CF} \pdfglyphtounicode{Idieresisacute}{1E2E} \pdfglyphtounicode{Idieresiscyrillic}{04E4} \pdfglyphtounicode{Idieresissmall}{F7EF} \pdfglyphtounicode{Idot}{0130} \pdfglyphtounicode{Idotaccent}{0130} \pdfglyphtounicode{Idotbelow}{1ECA} \pdfglyphtounicode{Iebrevecyrillic}{04D6} \pdfglyphtounicode{Iecyrillic}{0415} \pdfglyphtounicode{Ifraktur}{2111} \pdfglyphtounicode{Igrave}{00CC} \pdfglyphtounicode{Igravesmall}{F7EC} \pdfglyphtounicode{Ihookabove}{1EC8} \pdfglyphtounicode{Iicyrillic}{0418} \pdfglyphtounicode{Iinvertedbreve}{020A} \pdfglyphtounicode{Iishortcyrillic}{0419} \pdfglyphtounicode{Imacron}{012A} \pdfglyphtounicode{Imacroncyrillic}{04E2} \pdfglyphtounicode{Imonospace}{FF29} \pdfglyphtounicode{Iniarmenian}{053B} \pdfglyphtounicode{Iocyrillic}{0401} \pdfglyphtounicode{Iogonek}{012E} \pdfglyphtounicode{Iota}{0399} \pdfglyphtounicode{Iotaafrican}{0196} \pdfglyphtounicode{Iotadieresis}{03AA} \pdfglyphtounicode{Iotatonos}{038A} \pdfglyphtounicode{Ismall}{F769} \pdfglyphtounicode{Istroke}{0197} \pdfglyphtounicode{Itilde}{0128} \pdfglyphtounicode{Itildebelow}{1E2C} \pdfglyphtounicode{Izhitsacyrillic}{0474} \pdfglyphtounicode{Izhitsadblgravecyrillic}{0476} \pdfglyphtounicode{J}{004A} \pdfglyphtounicode{Jaarmenian}{0541} \pdfglyphtounicode{Jcircle}{24BF} \pdfglyphtounicode{Jcircumflex}{0134} \pdfglyphtounicode{Jecyrillic}{0408} \pdfglyphtounicode{Jheharmenian}{054B} \pdfglyphtounicode{Jmonospace}{FF2A} \pdfglyphtounicode{Jsmall}{F76A} \pdfglyphtounicode{K}{004B} \pdfglyphtounicode{KBsquare}{3385} \pdfglyphtounicode{KKsquare}{33CD} \pdfglyphtounicode{Kabashkircyrillic}{04A0} \pdfglyphtounicode{Kacute}{1E30} \pdfglyphtounicode{Kacyrillic}{041A} \pdfglyphtounicode{Kadescendercyrillic}{049A} \pdfglyphtounicode{Kahookcyrillic}{04C3} \pdfglyphtounicode{Kappa}{039A} \pdfglyphtounicode{Kastrokecyrillic}{049E} \pdfglyphtounicode{Kaverticalstrokecyrillic}{049C} \pdfglyphtounicode{Kcaron}{01E8} \pdfglyphtounicode{Kcedilla}{0136} \pdfglyphtounicode{Kcircle}{24C0} \pdfglyphtounicode{Kcommaaccent}{0136} \pdfglyphtounicode{Kdotbelow}{1E32} \pdfglyphtounicode{Keharmenian}{0554} \pdfglyphtounicode{Kenarmenian}{053F} \pdfglyphtounicode{Khacyrillic}{0425} \pdfglyphtounicode{Kheicoptic}{03E6} \pdfglyphtounicode{Khook}{0198} \pdfglyphtounicode{Kjecyrillic}{040C} \pdfglyphtounicode{Klinebelow}{1E34} \pdfglyphtounicode{Kmonospace}{FF2B} \pdfglyphtounicode{Koppacyrillic}{0480} \pdfglyphtounicode{Koppagreek}{03DE} \pdfglyphtounicode{Ksicyrillic}{046E} \pdfglyphtounicode{Ksmall}{F76B} \pdfglyphtounicode{L}{004C} \pdfglyphtounicode{LJ}{01C7} \pdfglyphtounicode{LL}{F6BF} \pdfglyphtounicode{Lacute}{0139} \pdfglyphtounicode{Lambda}{039B} \pdfglyphtounicode{Lcaron}{013D} \pdfglyphtounicode{Lcedilla}{013B} \pdfglyphtounicode{Lcircle}{24C1} \pdfglyphtounicode{Lcircumflexbelow}{1E3C} \pdfglyphtounicode{Lcommaaccent}{013B} \pdfglyphtounicode{Ldot}{013F} \pdfglyphtounicode{Ldotaccent}{013F} \pdfglyphtounicode{Ldotbelow}{1E36} \pdfglyphtounicode{Ldotbelowmacron}{1E38} \pdfglyphtounicode{Liwnarmenian}{053C} \pdfglyphtounicode{Lj}{01C8} \pdfglyphtounicode{Ljecyrillic}{0409} \pdfglyphtounicode{Llinebelow}{1E3A} \pdfglyphtounicode{Lmonospace}{FF2C} \pdfglyphtounicode{Lslash}{0141} \pdfglyphtounicode{Lslashsmall}{F6F9} \pdfglyphtounicode{Lsmall}{F76C} \pdfglyphtounicode{M}{004D} \pdfglyphtounicode{MBsquare}{3386} \pdfglyphtounicode{Macron}{F6D0} \pdfglyphtounicode{Macronsmall}{F7AF} \pdfglyphtounicode{Macute}{1E3E} \pdfglyphtounicode{Mcircle}{24C2} \pdfglyphtounicode{Mdotaccent}{1E40} \pdfglyphtounicode{Mdotbelow}{1E42} \pdfglyphtounicode{Menarmenian}{0544} \pdfglyphtounicode{Mmonospace}{FF2D} \pdfglyphtounicode{Msmall}{F76D} \pdfglyphtounicode{Mturned}{019C} \pdfglyphtounicode{Mu}{039C} \pdfglyphtounicode{N}{004E} \pdfglyphtounicode{NJ}{01CA} \pdfglyphtounicode{Nacute}{0143} \pdfglyphtounicode{Ncaron}{0147} \pdfglyphtounicode{Ncedilla}{0145} \pdfglyphtounicode{Ncircle}{24C3} \pdfglyphtounicode{Ncircumflexbelow}{1E4A} \pdfglyphtounicode{Ncommaaccent}{0145} \pdfglyphtounicode{Ndotaccent}{1E44} \pdfglyphtounicode{Ndotbelow}{1E46} \pdfglyphtounicode{Nhookleft}{019D} \pdfglyphtounicode{Nineroman}{2168} \pdfglyphtounicode{Nj}{01CB} \pdfglyphtounicode{Njecyrillic}{040A} \pdfglyphtounicode{Nlinebelow}{1E48} \pdfglyphtounicode{Nmonospace}{FF2E} \pdfglyphtounicode{Nowarmenian}{0546} \pdfglyphtounicode{Nsmall}{F76E} \pdfglyphtounicode{Ntilde}{00D1} \pdfglyphtounicode{Ntildesmall}{F7F1} \pdfglyphtounicode{Nu}{039D} \pdfglyphtounicode{O}{004F} \pdfglyphtounicode{OE}{0152} \pdfglyphtounicode{OEsmall}{F6FA} \pdfglyphtounicode{Oacute}{00D3} \pdfglyphtounicode{Oacutesmall}{F7F3} \pdfglyphtounicode{Obarredcyrillic}{04E8} \pdfglyphtounicode{Obarreddieresiscyrillic}{04EA} \pdfglyphtounicode{Obreve}{014E} \pdfglyphtounicode{Ocaron}{01D1} \pdfglyphtounicode{Ocenteredtilde}{019F} \pdfglyphtounicode{Ocircle}{24C4} \pdfglyphtounicode{Ocircumflex}{00D4} \pdfglyphtounicode{Ocircumflexacute}{1ED0} \pdfglyphtounicode{Ocircumflexdotbelow}{1ED8} \pdfglyphtounicode{Ocircumflexgrave}{1ED2} \pdfglyphtounicode{Ocircumflexhookabove}{1ED4} \pdfglyphtounicode{Ocircumflexsmall}{F7F4} \pdfglyphtounicode{Ocircumflextilde}{1ED6} \pdfglyphtounicode{Ocyrillic}{041E} \pdfglyphtounicode{Odblacute}{0150} \pdfglyphtounicode{Odblgrave}{020C} \pdfglyphtounicode{Odieresis}{00D6} \pdfglyphtounicode{Odieresiscyrillic}{04E6} \pdfglyphtounicode{Odieresissmall}{F7F6} \pdfglyphtounicode{Odotbelow}{1ECC} \pdfglyphtounicode{Ogoneksmall}{F6FB} \pdfglyphtounicode{Ograve}{00D2} \pdfglyphtounicode{Ogravesmall}{F7F2} \pdfglyphtounicode{Oharmenian}{0555} \pdfglyphtounicode{Ohm}{2126} \pdfglyphtounicode{Ohookabove}{1ECE} \pdfglyphtounicode{Ohorn}{01A0} \pdfglyphtounicode{Ohornacute}{1EDA} \pdfglyphtounicode{Ohorndotbelow}{1EE2} \pdfglyphtounicode{Ohorngrave}{1EDC} \pdfglyphtounicode{Ohornhookabove}{1EDE} \pdfglyphtounicode{Ohorntilde}{1EE0} \pdfglyphtounicode{Ohungarumlaut}{0150} \pdfglyphtounicode{Oi}{01A2} \pdfglyphtounicode{Oinvertedbreve}{020E} \pdfglyphtounicode{Omacron}{014C} \pdfglyphtounicode{Omacronacute}{1E52} \pdfglyphtounicode{Omacrongrave}{1E50} \pdfglyphtounicode{Omega}{2126} \pdfglyphtounicode{Omegacyrillic}{0460} \pdfglyphtounicode{Omegagreek}{03A9} \pdfglyphtounicode{Omegaroundcyrillic}{047A} \pdfglyphtounicode{Omegatitlocyrillic}{047C} \pdfglyphtounicode{Omegatonos}{038F} \pdfglyphtounicode{Omicron}{039F} \pdfglyphtounicode{Omicrontonos}{038C} \pdfglyphtounicode{Omonospace}{FF2F} \pdfglyphtounicode{Oneroman}{2160} \pdfglyphtounicode{Oogonek}{01EA} \pdfglyphtounicode{Oogonekmacron}{01EC} \pdfglyphtounicode{Oopen}{0186} \pdfglyphtounicode{Oslash}{00D8} \pdfglyphtounicode{Oslashacute}{01FE} \pdfglyphtounicode{Oslashsmall}{F7F8} \pdfglyphtounicode{Osmall}{F76F} \pdfglyphtounicode{Ostrokeacute}{01FE} \pdfglyphtounicode{Otcyrillic}{047E} \pdfglyphtounicode{Otilde}{00D5} \pdfglyphtounicode{Otildeacute}{1E4C} \pdfglyphtounicode{Otildedieresis}{1E4E} \pdfglyphtounicode{Otildesmall}{F7F5} \pdfglyphtounicode{P}{0050} \pdfglyphtounicode{Pacute}{1E54} \pdfglyphtounicode{Pcircle}{24C5} \pdfglyphtounicode{Pdotaccent}{1E56} \pdfglyphtounicode{Pecyrillic}{041F} \pdfglyphtounicode{Peharmenian}{054A} \pdfglyphtounicode{Pemiddlehookcyrillic}{04A6} \pdfglyphtounicode{Phi}{03A6} \pdfglyphtounicode{Phook}{01A4} \pdfglyphtounicode{Pi}{03A0} \pdfglyphtounicode{Piwrarmenian}{0553} \pdfglyphtounicode{Pmonospace}{FF30} \pdfglyphtounicode{Psi}{03A8} \pdfglyphtounicode{Psicyrillic}{0470} \pdfglyphtounicode{Psmall}{F770} \pdfglyphtounicode{Q}{0051} \pdfglyphtounicode{Qcircle}{24C6} \pdfglyphtounicode{Qmonospace}{FF31} \pdfglyphtounicode{Qsmall}{F771} \pdfglyphtounicode{R}{0052} \pdfglyphtounicode{Raarmenian}{054C} \pdfglyphtounicode{Racute}{0154} \pdfglyphtounicode{Rcaron}{0158} \pdfglyphtounicode{Rcedilla}{0156} \pdfglyphtounicode{Rcircle}{24C7} \pdfglyphtounicode{Rcommaaccent}{0156} \pdfglyphtounicode{Rdblgrave}{0210} \pdfglyphtounicode{Rdotaccent}{1E58} \pdfglyphtounicode{Rdotbelow}{1E5A} \pdfglyphtounicode{Rdotbelowmacron}{1E5C} \pdfglyphtounicode{Reharmenian}{0550} \pdfglyphtounicode{Rfraktur}{211C} \pdfglyphtounicode{Rho}{03A1} \pdfglyphtounicode{Ringsmall}{F6FC} \pdfglyphtounicode{Rinvertedbreve}{0212} \pdfglyphtounicode{Rlinebelow}{1E5E} \pdfglyphtounicode{Rmonospace}{FF32} \pdfglyphtounicode{Rsmall}{F772} \pdfglyphtounicode{Rsmallinverted}{0281} \pdfglyphtounicode{Rsmallinvertedsuperior}{02B6} \pdfglyphtounicode{S}{0053} \pdfglyphtounicode{SF010000}{250C} \pdfglyphtounicode{SF020000}{2514} \pdfglyphtounicode{SF030000}{2510} \pdfglyphtounicode{SF040000}{2518} \pdfglyphtounicode{SF050000}{253C} \pdfglyphtounicode{SF060000}{252C} \pdfglyphtounicode{SF070000}{2534} \pdfglyphtounicode{SF080000}{251C} \pdfglyphtounicode{SF090000}{2524} \pdfglyphtounicode{SF100000}{2500} \pdfglyphtounicode{SF110000}{2502} \pdfglyphtounicode{SF190000}{2561} \pdfglyphtounicode{SF200000}{2562} \pdfglyphtounicode{SF210000}{2556} \pdfglyphtounicode{SF220000}{2555} \pdfglyphtounicode{SF230000}{2563} \pdfglyphtounicode{SF240000}{2551} \pdfglyphtounicode{SF250000}{2557} \pdfglyphtounicode{SF260000}{255D} \pdfglyphtounicode{SF270000}{255C} \pdfglyphtounicode{SF280000}{255B} \pdfglyphtounicode{SF360000}{255E} \pdfglyphtounicode{SF370000}{255F} \pdfglyphtounicode{SF380000}{255A} \pdfglyphtounicode{SF390000}{2554} \pdfglyphtounicode{SF400000}{2569} \pdfglyphtounicode{SF410000}{2566} \pdfglyphtounicode{SF420000}{2560} \pdfglyphtounicode{SF430000}{2550} \pdfglyphtounicode{SF440000}{256C} \pdfglyphtounicode{SF450000}{2567} \pdfglyphtounicode{SF460000}{2568} \pdfglyphtounicode{SF470000}{2564} \pdfglyphtounicode{SF480000}{2565} \pdfglyphtounicode{SF490000}{2559} \pdfglyphtounicode{SF500000}{2558} \pdfglyphtounicode{SF510000}{2552} \pdfglyphtounicode{SF520000}{2553} \pdfglyphtounicode{SF530000}{256B} \pdfglyphtounicode{SF540000}{256A} \pdfglyphtounicode{Sacute}{015A} \pdfglyphtounicode{Sacutedotaccent}{1E64} \pdfglyphtounicode{Sampigreek}{03E0} \pdfglyphtounicode{Scaron}{0160} \pdfglyphtounicode{Scarondotaccent}{1E66} \pdfglyphtounicode{Scaronsmall}{F6FD} \pdfglyphtounicode{Scedilla}{015E} \pdfglyphtounicode{Schwa}{018F} \pdfglyphtounicode{Schwacyrillic}{04D8} \pdfglyphtounicode{Schwadieresiscyrillic}{04DA} \pdfglyphtounicode{Scircle}{24C8} \pdfglyphtounicode{Scircumflex}{015C} \pdfglyphtounicode{Scommaaccent}{0218} \pdfglyphtounicode{Sdotaccent}{1E60} \pdfglyphtounicode{Sdotbelow}{1E62} \pdfglyphtounicode{Sdotbelowdotaccent}{1E68} \pdfglyphtounicode{Seharmenian}{054D} \pdfglyphtounicode{Sevenroman}{2166} \pdfglyphtounicode{Shaarmenian}{0547} \pdfglyphtounicode{Shacyrillic}{0428} \pdfglyphtounicode{Shchacyrillic}{0429} \pdfglyphtounicode{Sheicoptic}{03E2} \pdfglyphtounicode{Shhacyrillic}{04BA} \pdfglyphtounicode{Shimacoptic}{03EC} \pdfglyphtounicode{Sigma}{03A3} \pdfglyphtounicode{Sixroman}{2165} \pdfglyphtounicode{Smonospace}{FF33} \pdfglyphtounicode{Softsigncyrillic}{042C} \pdfglyphtounicode{Ssmall}{F773} \pdfglyphtounicode{Stigmagreek}{03DA} \pdfglyphtounicode{T}{0054} \pdfglyphtounicode{Tau}{03A4} \pdfglyphtounicode{Tbar}{0166} \pdfglyphtounicode{Tcaron}{0164} \pdfglyphtounicode{Tcedilla}{0162} \pdfglyphtounicode{Tcircle}{24C9} \pdfglyphtounicode{Tcircumflexbelow}{1E70} \pdfglyphtounicode{Tcommaaccent}{0162} \pdfglyphtounicode{Tdotaccent}{1E6A} \pdfglyphtounicode{Tdotbelow}{1E6C} \pdfglyphtounicode{Tecyrillic}{0422} \pdfglyphtounicode{Tedescendercyrillic}{04AC} \pdfglyphtounicode{Tenroman}{2169} \pdfglyphtounicode{Tetsecyrillic}{04B4} \pdfglyphtounicode{Theta}{0398} \pdfglyphtounicode{Thook}{01AC} \pdfglyphtounicode{Thorn}{00DE} \pdfglyphtounicode{Thornsmall}{F7FE} \pdfglyphtounicode{Threeroman}{2162} \pdfglyphtounicode{Tildesmall}{F6FE} \pdfglyphtounicode{Tiwnarmenian}{054F} \pdfglyphtounicode{Tlinebelow}{1E6E} \pdfglyphtounicode{Tmonospace}{FF34} \pdfglyphtounicode{Toarmenian}{0539} \pdfglyphtounicode{Tonefive}{01BC} \pdfglyphtounicode{Tonesix}{0184} \pdfglyphtounicode{Tonetwo}{01A7} \pdfglyphtounicode{Tretroflexhook}{01AE} \pdfglyphtounicode{Tsecyrillic}{0426} \pdfglyphtounicode{Tshecyrillic}{040B} \pdfglyphtounicode{Tsmall}{F774} \pdfglyphtounicode{Twelveroman}{216B} \pdfglyphtounicode{Tworoman}{2161} \pdfglyphtounicode{U}{0055} \pdfglyphtounicode{Uacute}{00DA} \pdfglyphtounicode{Uacutesmall}{F7FA} \pdfglyphtounicode{Ubreve}{016C} \pdfglyphtounicode{Ucaron}{01D3} \pdfglyphtounicode{Ucircle}{24CA} \pdfglyphtounicode{Ucircumflex}{00DB} \pdfglyphtounicode{Ucircumflexbelow}{1E76} \pdfglyphtounicode{Ucircumflexsmall}{F7FB} \pdfglyphtounicode{Ucyrillic}{0423} \pdfglyphtounicode{Udblacute}{0170} \pdfglyphtounicode{Udblgrave}{0214} \pdfglyphtounicode{Udieresis}{00DC} \pdfglyphtounicode{Udieresisacute}{01D7} \pdfglyphtounicode{Udieresisbelow}{1E72} \pdfglyphtounicode{Udieresiscaron}{01D9} \pdfglyphtounicode{Udieresiscyrillic}{04F0} \pdfglyphtounicode{Udieresisgrave}{01DB} \pdfglyphtounicode{Udieresismacron}{01D5} \pdfglyphtounicode{Udieresissmall}{F7FC} \pdfglyphtounicode{Udotbelow}{1EE4} \pdfglyphtounicode{Ugrave}{00D9} \pdfglyphtounicode{Ugravesmall}{F7F9} \pdfglyphtounicode{Uhookabove}{1EE6} \pdfglyphtounicode{Uhorn}{01AF} \pdfglyphtounicode{Uhornacute}{1EE8} \pdfglyphtounicode{Uhorndotbelow}{1EF0} \pdfglyphtounicode{Uhorngrave}{1EEA} \pdfglyphtounicode{Uhornhookabove}{1EEC} \pdfglyphtounicode{Uhorntilde}{1EEE} \pdfglyphtounicode{Uhungarumlaut}{0170} \pdfglyphtounicode{Uhungarumlautcyrillic}{04F2} \pdfglyphtounicode{Uinvertedbreve}{0216} \pdfglyphtounicode{Ukcyrillic}{0478} \pdfglyphtounicode{Umacron}{016A} \pdfglyphtounicode{Umacroncyrillic}{04EE} \pdfglyphtounicode{Umacrondieresis}{1E7A} \pdfglyphtounicode{Umonospace}{FF35} \pdfglyphtounicode{Uogonek}{0172} \pdfglyphtounicode{Upsilon}{03A5} \pdfglyphtounicode{Upsilon1}{03D2} \pdfglyphtounicode{Upsilonacutehooksymbolgreek}{03D3} \pdfglyphtounicode{Upsilonafrican}{01B1} \pdfglyphtounicode{Upsilondieresis}{03AB} \pdfglyphtounicode{Upsilondieresishooksymbolgreek}{03D4} \pdfglyphtounicode{Upsilonhooksymbol}{03D2} \pdfglyphtounicode{Upsilontonos}{038E} \pdfglyphtounicode{Uring}{016E} \pdfglyphtounicode{Ushortcyrillic}{040E} \pdfglyphtounicode{Usmall}{F775} \pdfglyphtounicode{Ustraightcyrillic}{04AE} \pdfglyphtounicode{Ustraightstrokecyrillic}{04B0} \pdfglyphtounicode{Utilde}{0168} \pdfglyphtounicode{Utildeacute}{1E78} \pdfglyphtounicode{Utildebelow}{1E74} \pdfglyphtounicode{V}{0056} \pdfglyphtounicode{Vcircle}{24CB} \pdfglyphtounicode{Vdotbelow}{1E7E} \pdfglyphtounicode{Vecyrillic}{0412} \pdfglyphtounicode{Vewarmenian}{054E} \pdfglyphtounicode{Vhook}{01B2} \pdfglyphtounicode{Vmonospace}{FF36} \pdfglyphtounicode{Voarmenian}{0548} \pdfglyphtounicode{Vsmall}{F776} \pdfglyphtounicode{Vtilde}{1E7C} \pdfglyphtounicode{W}{0057} \pdfglyphtounicode{Wacute}{1E82} \pdfglyphtounicode{Wcircle}{24CC} \pdfglyphtounicode{Wcircumflex}{0174} \pdfglyphtounicode{Wdieresis}{1E84} \pdfglyphtounicode{Wdotaccent}{1E86} \pdfglyphtounicode{Wdotbelow}{1E88} \pdfglyphtounicode{Wgrave}{1E80} \pdfglyphtounicode{Wmonospace}{FF37} \pdfglyphtounicode{Wsmall}{F777} \pdfglyphtounicode{X}{0058} \pdfglyphtounicode{Xcircle}{24CD} \pdfglyphtounicode{Xdieresis}{1E8C} \pdfglyphtounicode{Xdotaccent}{1E8A} \pdfglyphtounicode{Xeharmenian}{053D} \pdfglyphtounicode{Xi}{039E} \pdfglyphtounicode{Xmonospace}{FF38} \pdfglyphtounicode{Xsmall}{F778} \pdfglyphtounicode{Y}{0059} \pdfglyphtounicode{Yacute}{00DD} \pdfglyphtounicode{Yacutesmall}{F7FD} \pdfglyphtounicode{Yatcyrillic}{0462} \pdfglyphtounicode{Ycircle}{24CE} \pdfglyphtounicode{Ycircumflex}{0176} \pdfglyphtounicode{Ydieresis}{0178} \pdfglyphtounicode{Ydieresissmall}{F7FF} \pdfglyphtounicode{Ydotaccent}{1E8E} \pdfglyphtounicode{Ydotbelow}{1EF4} \pdfglyphtounicode{Yericyrillic}{042B} \pdfglyphtounicode{Yerudieresiscyrillic}{04F8} \pdfglyphtounicode{Ygrave}{1EF2} \pdfglyphtounicode{Yhook}{01B3} \pdfglyphtounicode{Yhookabove}{1EF6} \pdfglyphtounicode{Yiarmenian}{0545} \pdfglyphtounicode{Yicyrillic}{0407} \pdfglyphtounicode{Yiwnarmenian}{0552} \pdfglyphtounicode{Ymonospace}{FF39} \pdfglyphtounicode{Ysmall}{F779} \pdfglyphtounicode{Ytilde}{1EF8} \pdfglyphtounicode{Yusbigcyrillic}{046A} \pdfglyphtounicode{Yusbigiotifiedcyrillic}{046C} \pdfglyphtounicode{Yuslittlecyrillic}{0466} \pdfglyphtounicode{Yuslittleiotifiedcyrillic}{0468} \pdfglyphtounicode{Z}{005A} \pdfglyphtounicode{Zaarmenian}{0536} \pdfglyphtounicode{Zacute}{0179} \pdfglyphtounicode{Zcaron}{017D} \pdfglyphtounicode{Zcaronsmall}{F6FF} \pdfglyphtounicode{Zcircle}{24CF} \pdfglyphtounicode{Zcircumflex}{1E90} \pdfglyphtounicode{Zdot}{017B} \pdfglyphtounicode{Zdotaccent}{017B} \pdfglyphtounicode{Zdotbelow}{1E92} \pdfglyphtounicode{Zecyrillic}{0417} \pdfglyphtounicode{Zedescendercyrillic}{0498} \pdfglyphtounicode{Zedieresiscyrillic}{04DE} \pdfglyphtounicode{Zeta}{0396} \pdfglyphtounicode{Zhearmenian}{053A} \pdfglyphtounicode{Zhebrevecyrillic}{04C1} \pdfglyphtounicode{Zhecyrillic}{0416} \pdfglyphtounicode{Zhedescendercyrillic}{0496} \pdfglyphtounicode{Zhedieresiscyrillic}{04DC} \pdfglyphtounicode{Zlinebelow}{1E94} \pdfglyphtounicode{Zmonospace}{FF3A} \pdfglyphtounicode{Zsmall}{F77A} \pdfglyphtounicode{Zstroke}{01B5} \pdfglyphtounicode{a}{0061} \pdfglyphtounicode{aabengali}{0986} \pdfglyphtounicode{aacute}{00E1} \pdfglyphtounicode{aadeva}{0906} \pdfglyphtounicode{aagujarati}{0A86} \pdfglyphtounicode{aagurmukhi}{0A06} \pdfglyphtounicode{aamatragurmukhi}{0A3E} \pdfglyphtounicode{aarusquare}{3303} \pdfglyphtounicode{aavowelsignbengali}{09BE} \pdfglyphtounicode{aavowelsigndeva}{093E} \pdfglyphtounicode{aavowelsigngujarati}{0ABE} \pdfglyphtounicode{abbreviationmarkarmenian}{055F} \pdfglyphtounicode{abbreviationsigndeva}{0970} \pdfglyphtounicode{abengali}{0985} \pdfglyphtounicode{abopomofo}{311A} \pdfglyphtounicode{abreve}{0103} \pdfglyphtounicode{abreveacute}{1EAF} \pdfglyphtounicode{abrevecyrillic}{04D1} \pdfglyphtounicode{abrevedotbelow}{1EB7} \pdfglyphtounicode{abrevegrave}{1EB1} \pdfglyphtounicode{abrevehookabove}{1EB3} \pdfglyphtounicode{abrevetilde}{1EB5} \pdfglyphtounicode{acaron}{01CE} \pdfglyphtounicode{acircle}{24D0} \pdfglyphtounicode{acircumflex}{00E2} \pdfglyphtounicode{acircumflexacute}{1EA5} \pdfglyphtounicode{acircumflexdotbelow}{1EAD} \pdfglyphtounicode{acircumflexgrave}{1EA7} \pdfglyphtounicode{acircumflexhookabove}{1EA9} 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\pdfglyphtounicode{afii10075}{0439} \pdfglyphtounicode{afii10076}{043A} \pdfglyphtounicode{afii10077}{043B} \pdfglyphtounicode{afii10078}{043C} \pdfglyphtounicode{afii10079}{043D} \pdfglyphtounicode{afii10080}{043E} \pdfglyphtounicode{afii10081}{043F} \pdfglyphtounicode{afii10082}{0440} \pdfglyphtounicode{afii10083}{0441} \pdfglyphtounicode{afii10084}{0442} \pdfglyphtounicode{afii10085}{0443} \pdfglyphtounicode{afii10086}{0444} \pdfglyphtounicode{afii10087}{0445} \pdfglyphtounicode{afii10088}{0446} \pdfglyphtounicode{afii10089}{0447} \pdfglyphtounicode{afii10090}{0448} \pdfglyphtounicode{afii10091}{0449} \pdfglyphtounicode{afii10092}{044A} \pdfglyphtounicode{afii10093}{044B} \pdfglyphtounicode{afii10094}{044C} \pdfglyphtounicode{afii10095}{044D} \pdfglyphtounicode{afii10096}{044E} \pdfglyphtounicode{afii10097}{044F} \pdfglyphtounicode{afii10098}{0491} \pdfglyphtounicode{afii10099}{0452} \pdfglyphtounicode{afii10100}{0453} \pdfglyphtounicode{afii10101}{0454} \pdfglyphtounicode{afii10102}{0455} \pdfglyphtounicode{afii10103}{0456} \pdfglyphtounicode{afii10104}{0457} \pdfglyphtounicode{afii10105}{0458} \pdfglyphtounicode{afii10106}{0459} \pdfglyphtounicode{afii10107}{045A} \pdfglyphtounicode{afii10108}{045B} \pdfglyphtounicode{afii10109}{045C} \pdfglyphtounicode{afii10110}{045E} \pdfglyphtounicode{afii10145}{040F} \pdfglyphtounicode{afii10146}{0462} \pdfglyphtounicode{afii10147}{0472} \pdfglyphtounicode{afii10148}{0474} \pdfglyphtounicode{afii10192}{F6C6} \pdfglyphtounicode{afii10193}{045F} \pdfglyphtounicode{afii10194}{0463} \pdfglyphtounicode{afii10195}{0473} \pdfglyphtounicode{afii10196}{0475} \pdfglyphtounicode{afii10831}{F6C7} \pdfglyphtounicode{afii10832}{F6C8} \pdfglyphtounicode{afii10846}{04D9} \pdfglyphtounicode{afii299}{200E} \pdfglyphtounicode{afii300}{200F} \pdfglyphtounicode{afii301}{200D} \pdfglyphtounicode{afii57381}{066A} \pdfglyphtounicode{afii57388}{060C} \pdfglyphtounicode{afii57392}{0660} \pdfglyphtounicode{afii57393}{0661} \pdfglyphtounicode{afii57394}{0662} \pdfglyphtounicode{afii57395}{0663} \pdfglyphtounicode{afii57396}{0664} \pdfglyphtounicode{afii57397}{0665} \pdfglyphtounicode{afii57398}{0666} \pdfglyphtounicode{afii57399}{0667} \pdfglyphtounicode{afii57400}{0668} \pdfglyphtounicode{afii57401}{0669} \pdfglyphtounicode{afii57403}{061B} \pdfglyphtounicode{afii57407}{061F} \pdfglyphtounicode{afii57409}{0621} \pdfglyphtounicode{afii57410}{0622} \pdfglyphtounicode{afii57411}{0623} \pdfglyphtounicode{afii57412}{0624} \pdfglyphtounicode{afii57413}{0625} \pdfglyphtounicode{afii57414}{0626} \pdfglyphtounicode{afii57415}{0627} \pdfglyphtounicode{afii57416}{0628} \pdfglyphtounicode{afii57417}{0629} \pdfglyphtounicode{afii57418}{062A} \pdfglyphtounicode{afii57419}{062B} \pdfglyphtounicode{afii57420}{062C} \pdfglyphtounicode{afii57421}{062D} \pdfglyphtounicode{afii57422}{062E} \pdfglyphtounicode{afii57423}{062F} \pdfglyphtounicode{afii57424}{0630} \pdfglyphtounicode{afii57425}{0631} \pdfglyphtounicode{afii57426}{0632} \pdfglyphtounicode{afii57427}{0633} \pdfglyphtounicode{afii57428}{0634} \pdfglyphtounicode{afii57429}{0635} \pdfglyphtounicode{afii57430}{0636} \pdfglyphtounicode{afii57431}{0637} \pdfglyphtounicode{afii57432}{0638} \pdfglyphtounicode{afii57433}{0639} \pdfglyphtounicode{afii57434}{063A} \pdfglyphtounicode{afii57440}{0640} \pdfglyphtounicode{afii57441}{0641} \pdfglyphtounicode{afii57442}{0642} \pdfglyphtounicode{afii57443}{0643} \pdfglyphtounicode{afii57444}{0644} \pdfglyphtounicode{afii57445}{0645} \pdfglyphtounicode{afii57446}{0646} \pdfglyphtounicode{afii57448}{0648} \pdfglyphtounicode{afii57449}{0649} \pdfglyphtounicode{afii57450}{064A} \pdfglyphtounicode{afii57451}{064B} \pdfglyphtounicode{afii57452}{064C} \pdfglyphtounicode{afii57453}{064D} \pdfglyphtounicode{afii57454}{064E} \pdfglyphtounicode{afii57455}{064F} \pdfglyphtounicode{afii57456}{0650} \pdfglyphtounicode{afii57457}{0651} \pdfglyphtounicode{afii57458}{0652} \pdfglyphtounicode{afii57470}{0647} \pdfglyphtounicode{afii57505}{06A4} \pdfglyphtounicode{afii57506}{067E} \pdfglyphtounicode{afii57507}{0686} \pdfglyphtounicode{afii57508}{0698} \pdfglyphtounicode{afii57509}{06AF} \pdfglyphtounicode{afii57511}{0679} \pdfglyphtounicode{afii57512}{0688} \pdfglyphtounicode{afii57513}{0691} \pdfglyphtounicode{afii57514}{06BA} \pdfglyphtounicode{afii57519}{06D2} \pdfglyphtounicode{afii57534}{06D5} \pdfglyphtounicode{afii57636}{20AA} \pdfglyphtounicode{afii57645}{05BE} \pdfglyphtounicode{afii57658}{05C3} \pdfglyphtounicode{afii57664}{05D0} \pdfglyphtounicode{afii57665}{05D1} \pdfglyphtounicode{afii57666}{05D2} \pdfglyphtounicode{afii57667}{05D3} \pdfglyphtounicode{afii57668}{05D4} \pdfglyphtounicode{afii57669}{05D5} \pdfglyphtounicode{afii57670}{05D6} \pdfglyphtounicode{afii57671}{05D7} \pdfglyphtounicode{afii57672}{05D8} \pdfglyphtounicode{afii57673}{05D9} \pdfglyphtounicode{afii57674}{05DA} \pdfglyphtounicode{afii57675}{05DB} \pdfglyphtounicode{afii57676}{05DC} \pdfglyphtounicode{afii57677}{05DD} \pdfglyphtounicode{afii57678}{05DE} \pdfglyphtounicode{afii57679}{05DF} \pdfglyphtounicode{afii57680}{05E0} \pdfglyphtounicode{afii57681}{05E1} \pdfglyphtounicode{afii57682}{05E2} \pdfglyphtounicode{afii57683}{05E3} \pdfglyphtounicode{afii57684}{05E4} \pdfglyphtounicode{afii57685}{05E5} \pdfglyphtounicode{afii57686}{05E6} \pdfglyphtounicode{afii57687}{05E7} \pdfglyphtounicode{afii57688}{05E8} \pdfglyphtounicode{afii57689}{05E9} \pdfglyphtounicode{afii57690}{05EA} \pdfglyphtounicode{afii57694}{FB2A} \pdfglyphtounicode{afii57695}{FB2B} \pdfglyphtounicode{afii57700}{FB4B} \pdfglyphtounicode{afii57705}{FB1F} \pdfglyphtounicode{afii57716}{05F0} \pdfglyphtounicode{afii57717}{05F1} \pdfglyphtounicode{afii57718}{05F2} \pdfglyphtounicode{afii57723}{FB35} \pdfglyphtounicode{afii57793}{05B4} \pdfglyphtounicode{afii57794}{05B5} \pdfglyphtounicode{afii57795}{05B6} \pdfglyphtounicode{afii57796}{05BB} \pdfglyphtounicode{afii57797}{05B8} \pdfglyphtounicode{afii57798}{05B7} \pdfglyphtounicode{afii57799}{05B0} \pdfglyphtounicode{afii57800}{05B2} \pdfglyphtounicode{afii57801}{05B1} \pdfglyphtounicode{afii57802}{05B3} \pdfglyphtounicode{afii57803}{05C2} \pdfglyphtounicode{afii57804}{05C1} \pdfglyphtounicode{afii57806}{05B9} \pdfglyphtounicode{afii57807}{05BC} \pdfglyphtounicode{afii57839}{05BD} \pdfglyphtounicode{afii57841}{05BF} \pdfglyphtounicode{afii57842}{05C0} \pdfglyphtounicode{afii57929}{02BC} \pdfglyphtounicode{afii61248}{2105} \pdfglyphtounicode{afii61289}{2113} \pdfglyphtounicode{afii61352}{2116} \pdfglyphtounicode{afii61573}{202C} \pdfglyphtounicode{afii61574}{202D} \pdfglyphtounicode{afii61575}{202E} \pdfglyphtounicode{afii61664}{200C} \pdfglyphtounicode{afii63167}{066D} \pdfglyphtounicode{afii64937}{02BD} \pdfglyphtounicode{agrave}{00E0} \pdfglyphtounicode{agujarati}{0A85} \pdfglyphtounicode{agurmukhi}{0A05} \pdfglyphtounicode{ahiragana}{3042} \pdfglyphtounicode{ahookabove}{1EA3} \pdfglyphtounicode{aibengali}{0990} \pdfglyphtounicode{aibopomofo}{311E} \pdfglyphtounicode{aideva}{0910} \pdfglyphtounicode{aiecyrillic}{04D5} \pdfglyphtounicode{aigujarati}{0A90} \pdfglyphtounicode{aigurmukhi}{0A10} \pdfglyphtounicode{aimatragurmukhi}{0A48} \pdfglyphtounicode{ainarabic}{0639} \pdfglyphtounicode{ainfinalarabic}{FECA} \pdfglyphtounicode{aininitialarabic}{FECB} \pdfglyphtounicode{ainmedialarabic}{FECC} \pdfglyphtounicode{ainvertedbreve}{0203} \pdfglyphtounicode{aivowelsignbengali}{09C8} \pdfglyphtounicode{aivowelsigndeva}{0948} \pdfglyphtounicode{aivowelsigngujarati}{0AC8} \pdfglyphtounicode{akatakana}{30A2} \pdfglyphtounicode{akatakanahalfwidth}{FF71} \pdfglyphtounicode{akorean}{314F} \pdfglyphtounicode{alef}{05D0} \pdfglyphtounicode{alefarabic}{0627} \pdfglyphtounicode{alefdageshhebrew}{FB30} \pdfglyphtounicode{aleffinalarabic}{FE8E} \pdfglyphtounicode{alefhamzaabovearabic}{0623} \pdfglyphtounicode{alefhamzaabovefinalarabic}{FE84} \pdfglyphtounicode{alefhamzabelowarabic}{0625} \pdfglyphtounicode{alefhamzabelowfinalarabic}{FE88} \pdfglyphtounicode{alefhebrew}{05D0} \pdfglyphtounicode{aleflamedhebrew}{FB4F} \pdfglyphtounicode{alefmaddaabovearabic}{0622} \pdfglyphtounicode{alefmaddaabovefinalarabic}{FE82} \pdfglyphtounicode{alefmaksuraarabic}{0649} \pdfglyphtounicode{alefmaksurafinalarabic}{FEF0} \pdfglyphtounicode{alefmaksurainitialarabic}{FEF3} \pdfglyphtounicode{alefmaksuramedialarabic}{FEF4} \pdfglyphtounicode{alefpatahhebrew}{FB2E} \pdfglyphtounicode{alefqamatshebrew}{FB2F} \pdfglyphtounicode{aleph}{2135} \pdfglyphtounicode{allequal}{224C} \pdfglyphtounicode{alpha}{03B1} \pdfglyphtounicode{alphatonos}{03AC} \pdfglyphtounicode{amacron}{0101} \pdfglyphtounicode{amonospace}{FF41} \pdfglyphtounicode{ampersand}{0026} \pdfglyphtounicode{ampersandmonospace}{FF06} \pdfglyphtounicode{ampersandsmall}{F726} \pdfglyphtounicode{amsquare}{33C2} \pdfglyphtounicode{anbopomofo}{3122} \pdfglyphtounicode{angbopomofo}{3124} \pdfglyphtounicode{angkhankhuthai}{0E5A} \pdfglyphtounicode{angle}{2220} \pdfglyphtounicode{anglebracketleft}{3008} \pdfglyphtounicode{anglebracketleftvertical}{FE3F} \pdfglyphtounicode{anglebracketright}{3009} \pdfglyphtounicode{anglebracketrightvertical}{FE40} \pdfglyphtounicode{angleleft}{2329} \pdfglyphtounicode{angleright}{232A} \pdfglyphtounicode{angstrom}{212B} \pdfglyphtounicode{anoteleia}{0387} \pdfglyphtounicode{anudattadeva}{0952} \pdfglyphtounicode{anusvarabengali}{0982} \pdfglyphtounicode{anusvaradeva}{0902} \pdfglyphtounicode{anusvaragujarati}{0A82} \pdfglyphtounicode{aogonek}{0105} \pdfglyphtounicode{apaatosquare}{3300} \pdfglyphtounicode{aparen}{249C} \pdfglyphtounicode{apostrophearmenian}{055A} \pdfglyphtounicode{apostrophemod}{02BC} \pdfglyphtounicode{apple}{F8FF} \pdfglyphtounicode{approaches}{2250} \pdfglyphtounicode{approxequal}{2248} \pdfglyphtounicode{approxequalorimage}{2252} \pdfglyphtounicode{approximatelyequal}{2245} \pdfglyphtounicode{araeaekorean}{318E} \pdfglyphtounicode{araeakorean}{318D} \pdfglyphtounicode{arc}{2312} \pdfglyphtounicode{arighthalfring}{1E9A} \pdfglyphtounicode{aring}{00E5} \pdfglyphtounicode{aringacute}{01FB} \pdfglyphtounicode{aringbelow}{1E01} \pdfglyphtounicode{arrowboth}{2194} \pdfglyphtounicode{arrowdashdown}{21E3} \pdfglyphtounicode{arrowdashleft}{21E0} \pdfglyphtounicode{arrowdashright}{21E2} \pdfglyphtounicode{arrowdashup}{21E1} \pdfglyphtounicode{arrowdblboth}{21D4} \pdfglyphtounicode{arrowdbldown}{21D3} \pdfglyphtounicode{arrowdblleft}{21D0} \pdfglyphtounicode{arrowdblright}{21D2} \pdfglyphtounicode{arrowdblup}{21D1} \pdfglyphtounicode{arrowdown}{2193} \pdfglyphtounicode{arrowdownleft}{2199} \pdfglyphtounicode{arrowdownright}{2198} \pdfglyphtounicode{arrowdownwhite}{21E9} \pdfglyphtounicode{arrowheaddownmod}{02C5} \pdfglyphtounicode{arrowheadleftmod}{02C2} \pdfglyphtounicode{arrowheadrightmod}{02C3} \pdfglyphtounicode{arrowheadupmod}{02C4} \pdfglyphtounicode{arrowhorizex}{F8E7} \pdfglyphtounicode{arrowleft}{2190} \pdfglyphtounicode{arrowleftdbl}{21D0} \pdfglyphtounicode{arrowleftdblstroke}{21CD} \pdfglyphtounicode{arrowleftoverright}{21C6} \pdfglyphtounicode{arrowleftwhite}{21E6} \pdfglyphtounicode{arrowright}{2192} \pdfglyphtounicode{arrowrightdblstroke}{21CF} \pdfglyphtounicode{arrowrightheavy}{279E} \pdfglyphtounicode{arrowrightoverleft}{21C4} \pdfglyphtounicode{arrowrightwhite}{21E8} \pdfglyphtounicode{arrowtableft}{21E4} \pdfglyphtounicode{arrowtabright}{21E5} \pdfglyphtounicode{arrowup}{2191} \pdfglyphtounicode{arrowupdn}{2195} \pdfglyphtounicode{arrowupdnbse}{21A8} \pdfglyphtounicode{arrowupdownbase}{21A8} \pdfglyphtounicode{arrowupleft}{2196} \pdfglyphtounicode{arrowupleftofdown}{21C5} \pdfglyphtounicode{arrowupright}{2197} \pdfglyphtounicode{arrowupwhite}{21E7} \pdfglyphtounicode{arrowvertex}{F8E6} \pdfglyphtounicode{asciicircum}{005E} \pdfglyphtounicode{asciicircummonospace}{FF3E} \pdfglyphtounicode{asciitilde}{007E} \pdfglyphtounicode{asciitildemonospace}{FF5E} \pdfglyphtounicode{ascript}{0251} \pdfglyphtounicode{ascriptturned}{0252} \pdfglyphtounicode{asmallhiragana}{3041} \pdfglyphtounicode{asmallkatakana}{30A1} \pdfglyphtounicode{asmallkatakanahalfwidth}{FF67} \pdfglyphtounicode{asterisk}{002A} \pdfglyphtounicode{asteriskaltonearabic}{066D} \pdfglyphtounicode{asteriskarabic}{066D} \pdfglyphtounicode{asteriskmath}{2217} \pdfglyphtounicode{asteriskmonospace}{FF0A} \pdfglyphtounicode{asterisksmall}{FE61} \pdfglyphtounicode{asterism}{2042} \pdfglyphtounicode{asuperior}{F6E9} \pdfglyphtounicode{asymptoticallyequal}{2243} \pdfglyphtounicode{at}{0040} \pdfglyphtounicode{atilde}{00E3} \pdfglyphtounicode{atmonospace}{FF20} \pdfglyphtounicode{atsmall}{FE6B} \pdfglyphtounicode{aturned}{0250} \pdfglyphtounicode{aubengali}{0994} \pdfglyphtounicode{aubopomofo}{3120} \pdfglyphtounicode{audeva}{0914} \pdfglyphtounicode{augujarati}{0A94} \pdfglyphtounicode{augurmukhi}{0A14} \pdfglyphtounicode{aulengthmarkbengali}{09D7} \pdfglyphtounicode{aumatragurmukhi}{0A4C} \pdfglyphtounicode{auvowelsignbengali}{09CC} \pdfglyphtounicode{auvowelsigndeva}{094C} \pdfglyphtounicode{auvowelsigngujarati}{0ACC} \pdfglyphtounicode{avagrahadeva}{093D} \pdfglyphtounicode{aybarmenian}{0561} \pdfglyphtounicode{ayin}{05E2} \pdfglyphtounicode{ayinaltonehebrew}{FB20} \pdfglyphtounicode{ayinhebrew}{05E2} \pdfglyphtounicode{b}{0062} \pdfglyphtounicode{babengali}{09AC} \pdfglyphtounicode{backslash}{005C} \pdfglyphtounicode{backslashmonospace}{FF3C} \pdfglyphtounicode{badeva}{092C} \pdfglyphtounicode{bagujarati}{0AAC} \pdfglyphtounicode{bagurmukhi}{0A2C} \pdfglyphtounicode{bahiragana}{3070} \pdfglyphtounicode{bahtthai}{0E3F} \pdfglyphtounicode{bakatakana}{30D0} \pdfglyphtounicode{bar}{007C} \pdfglyphtounicode{barmonospace}{FF5C} \pdfglyphtounicode{bbopomofo}{3105} \pdfglyphtounicode{bcircle}{24D1} \pdfglyphtounicode{bdotaccent}{1E03} \pdfglyphtounicode{bdotbelow}{1E05} \pdfglyphtounicode{beamedsixteenthnotes}{266C} \pdfglyphtounicode{because}{2235} \pdfglyphtounicode{becyrillic}{0431} \pdfglyphtounicode{beharabic}{0628} \pdfglyphtounicode{behfinalarabic}{FE90} \pdfglyphtounicode{behinitialarabic}{FE91} \pdfglyphtounicode{behiragana}{3079} \pdfglyphtounicode{behmedialarabic}{FE92} \pdfglyphtounicode{behmeeminitialarabic}{FC9F} \pdfglyphtounicode{behmeemisolatedarabic}{FC08} \pdfglyphtounicode{behnoonfinalarabic}{FC6D} \pdfglyphtounicode{bekatakana}{30D9} \pdfglyphtounicode{benarmenian}{0562} \pdfglyphtounicode{bet}{05D1} \pdfglyphtounicode{beta}{03B2} \pdfglyphtounicode{betasymbolgreek}{03D0} \pdfglyphtounicode{betdagesh}{FB31} \pdfglyphtounicode{betdageshhebrew}{FB31} \pdfglyphtounicode{bethebrew}{05D1} \pdfglyphtounicode{betrafehebrew}{FB4C} \pdfglyphtounicode{bhabengali}{09AD} \pdfglyphtounicode{bhadeva}{092D} \pdfglyphtounicode{bhagujarati}{0AAD} \pdfglyphtounicode{bhagurmukhi}{0A2D} \pdfglyphtounicode{bhook}{0253} \pdfglyphtounicode{bihiragana}{3073} \pdfglyphtounicode{bikatakana}{30D3} \pdfglyphtounicode{bilabialclick}{0298} \pdfglyphtounicode{bindigurmukhi}{0A02} \pdfglyphtounicode{birusquare}{3331} \pdfglyphtounicode{blackcircle}{25CF} \pdfglyphtounicode{blackdiamond}{25C6} \pdfglyphtounicode{blackdownpointingtriangle}{25BC} \pdfglyphtounicode{blackleftpointingpointer}{25C4} \pdfglyphtounicode{blackleftpointingtriangle}{25C0} \pdfglyphtounicode{blacklenticularbracketleft}{3010} \pdfglyphtounicode{blacklenticularbracketleftvertical}{FE3B} \pdfglyphtounicode{blacklenticularbracketright}{3011} \pdfglyphtounicode{blacklenticularbracketrightvertical}{FE3C} \pdfglyphtounicode{blacklowerlefttriangle}{25E3} \pdfglyphtounicode{blacklowerrighttriangle}{25E2} \pdfglyphtounicode{blackrectangle}{25AC} \pdfglyphtounicode{blackrightpointingpointer}{25BA} \pdfglyphtounicode{blackrightpointingtriangle}{25B6} \pdfglyphtounicode{blacksmallsquare}{25AA} \pdfglyphtounicode{blacksmilingface}{263B} \pdfglyphtounicode{blacksquare}{25A0} \pdfglyphtounicode{blackstar}{2605} \pdfglyphtounicode{blackupperlefttriangle}{25E4} \pdfglyphtounicode{blackupperrighttriangle}{25E5} \pdfglyphtounicode{blackuppointingsmalltriangle}{25B4} \pdfglyphtounicode{blackuppointingtriangle}{25B2} \pdfglyphtounicode{blank}{2423} \pdfglyphtounicode{blinebelow}{1E07} \pdfglyphtounicode{block}{2588} \pdfglyphtounicode{bmonospace}{FF42} \pdfglyphtounicode{bobaimaithai}{0E1A} \pdfglyphtounicode{bohiragana}{307C} \pdfglyphtounicode{bokatakana}{30DC} \pdfglyphtounicode{bparen}{249D} \pdfglyphtounicode{bqsquare}{33C3} \pdfglyphtounicode{braceex}{F8F4} \pdfglyphtounicode{braceleft}{007B} \pdfglyphtounicode{braceleftbt}{F8F3} \pdfglyphtounicode{braceleftmid}{F8F2} \pdfglyphtounicode{braceleftmonospace}{FF5B} \pdfglyphtounicode{braceleftsmall}{FE5B} \pdfglyphtounicode{bracelefttp}{F8F1} \pdfglyphtounicode{braceleftvertical}{FE37} \pdfglyphtounicode{braceright}{007D} \pdfglyphtounicode{bracerightbt}{F8FE} \pdfglyphtounicode{bracerightmid}{F8FD} \pdfglyphtounicode{bracerightmonospace}{FF5D} \pdfglyphtounicode{bracerightsmall}{FE5C} \pdfglyphtounicode{bracerighttp}{F8FC} \pdfglyphtounicode{bracerightvertical}{FE38} \pdfglyphtounicode{bracketleft}{005B} \pdfglyphtounicode{bracketleftbt}{F8F0} \pdfglyphtounicode{bracketleftex}{F8EF} \pdfglyphtounicode{bracketleftmonospace}{FF3B} \pdfglyphtounicode{bracketlefttp}{F8EE} \pdfglyphtounicode{bracketright}{005D} \pdfglyphtounicode{bracketrightbt}{F8FB} \pdfglyphtounicode{bracketrightex}{F8FA} \pdfglyphtounicode{bracketrightmonospace}{FF3D} \pdfglyphtounicode{bracketrighttp}{F8F9} \pdfglyphtounicode{breve}{02D8} \pdfglyphtounicode{brevebelowcmb}{032E} \pdfglyphtounicode{brevecmb}{0306} \pdfglyphtounicode{breveinvertedbelowcmb}{032F} \pdfglyphtounicode{breveinvertedcmb}{0311} \pdfglyphtounicode{breveinverteddoublecmb}{0361} \pdfglyphtounicode{bridgebelowcmb}{032A} \pdfglyphtounicode{bridgeinvertedbelowcmb}{033A} \pdfglyphtounicode{brokenbar}{00A6} \pdfglyphtounicode{bstroke}{0180} \pdfglyphtounicode{bsuperior}{F6EA} \pdfglyphtounicode{btopbar}{0183} \pdfglyphtounicode{buhiragana}{3076} \pdfglyphtounicode{bukatakana}{30D6} \pdfglyphtounicode{bullet}{2022} \pdfglyphtounicode{bulletinverse}{25D8} \pdfglyphtounicode{bulletoperator}{2219} \pdfglyphtounicode{bullseye}{25CE} \pdfglyphtounicode{c}{0063} \pdfglyphtounicode{caarmenian}{056E} \pdfglyphtounicode{cabengali}{099A} \pdfglyphtounicode{cacute}{0107} \pdfglyphtounicode{cadeva}{091A} \pdfglyphtounicode{cagujarati}{0A9A} \pdfglyphtounicode{cagurmukhi}{0A1A} \pdfglyphtounicode{calsquare}{3388} \pdfglyphtounicode{candrabindubengali}{0981} \pdfglyphtounicode{candrabinducmb}{0310} \pdfglyphtounicode{candrabindudeva}{0901} \pdfglyphtounicode{candrabindugujarati}{0A81} \pdfglyphtounicode{capslock}{21EA} \pdfglyphtounicode{careof}{2105} \pdfglyphtounicode{caron}{02C7} \pdfglyphtounicode{caronbelowcmb}{032C} \pdfglyphtounicode{caroncmb}{030C} \pdfglyphtounicode{carriagereturn}{21B5} \pdfglyphtounicode{cbopomofo}{3118} \pdfglyphtounicode{ccaron}{010D} \pdfglyphtounicode{ccedilla}{00E7} \pdfglyphtounicode{ccedillaacute}{1E09} \pdfglyphtounicode{ccircle}{24D2} \pdfglyphtounicode{ccircumflex}{0109} \pdfglyphtounicode{ccurl}{0255} \pdfglyphtounicode{cdot}{010B} \pdfglyphtounicode{cdotaccent}{010B} \pdfglyphtounicode{cdsquare}{33C5} \pdfglyphtounicode{cedilla}{00B8} \pdfglyphtounicode{cedillacmb}{0327} \pdfglyphtounicode{cent}{00A2} \pdfglyphtounicode{centigrade}{2103} \pdfglyphtounicode{centinferior}{F6DF} \pdfglyphtounicode{centmonospace}{FFE0} \pdfglyphtounicode{centoldstyle}{F7A2} \pdfglyphtounicode{centsuperior}{F6E0} \pdfglyphtounicode{chaarmenian}{0579} \pdfglyphtounicode{chabengali}{099B} \pdfglyphtounicode{chadeva}{091B} \pdfglyphtounicode{chagujarati}{0A9B} \pdfglyphtounicode{chagurmukhi}{0A1B} \pdfglyphtounicode{chbopomofo}{3114} \pdfglyphtounicode{cheabkhasiancyrillic}{04BD} \pdfglyphtounicode{checkmark}{2713} \pdfglyphtounicode{checyrillic}{0447} \pdfglyphtounicode{chedescenderabkhasiancyrillic}{04BF} \pdfglyphtounicode{chedescendercyrillic}{04B7} \pdfglyphtounicode{chedieresiscyrillic}{04F5} \pdfglyphtounicode{cheharmenian}{0573} \pdfglyphtounicode{chekhakassiancyrillic}{04CC} \pdfglyphtounicode{cheverticalstrokecyrillic}{04B9} \pdfglyphtounicode{chi}{03C7} \pdfglyphtounicode{chieuchacirclekorean}{3277} \pdfglyphtounicode{chieuchaparenkorean}{3217} \pdfglyphtounicode{chieuchcirclekorean}{3269} \pdfglyphtounicode{chieuchkorean}{314A} \pdfglyphtounicode{chieuchparenkorean}{3209} \pdfglyphtounicode{chochangthai}{0E0A} \pdfglyphtounicode{chochanthai}{0E08} \pdfglyphtounicode{chochingthai}{0E09} \pdfglyphtounicode{chochoethai}{0E0C} \pdfglyphtounicode{chook}{0188} \pdfglyphtounicode{cieucacirclekorean}{3276} \pdfglyphtounicode{cieucaparenkorean}{3216} \pdfglyphtounicode{cieuccirclekorean}{3268} \pdfglyphtounicode{cieuckorean}{3148} \pdfglyphtounicode{cieucparenkorean}{3208} \pdfglyphtounicode{cieucuparenkorean}{321C} \pdfglyphtounicode{circle}{25CB} \pdfglyphtounicode{circlemultiply}{2297} \pdfglyphtounicode{circleot}{2299} \pdfglyphtounicode{circleplus}{2295} \pdfglyphtounicode{circlepostalmark}{3036} \pdfglyphtounicode{circlewithlefthalfblack}{25D0} \pdfglyphtounicode{circlewithrighthalfblack}{25D1} \pdfglyphtounicode{circumflex}{02C6} \pdfglyphtounicode{circumflexbelowcmb}{032D} \pdfglyphtounicode{circumflexcmb}{0302} \pdfglyphtounicode{clear}{2327} \pdfglyphtounicode{clickalveolar}{01C2} \pdfglyphtounicode{clickdental}{01C0} \pdfglyphtounicode{clicklateral}{01C1} \pdfglyphtounicode{clickretroflex}{01C3} \pdfglyphtounicode{club}{2663} \pdfglyphtounicode{clubsuitblack}{2663} \pdfglyphtounicode{clubsuitwhite}{2667} \pdfglyphtounicode{cmcubedsquare}{33A4} \pdfglyphtounicode{cmonospace}{FF43} \pdfglyphtounicode{cmsquaredsquare}{33A0} \pdfglyphtounicode{coarmenian}{0581} \pdfglyphtounicode{colon}{003A} \pdfglyphtounicode{colonmonetary}{20A1} \pdfglyphtounicode{colonmonospace}{FF1A} \pdfglyphtounicode{colonsign}{20A1} \pdfglyphtounicode{colonsmall}{FE55} \pdfglyphtounicode{colontriangularhalfmod}{02D1} \pdfglyphtounicode{colontriangularmod}{02D0} \pdfglyphtounicode{comma}{002C} \pdfglyphtounicode{commaabovecmb}{0313} \pdfglyphtounicode{commaaboverightcmb}{0315} \pdfglyphtounicode{commaaccent}{F6C3} \pdfglyphtounicode{commaarabic}{060C} \pdfglyphtounicode{commaarmenian}{055D} \pdfglyphtounicode{commainferior}{F6E1} \pdfglyphtounicode{commamonospace}{FF0C} \pdfglyphtounicode{commareversedabovecmb}{0314} \pdfglyphtounicode{commareversedmod}{02BD} \pdfglyphtounicode{commasmall}{FE50} \pdfglyphtounicode{commasuperior}{F6E2} \pdfglyphtounicode{commaturnedabovecmb}{0312} \pdfglyphtounicode{commaturnedmod}{02BB} \pdfglyphtounicode{compass}{263C} \pdfglyphtounicode{congruent}{2245} \pdfglyphtounicode{contourintegral}{222E} \pdfglyphtounicode{control}{2303} \pdfglyphtounicode{controlACK}{0006} \pdfglyphtounicode{controlBEL}{0007} \pdfglyphtounicode{controlBS}{0008} \pdfglyphtounicode{controlCAN}{0018} \pdfglyphtounicode{controlCR}{000D} \pdfglyphtounicode{controlDC1}{0011} \pdfglyphtounicode{controlDC2}{0012} \pdfglyphtounicode{controlDC3}{0013} \pdfglyphtounicode{controlDC4}{0014} \pdfglyphtounicode{controlDEL}{007F} \pdfglyphtounicode{controlDLE}{0010} \pdfglyphtounicode{controlEM}{0019} \pdfglyphtounicode{controlENQ}{0005} \pdfglyphtounicode{controlEOT}{0004} \pdfglyphtounicode{controlESC}{001B} \pdfglyphtounicode{controlETB}{0017} \pdfglyphtounicode{controlETX}{0003} \pdfglyphtounicode{controlFF}{000C} \pdfglyphtounicode{controlFS}{001C} \pdfglyphtounicode{controlGS}{001D} \pdfglyphtounicode{controlHT}{0009} \pdfglyphtounicode{controlLF}{000A} \pdfglyphtounicode{controlNAK}{0015} \pdfglyphtounicode{controlRS}{001E} \pdfglyphtounicode{controlSI}{000F} \pdfglyphtounicode{controlSO}{000E} \pdfglyphtounicode{controlSOT}{0002} \pdfglyphtounicode{controlSTX}{0001} \pdfglyphtounicode{controlSUB}{001A} \pdfglyphtounicode{controlSYN}{0016} \pdfglyphtounicode{controlUS}{001F} \pdfglyphtounicode{controlVT}{000B} \pdfglyphtounicode{copyright}{00A9} \pdfglyphtounicode{copyrightsans}{F8E9} \pdfglyphtounicode{copyrightserif}{F6D9} \pdfglyphtounicode{cornerbracketleft}{300C} \pdfglyphtounicode{cornerbracketlefthalfwidth}{FF62} \pdfglyphtounicode{cornerbracketleftvertical}{FE41} \pdfglyphtounicode{cornerbracketright}{300D} \pdfglyphtounicode{cornerbracketrighthalfwidth}{FF63} \pdfglyphtounicode{cornerbracketrightvertical}{FE42} \pdfglyphtounicode{corporationsquare}{337F} \pdfglyphtounicode{cosquare}{33C7} \pdfglyphtounicode{coverkgsquare}{33C6} \pdfglyphtounicode{cparen}{249E} \pdfglyphtounicode{cruzeiro}{20A2} \pdfglyphtounicode{cstretched}{0297} \pdfglyphtounicode{curlyand}{22CF} \pdfglyphtounicode{curlyor}{22CE} \pdfglyphtounicode{currency}{00A4} \pdfglyphtounicode{cyrBreve}{F6D1} \pdfglyphtounicode{cyrFlex}{F6D2} \pdfglyphtounicode{cyrbreve}{F6D4} \pdfglyphtounicode{cyrflex}{F6D5} \pdfglyphtounicode{d}{0064} \pdfglyphtounicode{daarmenian}{0564} \pdfglyphtounicode{dabengali}{09A6} \pdfglyphtounicode{dadarabic}{0636} \pdfglyphtounicode{dadeva}{0926} \pdfglyphtounicode{dadfinalarabic}{FEBE} \pdfglyphtounicode{dadinitialarabic}{FEBF} \pdfglyphtounicode{dadmedialarabic}{FEC0} \pdfglyphtounicode{dagesh}{05BC} \pdfglyphtounicode{dageshhebrew}{05BC} \pdfglyphtounicode{dagger}{2020} \pdfglyphtounicode{daggerdbl}{2021} \pdfglyphtounicode{dagujarati}{0AA6} \pdfglyphtounicode{dagurmukhi}{0A26} \pdfglyphtounicode{dahiragana}{3060} \pdfglyphtounicode{dakatakana}{30C0} \pdfglyphtounicode{dalarabic}{062F} \pdfglyphtounicode{dalet}{05D3} \pdfglyphtounicode{daletdagesh}{FB33} \pdfglyphtounicode{daletdageshhebrew}{FB33} % dalethatafpatah;05D3 05B2 % dalethatafpatahhebrew;05D3 05B2 % dalethatafsegol;05D3 05B1 % dalethatafsegolhebrew;05D3 05B1 \pdfglyphtounicode{dalethebrew}{05D3} % dalethiriq;05D3 05B4 % dalethiriqhebrew;05D3 05B4 % daletholam;05D3 05B9 % daletholamhebrew;05D3 05B9 % daletpatah;05D3 05B7 % daletpatahhebrew;05D3 05B7 % daletqamats;05D3 05B8 % daletqamatshebrew;05D3 05B8 % daletqubuts;05D3 05BB % daletqubutshebrew;05D3 05BB % daletsegol;05D3 05B6 % daletsegolhebrew;05D3 05B6 % daletsheva;05D3 05B0 % daletshevahebrew;05D3 05B0 % dalettsere;05D3 05B5 % dalettserehebrew;05D3 05B5 \pdfglyphtounicode{dalfinalarabic}{FEAA} \pdfglyphtounicode{dammaarabic}{064F} \pdfglyphtounicode{dammalowarabic}{064F} \pdfglyphtounicode{dammatanaltonearabic}{064C} \pdfglyphtounicode{dammatanarabic}{064C} \pdfglyphtounicode{danda}{0964} \pdfglyphtounicode{dargahebrew}{05A7} \pdfglyphtounicode{dargalefthebrew}{05A7} \pdfglyphtounicode{dasiapneumatacyrilliccmb}{0485} \pdfglyphtounicode{dblGrave}{F6D3} \pdfglyphtounicode{dblanglebracketleft}{300A} \pdfglyphtounicode{dblanglebracketleftvertical}{FE3D} \pdfglyphtounicode{dblanglebracketright}{300B} \pdfglyphtounicode{dblanglebracketrightvertical}{FE3E} \pdfglyphtounicode{dblarchinvertedbelowcmb}{032B} \pdfglyphtounicode{dblarrowleft}{21D4} \pdfglyphtounicode{dblarrowright}{21D2} \pdfglyphtounicode{dbldanda}{0965} \pdfglyphtounicode{dblgrave}{F6D6} \pdfglyphtounicode{dblgravecmb}{030F} \pdfglyphtounicode{dblintegral}{222C} \pdfglyphtounicode{dbllowline}{2017} \pdfglyphtounicode{dbllowlinecmb}{0333} \pdfglyphtounicode{dbloverlinecmb}{033F} \pdfglyphtounicode{dblprimemod}{02BA} \pdfglyphtounicode{dblverticalbar}{2016} \pdfglyphtounicode{dblverticallineabovecmb}{030E} \pdfglyphtounicode{dbopomofo}{3109} \pdfglyphtounicode{dbsquare}{33C8} \pdfglyphtounicode{dcaron}{010F} \pdfglyphtounicode{dcedilla}{1E11} \pdfglyphtounicode{dcircle}{24D3} \pdfglyphtounicode{dcircumflexbelow}{1E13} \pdfglyphtounicode{dcroat}{0111} \pdfglyphtounicode{ddabengali}{09A1} \pdfglyphtounicode{ddadeva}{0921} \pdfglyphtounicode{ddagujarati}{0AA1} \pdfglyphtounicode{ddagurmukhi}{0A21} \pdfglyphtounicode{ddalarabic}{0688} \pdfglyphtounicode{ddalfinalarabic}{FB89} \pdfglyphtounicode{dddhadeva}{095C} \pdfglyphtounicode{ddhabengali}{09A2} \pdfglyphtounicode{ddhadeva}{0922} \pdfglyphtounicode{ddhagujarati}{0AA2} \pdfglyphtounicode{ddhagurmukhi}{0A22} \pdfglyphtounicode{ddotaccent}{1E0B} \pdfglyphtounicode{ddotbelow}{1E0D} \pdfglyphtounicode{decimalseparatorarabic}{066B} \pdfglyphtounicode{decimalseparatorpersian}{066B} \pdfglyphtounicode{decyrillic}{0434} \pdfglyphtounicode{degree}{00B0} \pdfglyphtounicode{dehihebrew}{05AD} \pdfglyphtounicode{dehiragana}{3067} \pdfglyphtounicode{deicoptic}{03EF} \pdfglyphtounicode{dekatakana}{30C7} \pdfglyphtounicode{deleteleft}{232B} \pdfglyphtounicode{deleteright}{2326} \pdfglyphtounicode{delta}{03B4} \pdfglyphtounicode{deltaturned}{018D} \pdfglyphtounicode{denominatorminusonenumeratorbengali}{09F8} \pdfglyphtounicode{dezh}{02A4} \pdfglyphtounicode{dhabengali}{09A7} \pdfglyphtounicode{dhadeva}{0927} \pdfglyphtounicode{dhagujarati}{0AA7} \pdfglyphtounicode{dhagurmukhi}{0A27} \pdfglyphtounicode{dhook}{0257} \pdfglyphtounicode{dialytikatonos}{0385} \pdfglyphtounicode{dialytikatonoscmb}{0344} \pdfglyphtounicode{diamond}{2666} \pdfglyphtounicode{diamondsuitwhite}{2662} \pdfglyphtounicode{dieresis}{00A8} \pdfglyphtounicode{dieresisacute}{F6D7} \pdfglyphtounicode{dieresisbelowcmb}{0324} \pdfglyphtounicode{dieresiscmb}{0308} \pdfglyphtounicode{dieresisgrave}{F6D8} \pdfglyphtounicode{dieresistonos}{0385} \pdfglyphtounicode{dihiragana}{3062} \pdfglyphtounicode{dikatakana}{30C2} \pdfglyphtounicode{dittomark}{3003} \pdfglyphtounicode{divide}{00F7} \pdfglyphtounicode{divides}{2223} \pdfglyphtounicode{divisionslash}{2215} \pdfglyphtounicode{djecyrillic}{0452} \pdfglyphtounicode{dkshade}{2593} \pdfglyphtounicode{dlinebelow}{1E0F} \pdfglyphtounicode{dlsquare}{3397} \pdfglyphtounicode{dmacron}{0111} \pdfglyphtounicode{dmonospace}{FF44} \pdfglyphtounicode{dnblock}{2584} \pdfglyphtounicode{dochadathai}{0E0E} \pdfglyphtounicode{dodekthai}{0E14} \pdfglyphtounicode{dohiragana}{3069} \pdfglyphtounicode{dokatakana}{30C9} \pdfglyphtounicode{dollar}{0024} \pdfglyphtounicode{dollarinferior}{F6E3} \pdfglyphtounicode{dollarmonospace}{FF04} \pdfglyphtounicode{dollaroldstyle}{F724} \pdfglyphtounicode{dollarsmall}{FE69} \pdfglyphtounicode{dollarsuperior}{F6E4} \pdfglyphtounicode{dong}{20AB} \pdfglyphtounicode{dorusquare}{3326} \pdfglyphtounicode{dotaccent}{02D9} \pdfglyphtounicode{dotaccentcmb}{0307} \pdfglyphtounicode{dotbelowcmb}{0323} \pdfglyphtounicode{dotbelowcomb}{0323} \pdfglyphtounicode{dotkatakana}{30FB} \pdfglyphtounicode{dotlessi}{0131} \pdfglyphtounicode{dotlessj}{F6BE} \pdfglyphtounicode{dotlessjstrokehook}{0284} \pdfglyphtounicode{dotmath}{22C5} \pdfglyphtounicode{dottedcircle}{25CC} \pdfglyphtounicode{doubleyodpatah}{FB1F} \pdfglyphtounicode{doubleyodpatahhebrew}{FB1F} \pdfglyphtounicode{downtackbelowcmb}{031E} \pdfglyphtounicode{downtackmod}{02D5} \pdfglyphtounicode{dparen}{249F} \pdfglyphtounicode{dsuperior}{F6EB} \pdfglyphtounicode{dtail}{0256} \pdfglyphtounicode{dtopbar}{018C} \pdfglyphtounicode{duhiragana}{3065} \pdfglyphtounicode{dukatakana}{30C5} \pdfglyphtounicode{dz}{01F3} \pdfglyphtounicode{dzaltone}{02A3} \pdfglyphtounicode{dzcaron}{01C6} \pdfglyphtounicode{dzcurl}{02A5} \pdfglyphtounicode{dzeabkhasiancyrillic}{04E1} \pdfglyphtounicode{dzecyrillic}{0455} \pdfglyphtounicode{dzhecyrillic}{045F} \pdfglyphtounicode{e}{0065} \pdfglyphtounicode{eacute}{00E9} \pdfglyphtounicode{earth}{2641} \pdfglyphtounicode{ebengali}{098F} \pdfglyphtounicode{ebopomofo}{311C} \pdfglyphtounicode{ebreve}{0115} \pdfglyphtounicode{ecandradeva}{090D} \pdfglyphtounicode{ecandragujarati}{0A8D} \pdfglyphtounicode{ecandravowelsigndeva}{0945} \pdfglyphtounicode{ecandravowelsigngujarati}{0AC5} \pdfglyphtounicode{ecaron}{011B} \pdfglyphtounicode{ecedillabreve}{1E1D} \pdfglyphtounicode{echarmenian}{0565} \pdfglyphtounicode{echyiwnarmenian}{0587} \pdfglyphtounicode{ecircle}{24D4} \pdfglyphtounicode{ecircumflex}{00EA} \pdfglyphtounicode{ecircumflexacute}{1EBF} \pdfglyphtounicode{ecircumflexbelow}{1E19} \pdfglyphtounicode{ecircumflexdotbelow}{1EC7} \pdfglyphtounicode{ecircumflexgrave}{1EC1} \pdfglyphtounicode{ecircumflexhookabove}{1EC3} \pdfglyphtounicode{ecircumflextilde}{1EC5} \pdfglyphtounicode{ecyrillic}{0454} \pdfglyphtounicode{edblgrave}{0205} \pdfglyphtounicode{edeva}{090F} \pdfglyphtounicode{edieresis}{00EB} \pdfglyphtounicode{edot}{0117} \pdfglyphtounicode{edotaccent}{0117} \pdfglyphtounicode{edotbelow}{1EB9} \pdfglyphtounicode{eegurmukhi}{0A0F} \pdfglyphtounicode{eematragurmukhi}{0A47} \pdfglyphtounicode{efcyrillic}{0444} \pdfglyphtounicode{egrave}{00E8} \pdfglyphtounicode{egujarati}{0A8F} \pdfglyphtounicode{eharmenian}{0567} \pdfglyphtounicode{ehbopomofo}{311D} \pdfglyphtounicode{ehiragana}{3048} \pdfglyphtounicode{ehookabove}{1EBB} \pdfglyphtounicode{eibopomofo}{311F} \pdfglyphtounicode{eight}{0038} \pdfglyphtounicode{eightarabic}{0668} \pdfglyphtounicode{eightbengali}{09EE} \pdfglyphtounicode{eightcircle}{2467} \pdfglyphtounicode{eightcircleinversesansserif}{2791} \pdfglyphtounicode{eightdeva}{096E} \pdfglyphtounicode{eighteencircle}{2471} \pdfglyphtounicode{eighteenparen}{2485} \pdfglyphtounicode{eighteenperiod}{2499} \pdfglyphtounicode{eightgujarati}{0AEE} \pdfglyphtounicode{eightgurmukhi}{0A6E} \pdfglyphtounicode{eighthackarabic}{0668} \pdfglyphtounicode{eighthangzhou}{3028} \pdfglyphtounicode{eighthnotebeamed}{266B} \pdfglyphtounicode{eightideographicparen}{3227} \pdfglyphtounicode{eightinferior}{2088} \pdfglyphtounicode{eightmonospace}{FF18} \pdfglyphtounicode{eightoldstyle}{F738} \pdfglyphtounicode{eightparen}{247B} \pdfglyphtounicode{eightperiod}{248F} \pdfglyphtounicode{eightpersian}{06F8} \pdfglyphtounicode{eightroman}{2177} \pdfglyphtounicode{eightsuperior}{2078} \pdfglyphtounicode{eightthai}{0E58} \pdfglyphtounicode{einvertedbreve}{0207} \pdfglyphtounicode{eiotifiedcyrillic}{0465} \pdfglyphtounicode{ekatakana}{30A8} \pdfglyphtounicode{ekatakanahalfwidth}{FF74} \pdfglyphtounicode{ekonkargurmukhi}{0A74} \pdfglyphtounicode{ekorean}{3154} \pdfglyphtounicode{elcyrillic}{043B} \pdfglyphtounicode{element}{2208} \pdfglyphtounicode{elevencircle}{246A} \pdfglyphtounicode{elevenparen}{247E} \pdfglyphtounicode{elevenperiod}{2492} \pdfglyphtounicode{elevenroman}{217A} \pdfglyphtounicode{ellipsis}{2026} \pdfglyphtounicode{ellipsisvertical}{22EE} \pdfglyphtounicode{emacron}{0113} \pdfglyphtounicode{emacronacute}{1E17} \pdfglyphtounicode{emacrongrave}{1E15} \pdfglyphtounicode{emcyrillic}{043C} \pdfglyphtounicode{emdash}{2014} \pdfglyphtounicode{emdashvertical}{FE31} \pdfglyphtounicode{emonospace}{FF45} \pdfglyphtounicode{emphasismarkarmenian}{055B} \pdfglyphtounicode{emptyset}{2205} \pdfglyphtounicode{enbopomofo}{3123} \pdfglyphtounicode{encyrillic}{043D} \pdfglyphtounicode{endash}{2013} \pdfglyphtounicode{endashvertical}{FE32} \pdfglyphtounicode{endescendercyrillic}{04A3} \pdfglyphtounicode{eng}{014B} \pdfglyphtounicode{engbopomofo}{3125} \pdfglyphtounicode{enghecyrillic}{04A5} \pdfglyphtounicode{enhookcyrillic}{04C8} \pdfglyphtounicode{enspace}{2002} \pdfglyphtounicode{eogonek}{0119} \pdfglyphtounicode{eokorean}{3153} \pdfglyphtounicode{eopen}{025B} \pdfglyphtounicode{eopenclosed}{029A} \pdfglyphtounicode{eopenreversed}{025C} \pdfglyphtounicode{eopenreversedclosed}{025E} \pdfglyphtounicode{eopenreversedhook}{025D} \pdfglyphtounicode{eparen}{24A0} \pdfglyphtounicode{epsilon}{03B5} \pdfglyphtounicode{epsilontonos}{03AD} \pdfglyphtounicode{equal}{003D} \pdfglyphtounicode{equalmonospace}{FF1D} \pdfglyphtounicode{equalsmall}{FE66} \pdfglyphtounicode{equalsuperior}{207C} \pdfglyphtounicode{equivalence}{2261} \pdfglyphtounicode{erbopomofo}{3126} \pdfglyphtounicode{ercyrillic}{0440} \pdfglyphtounicode{ereversed}{0258} \pdfglyphtounicode{ereversedcyrillic}{044D} \pdfglyphtounicode{escyrillic}{0441} \pdfglyphtounicode{esdescendercyrillic}{04AB} \pdfglyphtounicode{esh}{0283} \pdfglyphtounicode{eshcurl}{0286} \pdfglyphtounicode{eshortdeva}{090E} \pdfglyphtounicode{eshortvowelsigndeva}{0946} \pdfglyphtounicode{eshreversedloop}{01AA} \pdfglyphtounicode{eshsquatreversed}{0285} \pdfglyphtounicode{esmallhiragana}{3047} \pdfglyphtounicode{esmallkatakana}{30A7} \pdfglyphtounicode{esmallkatakanahalfwidth}{FF6A} \pdfglyphtounicode{estimated}{212E} \pdfglyphtounicode{esuperior}{F6EC} \pdfglyphtounicode{eta}{03B7} \pdfglyphtounicode{etarmenian}{0568} \pdfglyphtounicode{etatonos}{03AE} \pdfglyphtounicode{eth}{00F0} \pdfglyphtounicode{etilde}{1EBD} \pdfglyphtounicode{etildebelow}{1E1B} \pdfglyphtounicode{etnahtafoukhhebrew}{0591} \pdfglyphtounicode{etnahtafoukhlefthebrew}{0591} \pdfglyphtounicode{etnahtahebrew}{0591} \pdfglyphtounicode{etnahtalefthebrew}{0591} \pdfglyphtounicode{eturned}{01DD} \pdfglyphtounicode{eukorean}{3161} \pdfglyphtounicode{euro}{20AC} \pdfglyphtounicode{evowelsignbengali}{09C7} \pdfglyphtounicode{evowelsigndeva}{0947} \pdfglyphtounicode{evowelsigngujarati}{0AC7} \pdfglyphtounicode{exclam}{0021} \pdfglyphtounicode{exclamarmenian}{055C} \pdfglyphtounicode{exclamdbl}{203C} \pdfglyphtounicode{exclamdown}{00A1} \pdfglyphtounicode{exclamdownsmall}{F7A1} \pdfglyphtounicode{exclammonospace}{FF01} \pdfglyphtounicode{exclamsmall}{F721} \pdfglyphtounicode{existential}{2203} \pdfglyphtounicode{ezh}{0292} \pdfglyphtounicode{ezhcaron}{01EF} \pdfglyphtounicode{ezhcurl}{0293} \pdfglyphtounicode{ezhreversed}{01B9} \pdfglyphtounicode{ezhtail}{01BA} \pdfglyphtounicode{f}{0066} \pdfglyphtounicode{fadeva}{095E} \pdfglyphtounicode{fagurmukhi}{0A5E} \pdfglyphtounicode{fahrenheit}{2109} \pdfglyphtounicode{fathaarabic}{064E} \pdfglyphtounicode{fathalowarabic}{064E} \pdfglyphtounicode{fathatanarabic}{064B} \pdfglyphtounicode{fbopomofo}{3108} \pdfglyphtounicode{fcircle}{24D5} \pdfglyphtounicode{fdotaccent}{1E1F} \pdfglyphtounicode{feharabic}{0641} \pdfglyphtounicode{feharmenian}{0586} \pdfglyphtounicode{fehfinalarabic}{FED2} \pdfglyphtounicode{fehinitialarabic}{FED3} \pdfglyphtounicode{fehmedialarabic}{FED4} \pdfglyphtounicode{feicoptic}{03E5} \pdfglyphtounicode{female}{2640} \pdfglyphtounicode{ff}{FB00} \pdfglyphtounicode{ffi}{FB03} \pdfglyphtounicode{ffl}{FB04} \pdfglyphtounicode{fi}{FB01} \pdfglyphtounicode{fifteencircle}{246E} \pdfglyphtounicode{fifteenparen}{2482} \pdfglyphtounicode{fifteenperiod}{2496} \pdfglyphtounicode{figuredash}{2012} \pdfglyphtounicode{filledbox}{25A0} \pdfglyphtounicode{filledrect}{25AC} \pdfglyphtounicode{finalkaf}{05DA} \pdfglyphtounicode{finalkafdagesh}{FB3A} \pdfglyphtounicode{finalkafdageshhebrew}{FB3A} \pdfglyphtounicode{finalkafhebrew}{05DA} % finalkafqamats;05DA 05B8 % finalkafqamatshebrew;05DA 05B8 % finalkafsheva;05DA 05B0 % finalkafshevahebrew;05DA 05B0 \pdfglyphtounicode{finalmem}{05DD} \pdfglyphtounicode{finalmemhebrew}{05DD} \pdfglyphtounicode{finalnun}{05DF} \pdfglyphtounicode{finalnunhebrew}{05DF} \pdfglyphtounicode{finalpe}{05E3} \pdfglyphtounicode{finalpehebrew}{05E3} \pdfglyphtounicode{finaltsadi}{05E5} \pdfglyphtounicode{finaltsadihebrew}{05E5} \pdfglyphtounicode{firsttonechinese}{02C9} \pdfglyphtounicode{fisheye}{25C9} \pdfglyphtounicode{fitacyrillic}{0473} \pdfglyphtounicode{five}{0035} \pdfglyphtounicode{fivearabic}{0665} \pdfglyphtounicode{fivebengali}{09EB} \pdfglyphtounicode{fivecircle}{2464} \pdfglyphtounicode{fivecircleinversesansserif}{278E} \pdfglyphtounicode{fivedeva}{096B} \pdfglyphtounicode{fiveeighths}{215D} \pdfglyphtounicode{fivegujarati}{0AEB} \pdfglyphtounicode{fivegurmukhi}{0A6B} \pdfglyphtounicode{fivehackarabic}{0665} \pdfglyphtounicode{fivehangzhou}{3025} \pdfglyphtounicode{fiveideographicparen}{3224} \pdfglyphtounicode{fiveinferior}{2085} \pdfglyphtounicode{fivemonospace}{FF15} \pdfglyphtounicode{fiveoldstyle}{F735} \pdfglyphtounicode{fiveparen}{2478} \pdfglyphtounicode{fiveperiod}{248C} \pdfglyphtounicode{fivepersian}{06F5} \pdfglyphtounicode{fiveroman}{2174} \pdfglyphtounicode{fivesuperior}{2075} \pdfglyphtounicode{fivethai}{0E55} \pdfglyphtounicode{fl}{FB02} \pdfglyphtounicode{florin}{0192} \pdfglyphtounicode{fmonospace}{FF46} \pdfglyphtounicode{fmsquare}{3399} \pdfglyphtounicode{fofanthai}{0E1F} \pdfglyphtounicode{fofathai}{0E1D} \pdfglyphtounicode{fongmanthai}{0E4F} \pdfglyphtounicode{forall}{2200} \pdfglyphtounicode{four}{0034} \pdfglyphtounicode{fourarabic}{0664} \pdfglyphtounicode{fourbengali}{09EA} \pdfglyphtounicode{fourcircle}{2463} \pdfglyphtounicode{fourcircleinversesansserif}{278D} \pdfglyphtounicode{fourdeva}{096A} \pdfglyphtounicode{fourgujarati}{0AEA} \pdfglyphtounicode{fourgurmukhi}{0A6A} \pdfglyphtounicode{fourhackarabic}{0664} \pdfglyphtounicode{fourhangzhou}{3024} \pdfglyphtounicode{fourideographicparen}{3223} \pdfglyphtounicode{fourinferior}{2084} \pdfglyphtounicode{fourmonospace}{FF14} \pdfglyphtounicode{fournumeratorbengali}{09F7} \pdfglyphtounicode{fouroldstyle}{F734} \pdfglyphtounicode{fourparen}{2477} \pdfglyphtounicode{fourperiod}{248B} \pdfglyphtounicode{fourpersian}{06F4} \pdfglyphtounicode{fourroman}{2173} \pdfglyphtounicode{foursuperior}{2074} \pdfglyphtounicode{fourteencircle}{246D} \pdfglyphtounicode{fourteenparen}{2481} \pdfglyphtounicode{fourteenperiod}{2495} \pdfglyphtounicode{fourthai}{0E54} \pdfglyphtounicode{fourthtonechinese}{02CB} \pdfglyphtounicode{fparen}{24A1} \pdfglyphtounicode{fraction}{2044} \pdfglyphtounicode{franc}{20A3} \pdfglyphtounicode{g}{0067} \pdfglyphtounicode{gabengali}{0997} \pdfglyphtounicode{gacute}{01F5} \pdfglyphtounicode{gadeva}{0917} \pdfglyphtounicode{gafarabic}{06AF} \pdfglyphtounicode{gaffinalarabic}{FB93} \pdfglyphtounicode{gafinitialarabic}{FB94} \pdfglyphtounicode{gafmedialarabic}{FB95} \pdfglyphtounicode{gagujarati}{0A97} \pdfglyphtounicode{gagurmukhi}{0A17} \pdfglyphtounicode{gahiragana}{304C} \pdfglyphtounicode{gakatakana}{30AC} \pdfglyphtounicode{gamma}{03B3} \pdfglyphtounicode{gammalatinsmall}{0263} \pdfglyphtounicode{gammasuperior}{02E0} \pdfglyphtounicode{gangiacoptic}{03EB} \pdfglyphtounicode{gbopomofo}{310D} \pdfglyphtounicode{gbreve}{011F} \pdfglyphtounicode{gcaron}{01E7} \pdfglyphtounicode{gcedilla}{0123} \pdfglyphtounicode{gcircle}{24D6} \pdfglyphtounicode{gcircumflex}{011D} \pdfglyphtounicode{gcommaaccent}{0123} \pdfglyphtounicode{gdot}{0121} \pdfglyphtounicode{gdotaccent}{0121} \pdfglyphtounicode{gecyrillic}{0433} \pdfglyphtounicode{gehiragana}{3052} \pdfglyphtounicode{gekatakana}{30B2} \pdfglyphtounicode{geometricallyequal}{2251} \pdfglyphtounicode{gereshaccenthebrew}{059C} \pdfglyphtounicode{gereshhebrew}{05F3} \pdfglyphtounicode{gereshmuqdamhebrew}{059D} \pdfglyphtounicode{germandbls}{00DF} \pdfglyphtounicode{gershayimaccenthebrew}{059E} \pdfglyphtounicode{gershayimhebrew}{05F4} \pdfglyphtounicode{getamark}{3013} \pdfglyphtounicode{ghabengali}{0998} \pdfglyphtounicode{ghadarmenian}{0572} \pdfglyphtounicode{ghadeva}{0918} \pdfglyphtounicode{ghagujarati}{0A98} \pdfglyphtounicode{ghagurmukhi}{0A18} \pdfglyphtounicode{ghainarabic}{063A} \pdfglyphtounicode{ghainfinalarabic}{FECE} \pdfglyphtounicode{ghaininitialarabic}{FECF} \pdfglyphtounicode{ghainmedialarabic}{FED0} \pdfglyphtounicode{ghemiddlehookcyrillic}{0495} \pdfglyphtounicode{ghestrokecyrillic}{0493} \pdfglyphtounicode{gheupturncyrillic}{0491} \pdfglyphtounicode{ghhadeva}{095A} \pdfglyphtounicode{ghhagurmukhi}{0A5A} \pdfglyphtounicode{ghook}{0260} \pdfglyphtounicode{ghzsquare}{3393} \pdfglyphtounicode{gihiragana}{304E} \pdfglyphtounicode{gikatakana}{30AE} \pdfglyphtounicode{gimarmenian}{0563} \pdfglyphtounicode{gimel}{05D2} \pdfglyphtounicode{gimeldagesh}{FB32} \pdfglyphtounicode{gimeldageshhebrew}{FB32} \pdfglyphtounicode{gimelhebrew}{05D2} \pdfglyphtounicode{gjecyrillic}{0453} \pdfglyphtounicode{glottalinvertedstroke}{01BE} \pdfglyphtounicode{glottalstop}{0294} \pdfglyphtounicode{glottalstopinverted}{0296} \pdfglyphtounicode{glottalstopmod}{02C0} \pdfglyphtounicode{glottalstopreversed}{0295} \pdfglyphtounicode{glottalstopreversedmod}{02C1} \pdfglyphtounicode{glottalstopreversedsuperior}{02E4} \pdfglyphtounicode{glottalstopstroke}{02A1} \pdfglyphtounicode{glottalstopstrokereversed}{02A2} \pdfglyphtounicode{gmacron}{1E21} \pdfglyphtounicode{gmonospace}{FF47} \pdfglyphtounicode{gohiragana}{3054} \pdfglyphtounicode{gokatakana}{30B4} \pdfglyphtounicode{gparen}{24A2} \pdfglyphtounicode{gpasquare}{33AC} \pdfglyphtounicode{gradient}{2207} \pdfglyphtounicode{grave}{0060} \pdfglyphtounicode{gravebelowcmb}{0316} \pdfglyphtounicode{gravecmb}{0300} \pdfglyphtounicode{gravecomb}{0300} \pdfglyphtounicode{gravedeva}{0953} \pdfglyphtounicode{gravelowmod}{02CE} \pdfglyphtounicode{gravemonospace}{FF40} \pdfglyphtounicode{gravetonecmb}{0340} \pdfglyphtounicode{greater}{003E} \pdfglyphtounicode{greaterequal}{2265} \pdfglyphtounicode{greaterequalorless}{22DB} \pdfglyphtounicode{greatermonospace}{FF1E} \pdfglyphtounicode{greaterorequivalent}{2273} \pdfglyphtounicode{greaterorless}{2277} \pdfglyphtounicode{greateroverequal}{2267} \pdfglyphtounicode{greatersmall}{FE65} \pdfglyphtounicode{gscript}{0261} \pdfglyphtounicode{gstroke}{01E5} \pdfglyphtounicode{guhiragana}{3050} \pdfglyphtounicode{guillemotleft}{00AB} \pdfglyphtounicode{guillemotright}{00BB} \pdfglyphtounicode{guilsinglleft}{2039} \pdfglyphtounicode{guilsinglright}{203A} \pdfglyphtounicode{gukatakana}{30B0} \pdfglyphtounicode{guramusquare}{3318} \pdfglyphtounicode{gysquare}{33C9} \pdfglyphtounicode{h}{0068} \pdfglyphtounicode{haabkhasiancyrillic}{04A9} \pdfglyphtounicode{haaltonearabic}{06C1} \pdfglyphtounicode{habengali}{09B9} \pdfglyphtounicode{hadescendercyrillic}{04B3} \pdfglyphtounicode{hadeva}{0939} \pdfglyphtounicode{hagujarati}{0AB9} \pdfglyphtounicode{hagurmukhi}{0A39} \pdfglyphtounicode{haharabic}{062D} \pdfglyphtounicode{hahfinalarabic}{FEA2} \pdfglyphtounicode{hahinitialarabic}{FEA3} \pdfglyphtounicode{hahiragana}{306F} \pdfglyphtounicode{hahmedialarabic}{FEA4} \pdfglyphtounicode{haitusquare}{332A} \pdfglyphtounicode{hakatakana}{30CF} \pdfglyphtounicode{hakatakanahalfwidth}{FF8A} \pdfglyphtounicode{halantgurmukhi}{0A4D} \pdfglyphtounicode{hamzaarabic}{0621} % hamzadammaarabic;0621 064F % hamzadammatanarabic;0621 064C % hamzafathaarabic;0621 064E % hamzafathatanarabic;0621 064B \pdfglyphtounicode{hamzalowarabic}{0621} % hamzalowkasraarabic;0621 0650 % hamzalowkasratanarabic;0621 064D % hamzasukunarabic;0621 0652 \pdfglyphtounicode{hangulfiller}{3164} \pdfglyphtounicode{hardsigncyrillic}{044A} \pdfglyphtounicode{harpoonleftbarbup}{21BC} \pdfglyphtounicode{harpoonrightbarbup}{21C0} \pdfglyphtounicode{hasquare}{33CA} \pdfglyphtounicode{hatafpatah}{05B2} \pdfglyphtounicode{hatafpatah16}{05B2} \pdfglyphtounicode{hatafpatah23}{05B2} \pdfglyphtounicode{hatafpatah2f}{05B2} \pdfglyphtounicode{hatafpatahhebrew}{05B2} \pdfglyphtounicode{hatafpatahnarrowhebrew}{05B2} \pdfglyphtounicode{hatafpatahquarterhebrew}{05B2} \pdfglyphtounicode{hatafpatahwidehebrew}{05B2} \pdfglyphtounicode{hatafqamats}{05B3} \pdfglyphtounicode{hatafqamats1b}{05B3} \pdfglyphtounicode{hatafqamats28}{05B3} \pdfglyphtounicode{hatafqamats34}{05B3} \pdfglyphtounicode{hatafqamatshebrew}{05B3} \pdfglyphtounicode{hatafqamatsnarrowhebrew}{05B3} \pdfglyphtounicode{hatafqamatsquarterhebrew}{05B3} \pdfglyphtounicode{hatafqamatswidehebrew}{05B3} \pdfglyphtounicode{hatafsegol}{05B1} \pdfglyphtounicode{hatafsegol17}{05B1} \pdfglyphtounicode{hatafsegol24}{05B1} \pdfglyphtounicode{hatafsegol30}{05B1} \pdfglyphtounicode{hatafsegolhebrew}{05B1} \pdfglyphtounicode{hatafsegolnarrowhebrew}{05B1} \pdfglyphtounicode{hatafsegolquarterhebrew}{05B1} \pdfglyphtounicode{hatafsegolwidehebrew}{05B1} \pdfglyphtounicode{hbar}{0127} \pdfglyphtounicode{hbopomofo}{310F} \pdfglyphtounicode{hbrevebelow}{1E2B} \pdfglyphtounicode{hcedilla}{1E29} \pdfglyphtounicode{hcircle}{24D7} \pdfglyphtounicode{hcircumflex}{0125} \pdfglyphtounicode{hdieresis}{1E27} \pdfglyphtounicode{hdotaccent}{1E23} \pdfglyphtounicode{hdotbelow}{1E25} \pdfglyphtounicode{he}{05D4} \pdfglyphtounicode{heart}{2665} \pdfglyphtounicode{heartsuitblack}{2665} \pdfglyphtounicode{heartsuitwhite}{2661} \pdfglyphtounicode{hedagesh}{FB34} \pdfglyphtounicode{hedageshhebrew}{FB34} \pdfglyphtounicode{hehaltonearabic}{06C1} \pdfglyphtounicode{heharabic}{0647} \pdfglyphtounicode{hehebrew}{05D4} \pdfglyphtounicode{hehfinalaltonearabic}{FBA7} \pdfglyphtounicode{hehfinalalttwoarabic}{FEEA} \pdfglyphtounicode{hehfinalarabic}{FEEA} \pdfglyphtounicode{hehhamzaabovefinalarabic}{FBA5} \pdfglyphtounicode{hehhamzaaboveisolatedarabic}{FBA4} \pdfglyphtounicode{hehinitialaltonearabic}{FBA8} \pdfglyphtounicode{hehinitialarabic}{FEEB} \pdfglyphtounicode{hehiragana}{3078} \pdfglyphtounicode{hehmedialaltonearabic}{FBA9} \pdfglyphtounicode{hehmedialarabic}{FEEC} \pdfglyphtounicode{heiseierasquare}{337B} \pdfglyphtounicode{hekatakana}{30D8} \pdfglyphtounicode{hekatakanahalfwidth}{FF8D} \pdfglyphtounicode{hekutaarusquare}{3336} \pdfglyphtounicode{henghook}{0267} \pdfglyphtounicode{herutusquare}{3339} \pdfglyphtounicode{het}{05D7} \pdfglyphtounicode{hethebrew}{05D7} \pdfglyphtounicode{hhook}{0266} \pdfglyphtounicode{hhooksuperior}{02B1} \pdfglyphtounicode{hieuhacirclekorean}{327B} \pdfglyphtounicode{hieuhaparenkorean}{321B} \pdfglyphtounicode{hieuhcirclekorean}{326D} \pdfglyphtounicode{hieuhkorean}{314E} \pdfglyphtounicode{hieuhparenkorean}{320D} \pdfglyphtounicode{hihiragana}{3072} \pdfglyphtounicode{hikatakana}{30D2} \pdfglyphtounicode{hikatakanahalfwidth}{FF8B} \pdfglyphtounicode{hiriq}{05B4} \pdfglyphtounicode{hiriq14}{05B4} \pdfglyphtounicode{hiriq21}{05B4} \pdfglyphtounicode{hiriq2d}{05B4} \pdfglyphtounicode{hiriqhebrew}{05B4} \pdfglyphtounicode{hiriqnarrowhebrew}{05B4} \pdfglyphtounicode{hiriqquarterhebrew}{05B4} \pdfglyphtounicode{hiriqwidehebrew}{05B4} \pdfglyphtounicode{hlinebelow}{1E96} \pdfglyphtounicode{hmonospace}{FF48} \pdfglyphtounicode{hoarmenian}{0570} \pdfglyphtounicode{hohipthai}{0E2B} \pdfglyphtounicode{hohiragana}{307B} \pdfglyphtounicode{hokatakana}{30DB} \pdfglyphtounicode{hokatakanahalfwidth}{FF8E} \pdfglyphtounicode{holam}{05B9} \pdfglyphtounicode{holam19}{05B9} \pdfglyphtounicode{holam26}{05B9} \pdfglyphtounicode{holam32}{05B9} \pdfglyphtounicode{holamhebrew}{05B9} \pdfglyphtounicode{holamnarrowhebrew}{05B9} \pdfglyphtounicode{holamquarterhebrew}{05B9} \pdfglyphtounicode{holamwidehebrew}{05B9} \pdfglyphtounicode{honokhukthai}{0E2E} \pdfglyphtounicode{hookabovecomb}{0309} \pdfglyphtounicode{hookcmb}{0309} \pdfglyphtounicode{hookpalatalizedbelowcmb}{0321} \pdfglyphtounicode{hookretroflexbelowcmb}{0322} \pdfglyphtounicode{hoonsquare}{3342} \pdfglyphtounicode{horicoptic}{03E9} \pdfglyphtounicode{horizontalbar}{2015} \pdfglyphtounicode{horncmb}{031B} \pdfglyphtounicode{hotsprings}{2668} \pdfglyphtounicode{house}{2302} \pdfglyphtounicode{hparen}{24A3} \pdfglyphtounicode{hsuperior}{02B0} \pdfglyphtounicode{hturned}{0265} \pdfglyphtounicode{huhiragana}{3075} \pdfglyphtounicode{huiitosquare}{3333} \pdfglyphtounicode{hukatakana}{30D5} \pdfglyphtounicode{hukatakanahalfwidth}{FF8C} \pdfglyphtounicode{hungarumlaut}{02DD} \pdfglyphtounicode{hungarumlautcmb}{030B} \pdfglyphtounicode{hv}{0195} \pdfglyphtounicode{hyphen}{002D} \pdfglyphtounicode{hypheninferior}{F6E5} \pdfglyphtounicode{hyphenmonospace}{FF0D} \pdfglyphtounicode{hyphensmall}{FE63} \pdfglyphtounicode{hyphensuperior}{F6E6} \pdfglyphtounicode{hyphentwo}{2010} \pdfglyphtounicode{i}{0069} \pdfglyphtounicode{iacute}{00ED} \pdfglyphtounicode{iacyrillic}{044F} \pdfglyphtounicode{ibengali}{0987} \pdfglyphtounicode{ibopomofo}{3127} \pdfglyphtounicode{ibreve}{012D} \pdfglyphtounicode{icaron}{01D0} \pdfglyphtounicode{icircle}{24D8} \pdfglyphtounicode{icircumflex}{00EE} \pdfglyphtounicode{icyrillic}{0456} \pdfglyphtounicode{idblgrave}{0209} \pdfglyphtounicode{ideographearthcircle}{328F} \pdfglyphtounicode{ideographfirecircle}{328B} \pdfglyphtounicode{ideographicallianceparen}{323F} \pdfglyphtounicode{ideographiccallparen}{323A} \pdfglyphtounicode{ideographiccentrecircle}{32A5} \pdfglyphtounicode{ideographicclose}{3006} \pdfglyphtounicode{ideographiccomma}{3001} \pdfglyphtounicode{ideographiccommaleft}{FF64} \pdfglyphtounicode{ideographiccongratulationparen}{3237} \pdfglyphtounicode{ideographiccorrectcircle}{32A3} \pdfglyphtounicode{ideographicearthparen}{322F} \pdfglyphtounicode{ideographicenterpriseparen}{323D} \pdfglyphtounicode{ideographicexcellentcircle}{329D} \pdfglyphtounicode{ideographicfestivalparen}{3240} \pdfglyphtounicode{ideographicfinancialcircle}{3296} \pdfglyphtounicode{ideographicfinancialparen}{3236} \pdfglyphtounicode{ideographicfireparen}{322B} \pdfglyphtounicode{ideographichaveparen}{3232} \pdfglyphtounicode{ideographichighcircle}{32A4} \pdfglyphtounicode{ideographiciterationmark}{3005} \pdfglyphtounicode{ideographiclaborcircle}{3298} \pdfglyphtounicode{ideographiclaborparen}{3238} \pdfglyphtounicode{ideographicleftcircle}{32A7} \pdfglyphtounicode{ideographiclowcircle}{32A6} \pdfglyphtounicode{ideographicmedicinecircle}{32A9} \pdfglyphtounicode{ideographicmetalparen}{322E} \pdfglyphtounicode{ideographicmoonparen}{322A} \pdfglyphtounicode{ideographicnameparen}{3234} \pdfglyphtounicode{ideographicperiod}{3002} \pdfglyphtounicode{ideographicprintcircle}{329E} \pdfglyphtounicode{ideographicreachparen}{3243} \pdfglyphtounicode{ideographicrepresentparen}{3239} \pdfglyphtounicode{ideographicresourceparen}{323E} \pdfglyphtounicode{ideographicrightcircle}{32A8} \pdfglyphtounicode{ideographicsecretcircle}{3299} \pdfglyphtounicode{ideographicselfparen}{3242} \pdfglyphtounicode{ideographicsocietyparen}{3233} \pdfglyphtounicode{ideographicspace}{3000} \pdfglyphtounicode{ideographicspecialparen}{3235} \pdfglyphtounicode{ideographicstockparen}{3231} \pdfglyphtounicode{ideographicstudyparen}{323B} \pdfglyphtounicode{ideographicsunparen}{3230} \pdfglyphtounicode{ideographicsuperviseparen}{323C} \pdfglyphtounicode{ideographicwaterparen}{322C} \pdfglyphtounicode{ideographicwoodparen}{322D} \pdfglyphtounicode{ideographiczero}{3007} \pdfglyphtounicode{ideographmetalcircle}{328E} \pdfglyphtounicode{ideographmooncircle}{328A} \pdfglyphtounicode{ideographnamecircle}{3294} \pdfglyphtounicode{ideographsuncircle}{3290} \pdfglyphtounicode{ideographwatercircle}{328C} \pdfglyphtounicode{ideographwoodcircle}{328D} \pdfglyphtounicode{ideva}{0907} \pdfglyphtounicode{idieresis}{00EF} \pdfglyphtounicode{idieresisacute}{1E2F} \pdfglyphtounicode{idieresiscyrillic}{04E5} \pdfglyphtounicode{idotbelow}{1ECB} \pdfglyphtounicode{iebrevecyrillic}{04D7} \pdfglyphtounicode{iecyrillic}{0435} \pdfglyphtounicode{ieungacirclekorean}{3275} \pdfglyphtounicode{ieungaparenkorean}{3215} \pdfglyphtounicode{ieungcirclekorean}{3267} \pdfglyphtounicode{ieungkorean}{3147} \pdfglyphtounicode{ieungparenkorean}{3207} \pdfglyphtounicode{igrave}{00EC} \pdfglyphtounicode{igujarati}{0A87} \pdfglyphtounicode{igurmukhi}{0A07} \pdfglyphtounicode{ihiragana}{3044} \pdfglyphtounicode{ihookabove}{1EC9} \pdfglyphtounicode{iibengali}{0988} \pdfglyphtounicode{iicyrillic}{0438} \pdfglyphtounicode{iideva}{0908} \pdfglyphtounicode{iigujarati}{0A88} \pdfglyphtounicode{iigurmukhi}{0A08} \pdfglyphtounicode{iimatragurmukhi}{0A40} \pdfglyphtounicode{iinvertedbreve}{020B} \pdfglyphtounicode{iishortcyrillic}{0439} \pdfglyphtounicode{iivowelsignbengali}{09C0} \pdfglyphtounicode{iivowelsigndeva}{0940} \pdfglyphtounicode{iivowelsigngujarati}{0AC0} \pdfglyphtounicode{ij}{0133} \pdfglyphtounicode{ikatakana}{30A4} \pdfglyphtounicode{ikatakanahalfwidth}{FF72} \pdfglyphtounicode{ikorean}{3163} \pdfglyphtounicode{ilde}{02DC} \pdfglyphtounicode{iluyhebrew}{05AC} \pdfglyphtounicode{imacron}{012B} \pdfglyphtounicode{imacroncyrillic}{04E3} \pdfglyphtounicode{imageorapproximatelyequal}{2253} \pdfglyphtounicode{imatragurmukhi}{0A3F} \pdfglyphtounicode{imonospace}{FF49} \pdfglyphtounicode{increment}{2206} \pdfglyphtounicode{infinity}{221E} \pdfglyphtounicode{iniarmenian}{056B} \pdfglyphtounicode{integral}{222B} \pdfglyphtounicode{integralbottom}{2321} \pdfglyphtounicode{integralbt}{2321} \pdfglyphtounicode{integralex}{F8F5} \pdfglyphtounicode{integraltop}{2320} \pdfglyphtounicode{integraltp}{2320} \pdfglyphtounicode{intersection}{2229} \pdfglyphtounicode{intisquare}{3305} \pdfglyphtounicode{invbullet}{25D8} \pdfglyphtounicode{invcircle}{25D9} \pdfglyphtounicode{invsmileface}{263B} \pdfglyphtounicode{iocyrillic}{0451} \pdfglyphtounicode{iogonek}{012F} \pdfglyphtounicode{iota}{03B9} \pdfglyphtounicode{iotadieresis}{03CA} \pdfglyphtounicode{iotadieresistonos}{0390} \pdfglyphtounicode{iotalatin}{0269} \pdfglyphtounicode{iotatonos}{03AF} \pdfglyphtounicode{iparen}{24A4} \pdfglyphtounicode{irigurmukhi}{0A72} \pdfglyphtounicode{ismallhiragana}{3043} \pdfglyphtounicode{ismallkatakana}{30A3} \pdfglyphtounicode{ismallkatakanahalfwidth}{FF68} \pdfglyphtounicode{issharbengali}{09FA} \pdfglyphtounicode{istroke}{0268} \pdfglyphtounicode{isuperior}{F6ED} \pdfglyphtounicode{iterationhiragana}{309D} \pdfglyphtounicode{iterationkatakana}{30FD} \pdfglyphtounicode{itilde}{0129} \pdfglyphtounicode{itildebelow}{1E2D} \pdfglyphtounicode{iubopomofo}{3129} \pdfglyphtounicode{iucyrillic}{044E} \pdfglyphtounicode{ivowelsignbengali}{09BF} \pdfglyphtounicode{ivowelsigndeva}{093F} \pdfglyphtounicode{ivowelsigngujarati}{0ABF} \pdfglyphtounicode{izhitsacyrillic}{0475} \pdfglyphtounicode{izhitsadblgravecyrillic}{0477} \pdfglyphtounicode{j}{006A} \pdfglyphtounicode{jaarmenian}{0571} \pdfglyphtounicode{jabengali}{099C} \pdfglyphtounicode{jadeva}{091C} \pdfglyphtounicode{jagujarati}{0A9C} \pdfglyphtounicode{jagurmukhi}{0A1C} \pdfglyphtounicode{jbopomofo}{3110} \pdfglyphtounicode{jcaron}{01F0} \pdfglyphtounicode{jcircle}{24D9} \pdfglyphtounicode{jcircumflex}{0135} \pdfglyphtounicode{jcrossedtail}{029D} \pdfglyphtounicode{jdotlessstroke}{025F} \pdfglyphtounicode{jecyrillic}{0458} \pdfglyphtounicode{jeemarabic}{062C} \pdfglyphtounicode{jeemfinalarabic}{FE9E} \pdfglyphtounicode{jeeminitialarabic}{FE9F} \pdfglyphtounicode{jeemmedialarabic}{FEA0} \pdfglyphtounicode{jeharabic}{0698} \pdfglyphtounicode{jehfinalarabic}{FB8B} \pdfglyphtounicode{jhabengali}{099D} \pdfglyphtounicode{jhadeva}{091D} \pdfglyphtounicode{jhagujarati}{0A9D} \pdfglyphtounicode{jhagurmukhi}{0A1D} \pdfglyphtounicode{jheharmenian}{057B} \pdfglyphtounicode{jis}{3004} \pdfglyphtounicode{jmonospace}{FF4A} \pdfglyphtounicode{jparen}{24A5} \pdfglyphtounicode{jsuperior}{02B2} \pdfglyphtounicode{k}{006B} \pdfglyphtounicode{kabashkircyrillic}{04A1} \pdfglyphtounicode{kabengali}{0995} \pdfglyphtounicode{kacute}{1E31} \pdfglyphtounicode{kacyrillic}{043A} \pdfglyphtounicode{kadescendercyrillic}{049B} \pdfglyphtounicode{kadeva}{0915} \pdfglyphtounicode{kaf}{05DB} \pdfglyphtounicode{kafarabic}{0643} \pdfglyphtounicode{kafdagesh}{FB3B} \pdfglyphtounicode{kafdageshhebrew}{FB3B} \pdfglyphtounicode{kaffinalarabic}{FEDA} \pdfglyphtounicode{kafhebrew}{05DB} \pdfglyphtounicode{kafinitialarabic}{FEDB} \pdfglyphtounicode{kafmedialarabic}{FEDC} \pdfglyphtounicode{kafrafehebrew}{FB4D} \pdfglyphtounicode{kagujarati}{0A95} \pdfglyphtounicode{kagurmukhi}{0A15} \pdfglyphtounicode{kahiragana}{304B} \pdfglyphtounicode{kahookcyrillic}{04C4} \pdfglyphtounicode{kakatakana}{30AB} \pdfglyphtounicode{kakatakanahalfwidth}{FF76} \pdfglyphtounicode{kappa}{03BA} \pdfglyphtounicode{kappasymbolgreek}{03F0} \pdfglyphtounicode{kapyeounmieumkorean}{3171} \pdfglyphtounicode{kapyeounphieuphkorean}{3184} \pdfglyphtounicode{kapyeounpieupkorean}{3178} \pdfglyphtounicode{kapyeounssangpieupkorean}{3179} \pdfglyphtounicode{karoriisquare}{330D} \pdfglyphtounicode{kashidaautoarabic}{0640} \pdfglyphtounicode{kashidaautonosidebearingarabic}{0640} \pdfglyphtounicode{kasmallkatakana}{30F5} \pdfglyphtounicode{kasquare}{3384} \pdfglyphtounicode{kasraarabic}{0650} \pdfglyphtounicode{kasratanarabic}{064D} \pdfglyphtounicode{kastrokecyrillic}{049F} \pdfglyphtounicode{katahiraprolongmarkhalfwidth}{FF70} \pdfglyphtounicode{kaverticalstrokecyrillic}{049D} \pdfglyphtounicode{kbopomofo}{310E} \pdfglyphtounicode{kcalsquare}{3389} \pdfglyphtounicode{kcaron}{01E9} \pdfglyphtounicode{kcedilla}{0137} \pdfglyphtounicode{kcircle}{24DA} \pdfglyphtounicode{kcommaaccent}{0137} \pdfglyphtounicode{kdotbelow}{1E33} \pdfglyphtounicode{keharmenian}{0584} \pdfglyphtounicode{kehiragana}{3051} \pdfglyphtounicode{kekatakana}{30B1} \pdfglyphtounicode{kekatakanahalfwidth}{FF79} \pdfglyphtounicode{kenarmenian}{056F} \pdfglyphtounicode{kesmallkatakana}{30F6} \pdfglyphtounicode{kgreenlandic}{0138} \pdfglyphtounicode{khabengali}{0996} \pdfglyphtounicode{khacyrillic}{0445} \pdfglyphtounicode{khadeva}{0916} \pdfglyphtounicode{khagujarati}{0A96} \pdfglyphtounicode{khagurmukhi}{0A16} \pdfglyphtounicode{khaharabic}{062E} \pdfglyphtounicode{khahfinalarabic}{FEA6} \pdfglyphtounicode{khahinitialarabic}{FEA7} \pdfglyphtounicode{khahmedialarabic}{FEA8} \pdfglyphtounicode{kheicoptic}{03E7} \pdfglyphtounicode{khhadeva}{0959} \pdfglyphtounicode{khhagurmukhi}{0A59} \pdfglyphtounicode{khieukhacirclekorean}{3278} \pdfglyphtounicode{khieukhaparenkorean}{3218} \pdfglyphtounicode{khieukhcirclekorean}{326A} \pdfglyphtounicode{khieukhkorean}{314B} \pdfglyphtounicode{khieukhparenkorean}{320A} \pdfglyphtounicode{khokhaithai}{0E02} \pdfglyphtounicode{khokhonthai}{0E05} \pdfglyphtounicode{khokhuatthai}{0E03} \pdfglyphtounicode{khokhwaithai}{0E04} \pdfglyphtounicode{khomutthai}{0E5B} \pdfglyphtounicode{khook}{0199} \pdfglyphtounicode{khorakhangthai}{0E06} \pdfglyphtounicode{khzsquare}{3391} \pdfglyphtounicode{kihiragana}{304D} \pdfglyphtounicode{kikatakana}{30AD} \pdfglyphtounicode{kikatakanahalfwidth}{FF77} \pdfglyphtounicode{kiroguramusquare}{3315} \pdfglyphtounicode{kiromeetorusquare}{3316} \pdfglyphtounicode{kirosquare}{3314} \pdfglyphtounicode{kiyeokacirclekorean}{326E} \pdfglyphtounicode{kiyeokaparenkorean}{320E} \pdfglyphtounicode{kiyeokcirclekorean}{3260} \pdfglyphtounicode{kiyeokkorean}{3131} \pdfglyphtounicode{kiyeokparenkorean}{3200} \pdfglyphtounicode{kiyeoksioskorean}{3133} \pdfglyphtounicode{kjecyrillic}{045C} \pdfglyphtounicode{klinebelow}{1E35} \pdfglyphtounicode{klsquare}{3398} \pdfglyphtounicode{kmcubedsquare}{33A6} \pdfglyphtounicode{kmonospace}{FF4B} \pdfglyphtounicode{kmsquaredsquare}{33A2} \pdfglyphtounicode{kohiragana}{3053} \pdfglyphtounicode{kohmsquare}{33C0} \pdfglyphtounicode{kokaithai}{0E01} \pdfglyphtounicode{kokatakana}{30B3} \pdfglyphtounicode{kokatakanahalfwidth}{FF7A} \pdfglyphtounicode{kooposquare}{331E} \pdfglyphtounicode{koppacyrillic}{0481} \pdfglyphtounicode{koreanstandardsymbol}{327F} \pdfglyphtounicode{koroniscmb}{0343} \pdfglyphtounicode{kparen}{24A6} \pdfglyphtounicode{kpasquare}{33AA} \pdfglyphtounicode{ksicyrillic}{046F} \pdfglyphtounicode{ktsquare}{33CF} \pdfglyphtounicode{kturned}{029E} \pdfglyphtounicode{kuhiragana}{304F} \pdfglyphtounicode{kukatakana}{30AF} \pdfglyphtounicode{kukatakanahalfwidth}{FF78} \pdfglyphtounicode{kvsquare}{33B8} \pdfglyphtounicode{kwsquare}{33BE} \pdfglyphtounicode{l}{006C} \pdfglyphtounicode{labengali}{09B2} \pdfglyphtounicode{lacute}{013A} \pdfglyphtounicode{ladeva}{0932} \pdfglyphtounicode{lagujarati}{0AB2} \pdfglyphtounicode{lagurmukhi}{0A32} \pdfglyphtounicode{lakkhangyaothai}{0E45} \pdfglyphtounicode{lamaleffinalarabic}{FEFC} \pdfglyphtounicode{lamalefhamzaabovefinalarabic}{FEF8} \pdfglyphtounicode{lamalefhamzaaboveisolatedarabic}{FEF7} \pdfglyphtounicode{lamalefhamzabelowfinalarabic}{FEFA} \pdfglyphtounicode{lamalefhamzabelowisolatedarabic}{FEF9} \pdfglyphtounicode{lamalefisolatedarabic}{FEFB} \pdfglyphtounicode{lamalefmaddaabovefinalarabic}{FEF6} \pdfglyphtounicode{lamalefmaddaaboveisolatedarabic}{FEF5} \pdfglyphtounicode{lamarabic}{0644} \pdfglyphtounicode{lambda}{03BB} \pdfglyphtounicode{lambdastroke}{019B} \pdfglyphtounicode{lamed}{05DC} \pdfglyphtounicode{lameddagesh}{FB3C} \pdfglyphtounicode{lameddageshhebrew}{FB3C} \pdfglyphtounicode{lamedhebrew}{05DC} % lamedholam;05DC 05B9 % lamedholamdagesh;05DC 05B9 05BC % lamedholamdageshhebrew;05DC 05B9 05BC % lamedholamhebrew;05DC 05B9 \pdfglyphtounicode{lamfinalarabic}{FEDE} \pdfglyphtounicode{lamhahinitialarabic}{FCCA} \pdfglyphtounicode{laminitialarabic}{FEDF} \pdfglyphtounicode{lamjeeminitialarabic}{FCC9} \pdfglyphtounicode{lamkhahinitialarabic}{FCCB} \pdfglyphtounicode{lamlamhehisolatedarabic}{FDF2} \pdfglyphtounicode{lammedialarabic}{FEE0} \pdfglyphtounicode{lammeemhahinitialarabic}{FD88} \pdfglyphtounicode{lammeeminitialarabic}{FCCC} % lammeemjeeminitialarabic;FEDF FEE4 FEA0 % lammeemkhahinitialarabic;FEDF FEE4 FEA8 \pdfglyphtounicode{largecircle}{25EF} \pdfglyphtounicode{lbar}{019A} \pdfglyphtounicode{lbelt}{026C} \pdfglyphtounicode{lbopomofo}{310C} \pdfglyphtounicode{lcaron}{013E} \pdfglyphtounicode{lcedilla}{013C} \pdfglyphtounicode{lcircle}{24DB} \pdfglyphtounicode{lcircumflexbelow}{1E3D} \pdfglyphtounicode{lcommaaccent}{013C} \pdfglyphtounicode{ldot}{0140} \pdfglyphtounicode{ldotaccent}{0140} \pdfglyphtounicode{ldotbelow}{1E37} \pdfglyphtounicode{ldotbelowmacron}{1E39} \pdfglyphtounicode{leftangleabovecmb}{031A} \pdfglyphtounicode{lefttackbelowcmb}{0318} \pdfglyphtounicode{less}{003C} \pdfglyphtounicode{lessequal}{2264} \pdfglyphtounicode{lessequalorgreater}{22DA} \pdfglyphtounicode{lessmonospace}{FF1C} \pdfglyphtounicode{lessorequivalent}{2272} \pdfglyphtounicode{lessorgreater}{2276} \pdfglyphtounicode{lessoverequal}{2266} \pdfglyphtounicode{lesssmall}{FE64} \pdfglyphtounicode{lezh}{026E} \pdfglyphtounicode{lfblock}{258C} \pdfglyphtounicode{lhookretroflex}{026D} \pdfglyphtounicode{lira}{20A4} \pdfglyphtounicode{liwnarmenian}{056C} \pdfglyphtounicode{lj}{01C9} \pdfglyphtounicode{ljecyrillic}{0459} \pdfglyphtounicode{ll}{F6C0} \pdfglyphtounicode{lladeva}{0933} \pdfglyphtounicode{llagujarati}{0AB3} \pdfglyphtounicode{llinebelow}{1E3B} \pdfglyphtounicode{llladeva}{0934} \pdfglyphtounicode{llvocalicbengali}{09E1} \pdfglyphtounicode{llvocalicdeva}{0961} \pdfglyphtounicode{llvocalicvowelsignbengali}{09E3} \pdfglyphtounicode{llvocalicvowelsigndeva}{0963} \pdfglyphtounicode{lmiddletilde}{026B} \pdfglyphtounicode{lmonospace}{FF4C} \pdfglyphtounicode{lmsquare}{33D0} \pdfglyphtounicode{lochulathai}{0E2C} \pdfglyphtounicode{logicaland}{2227} \pdfglyphtounicode{logicalnot}{00AC} \pdfglyphtounicode{logicalnotreversed}{2310} \pdfglyphtounicode{logicalor}{2228} \pdfglyphtounicode{lolingthai}{0E25} \pdfglyphtounicode{longs}{017F} \pdfglyphtounicode{lowlinecenterline}{FE4E} \pdfglyphtounicode{lowlinecmb}{0332} \pdfglyphtounicode{lowlinedashed}{FE4D} \pdfglyphtounicode{lozenge}{25CA} \pdfglyphtounicode{lparen}{24A7} \pdfglyphtounicode{lslash}{0142} \pdfglyphtounicode{lsquare}{2113} \pdfglyphtounicode{lsuperior}{F6EE} \pdfglyphtounicode{ltshade}{2591} \pdfglyphtounicode{luthai}{0E26} \pdfglyphtounicode{lvocalicbengali}{098C} \pdfglyphtounicode{lvocalicdeva}{090C} \pdfglyphtounicode{lvocalicvowelsignbengali}{09E2} \pdfglyphtounicode{lvocalicvowelsigndeva}{0962} \pdfglyphtounicode{lxsquare}{33D3} \pdfglyphtounicode{m}{006D} \pdfglyphtounicode{mabengali}{09AE} \pdfglyphtounicode{macron}{00AF} \pdfglyphtounicode{macronbelowcmb}{0331} \pdfglyphtounicode{macroncmb}{0304} \pdfglyphtounicode{macronlowmod}{02CD} \pdfglyphtounicode{macronmonospace}{FFE3} \pdfglyphtounicode{macute}{1E3F} \pdfglyphtounicode{madeva}{092E} \pdfglyphtounicode{magujarati}{0AAE} \pdfglyphtounicode{magurmukhi}{0A2E} \pdfglyphtounicode{mahapakhhebrew}{05A4} \pdfglyphtounicode{mahapakhlefthebrew}{05A4} \pdfglyphtounicode{mahiragana}{307E} \pdfglyphtounicode{maichattawalowleftthai}{F895} \pdfglyphtounicode{maichattawalowrightthai}{F894} \pdfglyphtounicode{maichattawathai}{0E4B} \pdfglyphtounicode{maichattawaupperleftthai}{F893} \pdfglyphtounicode{maieklowleftthai}{F88C} \pdfglyphtounicode{maieklowrightthai}{F88B} \pdfglyphtounicode{maiekthai}{0E48} \pdfglyphtounicode{maiekupperleftthai}{F88A} \pdfglyphtounicode{maihanakatleftthai}{F884} \pdfglyphtounicode{maihanakatthai}{0E31} \pdfglyphtounicode{maitaikhuleftthai}{F889} \pdfglyphtounicode{maitaikhuthai}{0E47} \pdfglyphtounicode{maitholowleftthai}{F88F} \pdfglyphtounicode{maitholowrightthai}{F88E} \pdfglyphtounicode{maithothai}{0E49} \pdfglyphtounicode{maithoupperleftthai}{F88D} \pdfglyphtounicode{maitrilowleftthai}{F892} \pdfglyphtounicode{maitrilowrightthai}{F891} \pdfglyphtounicode{maitrithai}{0E4A} \pdfglyphtounicode{maitriupperleftthai}{F890} \pdfglyphtounicode{maiyamokthai}{0E46} \pdfglyphtounicode{makatakana}{30DE} \pdfglyphtounicode{makatakanahalfwidth}{FF8F} \pdfglyphtounicode{male}{2642} \pdfglyphtounicode{mansyonsquare}{3347} \pdfglyphtounicode{maqafhebrew}{05BE} \pdfglyphtounicode{mars}{2642} \pdfglyphtounicode{masoracirclehebrew}{05AF} \pdfglyphtounicode{masquare}{3383} \pdfglyphtounicode{mbopomofo}{3107} \pdfglyphtounicode{mbsquare}{33D4} \pdfglyphtounicode{mcircle}{24DC} \pdfglyphtounicode{mcubedsquare}{33A5} \pdfglyphtounicode{mdotaccent}{1E41} \pdfglyphtounicode{mdotbelow}{1E43} \pdfglyphtounicode{meemarabic}{0645} \pdfglyphtounicode{meemfinalarabic}{FEE2} \pdfglyphtounicode{meeminitialarabic}{FEE3} \pdfglyphtounicode{meemmedialarabic}{FEE4} \pdfglyphtounicode{meemmeeminitialarabic}{FCD1} \pdfglyphtounicode{meemmeemisolatedarabic}{FC48} \pdfglyphtounicode{meetorusquare}{334D} \pdfglyphtounicode{mehiragana}{3081} \pdfglyphtounicode{meizierasquare}{337E} \pdfglyphtounicode{mekatakana}{30E1} \pdfglyphtounicode{mekatakanahalfwidth}{FF92} \pdfglyphtounicode{mem}{05DE} \pdfglyphtounicode{memdagesh}{FB3E} \pdfglyphtounicode{memdageshhebrew}{FB3E} \pdfglyphtounicode{memhebrew}{05DE} \pdfglyphtounicode{menarmenian}{0574} \pdfglyphtounicode{merkhahebrew}{05A5} \pdfglyphtounicode{merkhakefulahebrew}{05A6} \pdfglyphtounicode{merkhakefulalefthebrew}{05A6} \pdfglyphtounicode{merkhalefthebrew}{05A5} \pdfglyphtounicode{mhook}{0271} \pdfglyphtounicode{mhzsquare}{3392} \pdfglyphtounicode{middledotkatakanahalfwidth}{FF65} \pdfglyphtounicode{middot}{00B7} \pdfglyphtounicode{mieumacirclekorean}{3272} \pdfglyphtounicode{mieumaparenkorean}{3212} \pdfglyphtounicode{mieumcirclekorean}{3264} \pdfglyphtounicode{mieumkorean}{3141} \pdfglyphtounicode{mieumpansioskorean}{3170} \pdfglyphtounicode{mieumparenkorean}{3204} \pdfglyphtounicode{mieumpieupkorean}{316E} \pdfglyphtounicode{mieumsioskorean}{316F} \pdfglyphtounicode{mihiragana}{307F} \pdfglyphtounicode{mikatakana}{30DF} \pdfglyphtounicode{mikatakanahalfwidth}{FF90} \pdfglyphtounicode{minus}{2212} \pdfglyphtounicode{minusbelowcmb}{0320} \pdfglyphtounicode{minuscircle}{2296} \pdfglyphtounicode{minusmod}{02D7} \pdfglyphtounicode{minusplus}{2213} \pdfglyphtounicode{minute}{2032} \pdfglyphtounicode{miribaarusquare}{334A} \pdfglyphtounicode{mirisquare}{3349} \pdfglyphtounicode{mlonglegturned}{0270} \pdfglyphtounicode{mlsquare}{3396} \pdfglyphtounicode{mmcubedsquare}{33A3} \pdfglyphtounicode{mmonospace}{FF4D} \pdfglyphtounicode{mmsquaredsquare}{339F} \pdfglyphtounicode{mohiragana}{3082} \pdfglyphtounicode{mohmsquare}{33C1} \pdfglyphtounicode{mokatakana}{30E2} \pdfglyphtounicode{mokatakanahalfwidth}{FF93} \pdfglyphtounicode{molsquare}{33D6} \pdfglyphtounicode{momathai}{0E21} \pdfglyphtounicode{moverssquare}{33A7} \pdfglyphtounicode{moverssquaredsquare}{33A8} \pdfglyphtounicode{mparen}{24A8} \pdfglyphtounicode{mpasquare}{33AB} \pdfglyphtounicode{mssquare}{33B3} \pdfglyphtounicode{msuperior}{F6EF} \pdfglyphtounicode{mturned}{026F} \pdfglyphtounicode{mu}{00B5} \pdfglyphtounicode{mu1}{00B5} \pdfglyphtounicode{muasquare}{3382} \pdfglyphtounicode{muchgreater}{226B} \pdfglyphtounicode{muchless}{226A} \pdfglyphtounicode{mufsquare}{338C} \pdfglyphtounicode{mugreek}{03BC} \pdfglyphtounicode{mugsquare}{338D} \pdfglyphtounicode{muhiragana}{3080} \pdfglyphtounicode{mukatakana}{30E0} \pdfglyphtounicode{mukatakanahalfwidth}{FF91} \pdfglyphtounicode{mulsquare}{3395} \pdfglyphtounicode{multiply}{00D7} \pdfglyphtounicode{mumsquare}{339B} \pdfglyphtounicode{munahhebrew}{05A3} \pdfglyphtounicode{munahlefthebrew}{05A3} \pdfglyphtounicode{musicalnote}{266A} \pdfglyphtounicode{musicalnotedbl}{266B} \pdfglyphtounicode{musicflatsign}{266D} \pdfglyphtounicode{musicsharpsign}{266F} \pdfglyphtounicode{mussquare}{33B2} \pdfglyphtounicode{muvsquare}{33B6} \pdfglyphtounicode{muwsquare}{33BC} \pdfglyphtounicode{mvmegasquare}{33B9} \pdfglyphtounicode{mvsquare}{33B7} \pdfglyphtounicode{mwmegasquare}{33BF} \pdfglyphtounicode{mwsquare}{33BD} \pdfglyphtounicode{n}{006E} \pdfglyphtounicode{nabengali}{09A8} \pdfglyphtounicode{nabla}{2207} \pdfglyphtounicode{nacute}{0144} \pdfglyphtounicode{nadeva}{0928} \pdfglyphtounicode{nagujarati}{0AA8} \pdfglyphtounicode{nagurmukhi}{0A28} \pdfglyphtounicode{nahiragana}{306A} \pdfglyphtounicode{nakatakana}{30CA} \pdfglyphtounicode{nakatakanahalfwidth}{FF85} \pdfglyphtounicode{napostrophe}{0149} \pdfglyphtounicode{nasquare}{3381} \pdfglyphtounicode{nbopomofo}{310B} \pdfglyphtounicode{nbspace}{00A0} \pdfglyphtounicode{ncaron}{0148} \pdfglyphtounicode{ncedilla}{0146} \pdfglyphtounicode{ncircle}{24DD} \pdfglyphtounicode{ncircumflexbelow}{1E4B} \pdfglyphtounicode{ncommaaccent}{0146} \pdfglyphtounicode{ndotaccent}{1E45} \pdfglyphtounicode{ndotbelow}{1E47} \pdfglyphtounicode{nehiragana}{306D} \pdfglyphtounicode{nekatakana}{30CD} \pdfglyphtounicode{nekatakanahalfwidth}{FF88} \pdfglyphtounicode{newsheqelsign}{20AA} \pdfglyphtounicode{nfsquare}{338B} \pdfglyphtounicode{ngabengali}{0999} \pdfglyphtounicode{ngadeva}{0919} \pdfglyphtounicode{ngagujarati}{0A99} \pdfglyphtounicode{ngagurmukhi}{0A19} \pdfglyphtounicode{ngonguthai}{0E07} \pdfglyphtounicode{nhiragana}{3093} \pdfglyphtounicode{nhookleft}{0272} \pdfglyphtounicode{nhookretroflex}{0273} \pdfglyphtounicode{nieunacirclekorean}{326F} \pdfglyphtounicode{nieunaparenkorean}{320F} \pdfglyphtounicode{nieuncieuckorean}{3135} \pdfglyphtounicode{nieuncirclekorean}{3261} \pdfglyphtounicode{nieunhieuhkorean}{3136} \pdfglyphtounicode{nieunkorean}{3134} \pdfglyphtounicode{nieunpansioskorean}{3168} \pdfglyphtounicode{nieunparenkorean}{3201} \pdfglyphtounicode{nieunsioskorean}{3167} \pdfglyphtounicode{nieuntikeutkorean}{3166} \pdfglyphtounicode{nihiragana}{306B} \pdfglyphtounicode{nikatakana}{30CB} \pdfglyphtounicode{nikatakanahalfwidth}{FF86} \pdfglyphtounicode{nikhahitleftthai}{F899} \pdfglyphtounicode{nikhahitthai}{0E4D} \pdfglyphtounicode{nine}{0039} \pdfglyphtounicode{ninearabic}{0669} \pdfglyphtounicode{ninebengali}{09EF} \pdfglyphtounicode{ninecircle}{2468} \pdfglyphtounicode{ninecircleinversesansserif}{2792} \pdfglyphtounicode{ninedeva}{096F} \pdfglyphtounicode{ninegujarati}{0AEF} \pdfglyphtounicode{ninegurmukhi}{0A6F} \pdfglyphtounicode{ninehackarabic}{0669} \pdfglyphtounicode{ninehangzhou}{3029} \pdfglyphtounicode{nineideographicparen}{3228} \pdfglyphtounicode{nineinferior}{2089} \pdfglyphtounicode{ninemonospace}{FF19} \pdfglyphtounicode{nineoldstyle}{F739} \pdfglyphtounicode{nineparen}{247C} \pdfglyphtounicode{nineperiod}{2490} \pdfglyphtounicode{ninepersian}{06F9} \pdfglyphtounicode{nineroman}{2178} \pdfglyphtounicode{ninesuperior}{2079} \pdfglyphtounicode{nineteencircle}{2472} \pdfglyphtounicode{nineteenparen}{2486} \pdfglyphtounicode{nineteenperiod}{249A} \pdfglyphtounicode{ninethai}{0E59} \pdfglyphtounicode{nj}{01CC} \pdfglyphtounicode{njecyrillic}{045A} \pdfglyphtounicode{nkatakana}{30F3} \pdfglyphtounicode{nkatakanahalfwidth}{FF9D} \pdfglyphtounicode{nlegrightlong}{019E} \pdfglyphtounicode{nlinebelow}{1E49} \pdfglyphtounicode{nmonospace}{FF4E} \pdfglyphtounicode{nmsquare}{339A} \pdfglyphtounicode{nnabengali}{09A3} \pdfglyphtounicode{nnadeva}{0923} \pdfglyphtounicode{nnagujarati}{0AA3} \pdfglyphtounicode{nnagurmukhi}{0A23} \pdfglyphtounicode{nnnadeva}{0929} \pdfglyphtounicode{nohiragana}{306E} \pdfglyphtounicode{nokatakana}{30CE} \pdfglyphtounicode{nokatakanahalfwidth}{FF89} \pdfglyphtounicode{nonbreakingspace}{00A0} \pdfglyphtounicode{nonenthai}{0E13} \pdfglyphtounicode{nonuthai}{0E19} \pdfglyphtounicode{noonarabic}{0646} \pdfglyphtounicode{noonfinalarabic}{FEE6} \pdfglyphtounicode{noonghunnaarabic}{06BA} \pdfglyphtounicode{noonghunnafinalarabic}{FB9F} % noonhehinitialarabic;FEE7 FEEC \pdfglyphtounicode{nooninitialarabic}{FEE7} \pdfglyphtounicode{noonjeeminitialarabic}{FCD2} \pdfglyphtounicode{noonjeemisolatedarabic}{FC4B} \pdfglyphtounicode{noonmedialarabic}{FEE8} \pdfglyphtounicode{noonmeeminitialarabic}{FCD5} \pdfglyphtounicode{noonmeemisolatedarabic}{FC4E} \pdfglyphtounicode{noonnoonfinalarabic}{FC8D} \pdfglyphtounicode{notcontains}{220C} \pdfglyphtounicode{notelement}{2209} \pdfglyphtounicode{notelementof}{2209} \pdfglyphtounicode{notequal}{2260} \pdfglyphtounicode{notgreater}{226F} \pdfglyphtounicode{notgreaternorequal}{2271} \pdfglyphtounicode{notgreaternorless}{2279} \pdfglyphtounicode{notidentical}{2262} \pdfglyphtounicode{notless}{226E} \pdfglyphtounicode{notlessnorequal}{2270} \pdfglyphtounicode{notparallel}{2226} \pdfglyphtounicode{notprecedes}{2280} \pdfglyphtounicode{notsubset}{2284} \pdfglyphtounicode{notsucceeds}{2281} \pdfglyphtounicode{notsuperset}{2285} \pdfglyphtounicode{nowarmenian}{0576} \pdfglyphtounicode{nparen}{24A9} \pdfglyphtounicode{nssquare}{33B1} \pdfglyphtounicode{nsuperior}{207F} \pdfglyphtounicode{ntilde}{00F1} \pdfglyphtounicode{nu}{03BD} \pdfglyphtounicode{nuhiragana}{306C} \pdfglyphtounicode{nukatakana}{30CC} \pdfglyphtounicode{nukatakanahalfwidth}{FF87} \pdfglyphtounicode{nuktabengali}{09BC} \pdfglyphtounicode{nuktadeva}{093C} \pdfglyphtounicode{nuktagujarati}{0ABC} \pdfglyphtounicode{nuktagurmukhi}{0A3C} \pdfglyphtounicode{numbersign}{0023} \pdfglyphtounicode{numbersignmonospace}{FF03} \pdfglyphtounicode{numbersignsmall}{FE5F} \pdfglyphtounicode{numeralsigngreek}{0374} \pdfglyphtounicode{numeralsignlowergreek}{0375} \pdfglyphtounicode{numero}{2116} \pdfglyphtounicode{nun}{05E0} \pdfglyphtounicode{nundagesh}{FB40} \pdfglyphtounicode{nundageshhebrew}{FB40} \pdfglyphtounicode{nunhebrew}{05E0} \pdfglyphtounicode{nvsquare}{33B5} \pdfglyphtounicode{nwsquare}{33BB} \pdfglyphtounicode{nyabengali}{099E} \pdfglyphtounicode{nyadeva}{091E} \pdfglyphtounicode{nyagujarati}{0A9E} \pdfglyphtounicode{nyagurmukhi}{0A1E} \pdfglyphtounicode{o}{006F} \pdfglyphtounicode{oacute}{00F3} \pdfglyphtounicode{oangthai}{0E2D} \pdfglyphtounicode{obarred}{0275} \pdfglyphtounicode{obarredcyrillic}{04E9} \pdfglyphtounicode{obarreddieresiscyrillic}{04EB} \pdfglyphtounicode{obengali}{0993} \pdfglyphtounicode{obopomofo}{311B} \pdfglyphtounicode{obreve}{014F} \pdfglyphtounicode{ocandradeva}{0911} \pdfglyphtounicode{ocandragujarati}{0A91} \pdfglyphtounicode{ocandravowelsigndeva}{0949} \pdfglyphtounicode{ocandravowelsigngujarati}{0AC9} \pdfglyphtounicode{ocaron}{01D2} \pdfglyphtounicode{ocircle}{24DE} \pdfglyphtounicode{ocircumflex}{00F4} \pdfglyphtounicode{ocircumflexacute}{1ED1} \pdfglyphtounicode{ocircumflexdotbelow}{1ED9} \pdfglyphtounicode{ocircumflexgrave}{1ED3} \pdfglyphtounicode{ocircumflexhookabove}{1ED5} \pdfglyphtounicode{ocircumflextilde}{1ED7} \pdfglyphtounicode{ocyrillic}{043E} \pdfglyphtounicode{odblacute}{0151} \pdfglyphtounicode{odblgrave}{020D} \pdfglyphtounicode{odeva}{0913} \pdfglyphtounicode{odieresis}{00F6} \pdfglyphtounicode{odieresiscyrillic}{04E7} \pdfglyphtounicode{odotbelow}{1ECD} \pdfglyphtounicode{oe}{0153} \pdfglyphtounicode{oekorean}{315A} \pdfglyphtounicode{ogonek}{02DB} \pdfglyphtounicode{ogonekcmb}{0328} \pdfglyphtounicode{ograve}{00F2} \pdfglyphtounicode{ogujarati}{0A93} \pdfglyphtounicode{oharmenian}{0585} \pdfglyphtounicode{ohiragana}{304A} \pdfglyphtounicode{ohookabove}{1ECF} \pdfglyphtounicode{ohorn}{01A1} \pdfglyphtounicode{ohornacute}{1EDB} \pdfglyphtounicode{ohorndotbelow}{1EE3} \pdfglyphtounicode{ohorngrave}{1EDD} \pdfglyphtounicode{ohornhookabove}{1EDF} \pdfglyphtounicode{ohorntilde}{1EE1} \pdfglyphtounicode{ohungarumlaut}{0151} \pdfglyphtounicode{oi}{01A3} \pdfglyphtounicode{oinvertedbreve}{020F} \pdfglyphtounicode{okatakana}{30AA} \pdfglyphtounicode{okatakanahalfwidth}{FF75} \pdfglyphtounicode{okorean}{3157} \pdfglyphtounicode{olehebrew}{05AB} \pdfglyphtounicode{omacron}{014D} \pdfglyphtounicode{omacronacute}{1E53} \pdfglyphtounicode{omacrongrave}{1E51} \pdfglyphtounicode{omdeva}{0950} \pdfglyphtounicode{omega}{03C9} \pdfglyphtounicode{omega1}{03D6} \pdfglyphtounicode{omegacyrillic}{0461} \pdfglyphtounicode{omegalatinclosed}{0277} \pdfglyphtounicode{omegaroundcyrillic}{047B} \pdfglyphtounicode{omegatitlocyrillic}{047D} \pdfglyphtounicode{omegatonos}{03CE} \pdfglyphtounicode{omgujarati}{0AD0} \pdfglyphtounicode{omicron}{03BF} \pdfglyphtounicode{omicrontonos}{03CC} \pdfglyphtounicode{omonospace}{FF4F} \pdfglyphtounicode{one}{0031} \pdfglyphtounicode{onearabic}{0661} \pdfglyphtounicode{onebengali}{09E7} \pdfglyphtounicode{onecircle}{2460} \pdfglyphtounicode{onecircleinversesansserif}{278A} \pdfglyphtounicode{onedeva}{0967} \pdfglyphtounicode{onedotenleader}{2024} \pdfglyphtounicode{oneeighth}{215B} \pdfglyphtounicode{onefitted}{F6DC} \pdfglyphtounicode{onegujarati}{0AE7} \pdfglyphtounicode{onegurmukhi}{0A67} \pdfglyphtounicode{onehackarabic}{0661} \pdfglyphtounicode{onehalf}{00BD} \pdfglyphtounicode{onehangzhou}{3021} \pdfglyphtounicode{oneideographicparen}{3220} \pdfglyphtounicode{oneinferior}{2081} \pdfglyphtounicode{onemonospace}{FF11} \pdfglyphtounicode{onenumeratorbengali}{09F4} \pdfglyphtounicode{oneoldstyle}{F731} \pdfglyphtounicode{oneparen}{2474} \pdfglyphtounicode{oneperiod}{2488} \pdfglyphtounicode{onepersian}{06F1} \pdfglyphtounicode{onequarter}{00BC} \pdfglyphtounicode{oneroman}{2170} \pdfglyphtounicode{onesuperior}{00B9} \pdfglyphtounicode{onethai}{0E51} \pdfglyphtounicode{onethird}{2153} \pdfglyphtounicode{oogonek}{01EB} \pdfglyphtounicode{oogonekmacron}{01ED} \pdfglyphtounicode{oogurmukhi}{0A13} \pdfglyphtounicode{oomatragurmukhi}{0A4B} \pdfglyphtounicode{oopen}{0254} \pdfglyphtounicode{oparen}{24AA} \pdfglyphtounicode{openbullet}{25E6} \pdfglyphtounicode{option}{2325} \pdfglyphtounicode{ordfeminine}{00AA} \pdfglyphtounicode{ordmasculine}{00BA} \pdfglyphtounicode{orthogonal}{221F} \pdfglyphtounicode{oshortdeva}{0912} \pdfglyphtounicode{oshortvowelsigndeva}{094A} \pdfglyphtounicode{oslash}{00F8} \pdfglyphtounicode{oslashacute}{01FF} \pdfglyphtounicode{osmallhiragana}{3049} \pdfglyphtounicode{osmallkatakana}{30A9} \pdfglyphtounicode{osmallkatakanahalfwidth}{FF6B} \pdfglyphtounicode{ostrokeacute}{01FF} \pdfglyphtounicode{osuperior}{F6F0} \pdfglyphtounicode{otcyrillic}{047F} \pdfglyphtounicode{otilde}{00F5} \pdfglyphtounicode{otildeacute}{1E4D} \pdfglyphtounicode{otildedieresis}{1E4F} \pdfglyphtounicode{oubopomofo}{3121} \pdfglyphtounicode{overline}{203E} \pdfglyphtounicode{overlinecenterline}{FE4A} \pdfglyphtounicode{overlinecmb}{0305} \pdfglyphtounicode{overlinedashed}{FE49} \pdfglyphtounicode{overlinedblwavy}{FE4C} \pdfglyphtounicode{overlinewavy}{FE4B} \pdfglyphtounicode{overscore}{00AF} \pdfglyphtounicode{ovowelsignbengali}{09CB} \pdfglyphtounicode{ovowelsigndeva}{094B} \pdfglyphtounicode{ovowelsigngujarati}{0ACB} \pdfglyphtounicode{p}{0070} \pdfglyphtounicode{paampssquare}{3380} \pdfglyphtounicode{paasentosquare}{332B} \pdfglyphtounicode{pabengali}{09AA} \pdfglyphtounicode{pacute}{1E55} \pdfglyphtounicode{padeva}{092A} \pdfglyphtounicode{pagedown}{21DF} \pdfglyphtounicode{pageup}{21DE} \pdfglyphtounicode{pagujarati}{0AAA} \pdfglyphtounicode{pagurmukhi}{0A2A} \pdfglyphtounicode{pahiragana}{3071} \pdfglyphtounicode{paiyannoithai}{0E2F} \pdfglyphtounicode{pakatakana}{30D1} \pdfglyphtounicode{palatalizationcyrilliccmb}{0484} \pdfglyphtounicode{palochkacyrillic}{04C0} \pdfglyphtounicode{pansioskorean}{317F} \pdfglyphtounicode{paragraph}{00B6} \pdfglyphtounicode{parallel}{2225} \pdfglyphtounicode{parenleft}{0028} \pdfglyphtounicode{parenleftaltonearabic}{FD3E} \pdfglyphtounicode{parenleftbt}{F8ED} \pdfglyphtounicode{parenleftex}{F8EC} \pdfglyphtounicode{parenleftinferior}{208D} \pdfglyphtounicode{parenleftmonospace}{FF08} \pdfglyphtounicode{parenleftsmall}{FE59} \pdfglyphtounicode{parenleftsuperior}{207D} \pdfglyphtounicode{parenlefttp}{F8EB} \pdfglyphtounicode{parenleftvertical}{FE35} \pdfglyphtounicode{parenright}{0029} \pdfglyphtounicode{parenrightaltonearabic}{FD3F} \pdfglyphtounicode{parenrightbt}{F8F8} \pdfglyphtounicode{parenrightex}{F8F7} \pdfglyphtounicode{parenrightinferior}{208E} \pdfglyphtounicode{parenrightmonospace}{FF09} \pdfglyphtounicode{parenrightsmall}{FE5A} \pdfglyphtounicode{parenrightsuperior}{207E} \pdfglyphtounicode{parenrighttp}{F8F6} \pdfglyphtounicode{parenrightvertical}{FE36} \pdfglyphtounicode{partialdiff}{2202} \pdfglyphtounicode{paseqhebrew}{05C0} \pdfglyphtounicode{pashtahebrew}{0599} \pdfglyphtounicode{pasquare}{33A9} \pdfglyphtounicode{patah}{05B7} \pdfglyphtounicode{patah11}{05B7} \pdfglyphtounicode{patah1d}{05B7} \pdfglyphtounicode{patah2a}{05B7} \pdfglyphtounicode{patahhebrew}{05B7} \pdfglyphtounicode{patahnarrowhebrew}{05B7} \pdfglyphtounicode{patahquarterhebrew}{05B7} \pdfglyphtounicode{patahwidehebrew}{05B7} \pdfglyphtounicode{pazerhebrew}{05A1} \pdfglyphtounicode{pbopomofo}{3106} \pdfglyphtounicode{pcircle}{24DF} \pdfglyphtounicode{pdotaccent}{1E57} \pdfglyphtounicode{pe}{05E4} \pdfglyphtounicode{pecyrillic}{043F} \pdfglyphtounicode{pedagesh}{FB44} \pdfglyphtounicode{pedageshhebrew}{FB44} \pdfglyphtounicode{peezisquare}{333B} \pdfglyphtounicode{pefinaldageshhebrew}{FB43} \pdfglyphtounicode{peharabic}{067E} \pdfglyphtounicode{peharmenian}{057A} \pdfglyphtounicode{pehebrew}{05E4} \pdfglyphtounicode{pehfinalarabic}{FB57} \pdfglyphtounicode{pehinitialarabic}{FB58} \pdfglyphtounicode{pehiragana}{307A} \pdfglyphtounicode{pehmedialarabic}{FB59} \pdfglyphtounicode{pekatakana}{30DA} \pdfglyphtounicode{pemiddlehookcyrillic}{04A7} \pdfglyphtounicode{perafehebrew}{FB4E} \pdfglyphtounicode{percent}{0025} \pdfglyphtounicode{percentarabic}{066A} \pdfglyphtounicode{percentmonospace}{FF05} \pdfglyphtounicode{percentsmall}{FE6A} \pdfglyphtounicode{period}{002E} \pdfglyphtounicode{periodarmenian}{0589} \pdfglyphtounicode{periodcentered}{00B7} \pdfglyphtounicode{periodhalfwidth}{FF61} \pdfglyphtounicode{periodinferior}{F6E7} \pdfglyphtounicode{periodmonospace}{FF0E} \pdfglyphtounicode{periodsmall}{FE52} \pdfglyphtounicode{periodsuperior}{F6E8} \pdfglyphtounicode{perispomenigreekcmb}{0342} \pdfglyphtounicode{perpendicular}{22A5} \pdfglyphtounicode{perthousand}{2030} \pdfglyphtounicode{peseta}{20A7} \pdfglyphtounicode{pfsquare}{338A} \pdfglyphtounicode{phabengali}{09AB} \pdfglyphtounicode{phadeva}{092B} \pdfglyphtounicode{phagujarati}{0AAB} \pdfglyphtounicode{phagurmukhi}{0A2B} \pdfglyphtounicode{phi}{03C6} \pdfglyphtounicode{phi1}{03D5} \pdfglyphtounicode{phieuphacirclekorean}{327A} \pdfglyphtounicode{phieuphaparenkorean}{321A} \pdfglyphtounicode{phieuphcirclekorean}{326C} \pdfglyphtounicode{phieuphkorean}{314D} \pdfglyphtounicode{phieuphparenkorean}{320C} \pdfglyphtounicode{philatin}{0278} \pdfglyphtounicode{phinthuthai}{0E3A} \pdfglyphtounicode{phisymbolgreek}{03D5} \pdfglyphtounicode{phook}{01A5} \pdfglyphtounicode{phophanthai}{0E1E} \pdfglyphtounicode{phophungthai}{0E1C} \pdfglyphtounicode{phosamphaothai}{0E20} \pdfglyphtounicode{pi}{03C0} \pdfglyphtounicode{pieupacirclekorean}{3273} \pdfglyphtounicode{pieupaparenkorean}{3213} \pdfglyphtounicode{pieupcieuckorean}{3176} \pdfglyphtounicode{pieupcirclekorean}{3265} \pdfglyphtounicode{pieupkiyeokkorean}{3172} \pdfglyphtounicode{pieupkorean}{3142} \pdfglyphtounicode{pieupparenkorean}{3205} \pdfglyphtounicode{pieupsioskiyeokkorean}{3174} \pdfglyphtounicode{pieupsioskorean}{3144} \pdfglyphtounicode{pieupsiostikeutkorean}{3175} \pdfglyphtounicode{pieupthieuthkorean}{3177} \pdfglyphtounicode{pieuptikeutkorean}{3173} \pdfglyphtounicode{pihiragana}{3074} \pdfglyphtounicode{pikatakana}{30D4} \pdfglyphtounicode{pisymbolgreek}{03D6} \pdfglyphtounicode{piwrarmenian}{0583} \pdfglyphtounicode{plus}{002B} \pdfglyphtounicode{plusbelowcmb}{031F} \pdfglyphtounicode{pluscircle}{2295} \pdfglyphtounicode{plusminus}{00B1} \pdfglyphtounicode{plusmod}{02D6} \pdfglyphtounicode{plusmonospace}{FF0B} \pdfglyphtounicode{plussmall}{FE62} \pdfglyphtounicode{plussuperior}{207A} \pdfglyphtounicode{pmonospace}{FF50} \pdfglyphtounicode{pmsquare}{33D8} \pdfglyphtounicode{pohiragana}{307D} \pdfglyphtounicode{pointingindexdownwhite}{261F} \pdfglyphtounicode{pointingindexleftwhite}{261C} \pdfglyphtounicode{pointingindexrightwhite}{261E} \pdfglyphtounicode{pointingindexupwhite}{261D} \pdfglyphtounicode{pokatakana}{30DD} \pdfglyphtounicode{poplathai}{0E1B} \pdfglyphtounicode{postalmark}{3012} \pdfglyphtounicode{postalmarkface}{3020} \pdfglyphtounicode{pparen}{24AB} \pdfglyphtounicode{precedes}{227A} \pdfglyphtounicode{prescription}{211E} \pdfglyphtounicode{primemod}{02B9} \pdfglyphtounicode{primereversed}{2035} \pdfglyphtounicode{product}{220F} \pdfglyphtounicode{projective}{2305} \pdfglyphtounicode{prolongedkana}{30FC} \pdfglyphtounicode{propellor}{2318} \pdfglyphtounicode{propersubset}{2282} \pdfglyphtounicode{propersuperset}{2283} \pdfglyphtounicode{proportion}{2237} \pdfglyphtounicode{proportional}{221D} \pdfglyphtounicode{psi}{03C8} \pdfglyphtounicode{psicyrillic}{0471} \pdfglyphtounicode{psilipneumatacyrilliccmb}{0486} \pdfglyphtounicode{pssquare}{33B0} \pdfglyphtounicode{puhiragana}{3077} \pdfglyphtounicode{pukatakana}{30D7} \pdfglyphtounicode{pvsquare}{33B4} \pdfglyphtounicode{pwsquare}{33BA} \pdfglyphtounicode{q}{0071} \pdfglyphtounicode{qadeva}{0958} \pdfglyphtounicode{qadmahebrew}{05A8} \pdfglyphtounicode{qafarabic}{0642} \pdfglyphtounicode{qaffinalarabic}{FED6} \pdfglyphtounicode{qafinitialarabic}{FED7} \pdfglyphtounicode{qafmedialarabic}{FED8} \pdfglyphtounicode{qamats}{05B8} \pdfglyphtounicode{qamats10}{05B8} \pdfglyphtounicode{qamats1a}{05B8} \pdfglyphtounicode{qamats1c}{05B8} \pdfglyphtounicode{qamats27}{05B8} \pdfglyphtounicode{qamats29}{05B8} \pdfglyphtounicode{qamats33}{05B8} \pdfglyphtounicode{qamatsde}{05B8} \pdfglyphtounicode{qamatshebrew}{05B8} \pdfglyphtounicode{qamatsnarrowhebrew}{05B8} \pdfglyphtounicode{qamatsqatanhebrew}{05B8} \pdfglyphtounicode{qamatsqatannarrowhebrew}{05B8} \pdfglyphtounicode{qamatsqatanquarterhebrew}{05B8} \pdfglyphtounicode{qamatsqatanwidehebrew}{05B8} \pdfglyphtounicode{qamatsquarterhebrew}{05B8} \pdfglyphtounicode{qamatswidehebrew}{05B8} \pdfglyphtounicode{qarneyparahebrew}{059F} \pdfglyphtounicode{qbopomofo}{3111} \pdfglyphtounicode{qcircle}{24E0} \pdfglyphtounicode{qhook}{02A0} \pdfglyphtounicode{qmonospace}{FF51} \pdfglyphtounicode{qof}{05E7} \pdfglyphtounicode{qofdagesh}{FB47} \pdfglyphtounicode{qofdageshhebrew}{FB47} % qofhatafpatah;05E7 05B2 % qofhatafpatahhebrew;05E7 05B2 % qofhatafsegol;05E7 05B1 % qofhatafsegolhebrew;05E7 05B1 \pdfglyphtounicode{qofhebrew}{05E7} % qofhiriq;05E7 05B4 % qofhiriqhebrew;05E7 05B4 % qofholam;05E7 05B9 % qofholamhebrew;05E7 05B9 % qofpatah;05E7 05B7 % qofpatahhebrew;05E7 05B7 % qofqamats;05E7 05B8 % qofqamatshebrew;05E7 05B8 % qofqubuts;05E7 05BB % qofqubutshebrew;05E7 05BB % qofsegol;05E7 05B6 % qofsegolhebrew;05E7 05B6 % qofsheva;05E7 05B0 % qofshevahebrew;05E7 05B0 % qoftsere;05E7 05B5 % qoftserehebrew;05E7 05B5 \pdfglyphtounicode{qparen}{24AC} \pdfglyphtounicode{quarternote}{2669} \pdfglyphtounicode{qubuts}{05BB} \pdfglyphtounicode{qubuts18}{05BB} \pdfglyphtounicode{qubuts25}{05BB} \pdfglyphtounicode{qubuts31}{05BB} \pdfglyphtounicode{qubutshebrew}{05BB} \pdfglyphtounicode{qubutsnarrowhebrew}{05BB} \pdfglyphtounicode{qubutsquarterhebrew}{05BB} \pdfglyphtounicode{qubutswidehebrew}{05BB} \pdfglyphtounicode{question}{003F} \pdfglyphtounicode{questionarabic}{061F} \pdfglyphtounicode{questionarmenian}{055E} \pdfglyphtounicode{questiondown}{00BF} \pdfglyphtounicode{questiondownsmall}{F7BF} \pdfglyphtounicode{questiongreek}{037E} \pdfglyphtounicode{questionmonospace}{FF1F} \pdfglyphtounicode{questionsmall}{F73F} \pdfglyphtounicode{quotedbl}{0022} \pdfglyphtounicode{quotedblbase}{201E} \pdfglyphtounicode{quotedblleft}{201C} \pdfglyphtounicode{quotedblmonospace}{FF02} \pdfglyphtounicode{quotedblprime}{301E} \pdfglyphtounicode{quotedblprimereversed}{301D} \pdfglyphtounicode{quotedblright}{201D} \pdfglyphtounicode{quoteleft}{2018} \pdfglyphtounicode{quoteleftreversed}{201B} \pdfglyphtounicode{quotereversed}{201B} \pdfglyphtounicode{quoteright}{2019} \pdfglyphtounicode{quoterightn}{0149} \pdfglyphtounicode{quotesinglbase}{201A} \pdfglyphtounicode{quotesingle}{0027} \pdfglyphtounicode{quotesinglemonospace}{FF07} \pdfglyphtounicode{r}{0072} \pdfglyphtounicode{raarmenian}{057C} \pdfglyphtounicode{rabengali}{09B0} \pdfglyphtounicode{racute}{0155} \pdfglyphtounicode{radeva}{0930} \pdfglyphtounicode{radical}{221A} \pdfglyphtounicode{radicalex}{F8E5} \pdfglyphtounicode{radoverssquare}{33AE} \pdfglyphtounicode{radoverssquaredsquare}{33AF} \pdfglyphtounicode{radsquare}{33AD} \pdfglyphtounicode{rafe}{05BF} \pdfglyphtounicode{rafehebrew}{05BF} \pdfglyphtounicode{ragujarati}{0AB0} \pdfglyphtounicode{ragurmukhi}{0A30} \pdfglyphtounicode{rahiragana}{3089} \pdfglyphtounicode{rakatakana}{30E9} \pdfglyphtounicode{rakatakanahalfwidth}{FF97} \pdfglyphtounicode{ralowerdiagonalbengali}{09F1} \pdfglyphtounicode{ramiddlediagonalbengali}{09F0} \pdfglyphtounicode{ramshorn}{0264} \pdfglyphtounicode{ratio}{2236} \pdfglyphtounicode{rbopomofo}{3116} \pdfglyphtounicode{rcaron}{0159} \pdfglyphtounicode{rcedilla}{0157} \pdfglyphtounicode{rcircle}{24E1} \pdfglyphtounicode{rcommaaccent}{0157} \pdfglyphtounicode{rdblgrave}{0211} \pdfglyphtounicode{rdotaccent}{1E59} \pdfglyphtounicode{rdotbelow}{1E5B} \pdfglyphtounicode{rdotbelowmacron}{1E5D} \pdfglyphtounicode{referencemark}{203B} \pdfglyphtounicode{reflexsubset}{2286} \pdfglyphtounicode{reflexsuperset}{2287} \pdfglyphtounicode{registered}{00AE} \pdfglyphtounicode{registersans}{F8E8} \pdfglyphtounicode{registerserif}{F6DA} \pdfglyphtounicode{reharabic}{0631} \pdfglyphtounicode{reharmenian}{0580} \pdfglyphtounicode{rehfinalarabic}{FEAE} \pdfglyphtounicode{rehiragana}{308C} % rehyehaleflamarabic;0631 FEF3 FE8E 0644 \pdfglyphtounicode{rekatakana}{30EC} \pdfglyphtounicode{rekatakanahalfwidth}{FF9A} \pdfglyphtounicode{resh}{05E8} \pdfglyphtounicode{reshdageshhebrew}{FB48} % reshhatafpatah;05E8 05B2 % reshhatafpatahhebrew;05E8 05B2 % reshhatafsegol;05E8 05B1 % reshhatafsegolhebrew;05E8 05B1 \pdfglyphtounicode{reshhebrew}{05E8} % reshhiriq;05E8 05B4 % reshhiriqhebrew;05E8 05B4 % reshholam;05E8 05B9 % reshholamhebrew;05E8 05B9 % reshpatah;05E8 05B7 % reshpatahhebrew;05E8 05B7 % reshqamats;05E8 05B8 % reshqamatshebrew;05E8 05B8 % reshqubuts;05E8 05BB % reshqubutshebrew;05E8 05BB % reshsegol;05E8 05B6 % reshsegolhebrew;05E8 05B6 % reshsheva;05E8 05B0 % reshshevahebrew;05E8 05B0 % reshtsere;05E8 05B5 % reshtserehebrew;05E8 05B5 \pdfglyphtounicode{reversedtilde}{223D} \pdfglyphtounicode{reviahebrew}{0597} \pdfglyphtounicode{reviamugrashhebrew}{0597} \pdfglyphtounicode{revlogicalnot}{2310} \pdfglyphtounicode{rfishhook}{027E} \pdfglyphtounicode{rfishhookreversed}{027F} \pdfglyphtounicode{rhabengali}{09DD} \pdfglyphtounicode{rhadeva}{095D} \pdfglyphtounicode{rho}{03C1} \pdfglyphtounicode{rhook}{027D} \pdfglyphtounicode{rhookturned}{027B} \pdfglyphtounicode{rhookturnedsuperior}{02B5} \pdfglyphtounicode{rhosymbolgreek}{03F1} \pdfglyphtounicode{rhotichookmod}{02DE} \pdfglyphtounicode{rieulacirclekorean}{3271} \pdfglyphtounicode{rieulaparenkorean}{3211} \pdfglyphtounicode{rieulcirclekorean}{3263} \pdfglyphtounicode{rieulhieuhkorean}{3140} \pdfglyphtounicode{rieulkiyeokkorean}{313A} \pdfglyphtounicode{rieulkiyeoksioskorean}{3169} \pdfglyphtounicode{rieulkorean}{3139} \pdfglyphtounicode{rieulmieumkorean}{313B} \pdfglyphtounicode{rieulpansioskorean}{316C} \pdfglyphtounicode{rieulparenkorean}{3203} \pdfglyphtounicode{rieulphieuphkorean}{313F} \pdfglyphtounicode{rieulpieupkorean}{313C} \pdfglyphtounicode{rieulpieupsioskorean}{316B} \pdfglyphtounicode{rieulsioskorean}{313D} \pdfglyphtounicode{rieulthieuthkorean}{313E} \pdfglyphtounicode{rieultikeutkorean}{316A} \pdfglyphtounicode{rieulyeorinhieuhkorean}{316D} \pdfglyphtounicode{rightangle}{221F} \pdfglyphtounicode{righttackbelowcmb}{0319} \pdfglyphtounicode{righttriangle}{22BF} \pdfglyphtounicode{rihiragana}{308A} \pdfglyphtounicode{rikatakana}{30EA} \pdfglyphtounicode{rikatakanahalfwidth}{FF98} \pdfglyphtounicode{ring}{02DA} \pdfglyphtounicode{ringbelowcmb}{0325} \pdfglyphtounicode{ringcmb}{030A} \pdfglyphtounicode{ringhalfleft}{02BF} \pdfglyphtounicode{ringhalfleftarmenian}{0559} \pdfglyphtounicode{ringhalfleftbelowcmb}{031C} \pdfglyphtounicode{ringhalfleftcentered}{02D3} \pdfglyphtounicode{ringhalfright}{02BE} \pdfglyphtounicode{ringhalfrightbelowcmb}{0339} \pdfglyphtounicode{ringhalfrightcentered}{02D2} \pdfglyphtounicode{rinvertedbreve}{0213} \pdfglyphtounicode{rittorusquare}{3351} \pdfglyphtounicode{rlinebelow}{1E5F} \pdfglyphtounicode{rlongleg}{027C} \pdfglyphtounicode{rlonglegturned}{027A} \pdfglyphtounicode{rmonospace}{FF52} \pdfglyphtounicode{rohiragana}{308D} \pdfglyphtounicode{rokatakana}{30ED} \pdfglyphtounicode{rokatakanahalfwidth}{FF9B} \pdfglyphtounicode{roruathai}{0E23} \pdfglyphtounicode{rparen}{24AD} \pdfglyphtounicode{rrabengali}{09DC} \pdfglyphtounicode{rradeva}{0931} \pdfglyphtounicode{rragurmukhi}{0A5C} \pdfglyphtounicode{rreharabic}{0691} \pdfglyphtounicode{rrehfinalarabic}{FB8D} \pdfglyphtounicode{rrvocalicbengali}{09E0} \pdfglyphtounicode{rrvocalicdeva}{0960} \pdfglyphtounicode{rrvocalicgujarati}{0AE0} \pdfglyphtounicode{rrvocalicvowelsignbengali}{09C4} \pdfglyphtounicode{rrvocalicvowelsigndeva}{0944} \pdfglyphtounicode{rrvocalicvowelsigngujarati}{0AC4} \pdfglyphtounicode{rsuperior}{F6F1} \pdfglyphtounicode{rtblock}{2590} \pdfglyphtounicode{rturned}{0279} \pdfglyphtounicode{rturnedsuperior}{02B4} \pdfglyphtounicode{ruhiragana}{308B} \pdfglyphtounicode{rukatakana}{30EB} \pdfglyphtounicode{rukatakanahalfwidth}{FF99} \pdfglyphtounicode{rupeemarkbengali}{09F2} \pdfglyphtounicode{rupeesignbengali}{09F3} \pdfglyphtounicode{rupiah}{F6DD} \pdfglyphtounicode{ruthai}{0E24} \pdfglyphtounicode{rvocalicbengali}{098B} \pdfglyphtounicode{rvocalicdeva}{090B} \pdfglyphtounicode{rvocalicgujarati}{0A8B} \pdfglyphtounicode{rvocalicvowelsignbengali}{09C3} \pdfglyphtounicode{rvocalicvowelsigndeva}{0943} \pdfglyphtounicode{rvocalicvowelsigngujarati}{0AC3} \pdfglyphtounicode{s}{0073} \pdfglyphtounicode{sabengali}{09B8} \pdfglyphtounicode{sacute}{015B} \pdfglyphtounicode{sacutedotaccent}{1E65} \pdfglyphtounicode{sadarabic}{0635} \pdfglyphtounicode{sadeva}{0938} \pdfglyphtounicode{sadfinalarabic}{FEBA} \pdfglyphtounicode{sadinitialarabic}{FEBB} \pdfglyphtounicode{sadmedialarabic}{FEBC} \pdfglyphtounicode{sagujarati}{0AB8} \pdfglyphtounicode{sagurmukhi}{0A38} \pdfglyphtounicode{sahiragana}{3055} \pdfglyphtounicode{sakatakana}{30B5} \pdfglyphtounicode{sakatakanahalfwidth}{FF7B} \pdfglyphtounicode{sallallahoualayhewasallamarabic}{FDFA} \pdfglyphtounicode{samekh}{05E1} \pdfglyphtounicode{samekhdagesh}{FB41} \pdfglyphtounicode{samekhdageshhebrew}{FB41} \pdfglyphtounicode{samekhhebrew}{05E1} \pdfglyphtounicode{saraaathai}{0E32} \pdfglyphtounicode{saraaethai}{0E41} \pdfglyphtounicode{saraaimaimalaithai}{0E44} \pdfglyphtounicode{saraaimaimuanthai}{0E43} \pdfglyphtounicode{saraamthai}{0E33} \pdfglyphtounicode{saraathai}{0E30} \pdfglyphtounicode{saraethai}{0E40} \pdfglyphtounicode{saraiileftthai}{F886} \pdfglyphtounicode{saraiithai}{0E35} \pdfglyphtounicode{saraileftthai}{F885} \pdfglyphtounicode{saraithai}{0E34} \pdfglyphtounicode{saraothai}{0E42} \pdfglyphtounicode{saraueeleftthai}{F888} \pdfglyphtounicode{saraueethai}{0E37} \pdfglyphtounicode{saraueleftthai}{F887} \pdfglyphtounicode{sarauethai}{0E36} \pdfglyphtounicode{sarauthai}{0E38} \pdfglyphtounicode{sarauuthai}{0E39} \pdfglyphtounicode{sbopomofo}{3119} \pdfglyphtounicode{scaron}{0161} \pdfglyphtounicode{scarondotaccent}{1E67} \pdfglyphtounicode{scedilla}{015F} \pdfglyphtounicode{schwa}{0259} \pdfglyphtounicode{schwacyrillic}{04D9} \pdfglyphtounicode{schwadieresiscyrillic}{04DB} \pdfglyphtounicode{schwahook}{025A} \pdfglyphtounicode{scircle}{24E2} \pdfglyphtounicode{scircumflex}{015D} \pdfglyphtounicode{scommaaccent}{0219} \pdfglyphtounicode{sdotaccent}{1E61} \pdfglyphtounicode{sdotbelow}{1E63} \pdfglyphtounicode{sdotbelowdotaccent}{1E69} \pdfglyphtounicode{seagullbelowcmb}{033C} \pdfglyphtounicode{second}{2033} \pdfglyphtounicode{secondtonechinese}{02CA} \pdfglyphtounicode{section}{00A7} \pdfglyphtounicode{seenarabic}{0633} \pdfglyphtounicode{seenfinalarabic}{FEB2} \pdfglyphtounicode{seeninitialarabic}{FEB3} \pdfglyphtounicode{seenmedialarabic}{FEB4} \pdfglyphtounicode{segol}{05B6} \pdfglyphtounicode{segol13}{05B6} \pdfglyphtounicode{segol1f}{05B6} \pdfglyphtounicode{segol2c}{05B6} \pdfglyphtounicode{segolhebrew}{05B6} \pdfglyphtounicode{segolnarrowhebrew}{05B6} \pdfglyphtounicode{segolquarterhebrew}{05B6} \pdfglyphtounicode{segoltahebrew}{0592} \pdfglyphtounicode{segolwidehebrew}{05B6} \pdfglyphtounicode{seharmenian}{057D} \pdfglyphtounicode{sehiragana}{305B} \pdfglyphtounicode{sekatakana}{30BB} \pdfglyphtounicode{sekatakanahalfwidth}{FF7E} \pdfglyphtounicode{semicolon}{003B} \pdfglyphtounicode{semicolonarabic}{061B} \pdfglyphtounicode{semicolonmonospace}{FF1B} \pdfglyphtounicode{semicolonsmall}{FE54} \pdfglyphtounicode{semivoicedmarkkana}{309C} \pdfglyphtounicode{semivoicedmarkkanahalfwidth}{FF9F} \pdfglyphtounicode{sentisquare}{3322} \pdfglyphtounicode{sentosquare}{3323} \pdfglyphtounicode{seven}{0037} \pdfglyphtounicode{sevenarabic}{0667} \pdfglyphtounicode{sevenbengali}{09ED} \pdfglyphtounicode{sevencircle}{2466} \pdfglyphtounicode{sevencircleinversesansserif}{2790} \pdfglyphtounicode{sevendeva}{096D} \pdfglyphtounicode{seveneighths}{215E} \pdfglyphtounicode{sevengujarati}{0AED} \pdfglyphtounicode{sevengurmukhi}{0A6D} \pdfglyphtounicode{sevenhackarabic}{0667} \pdfglyphtounicode{sevenhangzhou}{3027} \pdfglyphtounicode{sevenideographicparen}{3226} \pdfglyphtounicode{seveninferior}{2087} \pdfglyphtounicode{sevenmonospace}{FF17} \pdfglyphtounicode{sevenoldstyle}{F737} \pdfglyphtounicode{sevenparen}{247A} \pdfglyphtounicode{sevenperiod}{248E} \pdfglyphtounicode{sevenpersian}{06F7} \pdfglyphtounicode{sevenroman}{2176} \pdfglyphtounicode{sevensuperior}{2077} \pdfglyphtounicode{seventeencircle}{2470} \pdfglyphtounicode{seventeenparen}{2484} \pdfglyphtounicode{seventeenperiod}{2498} \pdfglyphtounicode{seventhai}{0E57} \pdfglyphtounicode{sfthyphen}{00AD} \pdfglyphtounicode{shaarmenian}{0577} \pdfglyphtounicode{shabengali}{09B6} \pdfglyphtounicode{shacyrillic}{0448} \pdfglyphtounicode{shaddaarabic}{0651} \pdfglyphtounicode{shaddadammaarabic}{FC61} \pdfglyphtounicode{shaddadammatanarabic}{FC5E} \pdfglyphtounicode{shaddafathaarabic}{FC60} % shaddafathatanarabic;0651 064B \pdfglyphtounicode{shaddakasraarabic}{FC62} \pdfglyphtounicode{shaddakasratanarabic}{FC5F} \pdfglyphtounicode{shade}{2592} \pdfglyphtounicode{shadedark}{2593} \pdfglyphtounicode{shadelight}{2591} \pdfglyphtounicode{shademedium}{2592} \pdfglyphtounicode{shadeva}{0936} \pdfglyphtounicode{shagujarati}{0AB6} \pdfglyphtounicode{shagurmukhi}{0A36} \pdfglyphtounicode{shalshelethebrew}{0593} \pdfglyphtounicode{shbopomofo}{3115} \pdfglyphtounicode{shchacyrillic}{0449} \pdfglyphtounicode{sheenarabic}{0634} \pdfglyphtounicode{sheenfinalarabic}{FEB6} \pdfglyphtounicode{sheeninitialarabic}{FEB7} \pdfglyphtounicode{sheenmedialarabic}{FEB8} \pdfglyphtounicode{sheicoptic}{03E3} \pdfglyphtounicode{sheqel}{20AA} \pdfglyphtounicode{sheqelhebrew}{20AA} \pdfglyphtounicode{sheva}{05B0} \pdfglyphtounicode{sheva115}{05B0} \pdfglyphtounicode{sheva15}{05B0} \pdfglyphtounicode{sheva22}{05B0} \pdfglyphtounicode{sheva2e}{05B0} \pdfglyphtounicode{shevahebrew}{05B0} \pdfglyphtounicode{shevanarrowhebrew}{05B0} \pdfglyphtounicode{shevaquarterhebrew}{05B0} \pdfglyphtounicode{shevawidehebrew}{05B0} \pdfglyphtounicode{shhacyrillic}{04BB} \pdfglyphtounicode{shimacoptic}{03ED} \pdfglyphtounicode{shin}{05E9} \pdfglyphtounicode{shindagesh}{FB49} \pdfglyphtounicode{shindageshhebrew}{FB49} \pdfglyphtounicode{shindageshshindot}{FB2C} \pdfglyphtounicode{shindageshshindothebrew}{FB2C} \pdfglyphtounicode{shindageshsindot}{FB2D} \pdfglyphtounicode{shindageshsindothebrew}{FB2D} \pdfglyphtounicode{shindothebrew}{05C1} \pdfglyphtounicode{shinhebrew}{05E9} \pdfglyphtounicode{shinshindot}{FB2A} \pdfglyphtounicode{shinshindothebrew}{FB2A} \pdfglyphtounicode{shinsindot}{FB2B} \pdfglyphtounicode{shinsindothebrew}{FB2B} \pdfglyphtounicode{shook}{0282} \pdfglyphtounicode{sigma}{03C3} \pdfglyphtounicode{sigma1}{03C2} \pdfglyphtounicode{sigmafinal}{03C2} \pdfglyphtounicode{sigmalunatesymbolgreek}{03F2} \pdfglyphtounicode{sihiragana}{3057} \pdfglyphtounicode{sikatakana}{30B7} \pdfglyphtounicode{sikatakanahalfwidth}{FF7C} \pdfglyphtounicode{siluqhebrew}{05BD} \pdfglyphtounicode{siluqlefthebrew}{05BD} \pdfglyphtounicode{similar}{223C} \pdfglyphtounicode{sindothebrew}{05C2} \pdfglyphtounicode{siosacirclekorean}{3274} \pdfglyphtounicode{siosaparenkorean}{3214} \pdfglyphtounicode{sioscieuckorean}{317E} \pdfglyphtounicode{sioscirclekorean}{3266} \pdfglyphtounicode{sioskiyeokkorean}{317A} \pdfglyphtounicode{sioskorean}{3145} \pdfglyphtounicode{siosnieunkorean}{317B} \pdfglyphtounicode{siosparenkorean}{3206} \pdfglyphtounicode{siospieupkorean}{317D} \pdfglyphtounicode{siostikeutkorean}{317C} \pdfglyphtounicode{six}{0036} \pdfglyphtounicode{sixarabic}{0666} \pdfglyphtounicode{sixbengali}{09EC} \pdfglyphtounicode{sixcircle}{2465} \pdfglyphtounicode{sixcircleinversesansserif}{278F} \pdfglyphtounicode{sixdeva}{096C} \pdfglyphtounicode{sixgujarati}{0AEC} \pdfglyphtounicode{sixgurmukhi}{0A6C} \pdfglyphtounicode{sixhackarabic}{0666} \pdfglyphtounicode{sixhangzhou}{3026} \pdfglyphtounicode{sixideographicparen}{3225} \pdfglyphtounicode{sixinferior}{2086} \pdfglyphtounicode{sixmonospace}{FF16} \pdfglyphtounicode{sixoldstyle}{F736} \pdfglyphtounicode{sixparen}{2479} \pdfglyphtounicode{sixperiod}{248D} \pdfglyphtounicode{sixpersian}{06F6} \pdfglyphtounicode{sixroman}{2175} \pdfglyphtounicode{sixsuperior}{2076} \pdfglyphtounicode{sixteencircle}{246F} \pdfglyphtounicode{sixteencurrencydenominatorbengali}{09F9} \pdfglyphtounicode{sixteenparen}{2483} \pdfglyphtounicode{sixteenperiod}{2497} \pdfglyphtounicode{sixthai}{0E56} \pdfglyphtounicode{slash}{002F} \pdfglyphtounicode{slashmonospace}{FF0F} \pdfglyphtounicode{slong}{017F} \pdfglyphtounicode{slongdotaccent}{1E9B} \pdfglyphtounicode{smileface}{263A} \pdfglyphtounicode{smonospace}{FF53} \pdfglyphtounicode{sofpasuqhebrew}{05C3} \pdfglyphtounicode{softhyphen}{00AD} \pdfglyphtounicode{softsigncyrillic}{044C} \pdfglyphtounicode{sohiragana}{305D} \pdfglyphtounicode{sokatakana}{30BD} \pdfglyphtounicode{sokatakanahalfwidth}{FF7F} \pdfglyphtounicode{soliduslongoverlaycmb}{0338} \pdfglyphtounicode{solidusshortoverlaycmb}{0337} \pdfglyphtounicode{sorusithai}{0E29} \pdfglyphtounicode{sosalathai}{0E28} \pdfglyphtounicode{sosothai}{0E0B} \pdfglyphtounicode{sosuathai}{0E2A} \pdfglyphtounicode{space}{0020} \pdfglyphtounicode{spacehackarabic}{0020} \pdfglyphtounicode{spade}{2660} \pdfglyphtounicode{spadesuitblack}{2660} \pdfglyphtounicode{spadesuitwhite}{2664} \pdfglyphtounicode{sparen}{24AE} \pdfglyphtounicode{squarebelowcmb}{033B} \pdfglyphtounicode{squarecc}{33C4} \pdfglyphtounicode{squarecm}{339D} \pdfglyphtounicode{squarediagonalcrosshatchfill}{25A9} \pdfglyphtounicode{squarehorizontalfill}{25A4} \pdfglyphtounicode{squarekg}{338F} \pdfglyphtounicode{squarekm}{339E} \pdfglyphtounicode{squarekmcapital}{33CE} \pdfglyphtounicode{squareln}{33D1} \pdfglyphtounicode{squarelog}{33D2} \pdfglyphtounicode{squaremg}{338E} \pdfglyphtounicode{squaremil}{33D5} \pdfglyphtounicode{squaremm}{339C} \pdfglyphtounicode{squaremsquared}{33A1} \pdfglyphtounicode{squareorthogonalcrosshatchfill}{25A6} \pdfglyphtounicode{squareupperlefttolowerrightfill}{25A7} \pdfglyphtounicode{squareupperrighttolowerleftfill}{25A8} \pdfglyphtounicode{squareverticalfill}{25A5} \pdfglyphtounicode{squarewhitewithsmallblack}{25A3} \pdfglyphtounicode{srsquare}{33DB} \pdfglyphtounicode{ssabengali}{09B7} \pdfglyphtounicode{ssadeva}{0937} \pdfglyphtounicode{ssagujarati}{0AB7} \pdfglyphtounicode{ssangcieuckorean}{3149} \pdfglyphtounicode{ssanghieuhkorean}{3185} \pdfglyphtounicode{ssangieungkorean}{3180} \pdfglyphtounicode{ssangkiyeokkorean}{3132} \pdfglyphtounicode{ssangnieunkorean}{3165} \pdfglyphtounicode{ssangpieupkorean}{3143} \pdfglyphtounicode{ssangsioskorean}{3146} \pdfglyphtounicode{ssangtikeutkorean}{3138} \pdfglyphtounicode{ssuperior}{F6F2} \pdfglyphtounicode{sterling}{00A3} \pdfglyphtounicode{sterlingmonospace}{FFE1} \pdfglyphtounicode{strokelongoverlaycmb}{0336} \pdfglyphtounicode{strokeshortoverlaycmb}{0335} \pdfglyphtounicode{subset}{2282} \pdfglyphtounicode{subsetnotequal}{228A} \pdfglyphtounicode{subsetorequal}{2286} \pdfglyphtounicode{succeeds}{227B} \pdfglyphtounicode{suchthat}{220B} \pdfglyphtounicode{suhiragana}{3059} \pdfglyphtounicode{sukatakana}{30B9} \pdfglyphtounicode{sukatakanahalfwidth}{FF7D} \pdfglyphtounicode{sukunarabic}{0652} \pdfglyphtounicode{summation}{2211} \pdfglyphtounicode{sun}{263C} \pdfglyphtounicode{superset}{2283} \pdfglyphtounicode{supersetnotequal}{228B} \pdfglyphtounicode{supersetorequal}{2287} \pdfglyphtounicode{svsquare}{33DC} \pdfglyphtounicode{syouwaerasquare}{337C} \pdfglyphtounicode{t}{0074} \pdfglyphtounicode{tabengali}{09A4} \pdfglyphtounicode{tackdown}{22A4} \pdfglyphtounicode{tackleft}{22A3} \pdfglyphtounicode{tadeva}{0924} \pdfglyphtounicode{tagujarati}{0AA4} \pdfglyphtounicode{tagurmukhi}{0A24} \pdfglyphtounicode{taharabic}{0637} \pdfglyphtounicode{tahfinalarabic}{FEC2} \pdfglyphtounicode{tahinitialarabic}{FEC3} \pdfglyphtounicode{tahiragana}{305F} \pdfglyphtounicode{tahmedialarabic}{FEC4} \pdfglyphtounicode{taisyouerasquare}{337D} \pdfglyphtounicode{takatakana}{30BF} \pdfglyphtounicode{takatakanahalfwidth}{FF80} \pdfglyphtounicode{tatweelarabic}{0640} \pdfglyphtounicode{tau}{03C4} \pdfglyphtounicode{tav}{05EA} \pdfglyphtounicode{tavdages}{FB4A} \pdfglyphtounicode{tavdagesh}{FB4A} \pdfglyphtounicode{tavdageshhebrew}{FB4A} \pdfglyphtounicode{tavhebrew}{05EA} \pdfglyphtounicode{tbar}{0167} \pdfglyphtounicode{tbopomofo}{310A} \pdfglyphtounicode{tcaron}{0165} \pdfglyphtounicode{tccurl}{02A8} \pdfglyphtounicode{tcedilla}{0163} \pdfglyphtounicode{tcheharabic}{0686} \pdfglyphtounicode{tchehfinalarabic}{FB7B} \pdfglyphtounicode{tchehinitialarabic}{FB7C} \pdfglyphtounicode{tchehmedialarabic}{FB7D} % tchehmeeminitialarabic;FB7C FEE4 \pdfglyphtounicode{tcircle}{24E3} \pdfglyphtounicode{tcircumflexbelow}{1E71} \pdfglyphtounicode{tcommaaccent}{0163} \pdfglyphtounicode{tdieresis}{1E97} \pdfglyphtounicode{tdotaccent}{1E6B} \pdfglyphtounicode{tdotbelow}{1E6D} \pdfglyphtounicode{tecyrillic}{0442} \pdfglyphtounicode{tedescendercyrillic}{04AD} \pdfglyphtounicode{teharabic}{062A} \pdfglyphtounicode{tehfinalarabic}{FE96} \pdfglyphtounicode{tehhahinitialarabic}{FCA2} \pdfglyphtounicode{tehhahisolatedarabic}{FC0C} \pdfglyphtounicode{tehinitialarabic}{FE97} \pdfglyphtounicode{tehiragana}{3066} \pdfglyphtounicode{tehjeeminitialarabic}{FCA1} \pdfglyphtounicode{tehjeemisolatedarabic}{FC0B} \pdfglyphtounicode{tehmarbutaarabic}{0629} \pdfglyphtounicode{tehmarbutafinalarabic}{FE94} \pdfglyphtounicode{tehmedialarabic}{FE98} \pdfglyphtounicode{tehmeeminitialarabic}{FCA4} \pdfglyphtounicode{tehmeemisolatedarabic}{FC0E} \pdfglyphtounicode{tehnoonfinalarabic}{FC73} \pdfglyphtounicode{tekatakana}{30C6} \pdfglyphtounicode{tekatakanahalfwidth}{FF83} \pdfglyphtounicode{telephone}{2121} \pdfglyphtounicode{telephoneblack}{260E} \pdfglyphtounicode{telishagedolahebrew}{05A0} \pdfglyphtounicode{telishaqetanahebrew}{05A9} \pdfglyphtounicode{tencircle}{2469} \pdfglyphtounicode{tenideographicparen}{3229} \pdfglyphtounicode{tenparen}{247D} \pdfglyphtounicode{tenperiod}{2491} \pdfglyphtounicode{tenroman}{2179} \pdfglyphtounicode{tesh}{02A7} \pdfglyphtounicode{tet}{05D8} \pdfglyphtounicode{tetdagesh}{FB38} \pdfglyphtounicode{tetdageshhebrew}{FB38} \pdfglyphtounicode{tethebrew}{05D8} \pdfglyphtounicode{tetsecyrillic}{04B5} \pdfglyphtounicode{tevirhebrew}{059B} \pdfglyphtounicode{tevirlefthebrew}{059B} \pdfglyphtounicode{thabengali}{09A5} \pdfglyphtounicode{thadeva}{0925} \pdfglyphtounicode{thagujarati}{0AA5} \pdfglyphtounicode{thagurmukhi}{0A25} \pdfglyphtounicode{thalarabic}{0630} \pdfglyphtounicode{thalfinalarabic}{FEAC} \pdfglyphtounicode{thanthakhatlowleftthai}{F898} \pdfglyphtounicode{thanthakhatlowrightthai}{F897} \pdfglyphtounicode{thanthakhatthai}{0E4C} \pdfglyphtounicode{thanthakhatupperleftthai}{F896} \pdfglyphtounicode{theharabic}{062B} \pdfglyphtounicode{thehfinalarabic}{FE9A} \pdfglyphtounicode{thehinitialarabic}{FE9B} \pdfglyphtounicode{thehmedialarabic}{FE9C} \pdfglyphtounicode{thereexists}{2203} \pdfglyphtounicode{therefore}{2234} \pdfglyphtounicode{theta}{03B8} \pdfglyphtounicode{theta1}{03D1} \pdfglyphtounicode{thetasymbolgreek}{03D1} \pdfglyphtounicode{thieuthacirclekorean}{3279} \pdfglyphtounicode{thieuthaparenkorean}{3219} \pdfglyphtounicode{thieuthcirclekorean}{326B} \pdfglyphtounicode{thieuthkorean}{314C} \pdfglyphtounicode{thieuthparenkorean}{320B} \pdfglyphtounicode{thirteencircle}{246C} \pdfglyphtounicode{thirteenparen}{2480} \pdfglyphtounicode{thirteenperiod}{2494} \pdfglyphtounicode{thonangmonthothai}{0E11} \pdfglyphtounicode{thook}{01AD} \pdfglyphtounicode{thophuthaothai}{0E12} \pdfglyphtounicode{thorn}{00FE} \pdfglyphtounicode{thothahanthai}{0E17} \pdfglyphtounicode{thothanthai}{0E10} \pdfglyphtounicode{thothongthai}{0E18} \pdfglyphtounicode{thothungthai}{0E16} \pdfglyphtounicode{thousandcyrillic}{0482} \pdfglyphtounicode{thousandsseparatorarabic}{066C} \pdfglyphtounicode{thousandsseparatorpersian}{066C} \pdfglyphtounicode{three}{0033} \pdfglyphtounicode{threearabic}{0663} \pdfglyphtounicode{threebengali}{09E9} \pdfglyphtounicode{threecircle}{2462} \pdfglyphtounicode{threecircleinversesansserif}{278C} \pdfglyphtounicode{threedeva}{0969} \pdfglyphtounicode{threeeighths}{215C} \pdfglyphtounicode{threegujarati}{0AE9} \pdfglyphtounicode{threegurmukhi}{0A69} \pdfglyphtounicode{threehackarabic}{0663} \pdfglyphtounicode{threehangzhou}{3023} \pdfglyphtounicode{threeideographicparen}{3222} \pdfglyphtounicode{threeinferior}{2083} \pdfglyphtounicode{threemonospace}{FF13} \pdfglyphtounicode{threenumeratorbengali}{09F6} \pdfglyphtounicode{threeoldstyle}{F733} \pdfglyphtounicode{threeparen}{2476} \pdfglyphtounicode{threeperiod}{248A} \pdfglyphtounicode{threepersian}{06F3} \pdfglyphtounicode{threequarters}{00BE} \pdfglyphtounicode{threequartersemdash}{F6DE} \pdfglyphtounicode{threeroman}{2172} \pdfglyphtounicode{threesuperior}{00B3} \pdfglyphtounicode{threethai}{0E53} \pdfglyphtounicode{thzsquare}{3394} \pdfglyphtounicode{tihiragana}{3061} \pdfglyphtounicode{tikatakana}{30C1} \pdfglyphtounicode{tikatakanahalfwidth}{FF81} \pdfglyphtounicode{tikeutacirclekorean}{3270} \pdfglyphtounicode{tikeutaparenkorean}{3210} \pdfglyphtounicode{tikeutcirclekorean}{3262} \pdfglyphtounicode{tikeutkorean}{3137} \pdfglyphtounicode{tikeutparenkorean}{3202} \pdfglyphtounicode{tilde}{02DC} \pdfglyphtounicode{tildebelowcmb}{0330} \pdfglyphtounicode{tildecmb}{0303} \pdfglyphtounicode{tildecomb}{0303} \pdfglyphtounicode{tildedoublecmb}{0360} \pdfglyphtounicode{tildeoperator}{223C} \pdfglyphtounicode{tildeoverlaycmb}{0334} \pdfglyphtounicode{tildeverticalcmb}{033E} \pdfglyphtounicode{timescircle}{2297} \pdfglyphtounicode{tipehahebrew}{0596} \pdfglyphtounicode{tipehalefthebrew}{0596} \pdfglyphtounicode{tippigurmukhi}{0A70} \pdfglyphtounicode{titlocyrilliccmb}{0483} \pdfglyphtounicode{tiwnarmenian}{057F} \pdfglyphtounicode{tlinebelow}{1E6F} \pdfglyphtounicode{tmonospace}{FF54} \pdfglyphtounicode{toarmenian}{0569} \pdfglyphtounicode{tohiragana}{3068} \pdfglyphtounicode{tokatakana}{30C8} \pdfglyphtounicode{tokatakanahalfwidth}{FF84} \pdfglyphtounicode{tonebarextrahighmod}{02E5} \pdfglyphtounicode{tonebarextralowmod}{02E9} \pdfglyphtounicode{tonebarhighmod}{02E6} \pdfglyphtounicode{tonebarlowmod}{02E8} \pdfglyphtounicode{tonebarmidmod}{02E7} \pdfglyphtounicode{tonefive}{01BD} \pdfglyphtounicode{tonesix}{0185} \pdfglyphtounicode{tonetwo}{01A8} \pdfglyphtounicode{tonos}{0384} \pdfglyphtounicode{tonsquare}{3327} \pdfglyphtounicode{topatakthai}{0E0F} \pdfglyphtounicode{tortoiseshellbracketleft}{3014} \pdfglyphtounicode{tortoiseshellbracketleftsmall}{FE5D} \pdfglyphtounicode{tortoiseshellbracketleftvertical}{FE39} \pdfglyphtounicode{tortoiseshellbracketright}{3015} \pdfglyphtounicode{tortoiseshellbracketrightsmall}{FE5E} \pdfglyphtounicode{tortoiseshellbracketrightvertical}{FE3A} \pdfglyphtounicode{totaothai}{0E15} \pdfglyphtounicode{tpalatalhook}{01AB} \pdfglyphtounicode{tparen}{24AF} \pdfglyphtounicode{trademark}{2122} \pdfglyphtounicode{trademarksans}{F8EA} \pdfglyphtounicode{trademarkserif}{F6DB} \pdfglyphtounicode{tretroflexhook}{0288} \pdfglyphtounicode{triagdn}{25BC} \pdfglyphtounicode{triaglf}{25C4} \pdfglyphtounicode{triagrt}{25BA} \pdfglyphtounicode{triagup}{25B2} \pdfglyphtounicode{ts}{02A6} \pdfglyphtounicode{tsadi}{05E6} \pdfglyphtounicode{tsadidagesh}{FB46} \pdfglyphtounicode{tsadidageshhebrew}{FB46} \pdfglyphtounicode{tsadihebrew}{05E6} \pdfglyphtounicode{tsecyrillic}{0446} \pdfglyphtounicode{tsere}{05B5} \pdfglyphtounicode{tsere12}{05B5} \pdfglyphtounicode{tsere1e}{05B5} \pdfglyphtounicode{tsere2b}{05B5} \pdfglyphtounicode{tserehebrew}{05B5} \pdfglyphtounicode{tserenarrowhebrew}{05B5} \pdfglyphtounicode{tserequarterhebrew}{05B5} \pdfglyphtounicode{tserewidehebrew}{05B5} \pdfglyphtounicode{tshecyrillic}{045B} \pdfglyphtounicode{tsuperior}{F6F3} \pdfglyphtounicode{ttabengali}{099F} \pdfglyphtounicode{ttadeva}{091F} \pdfglyphtounicode{ttagujarati}{0A9F} \pdfglyphtounicode{ttagurmukhi}{0A1F} \pdfglyphtounicode{tteharabic}{0679} \pdfglyphtounicode{ttehfinalarabic}{FB67} \pdfglyphtounicode{ttehinitialarabic}{FB68} \pdfglyphtounicode{ttehmedialarabic}{FB69} \pdfglyphtounicode{tthabengali}{09A0} \pdfglyphtounicode{tthadeva}{0920} \pdfglyphtounicode{tthagujarati}{0AA0} \pdfglyphtounicode{tthagurmukhi}{0A20} \pdfglyphtounicode{tturned}{0287} \pdfglyphtounicode{tuhiragana}{3064} \pdfglyphtounicode{tukatakana}{30C4} \pdfglyphtounicode{tukatakanahalfwidth}{FF82} \pdfglyphtounicode{tusmallhiragana}{3063} \pdfglyphtounicode{tusmallkatakana}{30C3} \pdfglyphtounicode{tusmallkatakanahalfwidth}{FF6F} \pdfglyphtounicode{twelvecircle}{246B} \pdfglyphtounicode{twelveparen}{247F} \pdfglyphtounicode{twelveperiod}{2493} \pdfglyphtounicode{twelveroman}{217B} \pdfglyphtounicode{twentycircle}{2473} \pdfglyphtounicode{twentyhangzhou}{5344} \pdfglyphtounicode{twentyparen}{2487} \pdfglyphtounicode{twentyperiod}{249B} \pdfglyphtounicode{two}{0032} \pdfglyphtounicode{twoarabic}{0662} \pdfglyphtounicode{twobengali}{09E8} \pdfglyphtounicode{twocircle}{2461} \pdfglyphtounicode{twocircleinversesansserif}{278B} \pdfglyphtounicode{twodeva}{0968} \pdfglyphtounicode{twodotenleader}{2025} \pdfglyphtounicode{twodotleader}{2025} \pdfglyphtounicode{twodotleadervertical}{FE30} \pdfglyphtounicode{twogujarati}{0AE8} \pdfglyphtounicode{twogurmukhi}{0A68} \pdfglyphtounicode{twohackarabic}{0662} \pdfglyphtounicode{twohangzhou}{3022} \pdfglyphtounicode{twoideographicparen}{3221} \pdfglyphtounicode{twoinferior}{2082} \pdfglyphtounicode{twomonospace}{FF12} \pdfglyphtounicode{twonumeratorbengali}{09F5} \pdfglyphtounicode{twooldstyle}{F732} \pdfglyphtounicode{twoparen}{2475} \pdfglyphtounicode{twoperiod}{2489} \pdfglyphtounicode{twopersian}{06F2} \pdfglyphtounicode{tworoman}{2171} \pdfglyphtounicode{twostroke}{01BB} \pdfglyphtounicode{twosuperior}{00B2} \pdfglyphtounicode{twothai}{0E52} \pdfglyphtounicode{twothirds}{2154} \pdfglyphtounicode{u}{0075} \pdfglyphtounicode{uacute}{00FA} \pdfglyphtounicode{ubar}{0289} \pdfglyphtounicode{ubengali}{0989} \pdfglyphtounicode{ubopomofo}{3128} \pdfglyphtounicode{ubreve}{016D} \pdfglyphtounicode{ucaron}{01D4} \pdfglyphtounicode{ucircle}{24E4} \pdfglyphtounicode{ucircumflex}{00FB} \pdfglyphtounicode{ucircumflexbelow}{1E77} \pdfglyphtounicode{ucyrillic}{0443} \pdfglyphtounicode{udattadeva}{0951} \pdfglyphtounicode{udblacute}{0171} \pdfglyphtounicode{udblgrave}{0215} \pdfglyphtounicode{udeva}{0909} \pdfglyphtounicode{udieresis}{00FC} \pdfglyphtounicode{udieresisacute}{01D8} \pdfglyphtounicode{udieresisbelow}{1E73} \pdfglyphtounicode{udieresiscaron}{01DA} \pdfglyphtounicode{udieresiscyrillic}{04F1} \pdfglyphtounicode{udieresisgrave}{01DC} \pdfglyphtounicode{udieresismacron}{01D6} \pdfglyphtounicode{udotbelow}{1EE5} \pdfglyphtounicode{ugrave}{00F9} \pdfglyphtounicode{ugujarati}{0A89} \pdfglyphtounicode{ugurmukhi}{0A09} \pdfglyphtounicode{uhiragana}{3046} \pdfglyphtounicode{uhookabove}{1EE7} \pdfglyphtounicode{uhorn}{01B0} \pdfglyphtounicode{uhornacute}{1EE9} \pdfglyphtounicode{uhorndotbelow}{1EF1} \pdfglyphtounicode{uhorngrave}{1EEB} \pdfglyphtounicode{uhornhookabove}{1EED} \pdfglyphtounicode{uhorntilde}{1EEF} \pdfglyphtounicode{uhungarumlaut}{0171} \pdfglyphtounicode{uhungarumlautcyrillic}{04F3} \pdfglyphtounicode{uinvertedbreve}{0217} \pdfglyphtounicode{ukatakana}{30A6} \pdfglyphtounicode{ukatakanahalfwidth}{FF73} \pdfglyphtounicode{ukcyrillic}{0479} \pdfglyphtounicode{ukorean}{315C} \pdfglyphtounicode{umacron}{016B} \pdfglyphtounicode{umacroncyrillic}{04EF} \pdfglyphtounicode{umacrondieresis}{1E7B} \pdfglyphtounicode{umatragurmukhi}{0A41} \pdfglyphtounicode{umonospace}{FF55} \pdfglyphtounicode{underscore}{005F} \pdfglyphtounicode{underscoredbl}{2017} \pdfglyphtounicode{underscoremonospace}{FF3F} \pdfglyphtounicode{underscorevertical}{FE33} \pdfglyphtounicode{underscorewavy}{FE4F} \pdfglyphtounicode{union}{222A} \pdfglyphtounicode{universal}{2200} \pdfglyphtounicode{uogonek}{0173} \pdfglyphtounicode{uparen}{24B0} \pdfglyphtounicode{upblock}{2580} \pdfglyphtounicode{upperdothebrew}{05C4} \pdfglyphtounicode{upsilon}{03C5} \pdfglyphtounicode{upsilondieresis}{03CB} \pdfglyphtounicode{upsilondieresistonos}{03B0} \pdfglyphtounicode{upsilonlatin}{028A} \pdfglyphtounicode{upsilontonos}{03CD} \pdfglyphtounicode{uptackbelowcmb}{031D} \pdfglyphtounicode{uptackmod}{02D4} \pdfglyphtounicode{uragurmukhi}{0A73} \pdfglyphtounicode{uring}{016F} \pdfglyphtounicode{ushortcyrillic}{045E} \pdfglyphtounicode{usmallhiragana}{3045} \pdfglyphtounicode{usmallkatakana}{30A5} \pdfglyphtounicode{usmallkatakanahalfwidth}{FF69} \pdfglyphtounicode{ustraightcyrillic}{04AF} \pdfglyphtounicode{ustraightstrokecyrillic}{04B1} \pdfglyphtounicode{utilde}{0169} \pdfglyphtounicode{utildeacute}{1E79} \pdfglyphtounicode{utildebelow}{1E75} \pdfglyphtounicode{uubengali}{098A} \pdfglyphtounicode{uudeva}{090A} \pdfglyphtounicode{uugujarati}{0A8A} \pdfglyphtounicode{uugurmukhi}{0A0A} \pdfglyphtounicode{uumatragurmukhi}{0A42} \pdfglyphtounicode{uuvowelsignbengali}{09C2} \pdfglyphtounicode{uuvowelsigndeva}{0942} \pdfglyphtounicode{uuvowelsigngujarati}{0AC2} \pdfglyphtounicode{uvowelsignbengali}{09C1} \pdfglyphtounicode{uvowelsigndeva}{0941} \pdfglyphtounicode{uvowelsigngujarati}{0AC1} \pdfglyphtounicode{v}{0076} \pdfglyphtounicode{vadeva}{0935} \pdfglyphtounicode{vagujarati}{0AB5} \pdfglyphtounicode{vagurmukhi}{0A35} \pdfglyphtounicode{vakatakana}{30F7} \pdfglyphtounicode{vav}{05D5} \pdfglyphtounicode{vavdagesh}{FB35} \pdfglyphtounicode{vavdagesh65}{FB35} \pdfglyphtounicode{vavdageshhebrew}{FB35} \pdfglyphtounicode{vavhebrew}{05D5} \pdfglyphtounicode{vavholam}{FB4B} \pdfglyphtounicode{vavholamhebrew}{FB4B} \pdfglyphtounicode{vavvavhebrew}{05F0} \pdfglyphtounicode{vavyodhebrew}{05F1} \pdfglyphtounicode{vcircle}{24E5} \pdfglyphtounicode{vdotbelow}{1E7F} \pdfglyphtounicode{vecyrillic}{0432} \pdfglyphtounicode{veharabic}{06A4} \pdfglyphtounicode{vehfinalarabic}{FB6B} \pdfglyphtounicode{vehinitialarabic}{FB6C} \pdfglyphtounicode{vehmedialarabic}{FB6D} \pdfglyphtounicode{vekatakana}{30F9} \pdfglyphtounicode{venus}{2640} \pdfglyphtounicode{verticalbar}{007C} \pdfglyphtounicode{verticallineabovecmb}{030D} \pdfglyphtounicode{verticallinebelowcmb}{0329} \pdfglyphtounicode{verticallinelowmod}{02CC} \pdfglyphtounicode{verticallinemod}{02C8} \pdfglyphtounicode{vewarmenian}{057E} \pdfglyphtounicode{vhook}{028B} \pdfglyphtounicode{vikatakana}{30F8} \pdfglyphtounicode{viramabengali}{09CD} \pdfglyphtounicode{viramadeva}{094D} \pdfglyphtounicode{viramagujarati}{0ACD} \pdfglyphtounicode{visargabengali}{0983} \pdfglyphtounicode{visargadeva}{0903} \pdfglyphtounicode{visargagujarati}{0A83} \pdfglyphtounicode{vmonospace}{FF56} \pdfglyphtounicode{voarmenian}{0578} \pdfglyphtounicode{voicediterationhiragana}{309E} \pdfglyphtounicode{voicediterationkatakana}{30FE} \pdfglyphtounicode{voicedmarkkana}{309B} \pdfglyphtounicode{voicedmarkkanahalfwidth}{FF9E} \pdfglyphtounicode{vokatakana}{30FA} \pdfglyphtounicode{vparen}{24B1} \pdfglyphtounicode{vtilde}{1E7D} \pdfglyphtounicode{vturned}{028C} \pdfglyphtounicode{vuhiragana}{3094} \pdfglyphtounicode{vukatakana}{30F4} \pdfglyphtounicode{w}{0077} \pdfglyphtounicode{wacute}{1E83} \pdfglyphtounicode{waekorean}{3159} \pdfglyphtounicode{wahiragana}{308F} \pdfglyphtounicode{wakatakana}{30EF} \pdfglyphtounicode{wakatakanahalfwidth}{FF9C} \pdfglyphtounicode{wakorean}{3158} \pdfglyphtounicode{wasmallhiragana}{308E} \pdfglyphtounicode{wasmallkatakana}{30EE} \pdfglyphtounicode{wattosquare}{3357} \pdfglyphtounicode{wavedash}{301C} \pdfglyphtounicode{wavyunderscorevertical}{FE34} \pdfglyphtounicode{wawarabic}{0648} \pdfglyphtounicode{wawfinalarabic}{FEEE} \pdfglyphtounicode{wawhamzaabovearabic}{0624} \pdfglyphtounicode{wawhamzaabovefinalarabic}{FE86} \pdfglyphtounicode{wbsquare}{33DD} \pdfglyphtounicode{wcircle}{24E6} \pdfglyphtounicode{wcircumflex}{0175} \pdfglyphtounicode{wdieresis}{1E85} \pdfglyphtounicode{wdotaccent}{1E87} \pdfglyphtounicode{wdotbelow}{1E89} \pdfglyphtounicode{wehiragana}{3091} \pdfglyphtounicode{weierstrass}{2118} \pdfglyphtounicode{wekatakana}{30F1} \pdfglyphtounicode{wekorean}{315E} \pdfglyphtounicode{weokorean}{315D} \pdfglyphtounicode{wgrave}{1E81} \pdfglyphtounicode{whitebullet}{25E6} \pdfglyphtounicode{whitecircle}{25CB} \pdfglyphtounicode{whitecircleinverse}{25D9} \pdfglyphtounicode{whitecornerbracketleft}{300E} \pdfglyphtounicode{whitecornerbracketleftvertical}{FE43} \pdfglyphtounicode{whitecornerbracketright}{300F} \pdfglyphtounicode{whitecornerbracketrightvertical}{FE44} \pdfglyphtounicode{whitediamond}{25C7} \pdfglyphtounicode{whitediamondcontainingblacksmalldiamond}{25C8} \pdfglyphtounicode{whitedownpointingsmalltriangle}{25BF} \pdfglyphtounicode{whitedownpointingtriangle}{25BD} \pdfglyphtounicode{whiteleftpointingsmalltriangle}{25C3} \pdfglyphtounicode{whiteleftpointingtriangle}{25C1} \pdfglyphtounicode{whitelenticularbracketleft}{3016} \pdfglyphtounicode{whitelenticularbracketright}{3017} \pdfglyphtounicode{whiterightpointingsmalltriangle}{25B9} \pdfglyphtounicode{whiterightpointingtriangle}{25B7} \pdfglyphtounicode{whitesmallsquare}{25AB} \pdfglyphtounicode{whitesmilingface}{263A} \pdfglyphtounicode{whitesquare}{25A1} \pdfglyphtounicode{whitestar}{2606} \pdfglyphtounicode{whitetelephone}{260F} \pdfglyphtounicode{whitetortoiseshellbracketleft}{3018} \pdfglyphtounicode{whitetortoiseshellbracketright}{3019} \pdfglyphtounicode{whiteuppointingsmalltriangle}{25B5} \pdfglyphtounicode{whiteuppointingtriangle}{25B3} \pdfglyphtounicode{wihiragana}{3090} \pdfglyphtounicode{wikatakana}{30F0} \pdfglyphtounicode{wikorean}{315F} \pdfglyphtounicode{wmonospace}{FF57} \pdfglyphtounicode{wohiragana}{3092} \pdfglyphtounicode{wokatakana}{30F2} \pdfglyphtounicode{wokatakanahalfwidth}{FF66} \pdfglyphtounicode{won}{20A9} \pdfglyphtounicode{wonmonospace}{FFE6} \pdfglyphtounicode{wowaenthai}{0E27} \pdfglyphtounicode{wparen}{24B2} \pdfglyphtounicode{wring}{1E98} \pdfglyphtounicode{wsuperior}{02B7} \pdfglyphtounicode{wturned}{028D} \pdfglyphtounicode{wynn}{01BF} \pdfglyphtounicode{x}{0078} \pdfglyphtounicode{xabovecmb}{033D} \pdfglyphtounicode{xbopomofo}{3112} \pdfglyphtounicode{xcircle}{24E7} \pdfglyphtounicode{xdieresis}{1E8D} \pdfglyphtounicode{xdotaccent}{1E8B} \pdfglyphtounicode{xeharmenian}{056D} \pdfglyphtounicode{xi}{03BE} \pdfglyphtounicode{xmonospace}{FF58} \pdfglyphtounicode{xparen}{24B3} \pdfglyphtounicode{xsuperior}{02E3} \pdfglyphtounicode{y}{0079} \pdfglyphtounicode{yaadosquare}{334E} \pdfglyphtounicode{yabengali}{09AF} \pdfglyphtounicode{yacute}{00FD} \pdfglyphtounicode{yadeva}{092F} \pdfglyphtounicode{yaekorean}{3152} \pdfglyphtounicode{yagujarati}{0AAF} \pdfglyphtounicode{yagurmukhi}{0A2F} \pdfglyphtounicode{yahiragana}{3084} \pdfglyphtounicode{yakatakana}{30E4} \pdfglyphtounicode{yakatakanahalfwidth}{FF94} \pdfglyphtounicode{yakorean}{3151} \pdfglyphtounicode{yamakkanthai}{0E4E} \pdfglyphtounicode{yasmallhiragana}{3083} \pdfglyphtounicode{yasmallkatakana}{30E3} \pdfglyphtounicode{yasmallkatakanahalfwidth}{FF6C} \pdfglyphtounicode{yatcyrillic}{0463} \pdfglyphtounicode{ycircle}{24E8} \pdfglyphtounicode{ycircumflex}{0177} \pdfglyphtounicode{ydieresis}{00FF} \pdfglyphtounicode{ydotaccent}{1E8F} \pdfglyphtounicode{ydotbelow}{1EF5} \pdfglyphtounicode{yeharabic}{064A} \pdfglyphtounicode{yehbarreearabic}{06D2} \pdfglyphtounicode{yehbarreefinalarabic}{FBAF} \pdfglyphtounicode{yehfinalarabic}{FEF2} \pdfglyphtounicode{yehhamzaabovearabic}{0626} \pdfglyphtounicode{yehhamzaabovefinalarabic}{FE8A} \pdfglyphtounicode{yehhamzaaboveinitialarabic}{FE8B} \pdfglyphtounicode{yehhamzaabovemedialarabic}{FE8C} \pdfglyphtounicode{yehinitialarabic}{FEF3} \pdfglyphtounicode{yehmedialarabic}{FEF4} \pdfglyphtounicode{yehmeeminitialarabic}{FCDD} \pdfglyphtounicode{yehmeemisolatedarabic}{FC58} \pdfglyphtounicode{yehnoonfinalarabic}{FC94} \pdfglyphtounicode{yehthreedotsbelowarabic}{06D1} \pdfglyphtounicode{yekorean}{3156} \pdfglyphtounicode{yen}{00A5} \pdfglyphtounicode{yenmonospace}{FFE5} \pdfglyphtounicode{yeokorean}{3155} \pdfglyphtounicode{yeorinhieuhkorean}{3186} \pdfglyphtounicode{yerahbenyomohebrew}{05AA} \pdfglyphtounicode{yerahbenyomolefthebrew}{05AA} \pdfglyphtounicode{yericyrillic}{044B} \pdfglyphtounicode{yerudieresiscyrillic}{04F9} \pdfglyphtounicode{yesieungkorean}{3181} \pdfglyphtounicode{yesieungpansioskorean}{3183} \pdfglyphtounicode{yesieungsioskorean}{3182} \pdfglyphtounicode{yetivhebrew}{059A} \pdfglyphtounicode{ygrave}{1EF3} \pdfglyphtounicode{yhook}{01B4} \pdfglyphtounicode{yhookabove}{1EF7} \pdfglyphtounicode{yiarmenian}{0575} \pdfglyphtounicode{yicyrillic}{0457} \pdfglyphtounicode{yikorean}{3162} \pdfglyphtounicode{yinyang}{262F} \pdfglyphtounicode{yiwnarmenian}{0582} \pdfglyphtounicode{ymonospace}{FF59} \pdfglyphtounicode{yod}{05D9} \pdfglyphtounicode{yoddagesh}{FB39} \pdfglyphtounicode{yoddageshhebrew}{FB39} \pdfglyphtounicode{yodhebrew}{05D9} \pdfglyphtounicode{yodyodhebrew}{05F2} \pdfglyphtounicode{yodyodpatahhebrew}{FB1F} \pdfglyphtounicode{yohiragana}{3088} \pdfglyphtounicode{yoikorean}{3189} \pdfglyphtounicode{yokatakana}{30E8} \pdfglyphtounicode{yokatakanahalfwidth}{FF96} \pdfglyphtounicode{yokorean}{315B} \pdfglyphtounicode{yosmallhiragana}{3087} \pdfglyphtounicode{yosmallkatakana}{30E7} \pdfglyphtounicode{yosmallkatakanahalfwidth}{FF6E} \pdfglyphtounicode{yotgreek}{03F3} \pdfglyphtounicode{yoyaekorean}{3188} \pdfglyphtounicode{yoyakorean}{3187} \pdfglyphtounicode{yoyakthai}{0E22} \pdfglyphtounicode{yoyingthai}{0E0D} \pdfglyphtounicode{yparen}{24B4} \pdfglyphtounicode{ypogegrammeni}{037A} \pdfglyphtounicode{ypogegrammenigreekcmb}{0345} \pdfglyphtounicode{yr}{01A6} \pdfglyphtounicode{yring}{1E99} \pdfglyphtounicode{ysuperior}{02B8} \pdfglyphtounicode{ytilde}{1EF9} \pdfglyphtounicode{yturned}{028E} \pdfglyphtounicode{yuhiragana}{3086} \pdfglyphtounicode{yuikorean}{318C} \pdfglyphtounicode{yukatakana}{30E6} \pdfglyphtounicode{yukatakanahalfwidth}{FF95} \pdfglyphtounicode{yukorean}{3160} \pdfglyphtounicode{yusbigcyrillic}{046B} \pdfglyphtounicode{yusbigiotifiedcyrillic}{046D} \pdfglyphtounicode{yuslittlecyrillic}{0467} \pdfglyphtounicode{yuslittleiotifiedcyrillic}{0469} \pdfglyphtounicode{yusmallhiragana}{3085} \pdfglyphtounicode{yusmallkatakana}{30E5} \pdfglyphtounicode{yusmallkatakanahalfwidth}{FF6D} \pdfglyphtounicode{yuyekorean}{318B} \pdfglyphtounicode{yuyeokorean}{318A} \pdfglyphtounicode{yyabengali}{09DF} \pdfglyphtounicode{yyadeva}{095F} \pdfglyphtounicode{z}{007A} \pdfglyphtounicode{zaarmenian}{0566} \pdfglyphtounicode{zacute}{017A} \pdfglyphtounicode{zadeva}{095B} \pdfglyphtounicode{zagurmukhi}{0A5B} \pdfglyphtounicode{zaharabic}{0638} \pdfglyphtounicode{zahfinalarabic}{FEC6} \pdfglyphtounicode{zahinitialarabic}{FEC7} \pdfglyphtounicode{zahiragana}{3056} \pdfglyphtounicode{zahmedialarabic}{FEC8} \pdfglyphtounicode{zainarabic}{0632} \pdfglyphtounicode{zainfinalarabic}{FEB0} \pdfglyphtounicode{zakatakana}{30B6} \pdfglyphtounicode{zaqefgadolhebrew}{0595} \pdfglyphtounicode{zaqefqatanhebrew}{0594} \pdfglyphtounicode{zarqahebrew}{0598} \pdfglyphtounicode{zayin}{05D6} \pdfglyphtounicode{zayindagesh}{FB36} \pdfglyphtounicode{zayindageshhebrew}{FB36} \pdfglyphtounicode{zayinhebrew}{05D6} \pdfglyphtounicode{zbopomofo}{3117} \pdfglyphtounicode{zcaron}{017E} \pdfglyphtounicode{zcircle}{24E9} \pdfglyphtounicode{zcircumflex}{1E91} \pdfglyphtounicode{zcurl}{0291} \pdfglyphtounicode{zdot}{017C} \pdfglyphtounicode{zdotaccent}{017C} \pdfglyphtounicode{zdotbelow}{1E93} \pdfglyphtounicode{zecyrillic}{0437} \pdfglyphtounicode{zedescendercyrillic}{0499} \pdfglyphtounicode{zedieresiscyrillic}{04DF} \pdfglyphtounicode{zehiragana}{305C} \pdfglyphtounicode{zekatakana}{30BC} \pdfglyphtounicode{zero}{0030} \pdfglyphtounicode{zeroarabic}{0660} \pdfglyphtounicode{zerobengali}{09E6} \pdfglyphtounicode{zerodeva}{0966} \pdfglyphtounicode{zerogujarati}{0AE6} \pdfglyphtounicode{zerogurmukhi}{0A66} \pdfglyphtounicode{zerohackarabic}{0660} \pdfglyphtounicode{zeroinferior}{2080} \pdfglyphtounicode{zeromonospace}{FF10} \pdfglyphtounicode{zerooldstyle}{F730} \pdfglyphtounicode{zeropersian}{06F0} \pdfglyphtounicode{zerosuperior}{2070} \pdfglyphtounicode{zerothai}{0E50} \pdfglyphtounicode{zerowidthjoiner}{FEFF} \pdfglyphtounicode{zerowidthnonjoiner}{200C} \pdfglyphtounicode{zerowidthspace}{200B} \pdfglyphtounicode{zeta}{03B6} \pdfglyphtounicode{zhbopomofo}{3113} \pdfglyphtounicode{zhearmenian}{056A} \pdfglyphtounicode{zhebrevecyrillic}{04C2} \pdfglyphtounicode{zhecyrillic}{0436} \pdfglyphtounicode{zhedescendercyrillic}{0497} \pdfglyphtounicode{zhedieresiscyrillic}{04DD} \pdfglyphtounicode{zihiragana}{3058} \pdfglyphtounicode{zikatakana}{30B8} \pdfglyphtounicode{zinorhebrew}{05AE} \pdfglyphtounicode{zlinebelow}{1E95} \pdfglyphtounicode{zmonospace}{FF5A} \pdfglyphtounicode{zohiragana}{305E} \pdfglyphtounicode{zokatakana}{30BE} \pdfglyphtounicode{zparen}{24B5} \pdfglyphtounicode{zretroflexhook}{0290} \pdfglyphtounicode{zstroke}{01B6} \pdfglyphtounicode{zuhiragana}{305A} \pdfglyphtounicode{zukatakana}{30BA} % entries from zapfdingbats.txt: \pdfglyphtounicode{a100}{275E} \pdfglyphtounicode{a101}{2761} \pdfglyphtounicode{a102}{2762} \pdfglyphtounicode{a103}{2763} \pdfglyphtounicode{a104}{2764} \pdfglyphtounicode{a105}{2710} \pdfglyphtounicode{a106}{2765} \pdfglyphtounicode{a107}{2766} \pdfglyphtounicode{a108}{2767} \pdfglyphtounicode{a109}{2660} \pdfglyphtounicode{a10}{2721} \pdfglyphtounicode{a110}{2665} \pdfglyphtounicode{a111}{2666} \pdfglyphtounicode{a112}{2663} \pdfglyphtounicode{a117}{2709} \pdfglyphtounicode{a118}{2708} \pdfglyphtounicode{a119}{2707} \pdfglyphtounicode{a11}{261B} \pdfglyphtounicode{a120}{2460} \pdfglyphtounicode{a121}{2461} \pdfglyphtounicode{a122}{2462} \pdfglyphtounicode{a123}{2463} \pdfglyphtounicode{a124}{2464} \pdfglyphtounicode{a125}{2465} \pdfglyphtounicode{a126}{2466} \pdfglyphtounicode{a127}{2467} \pdfglyphtounicode{a128}{2468} \pdfglyphtounicode{a129}{2469} \pdfglyphtounicode{a12}{261E} \pdfglyphtounicode{a130}{2776} \pdfglyphtounicode{a131}{2777} \pdfglyphtounicode{a132}{2778} \pdfglyphtounicode{a133}{2779} \pdfglyphtounicode{a134}{277A} \pdfglyphtounicode{a135}{277B} \pdfglyphtounicode{a136}{277C} \pdfglyphtounicode{a137}{277D} \pdfglyphtounicode{a138}{277E} \pdfglyphtounicode{a139}{277F} \pdfglyphtounicode{a13}{270C} \pdfglyphtounicode{a140}{2780} \pdfglyphtounicode{a141}{2781} \pdfglyphtounicode{a142}{2782} \pdfglyphtounicode{a143}{2783} \pdfglyphtounicode{a144}{2784} \pdfglyphtounicode{a145}{2785} \pdfglyphtounicode{a146}{2786} \pdfglyphtounicode{a147}{2787} \pdfglyphtounicode{a148}{2788} \pdfglyphtounicode{a149}{2789} \pdfglyphtounicode{a14}{270D} \pdfglyphtounicode{a150}{278A} \pdfglyphtounicode{a151}{278B} \pdfglyphtounicode{a152}{278C} \pdfglyphtounicode{a153}{278D} \pdfglyphtounicode{a154}{278E} \pdfglyphtounicode{a155}{278F} \pdfglyphtounicode{a156}{2790} \pdfglyphtounicode{a157}{2791} \pdfglyphtounicode{a158}{2792} \pdfglyphtounicode{a159}{2793} \pdfglyphtounicode{a15}{270E} \pdfglyphtounicode{a160}{2794} \pdfglyphtounicode{a161}{2192} \pdfglyphtounicode{a162}{27A3} \pdfglyphtounicode{a163}{2194} \pdfglyphtounicode{a164}{2195} \pdfglyphtounicode{a165}{2799} \pdfglyphtounicode{a166}{279B} \pdfglyphtounicode{a167}{279C} \pdfglyphtounicode{a168}{279D} \pdfglyphtounicode{a169}{279E} \pdfglyphtounicode{a16}{270F} \pdfglyphtounicode{a170}{279F} \pdfglyphtounicode{a171}{27A0} \pdfglyphtounicode{a172}{27A1} \pdfglyphtounicode{a173}{27A2} \pdfglyphtounicode{a174}{27A4} \pdfglyphtounicode{a175}{27A5} \pdfglyphtounicode{a176}{27A6} \pdfglyphtounicode{a177}{27A7} \pdfglyphtounicode{a178}{27A8} \pdfglyphtounicode{a179}{27A9} \pdfglyphtounicode{a17}{2711} \pdfglyphtounicode{a180}{27AB} \pdfglyphtounicode{a181}{27AD} \pdfglyphtounicode{a182}{27AF} \pdfglyphtounicode{a183}{27B2} \pdfglyphtounicode{a184}{27B3} \pdfglyphtounicode{a185}{27B5} \pdfglyphtounicode{a186}{27B8} \pdfglyphtounicode{a187}{27BA} \pdfglyphtounicode{a188}{27BB} \pdfglyphtounicode{a189}{27BC} \pdfglyphtounicode{a18}{2712} \pdfglyphtounicode{a190}{27BD} \pdfglyphtounicode{a191}{27BE} \pdfglyphtounicode{a192}{279A} \pdfglyphtounicode{a193}{27AA} \pdfglyphtounicode{a194}{27B6} \pdfglyphtounicode{a195}{27B9} \pdfglyphtounicode{a196}{2798} \pdfglyphtounicode{a197}{27B4} \pdfglyphtounicode{a198}{27B7} \pdfglyphtounicode{a199}{27AC} \pdfglyphtounicode{a19}{2713} \pdfglyphtounicode{a1}{2701} \pdfglyphtounicode{a200}{27AE} \pdfglyphtounicode{a201}{27B1} \pdfglyphtounicode{a202}{2703} \pdfglyphtounicode{a203}{2750} \pdfglyphtounicode{a204}{2752} \pdfglyphtounicode{a205}{276E} \pdfglyphtounicode{a206}{2770} \pdfglyphtounicode{a20}{2714} \pdfglyphtounicode{a21}{2715} \pdfglyphtounicode{a22}{2716} \pdfglyphtounicode{a23}{2717} \pdfglyphtounicode{a24}{2718} \pdfglyphtounicode{a25}{2719} \pdfglyphtounicode{a26}{271A} \pdfglyphtounicode{a27}{271B} \pdfglyphtounicode{a28}{271C} \pdfglyphtounicode{a29}{2722} \pdfglyphtounicode{a2}{2702} \pdfglyphtounicode{a30}{2723} \pdfglyphtounicode{a31}{2724} \pdfglyphtounicode{a32}{2725} \pdfglyphtounicode{a33}{2726} \pdfglyphtounicode{a34}{2727} \pdfglyphtounicode{a35}{2605} \pdfglyphtounicode{a36}{2729} \pdfglyphtounicode{a37}{272A} \pdfglyphtounicode{a38}{272B} \pdfglyphtounicode{a39}{272C} \pdfglyphtounicode{a3}{2704} \pdfglyphtounicode{a40}{272D} \pdfglyphtounicode{a41}{272E} \pdfglyphtounicode{a42}{272F} \pdfglyphtounicode{a43}{2730} \pdfglyphtounicode{a44}{2731} \pdfglyphtounicode{a45}{2732} \pdfglyphtounicode{a46}{2733} \pdfglyphtounicode{a47}{2734} \pdfglyphtounicode{a48}{2735} \pdfglyphtounicode{a49}{2736} \pdfglyphtounicode{a4}{260E} \pdfglyphtounicode{a50}{2737} \pdfglyphtounicode{a51}{2738} \pdfglyphtounicode{a52}{2739} \pdfglyphtounicode{a53}{273A} \pdfglyphtounicode{a54}{273B} \pdfglyphtounicode{a55}{273C} \pdfglyphtounicode{a56}{273D} \pdfglyphtounicode{a57}{273E} \pdfglyphtounicode{a58}{273F} \pdfglyphtounicode{a59}{2740} \pdfglyphtounicode{a5}{2706} \pdfglyphtounicode{a60}{2741} \pdfglyphtounicode{a61}{2742} \pdfglyphtounicode{a62}{2743} \pdfglyphtounicode{a63}{2744} \pdfglyphtounicode{a64}{2745} \pdfglyphtounicode{a65}{2746} \pdfglyphtounicode{a66}{2747} \pdfglyphtounicode{a67}{2748} \pdfglyphtounicode{a68}{2749} \pdfglyphtounicode{a69}{274A} \pdfglyphtounicode{a6}{271D} \pdfglyphtounicode{a70}{274B} \pdfglyphtounicode{a71}{25CF} \pdfglyphtounicode{a72}{274D} \pdfglyphtounicode{a73}{25A0} \pdfglyphtounicode{a74}{274F} \pdfglyphtounicode{a75}{2751} \pdfglyphtounicode{a76}{25B2} \pdfglyphtounicode{a77}{25BC} \pdfglyphtounicode{a78}{25C6} \pdfglyphtounicode{a79}{2756} \pdfglyphtounicode{a7}{271E} \pdfglyphtounicode{a81}{25D7} \pdfglyphtounicode{a82}{2758} \pdfglyphtounicode{a83}{2759} \pdfglyphtounicode{a84}{275A} \pdfglyphtounicode{a85}{276F} \pdfglyphtounicode{a86}{2771} \pdfglyphtounicode{a87}{2772} \pdfglyphtounicode{a88}{2773} \pdfglyphtounicode{a89}{2768} \pdfglyphtounicode{a8}{271F} \pdfglyphtounicode{a90}{2769} \pdfglyphtounicode{a91}{276C} \pdfglyphtounicode{a92}{276D} \pdfglyphtounicode{a93}{276A} \pdfglyphtounicode{a94}{276B} \pdfglyphtounicode{a95}{2774} \pdfglyphtounicode{a96}{2775} \pdfglyphtounicode{a97}{275B} \pdfglyphtounicode{a98}{275C} \pdfglyphtounicode{a99}{275D} \pdfglyphtounicode{a9}{2720} % entries from texglyphlist.txt: % Delta;2206 \pdfglyphtounicode{Ifractur}{2111} \pdfglyphtounicode{FFsmall}{D804} \pdfglyphtounicode{FFIsmall}{D807} \pdfglyphtounicode{FFLsmall}{D808} \pdfglyphtounicode{FIsmall}{D805} \pdfglyphtounicode{FLsmall}{D806} \pdfglyphtounicode{Germandbls}{D800} \pdfglyphtounicode{Germandblssmall}{D803} \pdfglyphtounicode{Ng}{014A} % Omega;2126 \pdfglyphtounicode{Rfractur}{211C} \pdfglyphtounicode{SS}{D800} \pdfglyphtounicode{SSsmall}{D803} \pdfglyphtounicode{altselector}{D802} \pdfglyphtounicode{angbracketleft}{27E8} \pdfglyphtounicode{angbracketright}{27E9} \pdfglyphtounicode{arrowbothv}{2195} \pdfglyphtounicode{arrowdblbothv}{21D5} \pdfglyphtounicode{arrowleftbothalf}{21BD} \pdfglyphtounicode{arrowlefttophalf}{21BC} \pdfglyphtounicode{arrownortheast}{2197} \pdfglyphtounicode{arrownorthwest}{2196} \pdfglyphtounicode{arrowrightbothalf}{21C1} \pdfglyphtounicode{arrowrighttophalf}{21C0} \pdfglyphtounicode{arrowsoutheast}{2198} \pdfglyphtounicode{arrowsouthwest}{2199} \pdfglyphtounicode{ascendercompwordmark}{D80A} \pdfglyphtounicode{asteriskcentered}{2217} \pdfglyphtounicode{bardbl}{2225} \pdfglyphtounicode{capitalcompwordmark}{D809} \pdfglyphtounicode{ceilingleft}{2308} \pdfglyphtounicode{ceilingright}{2309} \pdfglyphtounicode{circlecopyrt}{20DD} \pdfglyphtounicode{circledivide}{2298} \pdfglyphtounicode{circledot}{2299} \pdfglyphtounicode{circleminus}{2296} \pdfglyphtounicode{coproduct}{2A3F} \pdfglyphtounicode{cwm}{200C} \pdfglyphtounicode{dblbracketleft}{27E6} \pdfglyphtounicode{dblbracketright}{27E7} % diamond;2662 % diamondmath;22C4 % dotlessj;0237 % emptyset;2205 \pdfglyphtounicode{emptyslot}{D801} \pdfglyphtounicode{epsilon1}{03F5} \pdfglyphtounicode{equivasymptotic}{224D} \pdfglyphtounicode{flat}{266D} \pdfglyphtounicode{floorleft}{230A} \pdfglyphtounicode{floorright}{230B} \pdfglyphtounicode{follows}{227B} \pdfglyphtounicode{followsequal}{227D} \pdfglyphtounicode{greatermuch}{226B} % heart;2661 \pdfglyphtounicode{interrobang}{203D} \pdfglyphtounicode{interrobangdown}{D80B} \pdfglyphtounicode{intersectionsq}{2293} \pdfglyphtounicode{latticetop}{22A4} \pdfglyphtounicode{lessmuch}{226A} \pdfglyphtounicode{lscript}{2113} \pdfglyphtounicode{natural}{266E} \pdfglyphtounicode{negationslash}{0338} \pdfglyphtounicode{ng}{014B} \pdfglyphtounicode{owner}{220B} \pdfglyphtounicode{pertenthousand}{2031} % phi;03D5 % phi1;03C6 \pdfglyphtounicode{pi1}{03D6} \pdfglyphtounicode{precedesequal}{227C} \pdfglyphtounicode{prime}{2032} \pdfglyphtounicode{rho1}{03F1} \pdfglyphtounicode{ringfitted}{D80D} \pdfglyphtounicode{sharp}{266F} \pdfglyphtounicode{similarequal}{2243} \pdfglyphtounicode{slurabove}{2322} \pdfglyphtounicode{slurbelow}{2323} \pdfglyphtounicode{star}{22C6} \pdfglyphtounicode{subsetsqequal}{2291} \pdfglyphtounicode{supersetsqequal}{2292} \pdfglyphtounicode{triangle}{25B3} \pdfglyphtounicode{triangleinv}{25BD} \pdfglyphtounicode{triangleleft}{25B9} \pdfglyphtounicode{triangleright}{25C3} \pdfglyphtounicode{turnstileleft}{22A2} \pdfglyphtounicode{turnstileright}{22A3} \pdfglyphtounicode{twelveudash}{D80C} \pdfglyphtounicode{unionmulti}{228E} \pdfglyphtounicode{unionsq}{2294} \pdfglyphtounicode{vector}{20D7} \pdfglyphtounicode{visualspace}{2423} \pdfglyphtounicode{wreathproduct}{2240} \pdfglyphtounicode{Dbar}{0110} \pdfglyphtounicode{compwordmark}{200C} \pdfglyphtounicode{dbar}{0111} \pdfglyphtounicode{rangedash}{2013} \pdfglyphtounicode{hyphenchar}{002D} \pdfglyphtounicode{punctdash}{2014} \pdfglyphtounicode{visiblespace}{2423} % entries from additional.tex: \pdfglyphtounicode{ff}{0066 0066} \pdfglyphtounicode{fi}{0066 0069} \pdfglyphtounicode{fl}{0066 006C} \pdfglyphtounicode{ffi}{0066 0066 0069} \pdfglyphtounicode{ffl}{0066 0066 006C} \pdfglyphtounicode{IJ}{0049 004A} \pdfglyphtounicode{ij}{0069 006A} \pdfglyphtounicode{longs}{0073} ================================================ FILE: common/header.tex ================================================ % % A header that lets you compile a chapter by itself, or inside a larger document. % Adapted from stackoverflow.com/questions/3655454/conditional-import-in-latex % % %Use \inbpdocument and \outbpdocument in your individual files, in place of \begin{document} and \end{document}. In your main file, put in a \def \ismaindoc {} before including or importing anything. % % David Duvenaud % June 2011 % % ====================================== % % \ifx\ismaindoc\undefined \newcommand{\inbpdocument}{ \def \ismaindoc {} % Use this header if we are compiling by ourselves. \documentclass[a4paper,11pt,authoryear,index]{common/PhDThesisPSnPDF} \input{common/official-preamble.tex} \input{common/preamble.tex} \input{common/thesis-info} \begin{document} } \newcommand{\outbpdocument}[1]{ % Fake chapters so references aren't broken \label{ch:intro} \label{ch:kernels} \label{ch:grammar} \label{ch:description} \label{ch:warped} \label{ch:additive} \label{ch:deep-limits} \label{ch:discussion} %\bibliographystyle{common/CUEDthesis} \bibliographystyle{plainnat} \bibliography{references.bib} \end{document} } \else %If we're inside another document, no need to re-start the document. \ifx\inbpdocument\undefined \newcommand{\inbpdocument}{} \newcommand{\outbpdocument}[1]{} \fi \fi ================================================ FILE: common/humble.tex ================================================ % HUMBLE WORDS: shown slightly smaller when in normal text % Christian Steinruecken % \makeatletter% %\def\@humbleformat#1{{\fontsize{}{1em}\selectfont #1}} %\def\@humbleformat#1{\textsmaller{#1}}% \newlength{\nonHumbleHeight} \def\@humbleformat#1{{\settoheight{\nonHumbleHeight}{#1}\resizebox{!}{0.94\nonHumbleHeight}{#1}}}% \def\@idxhumbleformat#1{{\relscale{0.95}{#1}}}% %\def\@humbleformat#1{{#1}}% \def\declareHumble#1#2{% \expandafter\def\csname #1\endcsname{\@humbleformat{#2}}% \expandafter\def\csname s#1\endcsname{{#2}}% \expandafter\def\csname idx#1\endcsname{{\@idxhumbleformat{#2}}}% }% \def\humble#1{\@humbleformat{#1}}% \def\idxhumble#1{\@idxhumbleformat{#1}}% \makeatother% % Convenient indexing for humble abbreviations \def\humbleindex#1#2{\index{#1@\idxhumble{#1}}} ================================================ FILE: common/jmb.bst ================================================ %% %% This is file `jmb.bst', %% generated with the docstrip utility. %% %% The original source files were: %% %% merlin.mbs (with options: `ay,nat,seq-lab,nm-rvv,jnrlst,keyxyr,dt-beg,yr-par,yrp-per,note-yr,vol-bf,vnum-x,volp-com,num-xser,blk-tit,ppx,ed,abr,ord,jabr,amper,and-xcom,etal-it') %% ---------------------------------------- %% *** JMB *** %% %% Copyright 1994-2000 Patrick W Daly % =============================================================== % IMPORTANT NOTICE: % This bibliographic style (bst) file has been generated from one or % more master bibliographic style (mbs) files, listed above. % % This generated file can be redistributed and/or modified under the terms % of the LaTeX Project Public License Distributed from CTAN % archives in directory macros/latex/base/lppl.txt; either % version 1 of the License, or any later version. % =============================================================== % Name and version information of the main mbs file: % \ProvidesFile{merlin.mbs}[2000/05/04 4.01 (PWD, AO, DPC)] % For use with BibTeX version 0.99a or later %------------------------------------------------------------------- % This bibliography style file is intended for texts in ENGLISH % This is an author-year citation style bibliography. As such, it is % non-standard LaTeX, and requires a special package file to function properly. % Such a package is natbib.sty by Patrick W. Daly % The form of the \bibitem entries is % \bibitem[Jones et al.(1990)]{key}... % \bibitem[Jones et al.(1990)Jones, Baker, and Smith]{key}... % The essential feature is that the label (the part in brackets) consists % of the author names, as they should appear in the citation, with the year % in parentheses following. There must be no space before the opening % parenthesis! % With natbib v5.3, a full list of authors may also follow the year. % In natbib.sty, it is possible to define the type of enclosures that is % really wanted (brackets or parentheses), but in either case, there must % be parentheses in the label. % The \cite command functions as follows: % \citet{key} ==>> Jones et al. (1990) % \citet*{key} ==>> Jones, Baker, and Smith (1990) % \citep{key} ==>> (Jones et al., 1990) % \citep*{key} ==>> (Jones, Baker, and Smith, 1990) % \citep[chap. 2]{key} ==>> (Jones et al., 1990, chap. 2) % \citep[e.g.][]{key} ==>> (e.g. Jones et al., 1990) % \citep[e.g.][p. 32]{key} ==>> (e.g. Jones et al., p. 32) % \citeauthor{key} ==>> Jones et al. % \citeauthor*{key} ==>> Jones, Baker, and Smith % \citeyear{key} ==>> 1990 %--------------------------------------------------------------------- ENTRY { address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year } {} { label extra.label sort.label short.list } INTEGERS { output.state before.all mid.sentence after.sentence after.block } FUNCTION {init.state.consts} { #0 'before.all := #1 'mid.sentence := #2 'after.sentence := #3 'after.block := } STRINGS { s t} FUNCTION {output.nonnull} { 's := output.state mid.sentence = { ", " * write$ } { output.state after.block = { add.period$ write$ newline$ "\newblock " write$ } { output.state before.all = 'write$ { add.period$ " " * write$ } if$ } if$ mid.sentence 'output.state := } if$ s } FUNCTION {output} { duplicate$ empty$ 'pop$ 'output.nonnull if$ } FUNCTION {output.check} { 't := duplicate$ empty$ { pop$ "empty " t * " in " * cite$ * warning$ } 'output.nonnull if$ } FUNCTION {fin.entry} { add.period$ write$ newline$ } FUNCTION {new.block} { output.state before.all = 'skip$ { after.block 'output.state := } if$ } FUNCTION {new.sentence} { output.state after.block = 'skip$ { output.state before.all = 'skip$ { after.sentence 'output.state := } if$ } if$ } FUNCTION {add.blank} { " " * before.all 'output.state := } FUNCTION {date.block} { new.sentence } FUNCTION {not} { { #0 } { #1 } if$ } FUNCTION {and} { 'skip$ { pop$ #0 } if$ } FUNCTION {or} { { pop$ #1 } 'skip$ if$ } FUNCTION {new.block.checkb} { empty$ swap$ empty$ and 'skip$ 'new.block if$ } FUNCTION {field.or.null} { duplicate$ empty$ { pop$ "" } 'skip$ if$ } FUNCTION {emphasize} { duplicate$ empty$ { pop$ "" } { "{\em " swap$ * "\/}" * } if$ } FUNCTION {scapify} { duplicate$ empty$ { pop$ "" } { "{\sc " swap$ * "}" * } if$ } FUNCTION {bolden} { duplicate$ empty$ { pop$ "" } { "{\bf " swap$ * "}" * } if$ } FUNCTION {tie.or.space.prefix} { duplicate$ text.length$ #3 < { "~" } { " " } if$ swap$ } FUNCTION {capitalize} { "u" change.case$ "t" change.case$ } FUNCTION {space.word} { " " swap$ * " " * } % Here are the language-specific definitions for explicit words. % Each function has a name bbl.xxx where xxx is the English word. % The language selected here is ENGLISH FUNCTION {bbl.and} { "and"} FUNCTION {bbl.etal} { "et~al." } FUNCTION {bbl.editors} { "eds." } FUNCTION {bbl.editor} { "ed." } FUNCTION {bbl.edby} { "edited by" } FUNCTION {bbl.edition} { "edn." } FUNCTION {bbl.volume} { "vol." } FUNCTION {bbl.of} { "of" } FUNCTION {bbl.number} { "no." } FUNCTION {bbl.nr} { "no." } FUNCTION {bbl.in} { "in" } FUNCTION {bbl.pages} { "" } FUNCTION {bbl.page} { "" } FUNCTION {bbl.chapter} { "chap." } FUNCTION {bbl.techrep} { "Tech. Rep." } FUNCTION {bbl.mthesis} { "Master's thesis" } FUNCTION {bbl.phdthesis} { "Ph.D. thesis" } FUNCTION {bbl.first} { "1st" } FUNCTION {bbl.second} { "2nd" } FUNCTION {bbl.third} { "3rd" } FUNCTION {bbl.fourth} { "4th" } FUNCTION {bbl.fifth} { "5th" } FUNCTION {bbl.st} { "st" } FUNCTION {bbl.nd} { "nd" } FUNCTION {bbl.rd} { "rd" } FUNCTION {bbl.th} { "th" } MACRO {jan} {"Jan."} MACRO {feb} {"Feb."} MACRO {mar} {"Mar."} MACRO {apr} {"Apr."} MACRO {may} {"May"} MACRO {jun} {"Jun."} MACRO {jul} {"Jul."} MACRO {aug} {"Aug."} MACRO {sep} {"Sep."} MACRO {oct} {"Oct."} MACRO {nov} {"Nov."} MACRO {dec} {"Dec."} FUNCTION {eng.ord} { duplicate$ "1" swap$ * #-2 #1 substring$ "1" = { bbl.th * } { duplicate$ #-1 #1 substring$ duplicate$ "1" = { pop$ bbl.st * } { duplicate$ "2" = { pop$ bbl.nd * } { "3" = { bbl.rd * } { bbl.th * } if$ } if$ } if$ } if$ } MACRO {acmcs} {"ACM Comput. Surv."} MACRO {acta} {"Acta Inf."} MACRO {cacm} {"Commun. ACM"} MACRO {ibmjrd} {"IBM J. Res. Dev."} MACRO {ibmsj} {"IBM Syst.~J."} MACRO {ieeese} {"IEEE Trans. Software Eng."} MACRO {ieeetc} {"IEEE Trans. Comput."} MACRO {ieeetcad} {"IEEE Trans. Comput. Aid. Des."} MACRO {ipl} {"Inf. Process. Lett."} MACRO {jacm} {"J.~ACM"} MACRO {jcss} {"J.~Comput. Syst. Sci."} MACRO {scp} {"Sci. Comput. Program."} MACRO {sicomp} {"SIAM J. Comput."} MACRO {tocs} {"ACM Trans. Comput. Syst."} MACRO {tods} {"ACM Trans. Database Syst."} MACRO {tog} {"ACM Trans. Graphic."} MACRO {toms} {"ACM Trans. Math. Software"} MACRO {toois} {"ACM Trans. Office Inf. Syst."} MACRO {toplas} {"ACM Trans. Progr. Lang. Syst."} MACRO {tcs} {"Theor. Comput. Sci."} FUNCTION {bibinfo.check} { swap$ duplicate$ missing$ { pop$ pop$ "" } { duplicate$ empty$ { swap$ pop$ } { swap$ pop$ } if$ } if$ } FUNCTION {bibinfo.warn} { swap$ duplicate$ missing$ { swap$ "missing " swap$ * " in " * cite$ * warning$ pop$ "" } { duplicate$ empty$ { swap$ "empty " swap$ * " in " * cite$ * warning$ } { swap$ pop$ } if$ } if$ } STRINGS { bibinfo} INTEGERS { nameptr namesleft numnames } FUNCTION {format.names} { 'bibinfo := duplicate$ empty$ 'skip$ { 's := "" 't := #1 'nameptr := s num.names$ 'numnames := numnames 'namesleft := { namesleft #0 > } { s nameptr "{vv~}{ll}{, f{.}.}{, jj}" format.name$ bibinfo bibinfo.check 't := nameptr #1 > { namesleft #1 > { ", " * t * } { s nameptr "{ll}" format.name$ duplicate$ "others" = { 't := } { pop$ } if$ t "others" = { " " * bbl.etal emphasize * } { "\&" space.word * t * } if$ } if$ } 't if$ nameptr #1 + 'nameptr := namesleft #1 - 'namesleft := } while$ } if$ } FUNCTION {format.names.ed} { 'bibinfo := duplicate$ empty$ 'skip$ { 's := "" 't := #1 'nameptr := s num.names$ 'numnames := numnames 'namesleft := { namesleft #0 > } { s nameptr "{f{.}.~}{vv~}{ll}{ jj}" format.name$ bibinfo bibinfo.check 't := nameptr #1 > { namesleft #1 > { ", " * t * } { s nameptr "{ll}" format.name$ duplicate$ "others" = { 't := } { pop$ } if$ t "others" = { " " * bbl.etal emphasize * } { "\&" space.word * t * } if$ } if$ } 't if$ nameptr #1 + 'nameptr := namesleft #1 - 'namesleft := } while$ } if$ } FUNCTION {format.key} { empty$ { key field.or.null } { "" } if$ } FUNCTION {format.authors} { author "author" format.names scapify } FUNCTION {get.bbl.editor} { editor num.names$ #1 > 'bbl.editors 'bbl.editor if$ } FUNCTION {format.editors} { editor "editor" format.names scapify duplicate$ empty$ 'skip$ { "," * " " * get.bbl.editor * } if$ } FUNCTION {format.note} { note empty$ { "" } { note #1 #1 substring$ duplicate$ "{" = 'skip$ { output.state mid.sentence = { "l" } { "u" } if$ change.case$ } if$ note #2 global.max$ substring$ * "note" bibinfo.check } if$ } FUNCTION {format.title} { title duplicate$ empty$ 'skip$ { "t" change.case$ } if$ "title" bibinfo.check } FUNCTION {format.full.names} {'s := "" 't := #1 'nameptr := s num.names$ 'numnames := numnames 'namesleft := { namesleft #0 > } { s nameptr "{vv~}{ll}" format.name$ 't := nameptr #1 > { namesleft #1 > { ", " * t * } { s nameptr "{ll}" format.name$ duplicate$ "others" = { 't := } { pop$ } if$ t "others" = { " " * bbl.etal emphasize * } { "\&" space.word * t * } if$ } if$ } 't if$ nameptr #1 + 'nameptr := namesleft #1 - 'namesleft := } while$ } FUNCTION {author.editor.key.full} { author empty$ { editor empty$ { key empty$ { cite$ #1 #3 substring$ } 'key if$ } { editor format.full.names } if$ } { author format.full.names } if$ } FUNCTION {author.key.full} { author empty$ { key empty$ { cite$ #1 #3 substring$ } 'key if$ } { author format.full.names } if$ } FUNCTION {editor.key.full} { editor empty$ { key empty$ { cite$ #1 #3 substring$ } 'key if$ } { editor format.full.names } if$ } FUNCTION {make.full.names} { type$ "book" = type$ "inbook" = or 'author.editor.key.full { type$ "proceedings" = 'editor.key.full 'author.key.full if$ } if$ } FUNCTION {output.bibitem} { newline$ "\bibitem[{" write$ label write$ ")" make.full.names duplicate$ short.list = { pop$ } { * } if$ "}]{" * write$ cite$ write$ "}" write$ newline$ "" before.all 'output.state := } FUNCTION {n.dashify} { 't := "" { t empty$ not } { t #1 #1 substring$ "-" = { t #1 #2 substring$ "--" = not { "--" * t #2 global.max$ substring$ 't := } { { t #1 #1 substring$ "-" = } { "-" * t #2 global.max$ substring$ 't := } while$ } if$ } { t #1 #1 substring$ * t #2 global.max$ substring$ 't := } if$ } while$ } FUNCTION {word.in} { bbl.in capitalize " " * } FUNCTION {format.date} { year "year" bibinfo.check duplicate$ empty$ { "empty year in " cite$ * "; set to ????" * warning$ pop$ "????" } 'skip$ if$ extra.label * before.all 'output.state := " (" swap$ * ")" * } FUNCTION {format.btitle} { title "title" bibinfo.check duplicate$ empty$ 'skip$ { emphasize } if$ } FUNCTION {either.or.check} { empty$ 'pop$ { "can't use both " swap$ * " fields in " * cite$ * warning$ } if$ } FUNCTION {format.bvolume} { volume empty$ { "" } { bbl.volume volume tie.or.space.prefix "volume" bibinfo.check * * series "series" bibinfo.check duplicate$ empty$ 'pop$ { swap$ bbl.of space.word * swap$ emphasize * } if$ "volume and number" number either.or.check } if$ } FUNCTION {format.number.series} { volume empty$ { number empty$ { series field.or.null } { series empty$ { number "number" bibinfo.check } { output.state mid.sentence = { bbl.number } { bbl.number capitalize } if$ number tie.or.space.prefix "number" bibinfo.check * * bbl.in space.word * series "series" bibinfo.check * } if$ } if$ } { "" } if$ } FUNCTION {is.num} { chr.to.int$ duplicate$ "0" chr.to.int$ < not swap$ "9" chr.to.int$ > not and } FUNCTION {extract.num} { duplicate$ 't := "" 's := { t empty$ not } { t #1 #1 substring$ t #2 global.max$ substring$ 't := duplicate$ is.num { s swap$ * 's := } { pop$ "" 't := } if$ } while$ s empty$ 'skip$ { pop$ s } if$ } FUNCTION {convert.edition} { extract.num "l" change.case$ 's := s "first" = s "1" = or { bbl.first 't := } { s "second" = s "2" = or { bbl.second 't := } { s "third" = s "3" = or { bbl.third 't := } { s "fourth" = s "4" = or { bbl.fourth 't := } { s "fifth" = s "5" = or { bbl.fifth 't := } { s #1 #1 substring$ is.num { s eng.ord 't := } { edition 't := } if$ } if$ } if$ } if$ } if$ } if$ t } FUNCTION {format.edition} { edition duplicate$ empty$ 'skip$ { convert.edition output.state mid.sentence = { "l" } { "t" } if$ change.case$ "edition" bibinfo.check " " * bbl.edition * } if$ } INTEGERS { multiresult } FUNCTION {multi.page.check} { 't := #0 'multiresult := { multiresult not t empty$ not and } { t #1 #1 substring$ duplicate$ "-" = swap$ duplicate$ "," = swap$ "+" = or or { #1 'multiresult := } { t #2 global.max$ substring$ 't := } if$ } while$ multiresult } FUNCTION {format.pages} { pages duplicate$ empty$ 'skip$ { duplicate$ multi.page.check { n.dashify } { } if$ "pages" bibinfo.check } if$ } FUNCTION {format.journal.pages} { pages duplicate$ empty$ 'pop$ { swap$ duplicate$ empty$ { pop$ pop$ format.pages } { ", " * swap$ n.dashify "pages" bibinfo.check * } if$ } if$ } FUNCTION {format.vol.num.pages} { volume field.or.null duplicate$ empty$ 'skip$ { "volume" bibinfo.check } if$ bolden format.journal.pages } FUNCTION {format.chapter.pages} { chapter empty$ 'format.pages { type empty$ { bbl.chapter } { type "l" change.case$ "type" bibinfo.check } if$ chapter tie.or.space.prefix "chapter" bibinfo.check * * pages empty$ 'skip$ { ", " * format.pages * } if$ } if$ } FUNCTION {format.booktitle} { booktitle "booktitle" bibinfo.check emphasize } FUNCTION {format.in.ed.booktitle} { format.booktitle duplicate$ empty$ 'skip$ { editor "editor" format.names.ed duplicate$ empty$ 'pop$ { "," * " " * get.bbl.editor ", " * * swap$ * } if$ word.in swap$ * } if$ } FUNCTION {format.thesis.type} { type duplicate$ empty$ 'pop$ { swap$ pop$ "t" change.case$ "type" bibinfo.check } if$ } FUNCTION {format.tr.number} { number "number" bibinfo.check type duplicate$ empty$ { pop$ bbl.techrep } 'skip$ if$ "type" bibinfo.check swap$ duplicate$ empty$ { pop$ "t" change.case$ } { tie.or.space.prefix * * } if$ } FUNCTION {format.article.crossref} { word.in " \cite{" * crossref * "}" * } FUNCTION {format.book.crossref} { volume duplicate$ empty$ { "empty volume in " cite$ * "'s crossref of " * crossref * warning$ pop$ word.in } { bbl.volume capitalize swap$ tie.or.space.prefix "volume" bibinfo.check * * bbl.of space.word * } if$ " \cite{" * crossref * "}" * } FUNCTION {format.incoll.inproc.crossref} { word.in " \cite{" * crossref * "}" * } FUNCTION {format.org.or.pub} { 't := "" address empty$ t empty$ and 'skip$ { t empty$ { address "address" bibinfo.check * } { t * address empty$ 'skip$ { ", " * address "address" bibinfo.check * } if$ } if$ } if$ } FUNCTION {format.publisher.address} { publisher "publisher" bibinfo.warn format.org.or.pub } FUNCTION {format.organization.address} { organization "organization" bibinfo.check format.org.or.pub } FUNCTION {article} { output.bibitem format.authors "author" output.check author format.key output format.date "year" output.check date.block format.title "title" output.check new.sentence crossref missing$ { journal "journal" bibinfo.check emphasize "journal" output.check format.vol.num.pages output } { format.article.crossref output.nonnull format.pages output } if$ format.note output fin.entry } FUNCTION {book} { output.bibitem author empty$ { format.editors "author and editor" output.check editor format.key output } { format.authors output.nonnull crossref missing$ { "author and editor" editor either.or.check } 'skip$ if$ } if$ format.date "year" output.check date.block format.btitle "title" output.check crossref missing$ { format.bvolume output new.sentence format.number.series output format.publisher.address output } { new.sentence format.book.crossref output.nonnull } if$ format.edition output format.note output fin.entry } FUNCTION {booklet} { output.bibitem format.authors output author format.key output format.date "year" output.check date.block format.title "title" output.check new.sentence howpublished "howpublished" bibinfo.check output address "address" bibinfo.check output format.note output fin.entry } FUNCTION {inbook} { output.bibitem author empty$ { format.editors "author and editor" output.check editor format.key output } { format.authors output.nonnull crossref missing$ { "author and editor" editor either.or.check } 'skip$ if$ } if$ format.date "year" output.check date.block format.btitle "title" output.check crossref missing$ { format.bvolume output format.chapter.pages "chapter and pages" output.check new.sentence format.number.series output format.publisher.address output } { format.chapter.pages "chapter and pages" output.check new.sentence format.book.crossref output.nonnull } if$ format.edition output format.note output fin.entry } FUNCTION {incollection} { output.bibitem format.authors "author" output.check author format.key output format.date "year" output.check date.block format.title "title" output.check new.sentence crossref missing$ { format.in.ed.booktitle "booktitle" output.check format.bvolume output format.number.series output format.chapter.pages output format.publisher.address output format.edition output } { format.incoll.inproc.crossref output.nonnull format.chapter.pages output } if$ format.note output fin.entry } FUNCTION {inproceedings} { output.bibitem format.authors "author" output.check author format.key output format.date "year" output.check date.block format.title "title" output.check new.sentence crossref missing$ { format.in.ed.booktitle "booktitle" output.check format.bvolume output format.number.series output format.pages output publisher empty$ { format.organization.address output } { organization "organization" bibinfo.check output format.publisher.address output } if$ } { format.incoll.inproc.crossref output.nonnull format.pages output } if$ format.note output fin.entry } FUNCTION {conference} { inproceedings } FUNCTION {manual} { output.bibitem format.authors output author format.key output format.date "year" output.check date.block format.btitle "title" output.check new.sentence organization "organization" bibinfo.check output address "address" bibinfo.check output format.edition output format.note output fin.entry } FUNCTION {mastersthesis} { output.bibitem format.authors "author" output.check author format.key output format.date "year" output.check date.block format.btitle "title" output.check new.sentence bbl.mthesis format.thesis.type output.nonnull school "school" bibinfo.warn output address "address" bibinfo.check output format.note output fin.entry } FUNCTION {misc} { output.bibitem format.authors output author format.key output format.date "year" output.check date.block format.title output new.sentence howpublished "howpublished" bibinfo.check output format.note output fin.entry } FUNCTION {phdthesis} { output.bibitem format.authors "author" output.check author format.key output format.date "year" output.check date.block format.btitle "title" output.check new.sentence bbl.phdthesis format.thesis.type output.nonnull school "school" bibinfo.warn output address "address" bibinfo.check output format.note output fin.entry } FUNCTION {proceedings} { output.bibitem format.editors output editor format.key output format.date "year" output.check date.block format.btitle "title" output.check format.bvolume output format.number.series output publisher empty$ { format.organization.address output } { organization "organization" bibinfo.check output format.publisher.address output } if$ format.note output fin.entry } FUNCTION {techreport} { output.bibitem format.authors "author" output.check author format.key output format.date "year" output.check date.block format.title "title" output.check new.sentence format.tr.number output.nonnull institution "institution" bibinfo.warn output address "address" bibinfo.check output format.note output fin.entry } FUNCTION {unpublished} { output.bibitem format.authors "author" output.check author format.key output format.date "year" output.check date.block format.title "title" output.check format.note "note" output.check fin.entry } FUNCTION {default.type} { misc } READ FUNCTION {sortify} { purify$ "l" change.case$ } INTEGERS { len } FUNCTION {chop.word} { 's := 'len := s #1 len substring$ = { s len #1 + global.max$ substring$ } 's if$ } FUNCTION {format.lab.names} { 's := "" 't := s #1 "{vv~}{ll}" format.name$ s num.names$ duplicate$ #2 > { pop$ " " * bbl.etal emphasize * } { #2 < 'skip$ { s #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" = { " " * bbl.etal emphasize * } { " \& " * s #2 "{vv~}{ll}" format.name$ * } if$ } if$ } if$ } FUNCTION {author.key.label} { author empty$ { key empty$ { cite$ #1 #3 substring$ } 'key if$ } { author format.lab.names } if$ } FUNCTION {author.editor.key.label} { author empty$ { editor empty$ { key empty$ { cite$ #1 #3 substring$ } 'key if$ } { editor format.lab.names } if$ } { author format.lab.names } if$ } FUNCTION {editor.key.label} { editor empty$ { key empty$ { cite$ #1 #3 substring$ } 'key if$ } { editor format.lab.names } if$ } FUNCTION {calc.short.authors} { type$ "book" = type$ "inbook" = or 'author.editor.key.label { type$ "proceedings" = 'editor.key.label 'author.key.label if$ } if$ 'short.list := } FUNCTION {calc.label} { calc.short.authors short.list "(" * year duplicate$ empty$ short.list key field.or.null = or { pop$ "" } 'skip$ if$ * 'label := } FUNCTION {sort.format.names} { 's := #1 'nameptr := "" s num.names$ 'numnames := numnames 'namesleft := { namesleft #0 > } { s nameptr "{vv{ } }{ll{ }}{ f{ }}{ jj{ }}" format.name$ 't := nameptr #1 > { " " * namesleft #1 = t "others" = and { "zzzzz" * } { numnames #2 > nameptr #2 = and { "zz" * year field.or.null * " " * } 'skip$ if$ t sortify * } if$ } { t sortify * } if$ nameptr #1 + 'nameptr := namesleft #1 - 'namesleft := } while$ } FUNCTION {sort.format.title} { 't := "A " #2 "An " #3 "The " #4 t chop.word chop.word chop.word sortify #1 global.max$ substring$ } FUNCTION {author.sort} { author empty$ { key empty$ { "to sort, need author or key in " cite$ * warning$ "" } { key sortify } if$ } { author sort.format.names } if$ } FUNCTION {author.editor.sort} { author empty$ { editor empty$ { key empty$ { "to sort, need author, editor, or key in " cite$ * warning$ "" } { key sortify } if$ } { editor sort.format.names } if$ } { author sort.format.names } if$ } FUNCTION {editor.sort} { editor empty$ { key empty$ { "to sort, need editor or key in " cite$ * warning$ "" } { key sortify } if$ } { editor sort.format.names } if$ } FUNCTION {presort} { calc.label label sortify " " * type$ "book" = type$ "inbook" = or 'author.editor.sort { type$ "proceedings" = 'editor.sort 'author.sort if$ } if$ #1 entry.max$ substring$ 'sort.label := sort.label * " " * title field.or.null sort.format.title * #1 entry.max$ substring$ 'sort.key$ := } ITERATE {presort} SORT STRINGS { last.label next.extra } INTEGERS { last.extra.num number.label } FUNCTION {initialize.extra.label.stuff} { #0 int.to.chr$ 'last.label := "" 'next.extra := #0 'last.extra.num := #0 'number.label := } FUNCTION {forward.pass} { last.label label = { last.extra.num #1 + 'last.extra.num := last.extra.num int.to.chr$ 'extra.label := } { "a" chr.to.int$ 'last.extra.num := "" 'extra.label := label 'last.label := } if$ number.label #1 + 'number.label := } FUNCTION {reverse.pass} { next.extra "b" = { "a" 'extra.label := } 'skip$ if$ extra.label 'next.extra := extra.label duplicate$ empty$ 'skip$ { "{\natexlab{" swap$ * "}}" * } if$ 'extra.label := label extra.label * 'label := } EXECUTE {initialize.extra.label.stuff} ITERATE {forward.pass} REVERSE {reverse.pass} FUNCTION {bib.sort.order} { sort.label " " * year field.or.null sortify * " " * title field.or.null sort.format.title * #1 entry.max$ substring$ 'sort.key$ := } ITERATE {bib.sort.order} SORT FUNCTION {begin.bib} { preamble$ empty$ 'skip$ { preamble$ write$ newline$ } if$ "\begin{thebibliography}{" number.label int.to.str$ * "}" * write$ newline$ "\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi" write$ newline$ } EXECUTE {begin.bib} EXECUTE {init.state.consts} ITERATE {call.type$} FUNCTION {end.bib} { newline$ "\end{thebibliography}" write$ newline$ } EXECUTE {end.bib} %% End of customized bst file %% %% End of file `jmb.bst'. ================================================ FILE: common/official-preamble.tex ================================================ %\usepackage{draftwatermark} %\SetWatermarkLightness{0.95} % ****************************************************************************** % ****************************** Custom Margin ********************************* % Add `custommargin' in the document class options to use this section % Set {innerside margin / outerside margin / topmargin / bottom margin} and % other page dimensions \ifsetMargin \else \RequirePackage[left=37mm,right=30mm,top=35mm,bottom=30mm]{geometry} \setFancyHdr % To apply fancy header after geometry package is loaded \fi %\chead{Unfinished draft} %\cfoot{\texttt{Unfinished draft - compiled on \today{} at \currenttime}} % ***************************************************************************** % ******************* Fonts (like different typewriter fonts etc.)************* % Add `customfont' in the document class option to use this section \ifsetFont \else % Set your custom font here and use `customfont' in options. Leave empty to % load computer modern font (default LaTeX font). \RequirePackage{libertine} \fi % ***************************************************************************** % *************************** Bibliography and References ******************** %\usepackage{cleveref} %Referencing without need to explicitly state fig /table % Add `custombib' in the document class option to use this section \ifsetBib % True, Bibliography option is chosen in class options \else % If custom bibliography style chosen then load bibstyle here \RequirePackage[square, sort, numbers, authoryear]{natbib} % CustomBib % If you would like to use biblatex for your reference management, as opposed to the default `natbibpackage` pass the option `custombib` in the document class. Comment out the previous line to make sure you don't load the natbib package. Uncomment the following lines and specify the location of references.bib file % \RequirePackage[backend=biber, style=numeric-comp, citestyle=numeric, sorting=nty, natbib=true]{biblatex} % \bibliography{References/references} %Location of references.bib only for biblatex \fi % changes the default name `Bibliography` -> `References' \renewcommand{\bibname}{References} % ***************************************************************************** % *************** Changing the Visual Style of Chapter Headings *************** % Uncomment the section below. Requires titlesec package. %\RequirePackage{titlesec} %\newcommand{\PreContentTitleFormat}{\titleformat{\chapter}[display]{\scshape\Large} %{\Large\filleft{\chaptertitlename} \Huge\thechapter} %{1ex}{} %[\vspace{1ex}\titlerule]} %\newcommand{\ContentTitleFormat}{\titleformat{\chapter}[display]{\scshape\huge} %{\Large\filleft{\chaptertitlename} \Huge\thechapter}{1ex} %{\titlerule\vspace{1ex}\filright} %[\vspace{1ex}\titlerule]} %\newcommand{\PostContentTitleFormat}{\PreContentTitleFormat} %\PreContentTitleFormat % ***************************************************************************** % **************************** Custom Packages ******************************** % ***************************************************************************** % ************************* Algorithms and Pseudocode ************************** %\usepackage{algpseudocode} % ********************Captions and Hyperreferencing / URL ********************** % Captions: This makes captions of figures use a boldfaced small font. %\RequirePackage[small,bf]{caption} \RequirePackage[labelsep=space,tableposition=top]{caption} %\renewcommand{\figurename}{Figure} %to support older versions of captions.sty \captionsetup{labelsep = colon,belowskip=12pt,aboveskip=4pt} % ************************ Formatting / Footnote ******************************* %\usepackage[perpage]{footmisc} %Range of footnote options % ****************************** Line Numbers ********************************** %\RequirePackage{lineno} %\linenumbers % ************************** Graphics and figures ***************************** %\usepackage{rotating} %\usepackage{wrapfig} %\usepackage{float} \usepackage{subfig} %note: subfig must be included after the `caption` package. % ********************************* Table ************************************** %\usepackage{longtable} %\usepackage{multicol} %\usepackage{multirow} %\usepackage{tabularx} % ***************************** Math and SI Units ****************************** \usepackage{amsfonts} \usepackage{amsmath} \usepackage{amssymb} %\usepackage{siunitx} % use this package module for SI units % ****************************************************************************** % ************************* User Defined Commands ****************************** % ****************************************************************************** % *********** To change the name of Table of Contents / LOF and LOT ************ %\renewcommand{\contentsname}{My Table of Contents} %\renewcommand{\listfigurename}{List of figures} %\renewcommand{\listtablename}{List of tables} % ********************** TOC depth and numbering depth ************************* \setcounter{secnumdepth}{2} \setcounter{tocdepth}{2} % ******************************* Nomenclature ********************************* % To change the name of the Nomenclature section, uncomment the following line %\renewcommand{\nomname}{Symbols} % ********************************* Appendix *********************************** % The default value of both \appendixtocname and \appendixpagename is `Appendices'. These names can all be changed via: %\renewcommand{\appendixtocname}{List of appendices} %\renewcommand{\appendixname}{Appndx} ================================================ FILE: common/preamble.tex ================================================ % All my custom preamble stuff. Shouldn't overlap with anything in official-preamble % Paths to figure and table directories. \newcommand{\symmetryfigsdir}{figures/symmetries} \newcommand{\topologyfiguresdir}{figures/topology} \newcommand{\infinitefiguresdir}{figures/infinite} \newcommand{\grammarfiguresdir}{figures/grammar} \newcommand{\introfigsdir}{figures/intro} \newcommand{\gplvmfiguresdir}{figures/gplvm} \newcommand{\warpedfiguresdir}{figures/warped-mixtures} \newcommand{\deeplimitsfiguresdir}{figures/deep-limits} \newcommand{\quadraturefigsdir}{figures/quadrature} \newcommand{\additivefigsdir}{figures/additive} \newcommand{\decompfigsdir}{figures/decomp} \newcommand{\examplefigsdir}{figures/worked-example} \usepackage{bm} % for warped mixtures - is this necessary? \usepackage{booktabs} \usepackage{tabularx} \usepackage{multirow} \usepackage{datetime} \renewcommand{\tabularxcolumn}[1]{>{\arraybackslash}m{#1}} \usepackage{relsize} \usepackage{graphicx} \usepackage{amsmath,amssymb,textcomp} \usepackage{nicefrac} \usepackage{amsthm} \usepackage{tikz} \usetikzlibrary{arrows} \usetikzlibrary{calc} \usepackage{nth} \usepackage{rotating} \usepackage{array} \usepackage{fp} \usepackage{cleveref} % Note: this package sometimes causes the page counter to reset. \crefname{equation}{equation}{equations} \crefname{figure}{figure}{figures} %\usepackage{common/sectsty} % Controls capitalization of all headers %\usepackage{stringstrings} %\usepackage[explicit]{titlesec} %\newcommand\SentenceCase[1]{% % \caselower[e]{#1}% % \capitalize[q]{\thestring}% %} %\titleformat{\section} % {\normalfont\Large\bfseries}{\thesection}{1em}{\SentenceCase{#1}\thestring} %\titleformat{\section} % The normal, unstarred version % {\Large\bfseries}{}{2ex} % {\thesection. \MakeSentenceCase{#1}} %\titleformat{name=\section,numberless} % The starred version; note the `numberless` key % {\Large\bfseries}{}{2ex} % {\MakeSentenceCase{#1}} \usepackage[hyperpageref]{backref} % Setup to show (pages 4 and 9) sort of thing in the bibliography - DD %\def\foo{\hspace{\fill}\mbox{}\linebreak[0]\hspace*{\fill}} %\def\foo{\parbox{3cm}{\hfill} %\def\foo{\parbox{3cm}{\hfill} %\newcommand\foo[1]{{\raggedleft{\hfill{\mbox{\hfill{#1}}}}}} \newcommand{\comfyfill}[1]{% = Thorsten Donig's \signed \unskip\hspace*{0.1em plus 1fill} \nolinebreak[3]% \hspace*{\fill}\mbox{#1} \parfillskip0pt\par } \newcommand\foo[1]{{\comfyfill{\mbox{#1}}}} %\newcommand\foo[1]{{\mbox{#1}}} \renewcommand*{\backref}[1]{} \renewcommand*{\backrefalt}[4]{% \ifcase #1 % % \or \foo{(page #2)}% \else \foo{(pages #2)}% \fi } \usepackage{stringstrings} %\newcommand{\headercase}{\ %\DeclareFieldFormat{titlecase}{\MakeSentenceCase{#1}} %% For submission, make all render blank. \input{common/commenting.tex} %\renewcommand{\LATER}[1]{} %\renewcommand{\fLATER}[1]{} %\renewcommand{\TBD}[1]{} %\renewcommand{\fTBD}[1]{} %\renewcommand{\PROBLEM}[1]{} %\renewcommand{\fPROBLEM}[1]{} %\renewcommand{\NA}[1]{} % HUMBLE WORDS: shown slightly smaller when in normal text % Thanks to Christian Steinruecken! \input{common/humble.tex} % TODO: Clean up duplicates \declareHumble{ANOVA}{ANOVA} \declareHumble{ARD}{ARD} \declareHumble{BIC}{BIC} \declareHumble{BMC}{BMC} \declareHumble{bq}{BQ} \declareHumble{CRP}{CRP} \declareHumble{dirpro}{DP} \declareHumble{HDMR}{HDMR} \declareHumble{GAM}{GAM} \declareHumble{GEM}{GEM} \declareHumble{GMM}{GMM} \declareHumble{gplvm}{GP-LVM} \declareHumble{gpml}{GPML} \declareHumble{GPML}{GPML} \declareHumble{gprn}{GPRN} \declareHumble{gpt}{GP} \declareHumble{gp}{GP} \declareHumble{HKL}{HKL} \declareHumble{HMC}{HMC} \declareHumble{ibp}{IBP} \declareHumble{iGMM}{iGMM} \declareHumble{iwmm}{iWMM} \declareHumble{kCP}{CP} \declareHumble{kCW}{CW} \declareHumble{kC}{C} \declareHumble{KDE}{KDE} \declareHumble{kLin}{Lin} \declareHumble{KPCA}{KPCA} \declareHumble{kPer}{Per} \declareHumble{kPerGen}{ZMPer} \declareHumble{kRQ}{RQ} \declareHumble{kSE}{SE} \declareHumble{kWN}{WN} \declareHumble{Lin}{Lin} \declareHumble{LBFGS}{L-BFGS} \declareHumble{LIBSVM}{LIBSVM} \declareHumble{MAP}{MAP} \declareHumble{mcmc}{MCMC} \declareHumble{MKL}{MKL} \declareHumble{MLP}{MLP} \declareHumble{MNIST}{MNIST} \declareHumble{MSE}{MSE} \declareHumble{OU}{OU} \declareHumble{Per}{Per} \declareHumble{RBF}{RBF} \declareHumble{RMSE}{RMSE} \declareHumble{RQ}{RQ} \declareHumble{SBQ}{SBQ} \declareHumble{seard}{SE-ARD} \declareHumble{sefull}{SE-\textnormal{full}} \declareHumble{SEGP}{SE-GP} \declareHumble{SE}{SE} \declareHumble{SNR}{SNR} \declareHumble{SSANOVA}{SS-ANOVA} \declareHumble{SVM}{SVM} \declareHumble{UCI}{UCI} \declareHumble{UMIST}{UMIST} \declareHumble{vbgplvm}{VB GP-LVM} \newcommand{\kSig}{\boldsymbol\sigma} \def\subexpr{{\cal S}} \def\baseker{{\cal B}} \def\numWinners{k} \def\ie{i.e.\ } \def\eg{e.g.\ } \def\etc{etc.\ } \let\oldemptyset\emptyset %\let\emptyset 0 % Unify notation between neural-net land and GP-land. \newcommand{\hphi}{h} \newcommand{\hPhi}{\vh} \newcommand{\walpha}{w} \newcommand{\wboldalpha}{\bw} \newcommand{\wcapalpha}{\vW} \newcommand{\lengthscale}{w} \newcommand{\layerindex}{\ell} \newcommand{\gpdrawbox}[1]{ \setlength\fboxsep{0pt} \hspace{-0.15in} \fbox{ \includegraphics[width=0.464\columnwidth]{\deeplimitsfiguresdir/deep_draws/deep_gp_sample_layer_#1} }} \newcommand{\procedurename}{ABCD} \newcommand{\genText}[1]{{\sf #1}} \newcommand{\asdf}{$^{\textnormal{th}}$} \newcommand{\binarysum}{\sum_{\bf{x} \in \{0,1\}^D}} \newcommand{\expect}{\mathbb{E}} \newcommand{\expectargs}[2]{\mathbb{E}_{#1} \left[ {#2} \right]} \newcommand{\var}{\mathbb{V}} \newcommand{\varianceargs}[2]{\mathbb{V}_{#1} \left[ {#2} \right]} \newcommand{\cov}{\operatorname{cov}} \newcommand{\Cov}{\operatorname{Cov}} \newcommand{\covargs}[2]{\Cov \left[ {#1}, {#2} \right]} \newcommand{\variance}{\mathbb{V}} \newcommand{\vecop}[1]{\operatorname{vec} \left( {#1} \right)} \newcommand{\covarianceargs}[2]{\Cov_{#1} \left[ {#2} \right]} \newcommand{\colvec}[2]{\left[ \begin{array}{c} {#1} \\ {#2} \end{array} \right]} \newcommand{\tbtmat}[4]{\left[ \begin{array}{cc} {#1} & {#2} \\ {#3} & {#4} \end{array} \right]} \newcommand{\acro}[1]{{\humble{#1}}} %\newcommand{\vect}[1]{\boldsymbol{#1}} \newcommand{\vect}[1]{{\bf{#1}}} \newcommand{\mat}[1]{\mathbf{#1}} \newcommand{\pderiv}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\npderiv}[2]{\nicefrac{\partial #1}{\partial #2}} \newcommand{\pha}{^{\phantom{:}}} \newcommand{\argmin}{\operatornamewithlimits{argmin}} \newcommand{\argmax}{\operatornamewithlimits{argmax}} % The following designed for probabilities with long arguments \newcommand{\Prob}[2]{P\!\left(\,#1\;\middle\vert\;#2\,\right)} \newcommand{\ProbF}[3]{P\!\left(\,#1\!=\!#2\;\middle\vert\;#3\,\right)} \newcommand{\p}[2]{p\!\left(#1\middle\vert#2\right)} \newcommand{\po}[1]{p\!\left(#1\right)} \newcommand{\pF}[3]{p\!\left(\,#1\!=\!#2\;\middle\vert\;#3\,\right)} \newcommand{\mean}[2]{{m}\!\left(#1\middle\vert#2\right)} \newcommand{\valpha}{\boldsymbol{\alpha}} \newcommand{\va}{\vect{a}} \newcommand{\vA}{\vect{A}} \newcommand{\vB}{\mat{B}} \newcommand{\vb}{\vect{b}} \newcommand{\vC}{\mat{C}} \newcommand{\vc}{\vect{c}} \newcommand{\vecf}{\boldsymbol{f}} \newcommand{\vell}{\vect{\ell}} \newcommand{\vepsilon}{\boldsymbol{\epsilon}} \newcommand{\veps}{\boldsymbol{\epsilon}} \newcommand{\ve}{\boldsymbol{\epsilon}} \newcommand{\vf}{\vecf} \newcommand{\vg}{\vect{g}} \newcommand{\vh}{\vect{h}} \newcommand{\vI}{\mat{I}} \newcommand{\vK}{\mat{K}} \newcommand{\vk}{\vect{k}} \newcommand{\vL}{\mat{L}} \newcommand{\vl}{\vect{l}} \newcommand{\vmu}{{\boldsymbol{\mu}}} \newcommand{\vone}{\vect{1}} \newcommand{\vphi}{{\boldsymbol{\phi}}} \newcommand{\vpi}{{\boldsymbol{\pi}}} \newcommand{\vq}{\vect{q}} \newcommand{\vR}{\mat{R}} \newcommand{\vr}{\vect{r}} \newcommand{\vsigma}{{\boldsymbol{\sigma}}} \newcommand{\vSigma}{\mat{\Sigma}} \newcommand{\vS}{\mat{S}} \newcommand{\vs}{\vect{s}} \newcommand{\vtheta}{{\boldsymbol{\theta}}} \newcommand{\vu}{\vect{u}} \newcommand{\vV}{\mat{V}} \newcommand{\vW}{\mat{W}} \newcommand{\vw}{\vect{w}} \newcommand{\vX}{\mat{X}} \newcommand{\vx}{\vect{x}} \newcommand{\vY}{\mat{Y}} \newcommand{\vy}{\vect{y}} \newcommand{\vzero}{\vect{0}} \newcommand{\vZ}{\mat{Z}} \newcommand{\vz}{\vect{z}} % deep gp notation \newcommand{\netweights}{w} \newcommand{\vnetweights}{\vw} \newcommand{\mnetweights}{\vW} \newcommand{\outweights}{\v} \newcommand{\voutweights}{\vv} \newcommand{\moutweights}{\vV} \newcommand{\unitparams}{\v} \newcommand{\vunitparams}{\vv} \newcommand{\munitparams}{\vV} \newcommand{\He}{\mathcal{H}} \newcommand{\normx}[2]{\left\|#1\right\|_{#2}} \newcommand{\Hnorm}[1]{\normx{#1}{\He}} \newcommand{\mmd}{{\rm MMD}} \newcommand{\mf}{\bar{\vf}} %\newcommand{\mf}{\mu} %{\bar{\ell}} \newcommand{\lf}{f} % Likelihood function \newcommand{\st}{_\star} % from simpler log-bq writeup \newcommand{\lftwo}{{\log \ell}} \newcommand{\mftwo}{{\bar \ell}} \newcommand{\loggp}{{\log\acro{GP}}}%| \bX, \vy )}} \newcommand{\loggpdist}{{\acro{GP}(\lftwo)}}%| \vX, \vy )}} \newcommand{\inv}{^{{\mathsmaller{-1}}}} \newcommand{\tohalf}{^{{\mathsmaller{\nicefrac{1}{2}}}}} \newcommand{\Normal}{\mathcal{N}} \newcommand{\N}[3]{\mathcal{N}\!\left(#1 \middle| #2,#3\right)} \newcommand{\Nt}[2]{\mathcal{N}\!\left(#1,#2\right)} \newcommand{\NT}[2]{\mathcal{N}\!\left(#1,#2\right)} \newcommand{\GPdist}[3]{\mathcal{GP}\!\left(#1 \, \middle| \, #2, #3 \right)} \newcommand{\GPdisttwo}[2]{\mathcal{GP}\!\left(\, #1, #2 \right)} \newcommand{\bN}[3]{\mathcal{N}\big(#1 \middle| #2,#3\big)} \newcommand{\boldN}[3]{\text{\textbf{\mathcal{N}}}\big(#1;#2,#3\big)} \newcommand{\ones}[1]{\mat{1}_{#1}} \newcommand{\eye}[1]{\mat{E}_{#1}} \newcommand{\tra}{^{\mathsf{T}}} %\newcommand{\tra}{^{\top}} %\mathsf{T} \newcommand{\trace}{\operatorname{tr}} \newcommand{\shift}{\operatorname{shift}} \renewcommand{\mod}{\operatorname{mod}} \newcommand{\deq}{:=} \newcommand{\oneofk}{\operatorname{one-of-k}} %\newcommand{\degree}{^\circ} \newcommand{\GPt}[2]{\mathcal{GP}\!\left(#1,#2\right)} %\newcommand{\GPt}[2]{\gp\!\left(#1,#2\right)} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\chol}{chol} \DeclareMathOperator{\diag}{diag} \newenvironment{narrow}[2]{% \begin{list}{}{% \setlength{\topsep}{0pt}% \setlength{\leftmargin}{#1}% \setlength{\rightmargin}{#2}% \setlength{\listparindent}{\parindent}% \setlength{\itemindent}{\parindent}% \setlength{\parsep}{\parskip}}% \item[]}{\end{list}} \newcommand{\dist}{\ \sim\ } \def\given{\,|\,} % Table stuff \newcolumntype{C}[1]{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}m{#1}} \newcolumntype{L}[1]{>{\raggedright\let\newline\\\arraybackslash\hspace{0pt}}m{#1}} \newcolumntype{R}[1]{>{\raggedleft\let\newline\\\arraybackslash\hspace{0pt}}m{#1}} \newcommand{\defeq}{\mathrel{:\mkern-0.25mu=}} \def\ie{i.e.\ } \def\eg{e.g.\ } \def\iid{i.i.d.\ } %\def\simiid{\sim_{\mbox{\tiny iid}}} \def\simiid{\overset{\mbox{\tiny iid}}{\sim}} \def\simind{\overset{\mbox{\tiny \textnormal{ind}}}{\sim}} \def\eqdist{\stackrel{\mbox{\tiny d}}{=}} %\newcommand{\distas}[1]{\mathbin{\overset{#1}{\kern \z@ \sim}}} %TODO: fix this - it worked outside the thesis! \newcommand{\distas}[1]{\mathbin{\overset{#1}{\sim}}} \def\Reals{\mathbb{R}} \def\Uniform{\mbox{\rm Uniform}} \def\Bernoulli{\mbox{\rm Bernoulli}} \def\GP{\mathcal{GP}} \def\GPLVM{\mathcal{GP-LVM}} % Kernel stuff \def\iva{\vect{\inputVar}} \def\ivaone{\inputVar} \def\inputVar{x} \def\InputVar{X} \def\InputSpace{\mathcal{X}} \def\outputVar{y} \def\OutputSpace{\mathcal{Y}} \def\function{f} \def\kernel{k} \def\KernelMatrix{K} \def\SumKernel{\sum} \def\ProductKernel{\prod} \def\expression{e} \def\feat{\vh} \newcommand{\kerntimes}{ \! \times \!} \newcommand{\kernplus}{ \, + \,} % Proof stuff \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{prop}[theorem]{Proposition} \newtheorem{proposition}{Proposition} \newtheorem*{cor}{Corollary} % For infinite bq \newcommand{\iv}{\theta} \newcommand{\viv}{\vtheta} % For intro chapter \newcommand{\funcval}{\vf(\vX)} \newcommand{\testpoint}{{\vx^\star}} \newcommand{\underwrite}[2]{{\underbrace{#1}_{\textnormal{#2}}}} % For kernel figures \newcommand{\fhbig}{2cm}% \newcommand{\fwbig}{3cm}% \newcommand{\kernpic}[1]{\includegraphics[height=\fhbig,width=\fwbig]{\grammarfiguresdir/structure_examples/#1}}% \newcommand{\kernpicr}[1]{\rotatebox{90}{\includegraphics[height=\fwbig,width=\fhbig]{\grammarfiguresdir/structure_examples/#1}}}% \newcommand{\addkernpic}[1]{{\includegraphics[height=\fhbig,width=\fwbig]{\grammarfiguresdir/additive_multi_d/#1}}}% \newcommand{\largeplus}{\tabbox{{\Large+}}}% \newcommand{\largeeq}{\tabbox{{\Large=}}}% \newcommand{\largetimes}{\tabbox{{\Large$\times$}}}% \newcommand{\fixedx}{$x$ (with $x' = 1$)}% % for warped mixtures \newcommand{\CLAS}{\vz} %cluster assignments \newcommand{\CLASi}{z} % individual cluster assignments ================================================ FILE: common/thesis-info.tex ================================================ % ************************ Thesis Information & Meta-data ********************** %% The title of the thesis %\title{Structured Gaussian Process Models} %\title{Automatic Model Construction \\ through \\ Structured Gaussian Processes} %\title{Automatic Model-Building \\ through \\ Structured Gaussian Processes} %\title{Automatic Modeling \\ with \\ Structured Gaussian Processes} \title{Automatic Model Construction \\ with Gaussian Processes} %\title{Automatic Model Construction} %\title{Automating Statistical Model Construction} %\texorpdfstring is used for PDF metadata. Usage: %\texorpdfstring{LaTeX_Version}{PDF Version (non-latex)} eg., %\texorpdfstring{$sigma$}{sigma} %% The full name of the author \author{David Kristjanson Duvenaud} %% Department (eg. Department of Engineering, Maths, Physics) %\dept{Department of Engineering} %% University and Crest \university{University of Cambridge} \crest{\includegraphics[width=0.25\textwidth]{misc/University_Crest}} %% You can redefine the submission text: % Default as per the University guidelines: This dissertation is submitted for % the degree of Doctor of Philosophy %\renewcommand{\submissiontext}{change the default text here if needed} %% Full title of the Degree \degree{Doctor of Philosophy} %% College affiliation (optional) \college{Pembroke College} %% Submission date \degreedate{June 2014} %% Meta information \subject{Machine Learning} \keywords{{LaTeX} {PhD Thesis} {Engineering} {University of Cambridge} {Machine Learning} {Gaussian processes} {Time Series}} ================================================ FILE: deeplimits.tex ================================================ \input{common/header.tex} \inbpdocument \chapter{Deep Gaussian Processes} \label{ch:deep-limits} \begin{quotation} ``I asked myself: On any given day, would I rather be wrestling with a sampler, or proving theorems?'' \hspace*{\fill} \emph{ -- Peter Orbanz}, personal communication \end{quotation} %A less structured, but more flexible alternative to the \gp{} models explored in this thesis For modeling high-dimensional functions, a popular alternative to the Gaussian process models explored earlier in this thesis are deep neural networks. When training deep neural networks, choosing appropriate architectures and regularization strategies is important for good predictive performance. In this chapter, we propose to study this problem by viewing deep neural networks as priors on functions. By viewing neural networks this way, one can analyze their properties without reference to any particular dataset, loss function, or training method. %Instead, we one ask what sorts of information-processing structures these priors give rise to, and check whether those structures are the same sort which we expect to find in useful models. %To shed light on this problem, we analyze the analogous problem of constructing useful priors on compositions of functions. As a starting point, we will relate neural networks to Gaussian processes, and examine a class of infinitely-wide, deep neural networks called \emph{deep Gaussian processes}: compositions of functions drawn from \gp{} priors. Deep \gp{}s are an attractive model class to study for several reasons. First, \citet{damianou2012deep} showed that the probabilistic nature of deep \gp{}s guards against overfitting. Second, \citet{hensman2014deep} showed that stochastic variational inference is possible in deep \gp{}s, allowing mini-batch training on large datasets. Third, the availability of an approximation to the marginal likelihood allows one to automatically tune the model architecture without the need for cross-validation. %Together, these results suggest that deep \gp{}s are a promising alternative to neural nets. %For the analysis in this chapter, Finally, deep \gp{}s are attractive from a model-analysis point of view, because they remove some of the details of finite neural networks. %, such as the number of hidden units Our analysis will show that in standard architectures, the representational capacity of standard deep networks tends to decrease as the number of layers increases, retaining only a single degree of freedom in the limit. We propose an alternate network architecture that connects the input to each layer that does not suffer from this pathology. We also examine \emph{deep kernels}, obtained by composing arbitrarily many fixed feature transforms. The ideas contained in this chapter were developed through discussions with Oren Rippel, Ryan Adams and Zoubin Ghahramani, and appear in \citet{DuvRipAdaGha14}. %\section{Relations to Neural Networks} %\section{Relating Neural Networks to Deep Gaussian\\ \mbox{Processes}} \section{Relating deep neural networks to deep \sgp{}s} \label{sec:relating} This section gives a precise definition of deep \gp{}s, reviews the precise relationship between neural networks and Gaussian processes, and gives two equivalent ways of constructing neural networks which give rise to deep \gp{}s. \subsection{Definition of deep \sgp{}s} We define a deep \gp{} as a distribution on functions constructed by composing functions drawn from \gp{} priors. An example of a deep \gp{} is a composition of vector-valued functions, with each function drawn independently from \gp{} priors: % \begin{align} \vf^{(1:L)}(\vx) = \vf^{(L)}(\vf^{(L-1)}(\dots \vf^{(2)}(\vf^{(1)}(\vx)) \dots)) \\ %\nonumber\\ \textnormal{with each} \quad f_d^{(\layerindex)} \simind \gp{} \left( 0, k^{(\layerindex)}_d(\vx, \vx') \right) \nonumber \label{eq:deep-gp} \end{align} % %\citet{damianou2012deep} also use the term to refer to deep latent-variable models i Multilayer neural networks also implement compositions of vector-valued functions, one per layer. Therefore, understanding general properties of function compositions might helps us to gain insight into deep neural networks. %\subsection{A neural net with one hidden layer} %\vspace{-0.05in} \subsection{Single-hidden-layer models} First, we relate neural networks to standard ``shallow'' Gaussian processes, using the standard neural network architecture known as the multi-layer perceptron (\MLP{})~\citep{rosenblatt1962principles}. In the typical definition of an \MLP{} with one hidden layer, the hidden unit activations are defined as: % \begin{align} \vh(\vx) = \sigma \left( \vb + \munitparams \vx \right) \end{align} % where $\vh$ are the hidden unit activations, $\vb$ is a bias vector, $\munitparams$ is a weight matrix and $\sigma$ is a one-dimensional nonlinear function, usually a sigmoid, applied element-wise. The output vector $\vf(\vx)$ is simply a weighted sum of these hidden unit activations: % \begin{align} \vf(\vx) = \mnetweights \sigma \left( \vb + \munitparams \vx \right) = \mnetweights \vh(\vx) \label{eq:one-layer-nn} \end{align} % where $\mnetweights$ is another weight matrix. %There exists a correspondence between one-layer \MLP{}s and \gp{}s \citet[chapter 2]{neal1995bayesian} showed that some neural networks with infinitely many hidden units, one hidden layer, and unknown weights correspond to Gaussian processes. More precisely, for any model of the form % \begin{align} f(\vx) = \frac{1}{K}{\mathbf \vnetweights}\tra \hPhi(\vx) = \frac{1}{K} \sum_{i=1}^K \netweights_i \hphi_i(\vx), \label{eq:one-layer-gp} \end{align} % with fixed% \footnote{The above derivation gives the same result if the parameters of the hidden units are random, since in the infinite limit, their distribution on outputs is always the same with probability one. However, to avoid confusion, we refer to layers with infinitely-many nodes as ``fixed''. } features $\left[ \hphi_1(\vx), \dots, \hphi_K(\vx) \right]\tra = \hPhi(\vx)$ and i.i.d. $\netweights$'s with zero mean and finite variance $\sigma^2$, the central limit theorem implies that as the number of features $K$ grows, any two function values $f(\vx)$ and $f(\vx')$ have a joint distribution approaching a Gaussian: % \begin{align} \lim_{K \to \infty} p\left( \colvec{f(\vx)}{f(\vx')} \right) = \Nt{\colvec{0}{0}}{ \frac{\sigma^2}{K} \left[ \begin{array}{cc} \sum_{i=1}^K \hphi_i(\vx)\hphi_i(\vx) & \sum_{i=1}^K \hphi_i(\vx)\hphi_i(\vx') \\ \sum_{i=1}^K \hphi_i(\vx')\hphi_i(\vx) & \sum_{i=1}^K \hphi_i(\vx')\hphi_i(\vx') \end{array} \right] } \end{align} A joint Gaussian distribution between any set of function values is the definition of a Gaussian process. %Thus we can say that $f \sim \GPt{0}{ The result is surprisingly general: it puts no constraints on the features (other than having uniformly bounded activation), nor does it require that the feature weights $\vnetweights$ be Gaussian distributed. An \MLP{} with a finite number of nodes also gives rise to a \gp{}, but only if the distribution on $\vnetweights$ is Gaussian. One can also work backwards to derive a one-layer \MLP{} from any \gp{}: Mercer's theorem, discussed in \cref{sec:mercer}, implies that any positive-definite kernel function corresponds to an inner product of features: $k(\vx, \vx') = \hPhi(\vx) \tra \hPhi(\vx')$. %An \MLP{} having a finite number of fixed hidden nodes gives rise to a \gp{} if and only if the weights $\vnetweights$ are jointly Gaussian distributed. Thus, in the one-hidden-layer case, the correspondence between \MLP{}s and \gp{}s is straightforward: the implicit features $\hPhi(\vx)$ of the kernel correspond to hidden units of an \MLP{}. % For tikz figures in deep limits \newcommand{\numdims}[0]{3} \newcommand{\numhidden}[0]{3} \newcommand{\upnodedist}[0]{1cm} \newcommand{\bardist}[0]{\hspace{-0.2cm}} \def\layersep{2.3cm} \def\nodesep{1.3cm} \def\nodesize{1cm} \newcommand{\neuronfunc}[2]{ \FPeval{\result}{clip(#1+#2)} \includegraphics[width=1cm, clip, trim=0mm 0mm 0mm 0mm]{../figures/deep-limits/two-d-draws/sqexp-draw-\result} } \tikzstyle{input neuron}=[neuron, fill=green!15] \tikzstyle{output neuron}=[neuron, fill=red!15] \tikzstyle{hidden neuron}=[neuron, fill=blue!15] \newcommand{\indfeat}{h} %\newcommand{\indfeat}{\phi} \begin{figure}[t] \begin{tabular}{c|c} Neural net corresponding to a \gp{} & Net corresponding to a \gp{} with a deep kernel \\ \\ \null\hspace{-0.25cm} \begin{tikzpicture}[shorten >=1pt,->,draw=black!50, node distance=\layersep] \tikzstyle{every pin edge}=[<-,shorten <=1pt] \tikzstyle{neuron}=[circle,fill=black!25,minimum size=17pt,inner sep=0pt] \tikzstyle{annot} = [text width=4em, text centered] % Draw the input layer nodes \foreach \name / \y in {1,...,\numdims} % This is the same as writing \foreach \name / \y in {1/1,2/2,3/3,4/4} \node[input neuron, minimum size=\nodesize %, pin=left:Input \#\y ] (I-\name) at (0,-\nodesep*\y) {$x_\y$}; % Draw the hidden layer nodes % Draw the hidden layer nodes \foreach \name / \y in {1,2} \path[yshift=0.5cm] node[hidden neuron, minimum size=\nodesize] (H-\name) at (\layersep,-\nodesep*\y) {$\indfeat_{\y}$}; \foreach \name / \y in {3} \path[yshift=0.5cm] node[hidden neuron, minimum size=\nodesize] (H-\name) at (\layersep,-\nodesep*4) {$\indfeat_\infty$}; % Draw the output layer node \foreach \name / \y in {1,...,\numdims} \node[output neuron, minimum size=\nodesize] (O-\name) at (2*\layersep,-\nodesep*\y) {$f_{\y}$}; % Connect every node in the input layer with every node in the hidden layer. \foreach \source in {1,...,\numdims} \foreach \dest in {1,...,\numhidden} \path (I-\source) edge (H-\dest); % Connect every node in the hidden layer with the output layer \foreach \source in {1,...,\numhidden} \foreach \dest in {1,...,\numdims} \path (H-\source) edge (O-\dest); % Annotate the layers \node[annot,below of=H-2, node distance=1.15cm] {$\vdots$}; \node[annot,above of=I-1, node distance=\upnodedist] {Inputs}; \node[annot,above of=H-1, node distance=\upnodedist] {Fixed}; \node[annot,above of=O-1, node distance=\upnodedist] {Random}; \end{tikzpicture} & \begin{tikzpicture}[shorten >=1pt,->,draw=black!50, node distance=\layersep] \tikzstyle{every pin edge}=[<-,shorten <=1pt] \tikzstyle{neuron}=[circle,fill=black!25,minimum size=17pt,inner sep=0pt] \tikzstyle{annot} = [text width=4em, text centered] % Draw the input layer nodes \foreach \name / \y in {1,...,\numdims} % This is the same as writing \foreach \name / \y in {1/1,2/2,3/3,4/4} \node[input neuron, minimum size=\nodesize %, pin=left:Input \#\y ] (I-\name) at (0,-\nodesep*\y) {$x_\y$}; % Draw the first hidden layer nodes \foreach \name / \y in {1,2}%...,\numhidden} \path[yshift=0.5cm] node[hidden neuron, minimum size=\nodesize] (H-\name) at (\layersep,-\nodesep*\y) {$\indfeat^{(1)}_{\y}$}; \foreach \name / \y in {3} \path[yshift=0.5cm] node[hidden neuron, minimum size=\nodesize] (H-\name) at (\layersep,-\nodesep*4) {$\indfeat^{(1)}_\infty$}; % Draw the secdond hidden layer nodes \foreach \name / \y in {1,2}%...,\numhidden} \path[yshift=0.5cm] node[hidden neuron, minimum size=\nodesize] (H2-\name) at (2*\layersep,-\nodesep*\y) {$\indfeat^{(2)}_{\y}$}; \foreach \name / \y in {3} \path[yshift=0.5cm] node[hidden neuron, minimum size=\nodesize] (H2-\name) at (2*\layersep,-\nodesep*4) {$\indfeat^{(2)}_\infty$}; % Draw the output layer node \foreach \name / \y in {1,...,\numdims} \node[output neuron, minimum size=\nodesize %,pin={[pin edge={->}]right:Output } ] (O-\name) at (3*\layersep,-\nodesep*\y) {$f_{\y}$}; % Connect every node in the input layer with every node in the % hidden layer. \foreach \source in {1,...,\numdims} \foreach \dest in {1,...,\numhidden} \path (I-\source) edge (H-\dest); \foreach \source in {1,...,\numhidden} \foreach \dest in {1,...,\numhidden} \path (H-\source) edge (H2-\dest); % Connect every node in the hidden layer with the output layer \foreach \source in {1,...,\numhidden} \foreach \dest in {1,...,\numdims} \path (H2-\source) edge (O-\dest); % Annotate the layers \node[annot,above of=I-1, node distance=\upnodedist] {Inputs}; \node[annot,below of=H-2, node distance=1.15cm] {$\vdots$}; \node[annot,above of=H-1, node distance=\upnodedist] {Fixed}; \node[annot,below of=H2-2, node distance=1.15cm] {$\vdots$}; \node[annot,above of=H2-1, node distance=\upnodedist] {Fixed}; \node[annot,above of=O-1, node distance=\upnodedist] {Random}; \end{tikzpicture} \end{tabular} \caption[Neural network architectures giving rise to \sgp{}s] { \emph{Left:} \gp{}s can be derived as a one-hidden-layer \MLP{} with infinitely many fixed hidden units having unknown weights. \emph{Right:} Multiple layers of fixed hidden units gives rise to a \gp{} with a deep kernel, but not a deep \gp{}. } \label{fig:gp-architectures} \end{figure} \subsection{Multiple hidden layers} Next, we examine infinitely-wide \MLP{}s having multiple hidden layers. There are several ways to construct such networks, giving rise to different priors on functions. In an \MLP{} with multiple hidden layers, activation of the $\layerindex$th layer units are given by % the recurrence % \begin{align} \vh^{(\layerindex)}(\vx) = \sigma \left( \vb^{(\layerindex)} + \munitparams^{(\layerindex)} \vh^{(\layerindex-1)}(\vx) \right) \;. \label{eq:nextlayer} \end{align} This architecture is shown on the right of \cref{fig:gp-architectures}. %In this model, each hidden layer's output feeds directly into the next layer's input, weighted by the corresponding element of $\munitparams^{(\layerindex)}$. % For example, if we extend the model given by \cref{eq:one-layer-nn} to have two layers of feature mappings, the resulting model becomes % \begin{align} f(\vx) = \frac{1}{K}{\mathbf \vnetweights}\tra \hPhi^{(2)}\left( \hPhi^{(1)}(\vx) \right) \;. % = \frac{1}{K} \sum_{i=1}^K \netweights_i \hphi_i(\vx), \label{eq:mutli-layer-nn} \end{align} If the features $\vh^{(1)}(\vx)$ and $\vh^{(2)}(\vx)$ are fixed with only the last-layer weights ${\vnetweights}$ unknown, this model corresponds to a shallow \gp{} with a \emph{deep kernel}, given by \begin{align} k(\vx, \vx') = \left[ \hPhi^{(2)} ( \hPhi^{(1)}(\vx) ) \right] \tra \hPhi^{(2)} (\hPhi^{(1)}(\vx') ) \, . \end{align} Deep kernels, explored in \cref{sec:deep_kernels}, imply a fixed representation as opposed to a prior over representations. Thus, unless we richly parameterize these kernels, their capacity to learn an appropriate representation will be limited in comparison to more flexible models such as deep neural networks or deep \gp{}s. %since only the output weights ${\vnetweights}$ can adapt to the problem which is what we wish to analyze in this chapter. \subsection{Two network architectures equivalent to deep \sgp{}s} \def\halfshift{0.0cm} \begin{figure}[t!] \centering \begin{tabular}{c} %\multicolumn{2}{c}{} \\ %\multicolumn{2}{c}{ A neural net with fixed activation functions corresponding to a 3-layer deep \gp{}\\ %$\vy = \vf^{(2)} \left( \vf^{(1)}(\vx) \right)$ \\ \\ %\multicolumn{2}{c}{ \begin{tikzpicture}[shorten >=1pt,->,draw=black!50, node distance=\layersep] \tikzstyle{every pin edge}=[<-,shorten <=1pt] \tikzstyle{neuron}=[circle,fill=black!25,minimum size=17pt,inner sep=0pt] \tikzstyle{annot} = [text width=4em, text centered] % Draw the input layer nodes \foreach \name / \y in {1,...,\numdims} % This is the same as writing \foreach \name / \y in {1/1,2/2,3/3,4/4} \node[input neuron, minimum size=\nodesize %, pin=left:Input \#\y ] (I-\name) at (0,-\nodesep*\y) {$x_\y$}; % Draw the hidden layer nodes \foreach \name / \y in {1,2}%...,\numhidden} \path[yshift=0.5cm] node[hidden neuron, minimum size=\nodesize] (H-\name) at (\layersep,-\nodesep*\y) {$\indfeat^{(1)}_{\y}$}; \foreach \name / \y in {3} \path[yshift=0.5cm] node[hidden neuron, minimum size=\nodesize] (H-\name) at (\layersep,-\nodesep*4) {$\indfeat^{(1)}_\infty$}; % Draw the hidden layer nodes \foreach \name / \y in {1,2}%...,\numhidden} \path[yshift=0.5cm] node[hidden neuron, minimum size=\nodesize] (H2-\name) at (3*\layersep,-\nodesep*\y) {$\indfeat^{(2)}_{\y}$}; \foreach \name / \y in {3} \path[yshift=0.5cm] node[hidden neuron, minimum size=\nodesize] (H2-\name) at (3*\layersep,-\nodesep*4) {$\indfeat^{(2)}_\infty$}; % Draw the hidden layer nodes \foreach \name / \y in {1,2}%...,\numhidden} \path[yshift=0.5cm] node[hidden neuron, minimum size=\nodesize] (H3-\name) at (5*\layersep,-\nodesep*\y) {$\indfeat^{(3)}_{\y}$}; \foreach \name / \y in {3} \path[yshift=0.5cm] node[hidden neuron, minimum size=\nodesize] (H3-\name) at (5*\layersep,-\nodesep*4) {$\indfeat^{(3)}_\infty$}; % Draw the output layer node \foreach \name / \y in {1,...,\numdims} \node[output neuron, minimum size=\nodesize %,pin={[pin edge={->}]right:Output } ] (O1-\name) at (2*\layersep,-\nodesep*\y) {$f^{(1)}_{\y}$}; % Draw the output layer node \foreach \name / \y in {1,...,\numdims} \node[output neuron, minimum size=\nodesize %,pin={[pin edge={->}]right:Output } ] (O2-\name) at (4*\layersep,-\nodesep*\y) {$f^{(2)}_{\y}$}; % Draw the output layer node \foreach \name / \y in {1,...,\numdims} \node[output neuron, minimum size=\nodesize %,pin={[pin edge={->}]right:Output } ] (O3-\name) at (6*\layersep,-\nodesep*\y) {$f^{(3)}_{\y}$}; % Connect every node in the input layer with every node in the % hidden layer. \foreach \source in {1,...,\numdims} \foreach \dest in {1,...,\numhidden} \path (I-\source) edge (H-\dest); \foreach \source in {1,...,\numhidden} \foreach \dest in {1,...,\numdims} \path (H-\source) edge (O1-\dest); \foreach \source in {1,...,\numdims} \foreach \dest in {1,...,\numhidden} \path (O1-\source) edge (H2-\dest); \foreach \source in {1,...,\numdims} \foreach \dest in {1,...,\numhidden} \path (O2-\source) edge (H3-\dest); \foreach \source in {1,...,\numhidden} \foreach \dest in {1,...,\numdims} \path (H3-\source) edge (O3-\dest); % Connect every node in the hidden layer with the output layer \foreach \source in {1,...,\numhidden} \foreach \dest in {1,...,\numdims} \path (H2-\source) edge (O2-\dest); % Annotate the layers \node[annot,above of=I-1, node distance=\upnodedist] {Inputs}; \node[annot,below of=I-3, node distance=\upnodedist] {$\vx$}; \node[annot,above of=H-1, node distance=\upnodedist] {Fixed}; \node[annot,below of=O1-3, node distance=\upnodedist] {$\vf^{(1)}(\vx)$}; \node[annot,above of=O1-1, node distance=\upnodedist] {Random}; \node[annot,above of=H2-1, node distance=\upnodedist] {Fixed}; \node[annot,below of=O2-3, node distance=\upnodedist] {$\vf^{(1:2)}(\vx)$}; \node[annot,above of=O2-1, node distance=\upnodedist] {Random}; \node[annot,above of=H3-1, node distance=\upnodedist] {Fixed}; \node[annot,above of=O3-1, node distance=\upnodedist] {Random}; \node[annot,below of=O3-3, node distance=\upnodedist] {$\vy$}; \node[annot,below of=H-2, node distance=1.15cm] {$\vdots$}; \node[annot,below of=H2-2, node distance=1.15cm] {$\vdots$}; \node[annot,below of=H3-2, node distance=1.15cm] {$\vdots$}; \end{tikzpicture} %} \\[0.5em] \hline \\[-0.5em] A net with nonparametric activation functions corresponding to a 3-layer deep \gp{}\\ \\ %\begin{tabular}{c} \begin{tikzpicture}[shorten >=1pt,->,draw=black!50, node distance=\layersep] % \tikzstyle{interarrows}=[->, thick, black!50] \tikzstyle{every pin edge}=[<-,shorten <=1pt] % \tikzstyle{input neuron}=[neuron, minimum size=\nodesize]; \tikzstyle{neuron}=[circle, draw = black, inner sep=0pt, line width = 1pt] \tikzstyle{input neuron}=[circle, minimum size=\nodesize, fill=green!15] \tikzstyle{output neuron}=[neuron, minimum size=\nodesize, text width = 1cm]; \tikzstyle{hidden neuron}=[neuron, minimum size=\nodesize, text width = 1cm]; \tikzstyle{annot} = [text width=4em, text centered] % Draw the input layer nodes \foreach \name / \y in {1,...,\numdims} \node[input neuron] (I-\name) at (0,-\nodesep*\y) {$x_\y$}; % Draw the hidden layer nodes \foreach \name / \y in {1,...,\numhidden} \path[yshift=\halfshift] node[hidden neuron] (H-\name) at (\layersep,-\nodesep*\y) { \neuronfunc{\y}{0}}; % Draw the hidden layer nodes \foreach \name / \y in {1,...,\numhidden} \path[yshift=\halfshift] node[hidden neuron] (H2-\name) at (2*\layersep,-\nodesep*\y) {\neuronfunc{\y}{4}}; % Draw the output layer node \foreach \name / \y in {1,...,\numdims} \node[output neuron] (O-\name) at (3*\layersep,-\nodesep*\y) {\neuronfunc{\y}{8}}; % Connect every node in the input layer with every node in the hidden layer. \foreach \source in {1,...,\numdims} \foreach \dest in {1,...,\numhidden} \path (I-\source) edge (H-\dest); \foreach \source in {1,...,\numhidden} \foreach \dest in {1,...,\numhidden} \path (H-\source) edge (H2-\dest); % Connect every node in the hidden layer with the output layer \foreach \source in {1,...,\numhidden} \foreach \dest in {1,...,\numdims} \path (H2-\source) edge (O-\dest); %arrows={-angle 90} % Annotate the layers \node[annot,above of=I-1, node distance=\upnodedist] {Inputs}; \node[annot,below of=I-\numdims, node distance=\upnodedist] {$\vx$}; \node[annot,above of=H-1, node distance=\upnodedist, text width = 2cm] {\gp{}}; \node[annot,above of=H2-1, node distance=\upnodedist, text width = 2cm] {\gp{}}; \node[annot,below of=H-\numhidden, node distance=\upnodedist, text width = 2cm] {$\vf^{(1)}(\vx)$}; \node[annot,below of=H2-\numhidden, node distance=\upnodedist, text width = 2cm] {$\vf^{(1:2)}(\vx)$}; \node[annot,above of=O-1, node distance=\upnodedist] {\gp{}}; \node[annot,below of=O-\numdims, node distance=\upnodedist, text width = 1cm] {$\vy$}; \end{tikzpicture} %\end{tabular} \end{tabular} \caption[Neural network architectures giving rise to deep \sgp{}s] { Two equivalent views of deep \gp{}s as neural networks. \emph{Top:} A neural network whose every other layer is a weighted sum of an infinite number of fixed hidden units, whose weights are initially unknown. \emph{Bottom:} A neural network with a finite number of hidden units, each with a different unknown non-parametric activation function. The activation functions are visualized by draws from 2-dimensional \gp{}s, although their input dimension will actually be the same as the output dimension of the previous layer. } \label{fig:deep-gp-architectures} \end{figure} There are two equivalent neural network architectures that correspond to deep \gp{}s: one having fixed nonlinearities, and another having \gp{}-distributed nonlinearities. To construct a neural network corresponding to a deep \gp{} using only fixed nonlinearities, one can start with the infinitely-wide deep \gp{} shown in \cref{fig:gp-architectures}(right), and introduce a finite set of nodes in between each infinitely-wide set of fixed basis functions. This architecture is shown in the top of \cref{fig:deep-gp-architectures}. The $D^{(\layerindex)}$ nodes $\vf^{(\layerindex)}(\vx)$ in between each fixed layer are weighted sums (with random weights) of the fixed hidden units of the layer below, and the next layer's hidden units depend only on these $D^{(\layerindex)}$ nodes. This alternating-layer architecture has an interpretation as a series of linear information bottlenecks. To see this, substitute \cref{eq:one-layer-nn} into \cref{eq:nextlayer} to get % \begin{align} \hPhi^{(\layerindex)}(\vx) = \sigma \left( \vb^{(\layerindex)} + \left[ \munitparams^{(\layerindex)} \mnetweights^{(\layerindex-1)} \right] \hPhi^{(\layerindex-1)}(\vx) \right) \end{align} % where $\mnetweights^{(\ell-1)}$ is the weight matrix connecting $\hPhi^{(\ell-1)}$ to $\vf^{(\ell - 1)}$. Thus, ignoring the intermediate outputs $\vf^{(\layerindex)}(\vx)$, a deep \gp{} is an infinitely-wide, deep \MLP{} with each pair of layers connected by random, rank-$D_\layerindex$ matrices given by $\munitparams^{(\layerindex)} \mnetweights^{(\layerindex-1)}$. The second, more direct way to construct a network architecture corresponding to a deep \gp{} is to integrate out all $\mnetweights^{(\layerindex)}$, and view deep \gp{}s as a neural network with a finite number of nonparametric, \gp{}-distributed basis functions at each layer, in which $\vf^{(1:\layerindex)}(\vx)$ represent the output of the hidden nodes at the $\layerindex^{th}$ layer. This second view lets us compare deep \gp{} models to standard neural net architectures more directly. \Cref{fig:gp-architectures}(bottom) shows an example of this architecture. %\subsection{Relating initialization strategies, regularization, and priors} %Traditional training of deep networks requires both an an initialization strategy, and a regularization method. In a Bayesian model, the prior implicitly defines both of these things. %For example, one feature of \gp{} models with a clear analogue to initialization strategies in deep networks is automatic relevance determination. %\paragraph{The effects of automatic relevance determination} One feature of Gaussian process models that doesn't have a clear anologue in the neural-net literature is automatic relevance determination (ARD). Recall that %In the ARD procedure, the lengthscales of each dimension are scaled to maximize the marginal likelihood of the GP. In standard one-layer GP regression, it has the effect of smoothing out the posterior over functions in directions in which the data does not change. %\paragraph{Sparse Initialization} %\cite{martens2010deep} used “sparse initialization”, in which each hidden unit had 15 non-zero incoming connections. %They say ``this allows the units to be both highly differentiated as well as unsaturated, avoiding the problem in dense initializations where the connection weights must all be scaled very small in order to prevent saturation, leading to poor differentiation between units''. % %While it is not clear if deep GPs have analogous problems with saturation of connection weights, we note that %An anologous initialization strategy would be to put a heavy-tailed prior on the lengthscales in the squared-exp kernel. %\subsection{What kinds of functions do deep GPs prefer?} %Isotropic kernels such as the squared-exp encode the assumption that the function varies independently in all directions. %This implies that using an isotropic kernel to model a function that does \emph{not} vary in all direction independently is 'wasting marginal likelihood' by not making strong enough assumptions. % %Thus, deep GPs prefer to have hidden layers on whose outputs the higher-layer functions vary independently. In this way, we can say that the deep GP model prefers independent features. %Combining these two properties of isotropic kernels, we can say that a deep GP will attempt to transform its input into a representation with as few degrees of freedom as possible, and for which the output function varies independently across each dimension. \section{Characterizing deep Gaussian process priors} \label{sec:characterizing-deep-gps} This section develops several theoretical results characterizing the behavior of deep \gp{}s as a function of their depth. Specifically, we show that the size of the derivative of a one-dimensional deep \gp{} becomes log-normal distributed as the network becomes deeper. We also show that the Jacobian of a multivariate deep \gp{} is a product of independent Gaussian matrices having independent entries. These results will allow us to identify a pathology that emerges in very deep networks in \cref{sec:formalizing-pathology}. \subsection{One-dimensional asymptotics} \label{sec:1d} %One way to understand the properties of functions drawn from deep \gp{}s and deep networks is by looking at the distribution of the derivative of these functions. We first focus on the one-dimensional case. In this section, we derive the limiting distribution of the derivative of an arbitrarily deep, one-dimensional \gp{} having a squared-exp kernel: %in order to shed light on the ways in which a deep \gp{}s are non-Gaussian, and the ways in which lower layers affect the computation performed by higher layers. % \newcommand{\onedsamplepic}[1]{ \includegraphics[trim=2mm 5mm 7mm 6.4mm, clip, width=0.235\columnwidth]{\deeplimitsfiguresdir/1d_samples/latent_seed_0_1d_large/layer-#1}}% \newcommand{\onedsamplepiccon}[1]{ \includegraphics[trim=2mm 5mm 7mm 6.4mm, clip, width=0.235\columnwidth]{\deeplimitsfiguresdir/1d_samples/latent_seed_0_1d_large_connected/layer-#1}}% % \begin{figure} \centering \setlength{\tabcolsep}{1.5pt} \begin{tabular}{ccccc} & 1 Layer & 2 Layers & 5 Layers & 10 Layers \\ \raisebox{0.6cm}{\rotatebox{90}{$f^{(1:\ell)}(x)$}} & \onedsamplepic{1} & \onedsamplepic{2} & \onedsamplepic{5} & \onedsamplepic{10} \\[-3pt] & $x$ & $x$ & $x$ & $x$ \end{tabular} \caption[A one-dimensional draw from a deep \sgp{} prior] {A function drawn from a one-dimensional deep \gp{} prior, shown at different depths. The $x$-axis is the same for all plots. After a few layers, the functions begin to be either nearly flat, or quickly-varying, everywhere. This is a consequence of the distribution on derivatives becoming heavy-tailed. As well, the function values at each layer tend to cluster around the same few values as the depth increases. This happens because once the function values in different regions are mapped to the same value in an intermediate layer, there is no way for them to be mapped to different values in later layers.} \label{fig:deep_draw_1d} \end{figure} % \begin{align} \kSE(x, x') = \sigma^2 \exp \left( \frac{-(x - x')^2}{2\lengthscale^2} \right) \;. \label{eq:se_kernel} \end{align} % The parameter $\sigma^2$ controls the variance of functions drawn from the prior, and the lengthscale parameter $\lengthscale$ controls the smoothness. The derivative of a \gp{} with a squared-exp kernel is point-wise distributed as $\Nt{0}{\nicefrac{\sigma^2}{\lengthscale^2}}$. Intuitively, a draw from a \gp{} is likely to have large derivatives if the kernel has high variance and small lengthscales. By the chain rule, the derivative of a one-dimensional deep \gp{} is simply a product of the derivatives of each layer, which are drawn independently by construction. The distribution of the absolute value of this derivative is a product of half-normals, each with mean $\sqrt{\nicefrac{2 \sigma^2}{\pi \lengthscale^2}}$. % %\begin{align} %\frac{\partial f(x)}{\partial x} & \distas{iid} \Nt{0}{\frac{\sigma^2_f}{\lengthscale^2}} \\ %\implies %\left| \frac{\partial f(x)}{\partial x} \right| & \distas{iid} \textnormal{half}\Nt{\sqrt{\frac{2 \sigma^2_f}{\pi\lengthscale^2}}}{\frac{\sigma^2_f}{\lengthscale^2} \left( 1 - \frac{2}{\pi} \right)} %\end{align} % If one chooses kernel parameters such that ${\nicefrac{\sigma^2}{\lengthscale^2} = \nicefrac{\pi}{2}}$, then the expected magnitude of the derivative remains constant regardless of the depth. %then ${\expectargs{}{\left| \nicefrac{\partial f(x)}{\partial x} \right|} = 1}$, and so ${\expectargs{}{\left| \nicefrac{\partial f^{(1:L)}(x)}{\partial x} \right|} = 1}$ for any $L$. % If $\nicefrac{\sigma^2_f}{\lengthscale^2}$ is less than $\nicefrac{\pi}{2}$, the expected derivative magnitude goes to zero, and if it is greater, the expected magnitude goes to infinity as a function of $L$. The distribution of the log of the magnitude of the derivatives has finite moments: % \begin{align} m_{\log} = \expectargs{}{\log \left| \frac{\partial f(x)}{\partial x} \right|} & = 2 \log \left( \frac{\sigma}{\lengthscale} \right) - \log2 - \gamma \nonumber \\ %\varianceargs{}{\log \left| \frac{\partial f(x)}{\partial x} \right|} & = \frac{1}{4}\left[ \pi^2 + 2 \log^2 2 - 2 \gamma( \gamma + log4 ) + 8 \left( \gamma + \log2 - \log \left(\frac{\sigma_f}{\lengthscale}\right)\right) \log\left(\frac{\sigma_f}{\lengthscale}\right) \right] v_{\log} = \varianceargs{}{\log \left| \frac{\partial f(x)}{\partial x} \right|} & = \frac{\pi^2}{4} + \frac{\log^2 2}{2} - \gamma^2 - \gamma \log4 + 2 \log\left(\frac{\sigma}{\lengthscale}\right) \left[ \gamma + \log2 - \log \left(\frac{\sigma}{\lengthscale}\right) \right] \end{align} % where $\gamma \approxeq 0.5772$ is Euler's constant. Since the second moment is finite, by the central limit theorem, the limiting distribution of the size of the gradient approaches a log-normal as L grows: % \begin{align} \log \left| \frac{\partial f^{(1:L)}(x)}{\partial x} \right| = \log \prod_{\layerindex=1}^L \left| \frac{\partial f^{(\layerindex)}(x)}{\partial x} \right| = \sum_{\layerindex=1}^L \log \left| \frac{\partial f^{(\layerindex)}(x)}{\partial x} \right| % \implies %\log \left| \frac{\partial f^{1:L}(x)}{\partial x} \right| & \distas{L \rightarrow \infty} \Nt{L \sqrt{\frac{2 \sigma^2_f}{\pi\lengthscale^2}}}{L \frac{\sigma^2_f}{\lengthscale^2} \left( 1 - \frac{2}{\pi} \right)} %\log \left| \frac{\partial f^{(1:L)}(x)}{\partial x} \right| & \distas{L \rightarrow \infty} \Nt{ L m_{\log} }{L^2 v_{\log}} \end{align} % Even if the expected magnitude of the derivative remains constant, the variance of the log-normal distribution grows without bound as the depth increases. Because the log-normal distribution is heavy-tailed and its domain is bounded below by zero, the derivative will become very small almost everywhere, with rare but very large jumps. \Cref{fig:deep_draw_1d} shows this behavior in a draw from a 1D deep \gp{} prior. This figure also shows that once the derivative in one region of the input space becomes very large or very small, it is likely to remain that way in subsequent layers. % %Does it convege to some sort of jump process? %Note that there is another limit - if $\nicefrac{\sigma^2_f}{\lengthscale_{d_1}^2} < \nicefrac{\pi}{2}$, then the mean of the size of the gradient goes to zero, but the variance approaches a finite constant. %\subsection{The Jacobian of deep GP is a product of independent normal matrices} \subsection{Distribution of the Jacobian} \label{sec:theorem} Next, we characterize the distribution on Jacobians of multivariate functions drawn from deep \gp{} priors, finding them to be products of independent Gaussian matrices with independent entries. \begin{lemma} \label{thm:deriv-ind} The partial derivatives of a function mapping $\mathbb{R}^D \rightarrow \mathbb{R}$ drawn from a \gp{} prior with a product kernel are independently Gaussian distributed. \end{lemma} % %\vspace{-0.25in} \begin{proof} Because differentiation is a linear operator, the derivatives of a function drawn from a \gp{} prior are also jointly Gaussian distributed. The covariance between partial derivatives with respect to input dimensions $d_1$ and $d_2$ of vector $\vx$ are given by \citet{Solak03derivativeobservations}: % \begin{align} \cov \left( \frac{\partial f(\vx)}{\partial x_{d_1}}, \frac{\partial f(\vx)}{\partial x_{d_2}} \right) = \frac{\partial^2 k(\vx, \vx')}{\partial x_{d_1} \partial x_{d_2}'} \bigg|_{\vx=\vx'} \label{eq:deriv-kernel} \end{align} % If our kernel is a product over individual dimensions $k(\vx, \vx') = \prod_d^D k_d(x_d, x_d')$, %as in the case of the squared-exp kernel, then the off-diagonal entries are zero, implying that all elements are independent. \end{proof} %\vspace{-0.1in} For example, in the case of the multivariate squared-exp kernel, the covariance between derivatives has the form: % \begin{align} %k_{SE}(\vx, \vx') = & f(\vx) \sim \textnormal{GP}\left( 0, \sigma^2 \prod_{d=1}^D \exp \left(-\frac{1}{2} \frac{(x_d - x_d')^2}{\lengthscale_d^2} \right) \right) \nonumber \\ & \implies \cov \left( \frac{\partial f(\vx)}{\partial x_{d_1}}, \frac{\partial f(\vx)}{\partial x_{d_2}} \right) = %\frac{\partial^2 k_{SE}(\vx, \vx')}{\partial x_{d_1} \partial x_{d_2}'} \bigg|_{\vx=\vx'} = \begin{cases} \frac{\sigma^2}{\lengthscale_{d_1}^2} & \mbox{if } d_1 = d_2 \\ 0 & \mbox{if } d_1 \neq d_2 \end{cases} \end{align} \begin{lemma} \label{thm:matrix} The Jacobian of a set of $D$ functions $\mathbb{R}^D \rightarrow \mathbb{R}$ drawn from independent \gp{} priors, each having product kernel is a $D \times D$ matrix of independent Gaussian R.V.'s \end{lemma} % %\vspace{-0.2in} \begin{proof} The Jacobian of the vector-valued function $\vf(\vx)$ is a matrix $J$ with elements ${J_{ij}(\vx) = \frac{ \partial f_i (\vx) }{\partial x_j}}$. % %\begin{align} %J_{ij} = \dfrac{\partial f_i (\vx) }{\partial x_j} %\end{align} %\begin{align} %\Jx^\layerindex(\vx) =\begin{bmatrix} \dfrac{\partial f^\layerindex_1 (\vx) }{\partial x_1} & \cdots & \dfrac{\partial f^\layerindex_1 (\vx)}{\partial x_D} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial f^\layerindex_D (\vx)}{\partial x_1} & \cdots & \dfrac{\partial f^\layerindex_D (\vx)}{\partial x_D} \end{bmatrix} %\end{align} % % Because the \gp{}s on each output dimension $f_1(\vx), f_2(\vx), \dots, f_D(\vx)$ are independent by construction, it follows that each row of $J$ is independent. Lemma \ref{thm:deriv-ind} shows that the elements of each row are independent Gaussian. Thus all entries in the Jacobian of a \gp{}-distributed transform are independent Gaussian R.V.'s. \end{proof} \begin{theorem} \label{thm:prodjacob} The Jacobian of a deep \gp{} with a product kernel is a product of independent Gaussian matrices, with each entry in each matrix being drawn independently. \end{theorem} % %\vspace{-0.2in} \begin{proof} When composing $L$ different functions, we denote the \emph{immediate} Jacobian of the function mapping from layer $\layerindex -1$ to layer $\layerindex$ as $J^{(\layerindex)}(\vx)$, and the Jacobian of the entire composition of $L$ functions by $J^{(1:L)}(\vx)$. % By the multivariate chain rule, the Jacobian of a composition of functions is given by the product of the immediate Jacobian matrices of each function. % Thus the Jacobian of the composed (deep) function $\vf^{(L)}(\vf^{(L-1)}(\dots \vf^{(3)}( \vf^{(2)}( \vf^{(1)}(\vx)))\dots))$ is % % $J^L(f^{(L-1)}(\vx) \ldots J^2(f^{(1)}(\vx))J^1(\vx)$ % \begin{align} J^{(1:L)}(\vx) % = \frac{\partial \fdeep(x) }{\partial x} %= \frac{\partial f^1(x) }{\partial x} \frac{\partial f^{(2)}(x) }{\partial f^{(1)}(x)} \cdots \frac{\partial f^L(x) }{\partial f^({L-1})(x)} %= \prod_{\layerindex = 1}^{L} J^L (x) = J^{(L)} J^{(L-1)} \dots J^{(3)} J^{(2)} J^{(1)}. %= J^{(L)} J^{(L-1)} \dots J^{(3)}(\vf^{(1:2)}(\vx)) J^{(2)}(\vf^{(1)}(\vx)) J^{(1)}(\vx). \label{eq:jacobian-of-deep-gp} \end{align} % By lemma \ref{thm:matrix}, each ${J^{(\layerindex)}_{i,j} \simind \mathcal{N}}$, so the complete Jacobian is a product of independent Gaussian matrices, with each entry of each matrix drawn independently. \end{proof} %\Cref{thm:prodjacob} This result allows us to analyze the representational properties of a deep Gaussian process by examining the properties of products of independent Gaussian matrices. \section{Formalizing a pathology} \label{sec:formalizing-pathology} A common use of deep neural networks is building useful representations of data manifolds. What properties make a representation useful? \citet{rifai2011higher} argued that good representations of data manifolds are invariant in directions orthogonal to the data manifold. They also argued that, conversely, a good representation must also change in directions tangent to the data manifold, in order to preserve relevant information. \Cref{fig:hidden} visualizes a representation having these two properties. %\newcolumntype{L}[1]{>{\raggedright\let\newline\\\arraybackslash\hspace{0pt}}m{#1}} \begin{figure}[h] \centering %\begin{tabular}{L{\columnwidth}} %\includegraphics[width=0.45\columnwidth]{\deeplimitsfiguresdir/hidden_good} & \begin{tikzpicture}[pile/.style={thick, ->, >=stealth'}] \node[anchor=south west,inner sep=0] at (0,0) { \includegraphics[clip, trim = 0cm 12cm 0cm 0.0cm, width=0.6\columnwidth]{\deeplimitsfiguresdir/hidden_good} }; \coordinate (D) at (1.6,1.5); \coordinate (Do) at (2.5, 0.8); \coordinate (Dt) at (3,3); \draw[pile] (D) -- (Dt) node[above, text width=5em] { tangent }; \draw[pile] (D) -- (Do) node[right, text width=5em] { orthogonal }; \end{tikzpicture} %A noise-tolerant representation of \\ a one-dimensional manifold (white) %\includegraphics[clip, trim = 0cm 12cm 0cm 0.0cm, width=0.9\columnwidth]{\deeplimitsfiguresdir/hidden_bad} \\ %b) A na\"{i}ve representation (colors) \\ of a one-dimensional manifold (white) %\end{tabular} \caption[Desirable properties of representations of manifolds] {Representing a 1-D data manifold. %A representation is a function mapping the input space to some set of outputs. Colors are a function of the computed representation of the input space. The representation (blue \& green) changes little in directions orthogonal to the manifold (white), making it robust to noise in those directions. The representation also varies in directions tangent to the data manifold, preserving information for later layers. %, and reducing the number of parameters needed to represent a datapoint. %Representation b) changes in all directions, preserving potentially useless information.}% The representation on the right might be useful if the data were spread out in over plane. } \label{fig:hidden} \end{figure} As in \citet{rifai2011contractive}, we characterize the representational properties of a function by the singular value spectrum of the Jacobian. The number of relatively large singular values of the Jacobian indicate the number of directions in data-space in which the representation varies significantly. % \newcommand{\spectrumpic}[1]{ \includegraphics[trim=4mm 1mm 4mm 2.5mm, clip, width=0.475\columnwidth]{\deeplimitsfiguresdir/spectrum/layer-#1}}% \begin{figure} \centering \begin{tabular}{ccc} & 2 Layers & 6 Layers \\ \begin{sideways} { \quad Normalized singular value} \end{sideways} & \hspace{-0.2in} \spectrumpic{2} & \hspace{-0.1in} \spectrumpic{6} \\ & { Singular value index} & { Singular value index} \end{tabular} \caption[Distribution of singular values of the Jacobian of a deep \sgp{}] {%The normalized singular value spectrum of the Jacobian of a deep \gp{} warping $\Reals^5 \to \Reals^5$. The distribution of normalized singular values of the Jacobian of a function drawn from a 5-dimensional deep \gp{} prior 25 layers deep~(\emph{Left}) and 50 layers deep~(\emph{Right}). As nets get deeper, the largest singular value tends to become much larger than the others. This implies that with high probability, these functions vary little in all directions but one, making them unsuitable for computing representations of manifolds of more than one dimension. %As depth increases, the distribution on singular values also becomes heavy-tailed. } \label{fig:deep_spectrum} \end{figure}% % \begin{figure}% \centering \begin{tabular}{cc} No transformation: $p(\vx)$ & 1 Layer: $p \left( \vf^{(1)}(\vx) \right)$ \\ \gpdrawbox{1} & \gpdrawbox{2} \\ 4 Layers: $p \left( \vf^{(1:4)}(\vx) \right)$ & 6 Layers: $p \left( \vf^{(1:6)}(\vx) \right)$ \\ \gpdrawbox{4} & \gpdrawbox{6} \end{tabular} \caption[Points warped by a draw from a deep \sgp{}] {Points warped by a function drawn from a deep \gp{} prior. \emph{Top left:} Points drawn from a 2-dimensional Gaussian distribution, color-coded by their location. \emph{Subsequent panels:} Those same points, successively warped by compositions of functions drawn from a \gp{} prior. As the number of layers increases, the density concentrates along one-dimensional filaments. Warpings using random finite neural networks exhibit the same pathology, but also tend to concentrate density into 0-dimensional manifolds (points) due to saturation of all of the hidden units.} \label{fig:filamentation} \end{figure}% % \Cref{fig:deep_spectrum} shows the distribution of the singular value spectrum of draws from 5-dimensional deep \gp{}s of different depths.\footnote{\citet{rifai2011contractive} analyzed the Jacobian at location of the training points, but because the priors we are examining are stationary, the distribution of the Jacobian is identical everywhere.} As the nets gets deeper, the largest singular value tends to dominate, implying there is usually only one effective degree of freedom in the representations being computed. \Cref{fig:filamentation} demonstrates a related pathology that arises when composing functions to produce a deep density model. The density in the observed space eventually becomes locally concentrated onto one-dimensional manifolds, or \emph{filaments}. This again suggests that, when the width of the network is relatively small, deep compositions of independent functions are unsuitable for modeling manifolds whose underlying dimensionality is greater than one. %for file in `ls`; do convert $file $file; done \newcommand{\mappic}[1]{ \includegraphics[width=0.475\columnwidth]{\deeplimitsfiguresdir/map/latent_coord_map_layer_#1} } \newcommand{\mappiccon}[1]{ \includegraphics[width=0.475\columnwidth]{\deeplimitsfiguresdir/map_connected/latent_coord_map_layer_#1} } \begin{figure} \centering \begin{tabular}{cc} \hspace{-0.15in} Identity Map: $\vy = \vx$ & \hspace{-0.15in} 1 Layer: $\vy = \vf^{(1)}(\vx)$ \\ \hspace{-0.15in} \mappic{0} & \mappic{1} \\ \hspace{-0.15in} 10 Layers: $\vy = \vf^{(1:10)}(\vx)$ & \hspace{-0.15in} 40 Layers: $\vy = \vf^{(1:40)}(\vx)$ \\ \hspace{-0.15in} \mappic{10} & \mappic{40} \end{tabular} \caption[Visualization of a feature map drawn from a deep \sgp{}] {A visualization of the feature map implied by a function $\vf$ drawn from a deep \gp{}. Colors are a function of the 2D representation $\vy = \vf(\vx)$ that each point is mapped to. % by the deep function. %that each point is mapped to after being warped by a deep \gp{}. The number of directions in which the color changes rapidly corresponds to the number of large singular values in the Jacobian. Just as the densities in \cref{fig:filamentation} became locally one-dimensional, there is usually only one direction that one can move $\vx$ in locally to change $\vy$. This means that $\vf$ is unlikely to be a suitable representation for decision tasks that depend on more than one aspect of $\vx$. Also note that the overall shape of the mapping remains the same as the number of layers increase. For example, a roughly circular shape remains in the top-left corner even after 40 independent warpings.} \label{fig:deep_map} \end{figure} % To visualize this pathology in another way, \cref{fig:deep_map} illustrates a color-coding of the representation computed by a deep \gp{}, evaluated at each point in the input space. After 10 layers, we can see that locally, there is usually only one direction that one can move in $\vx$-space in order to change the value of the computed representation, or to cross a decision boundary. This means that such representations are likely to be unsuitable for decision tasks that depend on more than one property of the input. To what extent are these pathologies present in the types of neural networks commonly used in practice? In simulations, we found that for deep functions with a fixed hidden dimension $D$, the singular value spectrum remained relatively flat for hundreds of layers as long as $D > 100$. Thus, these pathologies are unlikely to severely effect the relatively shallow, wide networks most commonly used in practice. \section{Fixing the pathology} \label{sec:fix} As suggested by \citet[chapter 2]{neal1995bayesian}, we can fix the pathologies exhibited in figures \cref{fig:filamentation} and \ref{fig:deep_map} by simply making each layer depend not only on the output of the previous layer, but also on the original input $\vx$. We refer to these models as \emph{input-connected} networks, and denote deep functions having this architecture with the subscript $C$, as in $f_C(\vx)$. Formally, this functional dependence can be written as \begin{align} \vf_C^{(1:L)}(\vx) = \vf^{(L)} \left( \vf_C^{(1:L-1)}(\vx), \vx \right), \quad \forall L \end{align} % \Cref{fig:input-connected} shows a graphical representation of the two connectivity architectures. \begin{figure}[h]% \def\nodeseptwo{1.9cm}% \def\nodesize{.35cm}% \def\numhiddentwo{3}% \centering \begin{tabular}{ccc} a) Standard \MLP{} connectivity & & b) Input-connected architecture\\ \hspace{-4mm} \begin{tikzpicture}[draw=black!80] \tikzstyle{neuron}=[circle,minimum size=17pt, draw = black!80, fill = white, thick] \tikzstyle{input neuron}=[neuron, fill=green!50]; \tikzstyle{output neuron}=[neuron, fill=red!50]; \tikzstyle{hidden neuron}=[neuron, fill=blue!50]; \tikzstyle{pile} =[thick, ->, >=stealth', shorten <=7pt, shorten >=8pt]; % Define the input layer node \coordinate (I) at (0, 0); % Define the hidden layer nodes \foreach \name / \y in {1,...,\numhiddentwo} { \coordinate (H-\name) at (\nodeseptwo*\y, 0); } \path[pile] (I) edge (H-1) {}; % Connect every node \foreach \name in {2,...,\numhiddentwo} { \pgfmathsetmacro\hindex{\name - 1} \path[pile] (H-\hindex) edge (H-\name) {}; } \draw (I) node[neuron] {}; \draw (I) node[below = 0.5cm] {$\vx$}; % Draw the hidden layer nodes \foreach \name / \y in {1,...,\numhiddentwo} { \draw (H-\name) node[neuron] {}; \draw (H-\name) node[below = 0.34cm] {$\vf^{(\y)}(\vx)$}; } \end{tikzpicture} & \hspace{0.5cm} & \begin{tikzpicture}[draw=black!80] \tikzstyle{neuron}=[circle,minimum size=17pt, draw = black!80, fill = white, thick] \tikzstyle{input neuron}=[neuron, fill=green!50]; \tikzstyle{output neuron}=[neuron, fill=red!50]; \tikzstyle{hidden neuron}=[neuron, fill=blue!50]; \tikzstyle{pile} =[thick, ->, >=stealth', shorten <=7pt, shorten >=8pt]; % Define the input layer node \coordinate (I) at (0, 0); % Define the hidden layer nodes \foreach \name / \y in {1,...,\numhiddentwo} { \coordinate (H-\name) at (\nodeseptwo*\y, 0); } % Connect every node \path[pile] (I) edge (H-1) {}; \foreach \name in {2,...,\numhiddentwo} { \pgfmathsetmacro\hindex{\name - 1} \path[pile] (H-\hindex) edge (H-\name) {}; \path[pile] (I) edge [bend left] (H-\name) {}; } \draw (I) node[neuron] {}; \draw (I) node[below = 0.5cm] {$\vx$}; % Draw the hidden layer nodes \foreach \name / \y in {1,...,\numhiddentwo} { \draw (H-\name) node[neuron] {}; \draw (H-\name) node[below = 0.34cm] {$\vf_C^{(\y)}(\vx)$}; } \end{tikzpicture} \end{tabular} \caption[Two different architectures for deep neural networks] {Two different architectures for deep neural networks. \emph{Left:} The standard architecture connects each layer's outputs to the next layer's inputs. \emph{Right:} The input-connected architecture also connects the original input $\vx$ to each layer.} \label{fig:input-connected} \end{figure}% Similar connections between non-adjacent layers can also be found the primate visual cortex \citep{maunsell1983connections}. Visualizations of the resulting prior on functions are shown in \cref{fig:deep_draw_1d_connected,fig:no_filamentation,fig:deep_map_connected}. \begin{figure}[h] \centering \setlength{\tabcolsep}{1.5pt} \begin{tabular}{ccccc} & 1 Layer & 2 Layers & 5 Layers & 10 Layers \\ \raisebox{0.6cm}{\rotatebox{90}{$f_C^{(1:L)}(x)$}} & \onedsamplepiccon{1} & \onedsamplepiccon{2} & \onedsamplepiccon{5} & \onedsamplepiccon{10} \\[-3pt] & $x$ & $x$ & $x$ & $x$ \end{tabular} \caption[A draw from a 1D deep \sgp{} prior with each layer connected to the input] {A draw from a 1D deep \gp{} prior having each layer also connected to the input. The $x$-axis is the same for all plots. Even after many layers, the functions remain relatively smooth in some regions, while varying rapidly in other regions. Compare to standard-connectivity deep \gp{} draws shown in \cref{fig:deep_draw_1d}.} \label{fig:deep_draw_1d_connected} \end{figure} % \newcommand{\gpdrawboxcon}[1]{ \setlength\fboxsep{0pt} \hspace{-0.2in} \fbox{ \includegraphics[width=0.464\columnwidth]{\deeplimitsfiguresdir/deep_draws_connected/deep_sample_connected_layer#1} }} % \begin{figure} \centering \begin{tabular}{cc} 3 Connected layers: $p \left( \vf_C^{(1:3)}(\vx) \right)$ & 6 Connected layers: $p \left( \vf_C^{(1:6)}(\vx) \right)$ \\ \gpdrawboxcon{3} & \gpdrawboxcon{6} \end{tabular} \caption[Points warped by a draw from an input-connected deep \sgp{}] {Points warped by a draw from a deep \sgp{} with each layer connected to the input $\vx$. %\emph{Top left:} Points drawn from a 2-dimensional Gaussian distribution, color-coded by their location. %\emph{Subsequent panels:} Those same points, successively warped by functions drawn from a \gp{} prior. As depth increases, the density becomes more complex without concentrating only along one-dimensional filaments.} \label{fig:no_filamentation} \end{figure} % \begin{figure} \centering \newcommand{\spectrumpiccon}[1]{ \includegraphics[trim=4mm 1mm 4mm 2.5mm, clip, width=0.475\columnwidth]{\deeplimitsfiguresdir/spectrum/con-layer-#1}} \begin{tabular}{ccc} & 25 layers & 50 layers \\ \hspace{-0.5cm} \begin{sideways} {\quad Normalized singular value} \end{sideways} & \hspace{-0.2in} \spectrumpiccon{25} & \hspace{-0.16in} \spectrumpiccon{50} \\ & {Singular value number} & {Singular value number} \end{tabular} \caption[Distribution of singular values of an input-connected deep \sgp{}] {The distribution of singular values drawn from 5-dimensional input-connected deep \gp{} priors, 25 layers deep (\emph{Left}) and 50 layers deep (\emph{Right}). Compared to the standard architecture, the singular values are more likely to remain the same size as one another, meaning that the model outputs are more often sensitive to several directions of variation in the input.} \label{fig:good_spectrum} \end{figure} % % \begin{figure} \centering \begin{tabular}{cc} \hspace{-0.15in} Identity map: $\vy = \vx$ & \hspace{-0.15in} 2 Connected layers: $\vy = \vf^{(1:2)}(\vx)$ \\ \hspace{-0.15in} \mappic{0} & \mappiccon{2} \\ \hspace{-0.15in} 10 Connected layers: $\vy = \vf^{(1:10)}(\vx)$ & \hspace{-0.15in} 20 Connected layers: $\vy = \vf^{(1:20)}(\vx)$ \\ \hspace{-0.15in} \mappiccon{10} & \mappiccon{20} \end{tabular} \caption[Feature map of an input-connected deep \sgp{}] {The feature map implied by a function $\vf$ drawn from a deep \gp{} prior with each layer also connected to the input $\vx$, visualized at various depths. Compare to the map shown in \cref{fig:deep_map}. %Just as the densities in \cref{fig:no_filamentation} remained locally two-dimensional even after many warpings, In the mapping shown here there are sometimes two directions that one can move locally in $\vx$ to in order to change the value of $\vf(\vx)$. This means that the input-connected prior puts significant probability mass on a greater variety of types of representations, some of which depend on all aspects of the input. } \label{fig:deep_map_connected} \end{figure} The Jacobian of an input-connected deep function is defined by the recurrence % \newcommand{\sbi}[2]{\left[ \! \begin{array}{c} #1 \\ #2 \end{array} \! \right]} %\newcommand{\sbi}[2]{\left[ #1 \quad \!\! #2 \right]} \begin{align} {J_C^{(1:L)} = J^{(L)} \sbi{ J_C^{(1:L-1)}}{I_D}}. \end{align} % %So the entire Jacobian has the form: % %\begin{align} %J^{1:L}(x) = J^L \sbi{ J^{L-1} \sbi{ \dots J^{4} \sbi{ J^{3} \sbi{ J^2 J^1 }{ I_D }}{ I_D } \dots }{ I_D \\ %\vdots }}{ I_D} %\end{align} % where $I_D$ is a $D$-dimensional identity matrix. Thus the Jacobian of an input-connected deep \gp{} is a product of independent Gaussian matrices each with an identity matrix appended. \Cref{fig:good_spectrum} shows that with this architecture, even 50-layer deep \gp{}s have well-behaved singular value spectra. The pathology examined in this section is an example of the sort of analysis made possible by a well-defined prior on functions. %As explained in \cref{sec:relating}, \gp{}s are examples of a particular limit of Bayesian neural networks. The figures and analysis done in this section could be done using Bayesian neural networks with finite numbers of nodes, but would be more difficult. In particular, care would need to be taken to ensure that the networks do not produce degenerate mappings due to saturation of the hidden units. \section{Deep kernels} \label{sec:deep_kernels} \cite{ bengio2006curse} showed that kernel machines have limited generalization ability when they use ``local'' kernels such as the squared-exp. However, as shown in \cref{ch:kernels,ch:grammar,ch:additive}, structured, non-local kernels can %be constructed through addition and multiplication which allow extrapolation. Another way to build non-local kernels is by composing fixed feature maps, creating \emph{deep kernels}. To return to an example given in \cref{sec:expressing-symmetries}, periodic kernels can be viewed as a 2-layer-deep kernel, in which the first layer maps $x \rightarrow [\sin(x), \cos(x)]$, and the second layer maps through basis functions corresponding to the implicitly feature map giving rise to an \kSE{} kernel. This section builds on the work of \citet{cho2009kernel}, who derived several kinds of deep kernels by composing multiple layers of feature mappings. %In addition to analyzing \MLP{}s with random weights, we can also analyze fixed feature mappings with different connectivity architectures. %We can construct other useful kernels by composing fixed feature maps several times, creating deep kernels. %Given a kernel $k_1(\vx, \vx') = \hPhi(\vx) \tra \hPhi(\vx')$, In principle, one can compose the implicit feature maps of any two kernels $k_a$ and $k_b$ to get a new kernel, which we denote by $\left( k_b \circ k_a \right)$: % \begin{align} k_a(\vx, \vx') & = \hPhi_a(\vx) \tra \hPhi_a(\vx') \\ k_b(\vx, \vx') & = \hPhi_b(\vx) \tra \hPhi_b(\vx') \\ \left( k_b \circ k_a \right)(\vx, \vx') & = k_b \big(\hPhi_a(\vx), \hPhi_a(\vx') \big) = \left[ \hPhi_b \left( \hPhi_a(\vx) \right)\right] \tra \hPhi_b \left(\hPhi_a(\vx') \right) \end{align} % However, this composition might not always have a closed form if the number of hidden features of either kernel is infinite. Fortunately, composing the squared-exp (\kSE{}) kernel with the implicit mapping given by any other kernel has a simple closed form. If $k(\vx, \vx') = \hPhi(\vx)\tra \hPhi(\vx')$, then %, this composition operation has a closed form for any starting kernel:% for any set of starting features $\hPhi_n(\vx)$: % \begin{align} %k_1(\vx, \vx') & = \exp \left( -\frac{1}{2} ||\vx - \vx'||_2^2 \right) \\ \left( \kSE \circ k \right) \left( \vx, \vx' \right) & = k_{SE} \big( \hPhi(\vx), \hPhi(\vx') \big) \\ %& = \left( \hPhi^{SE} \left(\hPhi^{1}(\vx) \right) \right) \tra \hPhi^{SE} \left( \hPhi^{1}(\vx') \right) \\ & = \exp \left( -\frac{1}{2} || \hPhi(\vx) - \hPhi(\vx')||_2^2 \right) \\ %k_{n+1}(\vx, \vx') %& = \exp \left( -\frac{1}{2} \sum_i \left[ \hphi_n^{(i)}(\vx) - \hphi_n^{(i)}(\vx') \right]^2 \right) \\ & = \exp\left ( -\frac{1}{2} \left[ \hPhi(\vx) \tra \hPhi(\vx) - 2 \hPhi(\vx) \tra \hPhi(\vx') + \hPhi(\vx') \tra \hPhi(\vx') \right] \right) \\ %k_2(\vx, \vx') & = \exp \left( -\frac{1}{2} \left[ \sum_i \hphi_i(\vx)^2 - 2 \sum_i \hphi_i(\vx) \hphi_i(\vx') + \sum_i \hphi_i(\vx')^2 \right] \right) \\ %k_{n+1}(\vx, \vx') & = \exp \left( -\frac{1}{2} \big[ k(\vx, \vx) - 2 k(\vx, \vx') + k(\vx', \vx') \big] \right) \; . %k_{n+1}(\vx, \vx') %& = \exp \left( k_1(\vx, \vx') - 1 \right) \qquad \textnormal{(if $k_1(\vx, \vx) = 1$)} \nonumber \end{align} % %\begin{figure} %\centering %\begin{tabular}{ccc} %\includegraphics[width=0.5\columnwidth, clip, trim = 0cm 0cm 0cm 0.61cm]{\deeplimitsfiguresdir/deep_kernel} & %\includegraphics[width=0.5\columnwidth, clip, trim = 0cm 0cm 0cm 0.61cm]{\deeplimitsfiguresdir/deep_kernel_draws} \\ %Kernel derived from iterated feature transforms & Draws from the corresponding kernel %\end{tabular} %\caption{A degenerate kernel produced by repeatedly applying a feature transform.} %\label{fig:deep_kernel} %\end{figure} % %Thus, if $k_1(x,y) = e^{-||x - y||2}$, then the two-layer kernel is simply $k_2(x,y) = e^{k_1(x, y) - 1}$. This formula is true for every layer: $k_{n+1}(x,y) = e^{k_n(x, y) - 1}$. % %Note that nothing in this derivation depends on details of $k_n$, except that $k_n( \vx, \vx) = 1$. %Note that this result holds for any base kernel $k_n$, as long as $k_n( \vx, \vx) = 1$. % Because this is true for $k_2$ as well, this recursion holds in general, and we have that $k_{n+1}(x,y) = e^{k_n(x, y) - 1}$. This formula expresses the composed kernel $(\kSE \circ k)$ exactly in terms of evaluations of the original kernel $k$. \subsection{Infinitely deep kernels} What happens when one repeatedly composes feature maps many times, starting with the squared-exp kernel? If the output variance of the \kSE{} is normalized to ${k(\vx,\vx) = 1}$, then the infinite limit of composition with \kSE{} converges to ${\left(\kSE \circ \kSE \circ \ldots \circ \kSE\right)(\vx,\vx') = 1}$ for all pairs of inputs. A constant covariance corresponds to a prior on constant functions ${f(\vx) = c}$. %Figure \ref{fig:deep_kernel_connected} shows this kernel at different depths, including the degenerate limit. % %One interpretation of why repeated feature transforms lead to this degenerate prior is that each layer can only lose information about the previous set of features. %In the limit, the transformed features contain no information about the original input $\vx$. Since the function doesn't depend on its input, it must be the same everywhere. %\subsubsection{A non-degenerate construction} As above, we can overcome this degeneracy by connecting the input $\vx$ to each layer. To do so, we concatenate the composed feature vector at each layer, $\hPhi^{(1:\ell)}(\vx)$, with the input vector $\vx$ to produce an input-connected deep kernel $k_C^{(1:L)}$, defined by: % \begin{align} %k_1(\vx, \vx') & = \exp \left( -\frac{1}{2} ||\vx - \vx'||_2^2 \right) \\ k_C^{(1:\ell + 1)}(\vx, \vx') & = \exp \left( -\frac{1}{2} \left|\left| \left[ \! \begin{array}{c} \hPhi^{(1:\ell)}(\vx) \\ \vx \end{array} \! \right] - \left[ \! \begin{array}{c} \hPhi^{(1:\ell)}(\vx') \\ \vx' \end{array} \! \right] \right| \right|_2^2 \right) \\ %k_{n+1}(\vx, \vx') %& = \exp \left( -\frac{1}{2} \sum_i \left[ \hphi_i(\vx) - \hphi_i(\vx') \right]^2 -\frac{1}{2} || \vx - \vx' ||_2^2 \right) \\ %k_{n+1}(\vx, \vx') & = \exp\left ( -\frac{1}{2} \sum_i \left[ \hphi_i(\vx)^2 - 2 \hphi_i(\vx) \hphi_i(\vx') + \hphi_i(\vx')^2 \right] -\frac{1}{2} || \vx - \vx' ||_2^2 \right) \\ %k_2(\vx, \vx') & = \exp \left( -\frac{1}{2} \left[ \sum_i \hphi_i(\vx)^2 - 2 \sum_i \hphi_i(\vx) \hphi_i(\vx') + \sum_i \hphi_i(\vx')^2 \right] \right) \\ %k_2(\vx, \vx') & = \exp \left( -\frac{1}{2} \left[ k_1(\vx, \vx) - 2 k_1(\vx, \vx') + k_1(\vx', \vx') \right] \right) \\ %k_{n+1}(\vx, \vx') %& = \exp \left( k_n(\vx, \vx') - 1 -\frac{1}{2} || \vx - \vx' ||_2^2 \right) & = \exp \Big( -\frac{1}{2} \big[ k_C^{(1:\ell)}(\vx, \vx) - 2 k_C^{(1:\ell)}(\vx, \vx') + k_C^{(1:\ell)}(\vx', \vx') {\color{black} - || \vx - \vx' ||_2^2} \big] \Big) \nonumber \end{align} % Starting with the squared-exp kernel, this repeated mapping satisfies \begin{align} k_C^{(1:\infty)}(\vx, \vx') - \log \left( k_C^{(1:\infty)}(\vx, \vx') \right) = 1 + \frac{1}{2} || \vx - \vx' ||_2^2 \,. \end{align} % \begin{figure} \centering \begin{tabular}{cc} Input-connected deep kernels & Draws from corresponding \gp{}s \\ \hspace{-0.3cm} \rotatebox{90}{$\qquad \qquad k(x, x')$} \includegraphics[width=0.465\columnwidth, clip, trim = 1.3cm 0.4cm 0.9cm 0.3cm]{\deeplimitsfiguresdir/deep_kernel_connected} & \hspace{-0.3cm} \rotatebox{90}{$\qquad \qquad f(x)$} \includegraphics[width=0.44\columnwidth, clip, trim = 1.29cm 0.1cm 0.9cm 0.35cm]{\deeplimitsfiguresdir/deep_kernel_connected_draws} \\ $x - x'$ & $x$ \end{tabular} \caption[Infinitely deep kernels]{ \emph{Left:} Input-connected deep kernels of different depths. By connecting the input $\vx$ to each layer, the kernel can still depend on its input even after arbitrarily many layers of composition. \emph{Right:} Draws from \gp{}s with deep input-connected kernels.} \label{fig:deep_kernel_connected} \end{figure} % The solution to this recurrence is related to the Lambert W function~\citep{corless1996lambertw} and has no closed form. In one input dimension, it has a similar shape to the Ornstein-Uhlenbeck covariance ${\OU(x,x') = \exp( -|x - x'| )}$ but with lighter tails. Samples from a \gp{} prior having this kernel are not differentiable, and are locally fractal. \Cref{fig:deep_kernel_connected} shows this kernel at different depths, as well as samples from the corresponding \gp{} priors. %\item This kernel has smaller correlation than the squared-exp everywhere except at $\vx = \vx'$. %\item The tails have the same form as the squared-exp. One can also consider two related connectivity architectures: one in which each layer is connected to the output layer, and another in which every layer is connected to all subsequent layers. It is easy to show that in the limit of infinite depth of composing \kSE{} kernels, both these architectures converge to $k(\vx, \vx') = \delta( \vx, \vx' )$, the white noise kernel. %\subsection{A fully-connected kernel} %However, connecting every layer to every subsequent layer leades to a pathology: %\begin{align} %k_1(\vx, \vx') & = \exp \left( -\frac{1}{2} ||\vx - \vx'||_2^2 \right) \\ %k_{n+1}(\vx, \vx') & = \exp \left( -\frac{1}{2} \left|\left| \left[ \! \begin{array}{c} \hPhi_n(\vx) \\ \hPhi_{n-1}(\vx') \end{array} \! \right] - \left[ \! \begin{array}{c} \hPhi_n(\vx') \\ \hPhi_{n-1}(\vx') \end{array} \! \right] \right| \right|_2^2 \right) \\ %k_{n+1}(\vx, \vx') & = \exp \left( -\frac{1}{2} \sum_i \left[ \hphi^{(i)}_n(\vx) - \hphi^{(i)}_n(\vx') \right]^2 -\frac{1}{2} \sum_i \left[ \hphi^{(i)}_{n-1}(\vx) - \hphi^{(i)}_{n-1}(\vx') \right]^2 \right)\\ %k_{n+1}(\vx, \vx') & = \exp\left ( -\frac{1}{2} \sum_i \left[ \hphi_i(\vx)^2 - 2 \hphi_i(\vx) \hphi_i(\vx') + \hphi_i(\vx')^2 \right] -\frac{1}{2} || \vx - \vx' ||_2^2 \right) \\ %k_2(\vx, \vx') & = \exp \left( -\frac{1}{2} \left[ \sum_i \hphi_i(\vx)^2 - 2 \sum_i \hphi_i(\vx) \hphi_i(\vx') + \sum_i \hphi_i(\vx')^2 \right] \right) \\ %k_2(\vx, \vx') & = \exp \left( -\frac{1}{2} \left[ k_1(\vx, \vx) - 2 k_1(\vx, \vx') + k_1(\vx', \vx') \right] \right) \\ %k_{n+1}(\vx, \vx') & = \exp \left( k_n(\vx, \vx') - 1 \right) \exp \left( k_{n-1}(\vx, \vx') - 1 \right) %k_{n+1}(\vx, \vx') & = \prod_{i=1}^n \prod_{j=1}^i \exp \left( k_i(\vx, \vx') - 1 \right) %\end{align} %Which has the solution $k_\infty = \delta( \vx = \vx' )$, a white noise kernel. %\subsection{Connecting every layer to the end} %There is a fourth possibilty (suggested by Carl), of connecting every layer to the output: %\begin{align} %k_n(\vx, \vx') & = \exp \left( -\frac{1}{2} || \hPhi_n(\vx) - \hPhi_n(\vx')||_2^2 \right) \\ %k_{n+1}(\vx, \vx') & = \exp \left( k_n(\vx, \vx') - 1 \right) \\ %k_{L}(\vx, \vx') & = \prod_{i=1}^L \exp \left( k_i(\vx, \vx') - 1 \right) %\end{align} %Which also has the solution $k_\infty = \delta( \vx = \vx' )$, a white noise kernel. \subsection{When are deep kernels useful models?} Kernels correspond to fixed feature maps, and so kernel learning is an example of implicit representation learning. %can compute %useful representations. As we saw in \cref{ch:kernels,ch:grammar}, kernels can capture rich structure and can enable many types of generalization. %, such as affine invariance in images \citep{kondor2008group}. %\cite{salakhutdinov2008using} used a deep neural network to learn feature transforms for kernels, which learn invariants in an unsupervised manner. We believe that the relatively uninteresting properties of the deep kernels derived in this section simply reflect the fact that an arbitrary computation, even if it is ``deep'', is not likely to give rise to a useful representation unless combined with learning. To put it another way, any fixed representation is unlikely to be useful unless it has been chosen specifically for the problem at hand. \section{Related work} \subsubsection{Deep Gaussian processes} \citet[chapter 2]{neal1995bayesian} explored properties of arbitrarily deep Bayesian neural networks, including those that would give rise to deep \gp{}s. He noted that infinitely deep random neural networks without extra connections to the input would be equivalent to a Markov chain, and therefore would lead to degenerate priors on functions. He also suggested connecting the input to each layer in order to fix this problem. Much of the analysis in this chapter can be seen as a more detailed investigation, and vindication, of these claims. The first instance of deep \gp{}s being used in practice was \citep{lawrence2007hierarchical}, who presented a model called ``hierarchical \gplvm{}s'', in which time was mapped through a composition of multiple \gp{}s to produce observations. %The term ``deep Gaussian processes'' first appeared in \citet{damianou2012deep}, where it referred to a latent variable model whose warping function was defined by a composition of \gp{}s. The term ``deep Gaussian processes'' was first used by \citet{damianou2012deep}, who developed a variational inference method, analyzed the effect of automatic relevance determination, and showed that deep \gp{}S could learn with relatively little data. %I propose that this model might be more clearly described as a ``deep \gplvm{}'', and that the term ``deep Gaussian process'' should be reserved for describing compositions of functions drawn from \gp{} priors. % They used the term ``deep \gp{}'' to refer both to supervised models (compositions of \gp{}s) and to unsupervised models (compositions of \gplvm{}s). This conflation may be reasonable, since the activations of the hidden layers are themselves latent variables, even in supervised settings: Depending on kernel parameters, each latent variable may or may not depend on the layer below. In general, supervised models can also be latent-variable models. For example, \citet{wang2012gaussian} investigated single-layer \gp{} regression models that had additional latent inputs. \subsubsection{Nonparametric neural networks} %Other Bayesian deep neural network models have been proposed by, for example, . % and Sum-product networks \cite{poon2011sum} \citet{adams2010learning} proposed a prior on arbitrarily deep Bayesian networks having an unknown and unbounded number of parametric hidden units in each layer. Their architecture has connections only between adjacent layers, and so may have similar pathologies to the one discussed in the chapter as the number of layers increases. \citet{wilson2012gaussian} introduced Gaussian process regression networks, which are defined as a matrix product of draws from \gp{}s priors, rather than a composition. These networks have the form: %The form of a \gprn{} is % \begin{align} \vy(\vx) & = \vW(\vx) \vf(\vx) \qquad \textnormal{where each} \; f_{d}, W_{d,j} \simiid \GPt{\vzero}{\kSE + \kWN} . \end{align} % We can easily define a ``deep'' Gaussian process regression network: % \begin{align} \vy(\vx) = \vW^{(3)}(\vx) \vW^{(2)}(\vx) \vW^{(1)}(\vx) \vf(\vx) \end{align} % which repeatedly adds and multiplies functions drawn from \gp{}s, in contrast to deep \gp{}s, which repeatedly compose functions. This prior on functions has a similar form to the Jacobian of a deep \gp{} (\cref{eq:jacobian-of-deep-gp}), and so might be amenable to a similar analysis to that of section \ref{sec:characterizing-deep-gps}. % %This model has a similar form to the Jacobian of deep \gp{}, (since each row of the Jacobian is jointly a \gp{} but with with a kernel given by \eqref{eq:deriv-kernel}) except that the Jacobian is evaluated at the output from previous layers: % %\begin{align} %\vy(\vx) = J^3(\vf^2(\vx)) J^2(\vf^1(\vx)) J^1(\vx) %\end{align} % \subsubsection{Information-preserving architectures} Deep density networks \citep{rippel2013high} are constructed through a series of parametric warpings of fixed dimension, with penalty terms encouraging the preservation of information about lower layers. This is another promising approach to fixing the pathology discussed in \cref{sec:formalizing-pathology}. \subsubsection{Recurrent networks} \citet{bengio1994learning} and \citet{pascanu2012understanding} analyzed a related problem with gradient-based learning in recurrent networks, the ``exploding-gradients'' problem. They noted that in recurrent neural networks, the size of the training gradient can grow or shrink exponentially as it is back-propagated, making gradient-based training difficult. \citet{hochreiter1997long} addressed the exploding-gradients problem by introducing hidden units designed to have stable gradients. This architecture is known as long short-term memory. \subsubsection{Deep kernels} The first systematic examination of deep kernels was done by \citet{cho2009kernel}, who derived closed-form composition rules for $\kSE$, polynomial, and arc-cosine kernels, and showed that deep arc-cosine kernels performed competitively in machine-vision applications when used in a \SVM{}. %\citet{cho2012kernel} \citet{hermans2012recurrent} constructed deep kernels in a time-series setting, constructing kernels corresponding to infinite-width \emph{recurrent} neural networks. They also proposed concatenating the implicit feature vectors from previous time-steps with the current inputs, resulting in an architecture analogous to the input-connected architecture proposed by \citet[chapter 2]{neal1995bayesian}. \subsubsection{Analyses of deep learning} \citet{montavon2010layer} performed a layer-wise analysis of deep networks, and noted that the performance of \MLP{}s degrades as the number of layers with random weights increases, a result consistent with the analysis of this chapter. The experiments of \citet{saxe2011random} suggested that most of the good performance of convolutional neural networks could be attributed to the architecture alone. Later, \citet{saxedynamics} looked at the dynamics of gradient-based training methods in deep \emph{linear} networks as a tractable approximation to standard deep (nonlinear) neural networks. %\section{Discussion} %\paragraph{Recursive learning method} %Just as layer-wise unsupervised pre-training encourages the projection of the data into a representation with independent features in the higher layers, so does the procedure outlined here. This is because the isotropic kernel does not penalize independence between different dimensions, only the number of dimensions. \subsubsection{Source code} Source code to produce all figures is available at \url{http://www.github.com/duvenaud/deep-limits}. This code is also capable of producing visualizations of mappings such as \cref{fig:deep_map,fig:deep_map_connected} using neural nets instead of \gp{}s at each layer. \section{Conclusions} %In this work, we established a number of propositions which help us gain insight into the properties of very deep models, and allow making informed choices regarding their architecture. This chapter demonstrated that well-defined priors allow explicit examination of the assumptions being made about functions being modeled by different neural network architectures. As an example of the sort of analysis made possible by this approach, we attempted to gain insight into the properties of deep neural networks by characterizing the sorts of functions likely to be obtained under different choices of priors on compositions of functions. %established a number of propositions which help us gain insight into the properties of very deep models, and allow making informed choices regarding their architecture. %First, we identified equivalences between multi-layer perceptions and deep \gp{}s --- namely, that a deep \gp{} can be written as an \MLP{}s with a finite number of nonparametric hidden units, or as an \MLP{} with infinitely-many parametric hidden units. First, we identified deep Gaussian processes as an easy-to-analyze model corresponding to multi-layer preceptrons having nonparametric activation functions. % We also showed several other connections between deep \MLP{}s and different Gaussian process models. %equivalences between multi-layer perceptions and deep \gp{}s --- namely, that a deep \gp{} can be written as an \MLP{}s with a finite number of nonparametric hidden units, or as an \MLP{} with infinitely-many parametric hidden units. % %Second, we characterized the derivatives and Jacobians of deep \gp{}s through products of independent Gaussian matrices. % can be characterized using random matrix theory, which we applied to establish results regarding the distribution over the Jacobian of the composition transformation. We then showed that representations based on repeated composition of independent functions exhibit a pathology where the representations becomes invariant to all but one direction of variation. % This leads to extremely restricted expressiveness of such deep models in the limit of increasing number of layers. Finally, we showed that this problem could be alleviated by connecting the input to each layer. %proposed a way to alleviate this problem: connecting the input to each layer of a deep representation allows us to construct priors on deep functions that do not exhibit the information-capacity pathology. % We also examined properties of deep kernels, corresponding to arbitrarily many compositions of fixed feature maps. %Finally, we derived models obtained by performing dropout on Gaussian processes, finding a tractable approximation to exact dropout in \gp{}s. Much recent work on deep networks has focused on weight initialization \citep{martens2010deep}, regularization \citep{lee2007sparse} and network architecture \citep{gens2013learning}. %However, the interactions between these different design decisions can be complex and difficult to characterize. %We propose to approach the design of deep architectures by examining the problem of assigning priors to nested compositions of functions. % Although inference in fully Bayesian models is generally challenging, well-defined priors allow us to encode our beliefs into models in a data-independent way. If we can identify priors that give our models desirable properties, these might in turn suggest regularization, initialization, and architecture choices that also provide such properties. Existing neural network practice also requires expensive tuning of model hyperparameters such as the number of layers, the size of each layer, and regularization penalties by cross-validation. One advantage of deep \gp{}s is that the approximate marginal likelihood allows a principled method for automatically determining such model choices. \outbpdocument{ %This will be ignored, it's just so that Gummi can find the bibliography. \bibliographystyle{plainnat} \bibliography{references.bib} qwerty } ================================================ FILE: description.tex ================================================ \input{common/header.tex} \inbpdocument \chapter{Automatic Model Description} \label{ch:description} %“One day I will find the right words, and they will be simple.” ― Jack Kerouac, The Dharma Bums %“You can make anything by writing.” ― C.S. Lewis %“The role of a writer is not to say what we can all say, but what we are unable to say.” ― Anaïs Nin %“Stories may well be lies, but they are good lies that say true things, and which can sometimes pay the rent.” ― Neil Gaiman %“When I write, I feel like an armless, legless man with a crayon in his mouth.” ― Kurt Vonnegut %If words are not things, or maps are not the actual territory, then, obviously, the only possible link between the objective world and the linguistic world is found in structure, and structure alone. — Alfred Korzybski %Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics (1958), 61. %... the only possible link between the objective world and the linguistic world is found in structure, and structure alone. — Alfred Korzybski, Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics (1958), 61. %“The first draft of anything is shit.” %― Ernest Hemingway %“Not a wasted word. This has been a main point to my literary thinking all my life.” %—Hunter S. Thompson %“We are all apprentices in a craft where no one ever becomes a master.” %—Ernest Hemingway \begin{quotation} ``Not a wasted word. This has been a main point to my literary thinking all my life.'' \hspace*{\fill} \emph{ -- Hunter S. Thompson} \end{quotation} The previous chapter showed how to automatically build structured models by searching through a language of kernels. It also showed how to decompose the resulting models into the different types of structure present, and how to visually illustrate the type of structure captured by each component. This chapter shows how automatically describe the resulting model structures using English text. The main idea is to describe every part of a given product of kernels as an adjective, or as a short phrase that modifies the description of a kernel. To see how this could work, recall that the model decomposition plots of \cref{ch:grammar} showed that most of the structure in each component was determined by that component's kernel. Even across different datasets, the meanings of individual parts of different kernels are consistent in some ways. For example, $\kPer$ indicates repeating structure, and $\kSE$ indicates smooth change over time. This chapter also presents a system that generates reports combining automatically generated text and plots which highlight interpretable features discovered in a data sets. A complete example of an automatically-generated report can be found in appendix~\ref{ch:example-solar}. The work appearing in this chapter was written in collaboration with James Robert Lloyd, Roger Grosse, Joshua B. Tenenbaum, and Zoubin Ghahramani, and was published in \citet{LloDuvGroetal14}. The procedure translating kernels into adjectives developed out of discussions between James and myself. James Lloyd wrote the code to automatically generate reports, and ran all of the experiments. The paper upon which this chapter is based was written mainly by both James Lloyd and I.%, and Zoubin Ghahramani, with many helpful contributions and suggestions from Roger Grosse and Josh Tenenbaum. \section{Generating descriptions of composite kernels} There are two main features of our language of \gp{} models that allow description to be performed automatically. First, any kernel expression in the language can be simplified into a sum of products. As discussed in \cref{sec:modeling-sums-of-functions}, a sum of kernels corresponds to a sum of functions, so each resulting product of kernels can be described separately, as part of a sum. Second, each kernel in a product modifies the resulting model in a consistent way. Therefore, one can describe a product of kernels by concatenating descriptions of the effect of each part of the product. One part of the product needs to be described using a noun, which is modified by the other parts. For example, one can describe the product of kernels $\kPer \kerntimes \kSE$ by representing $\kPer$ by a noun (``a periodic function'') modified by a phrase representing the effect of the $\kSE$ kernel (``whose shape varies smoothly over time''). To simplify the system, we restricted base kernels to the set $\{\kC$, $\kLin$, $\kWN$, $\kSE$, $\kPer$, and $\vsigma\}$. Recall that the sigmoidal kernel $\vsigma(x, x') = \sigma(x)\sigma(x')$ allows changepoints and change-windows. \subsection{Simplification rules} \label{sec:desc-simplification} In order to be able to use the same phrase to describe the effect of each base kernel in different circumstances, our system converts each kernel expression into a standard, simplified form. First, our system distributes all products of sums into sums of products. Then, it applies several simplification rules to the kernel expression: \begin{itemize} \item Products of two or more $\kSE$ kernels can be equivalently replaced by a single $\kSE$ with different parameters. \item Multiplying the white-noise kernel ($\kWN$) by any stationary kernel ($\kC$, $\kWN$, $\kSE$, or $\kPer$) gives another $\kWN$ kernel. \item Multiplying any kernel by the constant kernel ($\kC$) only changes the parameters of the original kernel, and so can be factored out of any product in which it appears. \end{itemize} After applying these rules, any composite kernel expressible by the grammar can be written as a sum of terms of the form: \begin{align} K \prod_m \kLin^{(m)} \prod_n \boldsymbol\sigma^{(n)}, \label{eq:sop} \end{align} where $K$ is one of $\{\kWN, \kC, \kSE, \prod_k \kPer^{(k)}\}$ or $\{\kSE \times \prod_k \kPer^{(k)}\}$, and $\prod_i\kernel^{(i)}$ denotes a product of kernels, each having different parameters. Superscripts denote different instances of the same kernel appearing in a product: $\SE^{(1)}$ can have different kernel parameters than $\kSE^{(2)}$. \subsection{Describing each part of a product of kernels} Each kernel in a product modifies the resulting \gp{} model in a consistent way. This allows one to describe the contribution of each kernel in a product as an adjective, or more generally as a modifier of a noun. We now describe how each of the kernels in our grammar modifies a \gp{} model: %, and how this effect can be described in natural language: \begin{itemize} \item {\bf Multiplication by $\kSE$} removes long range correlations from a model, since $\kSE(x,x')$ decreases monotonically to 0 as $|x - x'|$ increases. %This can be described as making an existing model's correlation structure `local' or `approximate'. This converts any global correlation structure into local correlation only. \item {\bf Multiplication by $\kLin$} is equivalent to multiplying the function being modeled by a linear function. If $f(x) \dist \gp{}(0, \kernel)$, then $x \kerntimes f(x) \dist \gp{}\left(0, \kLin \kerntimes k \right)$. This causes the standard deviation of the model to vary linearly, without affecting the correlation between function values. % and can be described as \eg `with linearly increasing standard deviation'. \item {\bf Multiplication by $\boldsymbol\sigma$} is equivalent to multiplying the function being modeled by a sigmoid, which means that the function goes to zero before or after some point. %This can be described as \eg `from [time]' or `until [time]'. \item {\bf Multiplication by $\kPer$} removes correlation between all pairs of function values not close to one period apart, allowing variation within each period, but maintaining correlation between periods. \item {\bf Multiplication by any kernel} modifies the covariance in the same way as multiplying by a function drawn from a corresponding \gp{} prior. This follows from the fact that if ${f_1(x) \dist \gp{}(0, \kernel_1)}$ and ${f_2(x) \dist \gp{}(0, \kernel_2)}$ then % \begin{align} {\textrm{Cov} \big[f_1(x)f_2(x), \; f_1(x')f_2(x') \big] = k_1(x,x') \kerntimes k_2(x,x')}. \end{align} % Put more plainly, a \gp{} whose covariance is a product of kernels has the same covariance as a product of two functions, each drawn from the corresponding \gp{} prior. However, the distribution of $f_1 \kerntimes f_2$ is not always \gp{} distributed -- it can have third and higher central moments as well. This identity can be used to generate a cumbersome ``worst-case'' description in cases where a more concise description of the effect of a kernel is not available. For example, it is used in our system to describe products of more than one periodic kernel. \end{itemize} \Cref{table:modifiers} gives the corresponding description of the effect of each type of kernel in a product, written as a post-modifier. % \begin{table}[h!] \centering \begin{tabular}{l|l} Kernel & Postmodifier phrase \\ \midrule $\kSE$ & whose shape changes smoothly \\ $\kPer$ & modulated by a periodic function \\ $\kLin$ & with linearly varying amplitude \\ $\prod_k \kLin^{(k)}$ & with polynomially varying amplitude \\ $\prod_k \boldsymbol{\sigma}^{(k)}$ & which applies until / from [changepoint] \\ \end{tabular} \caption[Descriptions of the effect of each kernel, written as a post-modifier]{ Descriptions of the effect of each kernel, written as a post-modifier. } \label{table:modifiers} \end{table} \Cref{table:nouns} gives the corresponding description of each kernel before it has been multiplied by any other, written as a noun phrase. % \begin{table}[h!] \centering \begin{tabular}{l|l} Kernel & Noun phrase \\ \midrule $\kWN$ & uncorrelated noise \\ $\kC$ & constant \\ $\kSE$ & smooth function \\ $\kPer$ & periodic function \\ $\kLin$ & linear function \\ $\prod_k \kLin^{(k)}$ & \{quadratic, cubic, quartic, \dots \} function %polynomial \\ \end{tabular} \caption[Noun phrase descriptions of each type of kernel]{ Noun phrase descriptions of each type of kernel.} \label{table:nouns} \end{table} \subsection{Combining descriptions into noun phrases} In order to build a noun phrase describing a product of kernels, our system chooses one kernel to act as the head noun, which is then modified by appending descriptions of the other kernels in the product. %described by the functions it encodes for when unmodified \eg `smooth function' for $\kSE$. %Modifiers corresponding to the other kernels in the product are then appended to this description, forming a noun phrase of the form: %\begin{align*} %\textnormal{Determiner} + \textnormal{Premodifiers} + \textnormal{Noun} + \textnormal{Postmodifiers} %\end{align*} As an example, a kernel of the form $\kPer \kerntimes \kLin \kerntimes \boldsymbol{\sigma}$ could be described as a \begin{align*} \underbrace{\kPer}_{\textnormal{\small periodic function}} \times \underbrace{\kLin}_{\textnormal{\small with linearly varying amplitude}} \times \underbrace{\boldsymbol{\sigma}}_{\textnormal{\small which applies until 1700.}} \end{align*} where $\kPer$ was chosen to be the head noun. In our system, the head noun is chosen according to the following ordering: \begin{align} \kPer, \kWN, \kSE, \kC, \prod_m \kLin^{(m)}, \prod_n \boldsymbol\sigma^{(n)} \label{eq:noun-ordering} \end{align} %\ie $\kPer$ is always chosen as the head noun when present. Combining \cref{table:modifiers,table:nouns} with ordering \ref{eq:noun-ordering} provides a general method to produce descriptions of sums and products of these base kernels. \subsubsection{Extensions and refinements} %\subsubsection{Refinements to the Descriptions} In practice, the system also incorporates a number of other rules which help to make the descriptions shorter, easier to parse, or clearer: %There are a number of ways in which the descriptions of the kernels can be made more interpretable and informative: \begin{itemize} % \item We incorporate rules for which kernel is chosen as the head noun can change the interpretability of a description. \item The system adds extra adjectives depending on kernel parameters. For example, an $\kSE$ with a relatively short lengthscale might be described as ``a rapidly-varying smooth function'' as opposed to just ``a smooth function''. \item Descriptions can include kernel parameters. For example, the system might write that a function is ``repeating with a period of 7 days''. \item Descriptions can include extra information about the model not contained in the kernel. For example, based on the posterior distribution over the function's slope, the system might write ``a linearly increasing function'' as opposed to ``a linear function''. \item Some kernels can be described through pre-modifiers. For example, the system might write ``an approximately periodic function'' as opposed to ``a periodic function whose shape changes smoothly''. \end{itemize} %The reports in the supplementary material and in \cref{sec:example-descriptions} include some of these refinements. %The parameters and design choices of these refinements have been chosen by our best judgement, but learning these parameters objectively from expert statisticians would be an interesting area for future study. \subsubsection{Ordering additive components} The reports generated by our system attempt to present the most interesting or important features of a dataset first. As a heuristic, the system orders components by always adding next the component which most reduces the 10-fold cross-validated mean absolute error. \subsection{Worked example} This section shows an example of our procedure describing a compound kernel containing every type of base kernel in our set: % \begin{align} \kSE \kerntimes (\kWN \kerntimes \kLin \kernplus \kCP(\kC, \kPer)). \end{align} % The kernel is first converted into a sum of products, and the changepoint is converted into sigmoidal kernels (recall the definition of changepoint kernels in \cref{sec:changepoint-definition}): % \begin{align} \kSE \kerntimes \kWN \kerntimes \kLin \kernplus \kSE \kerntimes \kC \kerntimes \boldsymbol{\sigma} \kernplus \kSE \kerntimes \kPer \kerntimes \boldsymbol{\bar\sigma} \end{align} % which is then simplified using the rules in \cref{sec:desc-simplification} to % \begin{align} \kWN \kerntimes \kLin \kernplus \kSE \kerntimes \boldsymbol{\sigma} \kernplus \kSE \kerntimes \kPer \kerntimes \boldsymbol{\bar\sigma}. \end{align} To describe the first component, $(\kWN \kerntimes \kLin)$, the head noun description for $\kWN$, ``uncorrelated noise'', is concatenated with a modifier for $\kLin$, ``with linearly increasing standard deviation''. The second component, $(\kSE \kerntimes \vsigma)$, is described as ``A smooth function with a lengthscale of [lengthscale] [units]'', corresponding to the $\kSE$, ``which applies until [changepoint]''. Finally, the third component, $(\kSE \kerntimes \kPer \kerntimes \boldsymbol{\bar\sigma})$, is described as ``An approximately periodic function with a period of [period] [units] which applies from [changepoint]''. \section{Example descriptions} \label{sec:example-descriptions} In this section, we demonstrate the ability of our procedure, \procedurename, to write intelligible descriptions of the structure present in two time series. The examples presented here describe models produced by the automatic search method presented in \cref{ch:grammar}. \subsection{Summarizing 400 years of solar activity} \label{sec:solar} First, we show excerpts from the report automatically generated on annual solar irradiation data from 1610 to 2011. This dataset is shown in \cref{fig:solar}. % \begin{figure}[ht!] \centering \includegraphics[trim=0.2cm 18.0cm 8cm 2cm, clip, width=0.6\columnwidth]{\grammarfiguresdir/solarpages/pg_0002-crop} \caption[Solar irradiance dataset] {Solar irradiance data~\citep{lean1995reconstruction}.} \label{fig:solar} \end{figure} This time series has two pertinent features: First, a roughly 11-year cycle of solar activity. Second, a period lasting from 1645 to 1715 having almost no variance. This flat region is known as to the Maunder minimum, a period in which sunspots were extremely rare \citep{lean1995reconstruction}. The Maunder minimum is an example of the type of structure that can be captured by change-windows. \begin{figure}[ht!] \centering \fbox{\includegraphics[trim=0cm 10.8cm 0cm 9.3cm, clip, width=0.98\columnwidth]{\grammarfiguresdir/solarpages/pg_0002-crop}} \caption[Automatically-generated description of the solar irradiance data set] {Automatically generated descriptions of the first four components discovered by \procedurename{} on the solar irradiance data set. The dataset has been decomposed into diverse structures having concise descriptions.} \label{fig:exec} \end{figure} The first section of each report generated by \procedurename{} is a summary of the structure found in the dataset. \Cref{fig:exec} shows natural-language summaries of the top four components discovered by \procedurename{} on the solar dataset. From these summaries, we can see that the system has identified the Maunder minimum (second component) and the 11-year solar cycle (fourth component). These components are visualized and described in figures~\ref{fig:maunder} and \ref{fig:periodic}, respectively. The third component, visualized in \cref{fig:smooth}, captures the smooth variation over time of the overall level of solar activity. \begin{figure}[]% \centering \fbox{ \begin{tabular}{@{}c@{}} \includegraphics[trim=0cm 10.5cm 0cm 0.7cm, clip, width=0.98\columnwidth]{\grammarfiguresdir/solarpages/pg_0005-crop} \\ \includegraphics[trim=0cm 4.75cm 0cm 3.12cm, clip, width=0.98\columnwidth]{\grammarfiguresdir/solarpages/pg_0005-crop} \end{tabular} } \caption[A component corresponding to the Maunder minimum] {Extract from an automatically-generated report describing the model component corresponding to the Maunder minimum.} \label{fig:maunder} %\end{figure} % %\begin{figure}[h!] \centering \fbox{ \begin{tabular}{@{}c@{}} \includegraphics[trim=0cm 10.5cm 0cm 1.05cm, clip, width=0.98\columnwidth]{\grammarfiguresdir/solarpages/pg_0006-crop} \\ \includegraphics[trim=0cm 4.75cm 0cm 3.9cm, clip, width=0.98\columnwidth]{\grammarfiguresdir/solarpages/pg_0006-crop} \end{tabular} } \caption[\procedurename{} isolating part of the signal explained by a slowly-varying trend] {Characterizing the medium-term smoothness of solar activity levels. By allowing other components to explain the periodicity, noise, and the Maunder minimum, \procedurename{} can isolate the part of the signal best explained by a slowly-varying trend.} \label{fig:smooth} %\end{figure} % %\begin{figure}[ht!] \centering \fbox{ \begin{tabular}{@{}c@{}} \includegraphics[trim=0cm 10.5cm 0cm 1.05cm, clip, width=0.98\columnwidth]{\grammarfiguresdir/solarpages/pg_0007-crop} \\ \includegraphics[trim=0cm 4.75cm 0cm 4.66cm, clip, width=0.98\columnwidth]{\grammarfiguresdir/solarpages/pg_0007-crop} \end{tabular} } \caption[Automatic description of the solar cycle] { This part of the report isolates and describes the approximately 11-year sunspot cycle, also noting its disappearance during the Maunder minimum.} % also noting its disappearance during the 16th century, a time known as the Maunder minimum \citep{lean1995reconstruction}.} \label{fig:periodic} \end{figure} % The complete report generated on this dataset can be found in appendix~\ref{ch:example-solar}. Each report also contains samples from the model posterior. \phantom{a} %\newpage \subsection{Describing changing noise levels} % in Air Traffic Data} \label{sec:airline} %\begin{figure}[ht!] %\centering %\includegraphics[trim=0.4cm 16.8cm 8cm 2cm, clip, width=0.5\columnwidth]{\grammarfiguresdir/airlinepages/pg_0002-crop} %\caption[International airline passenger monthly volume dataset] %{International airline passenger monthly volume \citep[e.g.][]{box2013time}.} %\label{fig:airline} %\end{figure} Next, we present excerpts of the description generated by our procedure on a model of international airline passenger counts over time, shown in \cref{fig:airline_decomp}. %The model described is the same as that shown in \cref{fig:airline_decomp}. %constructed by \procedurename{} has four components: $\kLin \kernplus \kSE \kerntimes \kPer \kerntimes \kLin \kernplus \kSE \kernplus \kWN \kerntimes \kLin$, with % High-level descriptions of the four components discovered are shown in \cref{fig:exec-airline}. \begin{figure}[ht!] \centering \fbox{\includegraphics[trim=0cm 9.5cm 0cm 9.25cm, clip, width=0.98\columnwidth]{\grammarfiguresdir/airlinepages/pg_0002-crop}} \caption[Short descriptions of the four components of the airline model] {Short descriptions of the four components of a model describing the airline dataset.} \label{fig:exec-airline} \end{figure} \begin{figure}[ht!] \centering \fbox{ \begin{tabular}{@{}c@{}} \includegraphics[trim=0cm 10.5cm 0cm 1cm, clip, width=0.98\columnwidth]{\grammarfiguresdir/airlinepages/pg_0004-crop} \\ \includegraphics[trim=0cm 4.7cm 0cm 4.3cm, clip, width=0.98\columnwidth]{\grammarfiguresdir/airlinepages/pg_0004-crop} \end{tabular} } \caption[Describing non-stationary periodicity in the airline data] {Describing non-stationary periodicity in the airline data.} \label{fig:lin_periodic} \end{figure} \begin{figure}[ht!] \centering \fbox{ \begin{tabular}{@{}c@{}} \includegraphics[trim=0cm 6.7cm 0cm 0.6cm, clip, width=0.98\columnwidth]{\grammarfiguresdir/airlinepages/pg_0006-crop} \\ \includegraphics[trim=0cm 0cm 0cm 3.98cm, clip, width=0.98\columnwidth]{\grammarfiguresdir/airlinepages/pg_0006-crop} \end{tabular} } \caption[Describing time-changing variance in the airline dataset] {Describing time-changing variance in the airline dataset.} \label{fig:heteroscedastic} \end{figure} The second component, shown in \cref{fig:lin_periodic}, is accurately described as approximately ($\kSE$) periodic ($\kPer$) with linearly growing amplitude ($\kLin$). The description of the fourth component, shown in \cref{fig:heteroscedastic}, expresses the fact that the scale of the unstructured noise in the model grows linearly with time. %By multiplying a white noise kernel by a linear kernel, the model is able to express heteroscedasticity (covariance which changes over time). The complete report generated on this dataset can be found in the supplementary material of \citet{LloDuvGroetal14}. Other example reports describing a wide variety of time-series can be found at \url{http://mlg.eng.cam.ac.uk/lloyd/abcdoutput/} \section{Related work} \label{sec:description-related-work} To the best of our knowledge, our procedure is the first example of automatic textual description of a nonparametric statistical model. However, systems with natural language output have been developed for automatic video description \citep{barbu2012video} and automated theorem proving \citep{ganesalingam2013fully}. Although not a description procedure, \citet{durrande2013gaussian} developed an analytic method for decomposing \gp{} posteriors into entirely periodic and entirely non-periodic parts, even when using non-periodic kernels. \section{Limitations of this approach} \label{sec:limitations-of-abcd} During development, we noted several difficulties with this overall approach: \begin{itemize} \item {\bf Some kernels are hard to describe.} For instance, we did not include the $\kRQ$ kernel in the text-generation procedure. This was done for several reasons. First, the $\kRQ$ kernel can be equivalently expressed as a scale mixture of $\kSE$ kernels, making it redundant in principle. Second, it was difficult to think of a clear and concise description for effect of the hyperparameter that controls the heaviness of the tails of the $\kRQ$ kernel. Third, a product of two $\kRQ$ kernels does not give another $\kRQ$ kernel, which raises the question of how to concisely describe products of $\kRQ$ kernels. \item {\bf Reliance on additivity.} Much of the modularity of the description procedure is due to the additive decomposition. %However, some types of structure do not allow additive decomposition. %For example, A multiplication of , or only decompose additively after being warped. However, additivity is lost under any nonlinear transformation of the output. Such warpings can be learned~\citep{snelson2004warped}, but descriptions of transformations of the data may not be as clear to the end user. \item {\bf Difficulty of expressing uncertainty.} A natural extension to the model search procedure would be to report a posterior distribution on structures and kernel parameters, rather than point estimates. Describing uncertainty about the hyperparameters of a particular structure may be feasible, but describing even a few most-probable structures might result in excessively long reports. \end{itemize} \subsubsection{Source code} Source code to perform all experiments is available at \\\url{http://www.github.com/jamesrobertlloyd/gpss-research}. \section{Conclusions} This chapter presented a system which automatically generates detailed reports describing statistical structure captured by a \gp{} model. The properties of \gp{}s and the kernels being used allow a modular description, avoiding an exponential blowup in the number of special cases that need to be considered. Combining this procedure with the model search of \cref{ch:grammar} gives a system combining all the elements of an automatic statistician listed in \cref{sec:ingredients}: an open-ended language of models, a method to search through model space, a model comparison procedure, and a model description procedure. Each particular element used in the system presented here is merely a proof-of-concept. However, even this simple prototype demonstrated the ability to discover and describe a variety of patterns in time series. %We hope that systems such as these will allow the use of \outbpdocument{ \bibliographystyle{plainnat} \bibliography{references.bib} } ================================================ FILE: discussion.tex ================================================ \input{common/header.tex} \inbpdocument \chapter{Discussion} \label{ch:discussion} This chapter summarizes the contributions of this thesis, articulates some of the questions raised by this work, and relates the kernel-based model-building procedure of \cref{ch:kernels,ch:grammar,ch:description} to the larger project of automating statistics and machine learning. %We with a summary of the contributions of this thesis. \section{Summary of contributions} The main contribution of this thesis was to develop a way to automate the construction of structured, interpretable nonparametric regression models using Gaussian processes. This was done in several parts: First, \cref{ch:kernels} presented a systematic overview of kernel construction techniques, and examined the resulting \gp{} priors. Next, \cref{ch:grammar} showed the viability of a search over an open-ended space of kernels, and showed that the corresponding \gp{} models could be automatically decomposed into diverse parts allowing visualization of the structure found in the data. \Cref{ch:description} showed that sometimes parts of kernels can be described modularly, allowing automatically written text to be included in detailed reports describing \gp{} models. An example report is included in appendix~\ref{ch:example-solar}. Together, these chapters demonstrate a proof-of-concept of what could be called an ``automatic statistician'' capable of the performing some of the model construction and analysis currently performed by experts. The second half of this thesis examined several extensions of Gaussian processes, all of which enabled the automatic determination of model choices that were previously set by trial and error or cross-validation. \Cref{ch:deep-limits} characterized and visualized deep Gaussian processes, related them to existing deep neural networks, and derived novel deep kernels. \Cref{ch:additive} investigated additive \gp{}s, %a family of models consisting of sums of functions of all subsets of input variables, and showed that they have the same covariance as a \gp{} using dropout. \Cref{ch:warped} extended the \gp{} latent variable model into a Bayesian clustering model which automatically infers the nonparametric shape of each cluster, as well as the number of clusters. % TODO: move to the relevant chapter %\section{Future Work} %How far can we push the kernel structure discovery procedure outlined in this thesis? %\subsubsection{Multiple Dimensions} %The experiments in \cref{ch:additive} showed that the structure search had very good performance in moderate dimensions. %\subsubsection{Input and Output Warpings} \section{Structured versus unstructured models} One question left unanswered by this thesis is: when to prefer the structured, kernel-based models described in \cref{ch:kernels,ch:grammar,ch:description,ch:additive} to the relatively unstructured deep \gp{} models described in \cref{ch:deep-limits}? This section considers some advantages and disadvantages of the two approaches. \begin{itemize} \item {\bf Difficulty of optimization.} The discrete nature of the space of composite kernel structures can be seen as both a blessing and a curse. Certainly, a mixed discrete-and-continuous search space requires more complex optimization procedures than the continuous-only optimization possible in deep \gp{}s. However, the discrete nature of the space of composite kernels offers the possibility of learning heuristics to suggest which types of structure to add, based on features of the dataset, or previous model fits. For example, finding periodic structure or growing variance in the residuals suggests adding periodic or linear components to the kernel, respectively. It is not clear whether such heuristics could easily be constructed for optimizing the variational parameters of a deep \gp{}. \item {\bf Long-range extrapolation.} Another open question is whether the inductive bias of deep \gp{}s can be made to allow the sorts of long-range extrapolation shown in \cref{ch:kernels,ch:grammar}. As an example, consider the problem of extrapolating a periodic function. A deep \gp{} could learn a latent representation similar to that of the periodic kernel, projecting into a basis equivalent to $[\sin(x), \cos(x)]$ in the first hidden layer. However, to extrapolate a periodic function, the $\sin$ and $\cos$ functions would have to repeat beyond the input range of the training data, which would not happen if each layer assumed only local smoothness. One obvious possibility is to marry the two approaches, building deep \gp{}s with structured kernels. However, we may lose some of the advantages of interpretability by this approach, and inference would become more difficult. Another point to consider is that, in high dimensions, the distinction between interpolation and extrapolation becomes less meaningful. If the training and test data both live on a low-dimensional manifold, then learning a suitable representation of that manifold may be sufficient for obtaining high predictive accuracy. \item {\bf Ease of interpretation.} Historically, the statistics community has put more emphasis on the interpretability and meaning of models than the machine learning community, which has focused more on predictive performance. To begin to automate the practice of statistics, developing model-description procedures for powerful open-ended model classes seems to be a necessary step. %promising direction.% with the most low-hanging fruit. At first glance, automatic model description may seem to require a decomposition of the model being described into discrete components, as in the additive decomposition demonstrated in \cref{ch:kernels,ch:grammar,ch:description}. %For example, \cref{ch:description} showed that composite kernels allow automatic visualization and desription of low-dimensional structure. On the other hand, most probabilistic models allow a form of summarization of high-dimensional structure through sampling from the posterior. For the specific case of deep \gp{}s, \citet{damianou2012deep} showed that these models allow summarization through examining the dimension of each latent layer, visualizing latent coordinates, and examining how the predictive distribution changes as one moves in the latent space. Perhaps more sophisticated procedures could also allow intelligible text-based descriptions of such models. \end{itemize} The warped mixture model of \cref{ch:warped} represents a compromise between these two approaches, combining a discrete clustering model with an unstructured warping function. However, the explicit clustering model may by unnecessary: the results of \citet{damianou2012deep} suggest that clustering can be automatically (but implicitly) accomplished by a sufficiently deep, unstructured \gp{}. \section{Approaches to automating model construction} \label{top-down-vs-bottom-up} This thesis is a small part of a larger push to automate the practice of model building and inference. Broadly speaking, this problem is being attacked from two directions. From the top-down, the probabilistic programming community is developing automatic inference engines for extremely broad classes of models~\citep{goodman2008church,mansinghka2014venture,Wood-AISTATS-2014,koller1997effective,milch20071,stan-software:2014} such as the class of all computable distributions~\citep{solomonoff1964formal, li2009introduction}. As discussed in \cref{sec:ingredients}, model construction procedures can usually be seen as a search through an open-ended model class. %For example, \citet{liang2010learning} %Exact inference in sufficiently powerful model classes is \#P-hard (cite), so approximate inference algorithms are necessary. Exhaustive search strategies have been constructed for the space of computable distributions~\citep{hutter2002fastest,schmidhuber2002speed,levin1973universal}, but they remain impractically slow. An alternative, bottom-up, approach is to design procedures which extend and combine existing model classes for which relatively efficient inference algorithms are already known. The language of models proposed in \cref{ch:grammar} is an example of this bottom-up approach. Another example is \citet{roger-grosse-thesis}, who built an open-ended language of matrix decomposition models and a corresponding compositional language of relatively efficient approximate inference algorithms. Similarly, \citet{christian-thesis} showed how to compose inference algorithms for sequence models. These approaches have the advantage that inference is usually feasible for any model in the language. However, extending these languages may require developing new inference algorithms. If sufficiently powerful building-blocks are composed, the line between the top-down and bottom-up approaches becomes blurred. For example, deep generative models~\citep{adams2010learning,damianou2012deep,rippel2013high,bengio2013generalized,salakhutdinov2009deep} could be considered an example of the bottom-up approach, since they compose individual model ``layers'' to produce more powerful models. However, large neural nets can capture enough different types of structure that they could also be seen as an example of the universalist, top-down approach. %As well, inference in these models has so far been done in a one-size-fits all approach, ignoring the particular structure of a given problem. \section{Conclusion} It seems clear that one way or another, large parts of the existing practice of model-building will eventually be automated. However, it remains to be seen which of the above model-building approaches will be most useful. I hope that this thesis will contribute to our understanding of the strengths and weaknesses of these different approaches, and towards the use of more powerful model classes by practitioners in other fields. % Benefit: systematically explores model space? %\section{Approaches to automating model description} %The machine learning community has so far focused largely on producing efficient inference strategies for powerful model classes, which is sufficient for improving predictive performance. %\section{Automating Statistics} %Automating the process of statistical modeling would have a tremendous impact on fields that currently rely on expert statisticians, machine learning researchers, and data scientists. %While fitting simple models (such as linear regression) is largely automated by standard software packages, there has been little work on the automatic construction of flexible but interpretable models. \outbpdocument{ \bibliographystyle{plainnat} \bibliography{references.bib} } \iffalse \section{Why assume zero mean?} \section{Why not learn the mean function instead?} One might ask: besides integrating over the magnitude, what is the advantage of moving the mean function into the covariance function? After all, mean functions are certainly more interpretable than a posterior distribution over functions. Instead of searching over a large class of covariance functions, which seems strange and unnatural, we might consider simply searching over a large class of structured mean functions, assuming a simple \iid noise model. This is the approach taken by practically every other regression technique: neural networks, decision trees, boosting, \etc. If we could integrate over a wide class of possible mean functions, we would have a very powerful learning an inference method. The problem faced by all of these methods is the well-known problem \emph{overfitting}. If we are forced to choose just a single function with which to make predictions, we must carefully control the flexibility of the model we learn, generally preferring ``simple'' functions, or to choose a function from a restricted set. If, on the other hand, we are allowed to keep in mind many possible explanations of the data, \emph{there is no need to penalize complexity}. [cite Occam's razor paper?] The power of putting structure into the covariance function is that doing so allows us to implicitly integrate over many functions, maintaining a posterior distribution over infinitely many functions, instead of choosing just one. In fact, each of the functions being considered can be infinitely complex, without causing any form of overfitting. For example, each of the samples shown in \cref{fig:gp-post} varies randomly over the whole real line, never repeating, each one requiring an infinite amount of information to describe. Choosing the one function which best fits the data will almost certainly cause overfitting. However, if we integrate over many such functions, we will end up with a posterior putting mass on only those functions which are compatible with the data. In other words, the parts of the function that we can determine from the data will be predicted with certainty, but the parts which are still uncertain will give rise to a wide range of predictions. To repeat: \emph{there is no need to assume that the function being modeled is simple, or to prefer simple explanations} in order to avoid overfitting, if we integrate over many possible explanations rather than choosing just the one. In Chapter~\ref{ch:grammar}, we will compare a system which estimates a parametric covariance function against one which estimates a parametric mean function. \section{Signal versus noise} In most derivations of Gaussian processes, the model is given as % \begin{align} y = f(\inputVar) + \epsilon, \quad \textnormal{where} \quad \epsilon \simiid \Nt{0}{\sigma^2_{\textnormal{noise}}} \end{align} However, $\epsilon$ can equivalently be thought of as another Gaussian process, and so this model can be written as $y(\inputVar) \sim \GP\left(0, k + \delta \right)$. The lack of a hard distinction between the noise model and the signal model raises the larger question: Which part of a model represents signal, and which represents noise? We believe that it depends on what you want to do with the model - there is no hard distinction between signal and noise in general. For example: often, we do not care about the short-term variations in a function, and only in the long-term trend. 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================================================ \input{common/header.tex} \inbpdocument \chapter{Automatic Model Construction} \label{ch:grammar} \begin{quotation} ``It would be very nice to have a formal apparatus that gives us some `optimal' way of recognizing unusual phenomena and inventing new classes of hypotheses that are most likely to contain the true one; but this remains an art for the creative human mind.'' \defcitealias{Jaynes85highlyinformative}{ -- E. T. Jaynes (1985)} \hspace*{\fill}\citetalias{Jaynes85highlyinformative} \end{quotation} In \cref{ch:kernels}, we saw that the choice of kernel determines the type of structure that can be learned by a \gp{} model, and that a wide variety of models could be constructed by adding and multiplying a few base kernels together. However, we did not answer the difficult question of which kernel to use for a given problem. Even for experts, choosing the kernel in \gp{} regression remains something of a black art. The contribution of this chapter is to show a way to automate the process of building kernels for \gp{} models. We do this by defining an open-ended space of kernels built by adding and multiplying together kernels from a fixed set. We then define a procedure to search over this space to find a kernel which matches the structure in the data. Searching over such a large, structured model class has two main benefits. First, this procedure has good predictive accuracy, since it tries out a large number of different regression models. Second, this procedure can sometimes discover interpretable structure in datasets. Because \gp{} posteriors can be decomposed (as in \cref{sec:concrete}), the resulting structures can be examined visually. In \cref{ch:description}, we also show how to automatically generate English-language descriptions of the resulting models. %\paragraph{Attribution} This chapter is based on work done in collaboration with James Robert Lloyd, Roger Grosse, Joshua B. Tenenbaum, and Zoubin Ghahramani. It was published in \citet{DuvLloGroetal13} and \citet{LloDuvGroetal14}. Myself, James Lloyd and Roger Grosse jointly developed the idea of searching through a grammar-based language of \gp{} models, inspired by \citet{grosse2012exploiting}, and wrote the first versions of the code together. James Lloyd ran most of the experiments, and I constructed most of the figures. %I produced all of the figures. %Zoubin Ghahramani and Josh Tenenbaum provided many conceptual insights, as well as suggestions about how the resulting procedure could be most fruitfully applied. %\paragraph{Automatic statistician:} The work appearing in chapter \ref{ch:description} was written in collaboration with James Robert Lloyd, Roger Grosse, Joshua B. Tenenbaum, Zoubin Ghahramani, and was published in . %The procedure translating kernels into adjectives grew out of discussions between James and myself. %James Lloyd wrote the code to automatically generate reports, and ran all of the experiments. %The text was written mainly by myself, James Lloyd, and Zoubin Ghahramani, with many helpful contributions and suggestions from Roger Grosse and Josh Tenenbaum. %Section \ref{sec:Structure} outlines some commonly used kernel families as well as ways in which they can be composed. %Our grammar over kernels and our proposed structure discovery algorithm are described in Section \ref{sec:Search}. %Section \ref{sec:related_work} situates our work in the context of other nonparametric regression, kernel learning, and structure discovery methods. %We evaluate our methods on synthetic datasets, time series analysis, and high-dimensional prediction problems in Sections \ref{sec:synthetic} through \ref{sec:quantitative}, respectively. \section{Ingredients of an automatic statistician} \label{sec:ingredients} \citet{gelman2013philblogpost} asks: ``How can an artificial intelligence do statistics? \dots It needs not just an inference engine, but also a way to construct new models and a way to check models. Currently, those steps are performed by humans, but the AI would have to do it itself''. % This section will discuss the different parts we think are required to build an artificial intelligence that can do statistics. \begin{enumerate} \item {\bf An open-ended language of models.} Many learning algorithms consider all models in a class of fixed size. For example, graphical model learning algorithms \citep{Friedman03,Eaton07_uai} search over different connectivity graphs for a given set of nodes. Such methods can be powerful, but human statisticians are sometimes capable of deriving \emph{novel} model classes when required. An automatic search through an open-ended class of models can achieve some of this flexibility, %growing the complexity of the model as needed, possibly combining existing structures in novel ways. \item {\bf A search through model space.} %An open-ended space of models cannot be searched exhaustively. Every procedure which eventually considers arbitrarily-complex models must start with relatively simple models before moving on to more complex ones. %Any model search procedure which visits every model eventually cannot start by guessing the Thus any search strategy capable of building arbitrarily complex models is likely to resemble an iterative model-building procedure. Just as human researchers iteratively refine their models, search procedures can propose new candidate models based on the results of previous model fits. \item {\bf A model comparison procedure.} Search strategies requires an objective to optimize. In this work, we use approximate marginal likelihood to compare models, penalizing complexity using the Bayesian Information Criterion as a heuristic. More generally, an automatic statistician needs to somehow check the models it has constructed. %, and formal procedures from model checking provide a way for it to do this. \citet{gelman2012philosophy} review the literature on model checking. \item {\bf A model description procedure.} Part of the value of statistical models comes from helping humans to understand a dataset or a phenomenon. Furthermore, a clear description of the statistical structure found in a dataset helps a user to notice when the dataset has errors, the wrong question was asked, the model-building procedure failed to capture known structure, a relevant piece of data or constraint is missing, or when a novel statistical structure has been found. \end{enumerate} In this chapter, we introduce a system containing simple examples of all the above ingredients. We call this system the automatic Bayesian covariance discovery (\procedurename{}) system. The next four sections of this chapter describe the mechanisms we use to incorporate these four ingredients into a limited example of an artificial intelligence which does statistics. \section{A language of regression models} \label{sec:improvements} As shown in \cref{ch:kernels}, one can construct a wide variety of kernel structures by adding and multiplying a small number of base kernels. We can therefore define a language of \gp{} regression models simply by specifying a language of kernels. \begin{figure}[ht!]% \centering \begin{tabular}{r|ccc} Kernel name: & Rational quadratic (\kRQ) & Cosine ($\cos$) & White noise (\kLin) \\[10pt] $k(x, x') =$ & $ \sigma_f^2 \left( 1 + \frac{(\inputVar - \inputVar')^2}{2\alpha\ell^2}\right)^{-\alpha}$ & $\sigma_f^2 \cos\left(2 \pi \frac{ (x - x')}{p}\right)$ & $\sigma_f^2 \delta(\inputVar - \inputVar')$ \\[14pt] \raisebox{1cm}{Plot of kernel:} & \kernpic{rq_kernel} & \kernpic{cos_kernel} & \kernpic{wn_kernel}\\ & $x -x'$ & $x -x'$ & $x -x'$ \\ %& & & \\ & \large $\downarrow$ & \large $\downarrow$ & \large $\downarrow$ \\ \raisebox{1cm}{\parbox{2.7cm}{\raggedleft Functions $f(x)$ sampled from \gp{} prior:}} & \kernpic{rq_kernel_draws_s4} & \kernpic{cos_kernel_draws_s1} & \kernpic{wn_kernel_draws_s1} \\[-2pt] & $x$ & $x$ & $x$ \\ Type of structure: & multiscale variation & sinusoidal & uncorrelated noise \end{tabular} \vspace{6pt} \caption[Another set of basic kernels] {New base kernels introduced in this chapter, and the types of structure they encode. Other types of kernels can be constructed by adding and multiplying base kernels together.} \label{fig:basic_kernels_two} \end{figure} % Our language of models is specified by a set of base kernels which capture different properties of functions, and a set of rules which combine kernels to yield other valid kernels. In this chapter, we will use such base kernels as white noise ($\kWN$), constant ($\kC$), linear ($\kLin$), squared-exponential ($\kSE$), rational-quadratic ($\kRQ$), sigmoidal ($\kSig$) and periodic ($\kPer$). We use a form of $\kPer$ due to James Lloyd (personal communication) which has its constant component removed, and $\cos(x - x')$ as a special case. \Cref{fig:basic_kernels_two} shows the new kernels introduced in this chapter. For precise definitions of all kernels, see appendix~\ref{sec:kernel-definitions}. To specify an open-ended language of structured kernels, we consider the set of all kernels that can be built by adding and multiplying these base kernels together, which we write in shorthand by: \begin{align} k_1 + k_2 = & \,\, k_1(\vx, \vx') + k_2(\vx, \vx') \\ k_1 \times k_2 = & \,\, k_1(\vx, \vx') \times k_2(\vx, \vx') \end{align} % The space of kernels constructable by adding and multiplying the above set of kernels contains many existing regression models. \Cref{table:motifs} lists some of these, which are discussed in more detail in \cref{sec:gpss-related-work}. % regression models that can be expressed by this language. \begin{table}[ht] \centering \begin{tabular}{l|l|l} Regression model & Kernel structure & Example of related work\\ \midrule Linear regression & $\kC \kernplus \kLin \kernplus \kWN$ & \\ Polynomial regression & $\kC \kernplus \prod \kLin \kernplus \kWN$ & \\ %Kernel ridge regression & $\kSE \kernplus \kWN$ & \\ Semi-parametric & $\kLin \kernplus \kSE \kernplus \kWN$ & \citet{ruppert2003semiparametric} \\ Multiple kernel learning & $\sum \kSE \kernplus \kWN$ & \citet{gonen2011multiple} \\ Fourier decomposition & $\kC \kernplus \sum \cos \kernplus \kWN$ & \\ Trend, cyclical, irregular & $\sum \kSE \kernplus \sum \kPer \kernplus \kWN$ & \citet{lind2006basic}\\ Sparse spectrum \gp{}s & $\sum \cos \kernplus \kWN$ & \citet{lazaro2010sparse} \\ Spectral mixture & $\sum \SE \kerntimes \cos \kernplus \kWN$ & \citet{WilAda13} \\ Changepoints & \eg $\kCP(\kSE, \kSE) \kernplus \kWN$ & \citet{garnett2010sequential} \\ Time-changing variance & \eg $\kSE \kernplus \kLin \kerntimes \kWN$ & \\ Interpretable + flexible & $ \sum_d \kSE_d \kernplus \prod_d \kSE_d$ & \citet{plate1999accuracy} \\ Additive \gp{}s & \eg $\prod_d (1 + \kSE_d) $ & \Cref{ch:additive} \end{tabular} \caption[Common regression models expressible in the kernel language] {Existing regression models expressible by sums and products of base kernels. $\cos(\cdot, \cdot)$ is a special case of our reparametrized $\kPer(\cdot, \cdot)$. } \label{table:motifs} \end{table} \section{A model search procedure} \label{sec:model-search-procedure} We explore this open-ended space of regression models using a simple greedy search. At each stage, we choose the highest scoring kernel, and propose modifying it by applying an operation to one of its parts, that combines or replaces that part with another base kernel. The basic operations we can perform on any part $k$ of a kernel are: % \begin{center} \begin{tabular}{rccl} \textnormal{Replacement:} & $\kernel$ & $\to$ & $\kernel'$\\ \textnormal{Addition:} & $\kernel$ & $\to$ & $(\kernel + \kernel')$\\ \textnormal{Multiplication:} & $\kernel$ & $\to$ & $(\kernel \times \kernel')$\\ \end{tabular} \end{center} % where $k'$ is a new base kernel. These operators can generate all possible algebraic expressions involving addition and multiplication of base kernels. To see this, observe that if we restricted the addition and multiplication rules to only apply to base kernels, we would obtain a grammar which generates the set of algebraic expressions. \begin{figure} \centering %\begin{tabular}{cc} %\begin{minipage}[t][14cm][t]{0.65\columnwidth} \newcommand{\treescale}{*1.5\columnwidth} \begin{tikzpicture} [sibling distance=0.15\treescale,-,thick, level distance=2cm] %\footnotesize \node[shape=rectangle,draw,thick,fill=blue!30] {No structure} child {node[shape=rectangle,draw,thick] {$\SE$} } child {node[shape=rectangle,draw,thick,fill=blue!30] {$\RQ$} [sibling distance=0.1\treescale] child {node[shape=rectangle,draw,thick] {$\SE$ + \RQ}} child {node {\ldots}} child {node[shape=rectangle,draw,thick,fill=blue!30] {$\Per + \RQ$} [sibling distance=0.15\treescale] child {node[shape=rectangle,draw,thick] {$\SE + \Per + \RQ$}} child {node {\ldots}} child {node[shape=rectangle,draw,thick,fill=blue!30] {$\SE \kerntimes (\Per + \RQ)$} [sibling distance=0.14\treescale] child {node {\ldots}} child {node {\ldots}} child {node {\ldots}} } child {node {\ldots}} } child {node {\ldots}} child {node[shape=rectangle,draw,thick] {$\Per \kerntimes \RQ$}} } child {node[shape=rectangle,draw,thick] {$\Lin$} } child {node[shape=rectangle,draw,thick] {$\Per$} }; \end{tikzpicture} \caption[A search tree over kernels]{An example of a search tree over kernel expressions. \Cref{fig:mauna_grow} shows the corresponding model increasing in sophistication as the kernel expression grows. } \label{fig:mauna_search_tree} \end{figure} \begin{figure} \centering \newcommand{\wmg}{0.31\columnwidth} % width maunu growth \newcommand{\hmg}{3.2cm} % height maunu growth \newcommand{\maunadecomp}[1]{\hspace{-0.15cm} \includegraphics[width=\wmg,height=\hmg, clip,trim=0mm 0mm 0mm 7mm]{\grammarfiguresdir/decomposition/11-Feb-v4-03-mauna2003-s_max_level_#1/03-mauna2003-s_all_small}} \begin{tabular}{ccc} Level 1: & Level 2: & Level 3: \\ \RQ & $\Per + \RQ$ & $\SE \kerntimes (\Per + \RQ )$ \\[0.5em] %Level 1: \RQ & Level 2: $\Per + \RQ$ & Level 3: $\SE \kerntimes (\Per + \RQ )$ \\[0.5em] \maunadecomp{0} & \maunadecomp{1} & \maunadecomp{2} \\[0.5em] \end{tabular} \caption[Progression of models as the search depth increases] {Posterior mean and variance for different depths of kernel search on the Mauna Loa dataset, described in \cref{sec:extrapolation}. The dashed line marks the end of the dataset. \emph{Left:} First, the function is only modeled as a locally smooth function, and the extrapolation is poor. \emph{Middle:} A periodic component is added, and the extrapolation improves. \emph{Right:} At depth 3, the kernel can capture most of the relevant structure, and is able to extrapolate reasonably. } \label{fig:mauna_grow} \end{figure} \Cref{fig:mauna_search_tree} shows an example search tree followed by our algorithm. \Cref{fig:mauna_grow} shows how the resulting model changes as the search is followed. In practice, we also include extra operators which propose commonly-occurring structures, such as changepoints. A complete list is contained in appendix~\ref{sec:search-operators}. Our search operators have rough parallels with strategies used by human researchers to construct regression models. In particular, \begin{itemize} \item One can look for structure in the residuals of a model, such as periodicity, and then extend the model to capture that structure. This corresponds to adding a new kernel to the existing structure. \item One can start with structure which is assumed to hold globally, such as linearity, but find that it only holds locally. This corresponds to multiplying a kernel structure by a local kernel such as $\kSE$. \item One can incorporate input dimensions incrementally, analogous to algorithms like boosting, back-fitting, or forward selection. This corresponds to adding or multiplying with kernels on dimensions not yet included in the model. \end{itemize} %We stop the procedure at a fixed depth (typically 10), and return the best-scoring kernel over all depths. %The fixed depth was enforcing solely for convencience - because the \subsubsection{Hyperparameter initialization} Unfortunately, optimizing the marginal likelihood over parameters is not a convex optimization problem, and the space can have many local optima. For example, in data having periodic structure, integer multiples of the true period (harmonics) are often local optima. We take advantage of our search procedure to provide reasonable initializations: all parameters which were part of the previous kernel are initialized to their previous values. Newly introduced parameters are initialized randomly. In the newly proposed kernel, all parameters are then optimized using conjugate gradients. This procedure is not guaranteed to find the global optimum, but it implements the commonly used heuristic of iteratively modeling residuals. \section{A model comparison procedure} Choosing a kernel requires a method for comparing models. We choose marginal likelihood as our criterion, since it balances the fit and complexity of a model \citep{rasmussen2001occam}. Conditioned on kernel parameters, the marginal likelihood of a \gp{} can be computed analytically by \cref{eq:gp_marg_lik}. %Given a parametric form of a kernel, we can also choose its parameters using marginal likelihood. %However, choosing kernel parameters by maximum likelihood (as opposed to integrating them out) raises the possibility of overfitting. In addition, if one compares \gp{} models by the maximum likelihood value obtained after optimizing their kernel parameters, then all else being equal, the model having more free parameters will be chosen. This introduces a bias in favor of more complex models. We could avoid overfitting by integrating the marginal likelihood over all free parameters, but this integral is difficult to do in general. Instead, we loosely approximate this integral using the Bayesian information criterion (\BIC{}) \citep{schwarz1978estimating}: % \begin{equation} \textrm{BIC}(M) = \log p(D \given M) - \frac{1}{2} |M| \log N \end{equation} % where $p(D|M)$ is the marginal likelihood of the data evaluated at the optimized kernel parameters, $|M|$ is the number of kernel parameters, and $N$ is the number of data points. \BIC{} simply penalizes the marginal likelihood in proportion to how many parameters the model has. %BIC trades off model fit against model complexity and implements what is known as ``Bayesian Occam's Razor'' \citep{rasmussen2001occam,mackay2003information}. Because BIC is a function of the number of parameters in a model, we did not count kernel parameters known to not affect the model. For example, when two kernels are multiplied, one of their output variance parameters becomes redundant, and can be ignored. The assumptions made by \BIC{} are clearly inappropriate for the model class being considered. For instance, \BIC{} assumes that the data are \iid given the model parameters, which is not true except under a white noise kernel. Other more sophisticated approximations are possible, such as Laplace's approximation. We chose to try \BIC{} first because of its simplicity, and it performed reasonably well in our experiments. \section{A model description procedure} As discussed in \cref{ch:kernels}, a \gp{} whose kernel is a sum of kernels can be viewed as a sum of functions drawn from different \gp{}s. We can always express any kernel structure as a sum of products of kernels by distributing all products of sums. For example, % \begin{align} \SE \kerntimes (\RQ \kernplus \Lin) = \SE \kerntimes \RQ \kernplus \SE \kerntimes \Lin \,. \end{align} % When all kernels in a product apply to the same dimension, we can use the formulas in \cref{sec:posterior-variance} to visualize the marginal posterior distribution of that component. This decomposition into additive components provides a method of visualizing \gp{} models which disentangles the different types of structure in the model. The following section shows examples of such decomposition plots. In \cref{ch:description}, we will extend this model visualization method to include automatically generated English text explaining types of structure discovered. \section{Structure discovery in time series} \label{sec:time_series} To investigate our method's ability to discover structure, we ran the kernel search on several time-series. In the following example, the search was run to depth 10, using \kSE{}, \kRQ{}, \kLin{}, \kPer{} and \kWN{} as base kernels. \begin{figure}[ht!] \newcommand{\wmgd}{0.5\columnwidth} % width mauna decomp \newcommand{\hmgd}{3.51cm} % height mauna decomp \newcommand{\mdrd}{\grammarfiguresdir/decomposition/11-Feb-03-mauna2003-s} % mauna decomp results dir \newcommand{\mbm}{\hspace{-0.2cm}} % move back \newcommand{\maunadgfx}[1]{\mbm \includegraphics[width=\wmgd,height=\hmgd, clip, trim=0mm 0mm 0mm 6mm]{\mdrd/03-mauna2003-s_#1}} \begin{tabular}{cc} \multicolumn{2}{c}{ \begin{tabular}{c} Complete model: $\kLin \kerntimes \kSE \kernplus \kSE \kerntimes ( \kPer \kernplus \RQ ) \kernplus \kWN$ \\ \maunadgfx{all} \end{tabular} } \\\\ %\multicolumn{2}{c}{Decomposition} \\ Long-term trend: $\Lin \kerntimes \SE$ & Yearly periodic: $\SE \kerntimes \Per$ \\ \maunadgfx{1} & \maunadgfx{2_zoom} \\[1em] Medium-term deviation: $\SE \kerntimes \RQ$ & Noise: \kWN \\ \maunadgfx{3} & \maunadgfx{resid} \end{tabular} \caption[Decomposition of the Mauna Loa model] {\emph{First row:} The full posterior on the Mauna Loa dataset, after a search of depth 10. \emph{Subsequent rows:} The automatic decomposition of the time series. The model is a sum of long-term, yearly periodic, medium-term components, and residual noise, respectively. The yearly periodic component has been rescaled for clarity.} \label{fig:mauna_decomp} \end{figure} \subsection{Mauna Loa atmospheric CO$\mathbf{_{2}}$} \label{sec:extrapolation} First, our method analyzed records of carbon dioxide levels recorded at the Mauna Loa observatory~\citep{co2data}. Since this dataset was analyzed in detail by \citet[chapter 5]{rasmussen38gaussian}, we can compare the kernel chosen by our method to a kernel constructed by human experts. \Cref{fig:mauna_grow} shows the posterior mean and variance on this dataset as the search depth increases. While the data can be smoothly interpolated by a model with only a single base kernel, the extrapolations improve dramatically as the increased search depth allows more structure to be included. \Cref{fig:mauna_decomp} shows the final model chosen by our method together with its decomposition into additive components. The final model exhibits plausible extrapolation and interpretable components: a long-term trend, annual periodicity, and medium-term deviations. These components have similar structure to the kernel hand-constructed by \citet[chapter 5]{rasmussen38gaussian}: % \begin{align} \underwrite{\kSE}{\small long-term trend} \kernplus \underwrite{\kSE \kerntimes \kPer}{\small yearly periodic} \kernplus \underwrite{\kRQ}{\small medium-term irregularities} \kernplus \underwrite{\kSE \kernplus \kWN}{\small short-term noise} \end{align} We also plot the residuals modeled by a white noise ($\kWN$) component, showing that there is little obvious structure left in the data. More generally, some components capture slowly-changing structure while others capture quickly-varying structure, but often there is no hard distinction between ``signal'' components and ``noise'' components. \begin{figure}[ht!] \centering \newcommand{\wagd}{0.48\columnwidth} % width airline decomp \newcommand{\hagd}{4cm} % height airline decomp \newcommand{\mb}{\hspace{-0.2cm}} % move back %\newcommand{\ard}{../figures/decomposition/11-Feb-01-airline-s} % airline results dir %\newcommand{\ard}{\grammarfiguresdir/decomposition/31-Jan-v301-airline-months} % airline results dir % % Level 5 of % https://github.com/jamesrobertlloyd/gpss-research/blob/master/results/2014-01-14-GPSS-add/01-airline_result.txt %SumKernel(SqExp, Product[NoiseKernel LinearKernel], ProductKernel[SqExp, PeriodicKernel, LinearKernel, ProductKernel[SqExpKernel, LinearKernel])]) \newcommand{\ard}{\grammarfiguresdir/decomposition/19-Jan-2014-airline-months} % airline results dir \newcommand{\airdgfx}[1]{\mb \includegraphics[width=\wagd,height=\hagd, clip, trim=10mm 0mm 14mm 8mm]{\ard/#1} } % \begin{tabular}{cc} \multicolumn{2}{c}{ \begin{tabular}{c} Complete Model: $\kSE \kerntimes \Lin \kernplus \kPer \kerntimes \kLin \kerntimes \kSE \kernplus \kLin \kerntimes \kSE \kernplus \kWN \kerntimes \kLin$ \\ \airdgfx{01-airline_all} \\ %{\mb \includegraphics[width=\wagd,height=\hagd, clip, trim=4.5mm 60mm 77mm 37.3mm]{\grammarfiguresdir/airlinepages/pg_0002-crop} } \end{tabular} } \\ Long-term trend: $\kSE \kerntimes \kLin$ & Yearly periodic: $\kPer \kerntimes \Lin \kerntimes \kSE$ \\ %{\mb \includegraphics[width=\wagd,height=\hagd, clip, trim=4.5mm 60mm 77mm 37.3mm]{\grammarfiguresdir/airlinepages/pg_0003-crop} } & \airdgfx{01-airline_1_extrap} & \airdgfx{01-airline_2_extrap} \\ %{\mb \includegraphics[width=\wagd,height=\hagd, clip, trim=4.5mm 60mm 77mm 43mm]{\grammarfiguresdir/airlinepages/pg_0004-crop} } \\ Medium-term deviation: $\kSE$ & Growing noise: $\kWN \kerntimes \Lin$ \\ \airdgfx{01-airline_3_extrap} & %{\mb \includegraphics[width=\wagd,height=\hagd, clip, trim=5mm 60mm 77mm 18.5mm]{\grammarfiguresdir/airlinepages/pg_0005-crop} } & %\airdgfx{01-airline_4_extrap} {\mb \includegraphics[width=\wagd,height=\hagd, clip, trim=5mm 8.5mm 7.8cm 69.5mm]{\grammarfiguresdir/airlinepages/pg_0005-crop} } \end{tabular} \caption[Decomposition of airline dataset model] {\emph{First row:} The airline dataset and posterior after a search of depth 10. \emph{Subsequent rows:} Additive decomposition of posterior into long-term smooth trend, yearly variation, and short-term deviations. Due to the linear kernel, the marginal variance grows over time, making this a heteroskedastic model. } \label{fig:airline_decomp} \end{figure} \subsection{Airline passenger counts} \Cref{fig:airline_decomp} shows the decomposition produced by applying our method to monthly totals of international airline passengers~\citep{box2013time}. We observe similar components to those in the Mauna Loa dataset: a long term trend, annual periodicity, and medium-term deviations. In addition, the composite kernel captures the near-linearity of the long-term trend, and the linearly growing amplitude of the annual oscillations. % Created by the command: % postprocessing.make_all_1d_figures(folder='../results/31-Jan-1d/', prefix='31-Jan-v3', rescale=False) The model search can be run without modification on multi-dimensional datasets (as in \cref{sec:synthetic,sec:additive-experiments}), but the resulting structures are more difficult to visualize. \section{Related work} \label{sec:gpss-related-work} \def\rwsheader{\subsubsection} \rwsheader{Building kernel functions by hand} \citet[chapter 5]{rasmussen38gaussian} devoted 4 pages to manually constructing a composite kernel to model the Mauna Loa dataset. %In the supplementary material, we include a report automatically generated by \procedurename{} for this dataset; our procedure chose a model similar to the one they constructed by hand. Other examples of papers whose main contribution is to manually construct and fit a composite \gp{} kernel are \citet{textperiodic13}, \citet{lloydgefcom2012}, and \citet{EdgarTelescope}. These papers show that experts are capable of constructing kernels, in one or two dimensions, of similar complexity to the ones shown in this chapter. However, a more systematic search can consider possibilities that might otherwise be missed. For example, the kernel structure $\kSE \kerntimes \kPer \kerntimes \kLin$, while appropriate for the airline dataset, had never been considered by the authors before it was chosen by the automatic search. \rwsheader{Nonparametric regression in high dimensions} Nonparametric regression methods such as splines, locally-weighted regression, and \gp{} regression are capable of learning arbitrary smooth functions from data. Unfortunately, they suffer from the curse of dimensionality: it is sometimes difficult for these models to generalize well in more than a few dimensions. Applying nonparametric methods in high-dimensional spaces can require imposing additional structure on the model. One such structure is additivity. Generalized additive models~\citep{hastie1990generalized} assume the regression function is a transformed sum of functions defined on the individual dimensions: $\expect[f(\vx)] = g\inv(\sum_{d=1}^D f_d(x_d))$. These models have a restricted form, but one which is interpretable and often generalizes well. Models in our grammar can capture similar structure through sums of base kernels along different dimensions, although we have not yet tried incorporating a warping function $g(\cdot)$. It is possible to extend additive models by adding more flexible interaction terms between dimensions. \Cref{ch:additive} considers \gp{} models whose kernel implicitly sums over all possible interactions of input variables. \citet{plate1999accuracy} constructs a special case of this model class, summing an \kSE{} kernel along each dimension (for interpretability) plus a single \seard{} kernel over all dimensions (for flexibility). Both types of model can be expressed in our grammar. A closely related procedure is smoothing-splines \ANOVA{} \citep{wahba1990spline, gu2002smoothing}. This model is a weighted sum of splines along each input dimension, all pairs of dimensions, and possibly higher-dimensional combinations. Because the number of terms to consider grows exponentially with the number of dimensions included in each term, in practice, only one- and two-dimensional terms are usually considered. Semi-parametric regression \citep[e.g.][]{ruppert2003semiparametric} attempts to combine interpretability with flexibility by building a composite model out of an interpretable, parametric part (such as linear regression) and a ``catch-all'' nonparametric part (such as a \gp{} having an \kSE{} kernel). This model class can be represented through kernels such as ${\kSE \kernplus \kLin}$. \rwsheader{Kernel learning} There is a large body of work attempting to construct rich kernels through a weighted sum of base kernels, called multiple kernel learning (\MKL{}) \citep[e.g.][]{gonen2011multiple, bach2004multiple}. These approaches usually have a convex objective function. However the component kernels, as well as their parameters, must be specified in advance. We compare to a Bayesian variant of \MKL{} in \cref{sec:numerical}, expressed as a restriction of our language of kernels. \citet{salakhutdinov2008using} use a deep neural network with unsupervised pre-training to learn an embedding $g(\vx)$ onto which a \gp{} with an $\kSE$ kernel is placed: ${\Cov{[f(\vx), f(\vx')]} = k(g(\vx), g(\vx'))}$. This is a flexible approach to kernel learning, but relies mainly on finding structure in the input density $p(\vx)$. Instead, we focus on domains where most of the interesting structure is in $f(\vx)$. Sparse spectrum \gp{}s \citep{lazaro2010sparse} approximate the spectral density of a stationary kernel function using sums of Dirac delta functions, which corresponds to kernels of the form $\sum \cos$. Similarly, \citet{WilAda13} introduced spectral mixture kernels, which approximate the spectral density using a mixture of Gaussians, corresponding to kernels of the form $\sum \kSE \kerntimes \cos$. Both groups demonstrated, using Bochner's theorem \citep{bochner1959lectures}, that these kernels can approximate any stationary covariance function. Our language of kernels includes both of these kernel classes (see \cref{table:motifs}). \rwsheader{Changepoints} There is a wide body of work on changepoint modeling. \citet{adams2007bayesian} developed a Bayesian online changepoint detection method which segments time-series into independent parts. This approach was extended by \citet{saatcci2010gaussian} to Gaussian process models. \citet{garnett2010sequential} developed a family of kernels which modeled changepoints occurring abruptly at a single point. The changepoint kernel (\kCP) presented in this work is a straightforward extension to smooth changepoints. \rwsheader{Equation learning} \cite{todorovski1997declarative}, \cite{washio1999discovering} and \cite{Schmidt2009b} learned parametric forms of functions, specifying time series or relations between quantities. In contrast, \procedurename{} learns a parametric form for the covariance function, allowing it to model functions which do not have a simple parametric form but still have high-level structure. An examination of the structure discovered by the automatic equation-learning software Eureqa \citep{Eureqa} on the airline and Mauna Loa datasets can be found in \citet{LloDuvGroetal14}. %\Cref{sec:eqn-learning-comp}. \rwsheader{Structure discovery through grammars} \citet{kemp2008discovery} learned the structural form of graphs that modeled human similarity judgements. Their grammar on graph structures includes planes, trees, and cylinders. Some of their discrete graph structures have continuous analogues in our language of models. For example, $\SE_1 \kerntimes \SE_2$ and $\SE_1 \kerntimes \Per_2$ can be seen as mapping the data onto a Euclidean surface and a cylinder, respectively. \Cref{sec:topological-manifolds} examined these structures in more detail. \citet{diosan2007evolving} and \citet{bing2010gp} learned composite kernels for support vector machines and relevance vector machines, respectively, using genetic search algorithms to optimize cross-validation error. Similarly, \citet{kronberger2013evolution} searched over composite kernels for \gp{}s using genetic programming, optimizing the unpenalized marginal likelihood. These methods explore similar languages of kernels to the one explored in this chapter. %However, each of these methods suffer either from a model selection criterion which is not differentiable with respect to kernel parameters (cross-validation of classification accuracy), or from one which does not penalize complexity of the resulting kernel expressions. It is not clear whether the complex genetic searches used by these methods offer advantages over the straightforward but na\"{i}ve greedy search used in this chapter. %However, \citet{kronberger2013evolution} observed overfitting %Our work goes beyond ths prior work by demonstrating the structure implied by composite kernels, employing a Bayesian search criterion, and allowing for the automatic discovery of interpretable structure from data. Our search criterion has the advantages of being both differentiable with respect to kernel parameters, and of trading off model fit and complexity automatically. These related works also did not explore the automatic model decomposition, summarization and description made possible by the use of \gp{} models. % and goes beyond this prior work by demonstrating the interpretability of the structure implied by composite kernels, and how such structure allows for extrapolation. \citet{grosse2012exploiting} performed a greedy search over a compositional model class for unsupervised learning, using a grammar of matrix decomposition models, and a greedy search procedure based on held-out predictive likelihood. This model class contains many existing unsupervised models as special cases, and was able to discover diverse forms of structure, such as co-clustering or sparse latent feature models, automatically from data. Our framework takes a similar approach, but in a supervised setting. Similarly, \citet{christian-thesis} showed to automatically perform inference in arbitrary compositions of discrete sequence models. More generally, \citet{dechter2013bootstrap} and \citet{liang2010learning} constructed grammars over programs, and automatically searched the resulting spaces. \section{Experiments} \label{sec:numerical} \subsection{Interpretability versus accuracy} BIC trades off model fit and complexity by penalizing the number of parameters in a kernel expression. This can result in \procedurename{} favoring kernel expressions with nested products of sums, producing descriptions involving many additive components after expanding out all terms. While these models typically have good predictive performance, their large number of components can make them less interpretable. We experimented with not allowing parentheses during the search, discouraging nested expressions. This was done by distributing all products immediately after each search operator was applied. We call this procedure \procedurename{}-interpretability, in contrast to the unrestricted version of the search, \procedurename{}-accuracy. \subsection{Predictive accuracy on time series} \label{sec:Predictive accuracy on time series} %\subsubsection{Datasets} We evaluated the performance of the algorithms listed below on 13 real time-series from various domains from the time series data library \citep{TSDL}. The pre-processed datasets used in our experiments are available at \\\url{http://github.com/jamesrobertlloyd/gpss-research/tree/master/data/tsdlr} % plots of the data can be found at the beginning of the reports in the supplementary material. \subsubsection{Algorithms} We compare \procedurename{} to equation learning using Eureqa \citep{Eureqa}, as well as six other regression algorithms: linear regression, \gp{} regression with a single $\kSE$ kernel (squared exponential), a Bayesian variant of multiple kernel learning (\MKL{}) \citep[e.g.][]{gonen2011multiple, bach2004multiple}, changepoint modeling \citep[e.g.][]{garnett2010sequential, saatcci2010gaussian, FoxDunson:NIPS2012}, spectral mixture kernels \citep{WilAda13} (spectral kernels), and trend-cyclical-irregular models \citep[e.g.][]{lind2006basic}. We set Eureqa's search objective to the default mean-absolute-error. All algorithms besides Eureqa can be expressed as restrictions of our modeling language (see \cref{table:motifs}), so we performed inference using the same search and objective function, with appropriate restrictions to the language. %For \MKL{}, trend-cyclical-irregular, and spectral kernels, the greedy search procedure of \procedurename{} corresponds to a forward-selection algorithm. %For squared-exponential and linear regression, the procedure corresponds to standard marginal likelihood optimization. %More advanced inference methods are sometimes used for changepoint modeling, but we use the same inference method for all algorithms for comparability. We restricted our experiments to regression algorithms for comparability; we did not include models which regress on previous values of times series, such as auto-regressive or moving-average models \citep[e.g.][]{box2013time}. Constructing a language of autoregressive time-series models would be an interesting area for future research. \subsubsection{Extrapolation experiments} To test extrapolation, we trained all algorithms on the first 90\% of the data, predicted the remaining 10\% and then computed the root mean squared error (\RMSE{}). The \RMSE{}s were then standardised by dividing by the smallest \RMSE{} for each data set, so the best performance on each data set has a value of 1. \begin{figure}[h!] \includegraphics[width=1.02\textwidth]{\grammarfiguresdir/comparison/box_extrap_wide} \caption[Extrapolation error of all methods on 13 time-series datasets] {Box plot (showing median and quartiles) of standardised extrapolation \RMSE{} (best performance = 1) on 13 time-series. Methods are ordered by median. } \label{fig:box_extrap_dist} \end{figure} \Cref{fig:box_extrap_dist} shows the standardised \RMSE{}s across algorithms. \procedurename{}-accuracy usually outperformed \procedurename{}-interpretability. Both algorithms had lower quartiles than all other methods. Overall, the model construction methods having richer languages of models performed better: \procedurename{} outperformed trend-cyclical-irregular, which outperformed Bayesian \MKL{}, which outperformed squared-exponential. Despite searching over a rich model class, Eureqa performed relatively poorly. This may be because few datasets are parsimoniously explained by a parametric equation, or because of the limited regularization ability of this procedure. Not shown on the plot are large outliers for spectral kernels, Eureqa, squared exponential and linear regression with normalized \RMSE{}s of 11, 493, 22 and 29 respectively. %All of these outliers occurred on a data set with a large discontinuity (see the call centre data in the supplementary material). %\subsubsection{Interpolation} %To test the ability of the methods to interpolate, we randomly divided each data set into equal amounts of training data and testing data. %The results are similar to those for extrapolation, and are included in \cref{sec:interpolation-appendix}. %\fTBD{Josh: Can you write about the Blessing of abstraction, and doing lots with a small amount of data? This might become apparent from plotting the extraplolations.} %\fTBD{RBG: We could make this point as part of the learning curves for extrapolation by marking the points at which each additional aspect of the structure is found.} \subsection{Multi-dimensional prediction} \procedurename{} can be applied to multidimensional regression problems without modification. An experimental comparison with other methods can be found in \cref{sec:additive-experiments}, where it has the best performance on every dataset.%outperforms a wide variety of multidimensional regression methods. \subsection{Structure recovery on synthetic data} \label{sec:synthetic} The structure found in the examples above may seem reasonable, but we may wonder to what extent \procedurename{} is consistent -- that is, does it recover all the structure in any given dataset? It is difficult to tell from predictive accuracy alone if the search procedure is finding the correct structure, especially in multiple dimensions. To address this question, we tested our method's ability to recover known structure on a set of synthetic datasets. For several composite kernel expressions, we constructed synthetic data by first sampling 300 locations uniformly at random, then sampling function values at those locations from a \gp{} prior. We then added \iid Gaussian noise to the functions at various signal-to-noise ratios (\SNR{}). \begin{table}[ht!] \caption[Kernels chosen on synthetic data] { Kernels chosen by \procedurename{} on synthetic data generated using known kernel structures. $D$ denotes the dimension of the function being modeled. \SNR{} indicates the signal-to-noise ratio. Dashes (--) indicate no structure was found. Each kernel implicitly has a \kWN{} kernel added to it. } \label{tbl:synthetic} \addtolength{\tabcolsep}{1pt} \setlength\extrarowheight{2pt} \begin{center} {\small \begin{tabular}{c c | c c c} True kernel & $D$ & \SNR{} = 10 & \SNR{} = 1 & \hspace{-1cm} \SNR{} = 0.1 \\ \hline $\SE \kernplus \RQ$ & 1 & $\SE$ & $\SE \kerntimes \Per$ & $\SE$ \\ $\Lin \kerntimes \Per$ & 1 & $\Lin \kerntimes \Per$ & $\Lin \kerntimes \Per$ & $\SE$ \\ $\SE_1 \kernplus \RQ_2$ & 2 & $\SE_1 \kernplus \SE_2$ & $\Lin_1 \kernplus \SE_2$ & $\Lin_1$ \\ $\SE_1 \kernplus \SE_2 \kerntimes \Per_1 \kernplus \SE_3$ & 3 & $\SE_1 \kernplus \SE_2 \kerntimes \Per_1 \kernplus \SE_3$ & $\SE_2 \kerntimes \Per_1 \kernplus \SE_3$ & -- \\ $\SE_1 \kerntimes \SE_2$ & 4 & $\SE_1 \kerntimes \SE_2$ & $\Lin_1 \kerntimes \SE_2$ & $\Lin_2$ \\ $\SE_1 \kerntimes \SE_2 \kernplus \SE_2 \kerntimes \SE_3$ & 4 & $\SE_1 \kerntimes \SE_2 \kernplus \SE_2 \kerntimes \SE_3$ & $\SE_1 \kernplus \SE_2 \kerntimes \SE_3$ & $\SE_1$ \\ \multirow{2}{*}{ $(\SE_1 \kernplus \SE_2) \kerntimes (\SE_3 \kernplus \SE_4)$ } & \multirow{2}{*}{4} & $(\SE_1 \kernplus \SE_2) \, \kerntimes$ \dots & $(\SE_1 \kernplus \SE_2) \, \kerntimes$ \dots & \multirow{2}{*}{--} \\ & & $(\SE_3\kerntimes\Lin_3\kerntimes\Lin_1 \kernplus \SE_4)$ & $\SE_3 \kerntimes \SE_4$ & \end{tabular} } \end{center} \end{table} \Cref{tbl:synthetic} shows the results. For the highest signal-to-noise ratio, \procedurename{} usually recovers the correct structure. The reported additional linear structure in the last row can be explained the fact that functions sampled from \kSE{} kernels with long length-scales occasionally have near-linear trends. As the noise increases, our method generally backs off to simpler structures rather than reporting spurious structure. \subsubsection{Source code} All \gp{} parameter optimization was performed by automated calls to the \GPML{} toolbox~\citep{GPML}. %, available at \url{http://www.gaussianprocess.org/gpml/code}. Source code to perform all experiments is available at \url{http://www.github.com/jamesrobertlloyd/gp-structure-search}. \section{Conclusion} This chapter presented a system which constructs a model from an open-ended language, and automatically generates plots decomposing the different types of structure present in the model. This was done by introducing a space of kernels defined by sums and products of a small number of base kernels. The set of models in this space includes many standard regression models. We proposed a search procedure for this space of kernels, and argued that this search process parallels the process of model-building by statisticians. We found that the learned structures enable relatively accurate extrapolation in time-series datasets. %, and are competitive with widely used kernel classes on a variety of prediction tasks. The learned kernels can yield decompositions of a signal into diverse and interpretable components, enabling model-checking by humans. We hope that this procedure has the potential to make powerful statistical model-building techniques accessible to non-experts. Some discussion of the limitations of this approach to model-building can be found in \cref{sec:limitations-of-abcd}, and discussion of this approach relative to other model-building approaches can be found in \cref{top-down-vs-bottom-up}. The next chapter will show how the model components found by \procedurename{} can be automatically described using English-language text. \iffalse \section{Future Work} \subsubsection{Structure search in classification} While we focus on Gaussian process regression, the kernel search method could be applied to other supervised learning problems such as classification or ordinal regression. %, or to other kinds of kernel architectures such as kernel \SVM{}s. However, it might be more difficult to discover structure from class labels alone, since they usually contain less information about the underlying process than regression targets. %However, whether or not such modularity and interpretability is present in other model classes is an open question. \subsubsection{Improving the search} The search procedure used in this chapter leaves much room for improvement. Presumably, some sort of model-based search would be worthwhile, since evaluating each kernel expression is relatively expensive. This raises the question of how to build an open-ended model of the relative marginal likelihood of \gp{} models. In order to use a \gp{} for this task, one would need to create a kernel over kernels, as in \citet{ong2002hyperkernels}. One interesting possibility is that the model of model performance could generate its own data offline. The system could generate synthetic regression datasets, then score many kernel structures on those datasets in order to improve its model of kernel performance. %\subsubsection{Building languages of other model classes} %The compositionality of diverse types of kernels is what allowed a rich language of \gp{} models. %Can we build similar languages of other model classes? %As noted in \cref{sec:gpss-related-work}, languages have already been constructed on several powerful model classes: matrix decompositions, sequence models, graphical models, and functional programs. %Building languages of models that support efficient search as well as interpretable decompositions could be a fruitful direction. %\citet{roger-grosse-thesis} demonstrated a very rich class of unsupervised models by composing matrix decompositions. %\citet{kemp2008discovery} and \citet{ellislearning} built grammars on graphical models. \subsubsection{Use in model-based optimization} The automatic model-building procedure described in this chapter could also be used in the inner loop of \gp{}-based Bayesian optimization routines, such as \citet{snoek2012practical}. Discovering structure such as additivity in a problem could be expected to decrease the number of samples needed. \subsubsection{Incorporating uncertainty in the structure} \fi \outbpdocument{ \bibliographystyle{plainnat} \bibliography{references.bib} } ================================================ FILE: intro.tex ================================================ \input{common/header.tex} \inbpdocument \chapter{Introduction} \label{ch:intro} %\begin{quotation} %``I only work on intractable nonparametrics - Gaussian processes don't count.'' \\ %\hspace*{\fill} Sinead Williamson, personal communication %\end{quotation} \begin{quotation} ``All models are wrong, but yours are stupid too.'' \\ \hspace*{\fill} \citet{mlhipster} \end{quotation} Prediction, extrapolation, and induction are all examples of learning a function from data. There are many ways to learn functions, but one particularly elegant way is by probabilistic inference. Probabilistic inference takes a group of hypotheses (a \emph{model}), and weights those hypotheses based on how well their predictions match the data. This approach is appealing for two reasons. First, keeping all hypotheses that match the data helps to guard against over-fitting. Second, comparing how well a dataset is fit by different models gives a way of finding which sorts of structure are present in that data. % models having different types of structure is a way to find which sorts of structure are present in a dataset. This thesis focuses on constructing models of functions. %The types of structure examined in this thesis Chapter \ref{ch:kernels} describes how to model functions having many different types of structure, such as additivity, symmetry, periodicity, changepoints, or combinations of these, using Gaussian processes (\gp{}s). Chapters \ref{ch:grammar} and \ref{ch:description} show how such models can be automatically constructed from data, and then automatically described. Later chapters explore several extensions of these models. %will describe how to model functions having many different types of structure, such as additivity, symmetry, periodicity, changepoints, or combinations of these, using Gaussian processes (\gp{}s). %To be able to learn a wide variety of structures, we would like to have an expressive language of models of functions. %We would like to be able to represent simple kinds of functions, such as linear functions or polynomials. %We would also like to have models of arbitrarily complex functions, specified in terms of high-level properties such as how smooth they are, whether they repeat over time, or which symmetries they have. %Chapters 2 and 3 will % %All of these models of function can be constructed using Gaussian processes (\gp{}s).%, a tractable class of models of functions. % %This chapter will introduce the basic properties of \gp{}s. %Chapter \ref{ch:kernels} will describe how to model these different types of structure using \gp{}s. This short chapter introduces the basic properties of \gp{}s, and provides an outline of the thesis. %Chapter \ref{ch:grammar} will show how searching over many \section{Gaussian process models} Gaussian processes are a simple and general class of models of functions. % \begin{figure}[t] \begin{centering} \begin{tabular}{cc} \hspace{-3mm}\raisebox{2cm}{$f(x)$} \includegraphics[width=0.46\textwidth]{\introfigsdir/fuzzy-0} & \includegraphics[width=0.46\textwidth]{\introfigsdir/fuzzy-1} \\ \hspace{-3mm}\raisebox{2cm}{$f(x)$} \includegraphics[width=0.46\textwidth]{\introfigsdir/fuzzy-2} & \includegraphics[width=0.46\textwidth]{\introfigsdir/fuzzy-3} \\[-1mm] $x$ & $x$ \end{tabular} \end{centering} \caption[A one-dimensional Gaussian process posterior] {A visual representation of a Gaussian process modeling a one-dimensional function. Different shades of red correspond to deciles of the predictive density at each input location. Coloured lines show samples from the process -- examples of some of the hypotheses included in the model. \emph{Top left:} A \gp{} not conditioned on any datapoints. \emph{Remaining plots:} The posterior after conditioning on different amounts of data. All plots have the same axes. } \label{fig:gp-post} \end{figure} % %Figure \ref{fig:gp-post} shows a Gaussian process distribution, as it is conditioned on more and more observations. To be precise, a \gp{} is any distribution over functions such that any finite set of function values $f(\vx_1), f(\vx_2), \ldots f(\vx_N)$ have a joint Gaussian distribution \citep[chapter 2]{rasmussen38gaussian}. A \gp{} model, before conditioning on data, is completely specified by its mean function, % \begin{align} %\mathbb{E}(f(x)) = \mu(x) \expectargs{}{f(\vx)} = \mu(\vx) \end{align} % and its covariance function, also called the \emph{kernel}: % \begin{align} \textrm{Cov} \left[f(\vx), f(\vx') \right] = \kernel(\vx,\vx') \end{align} % It is common practice to assume that the mean function is simply zero everywhere, since uncertainty about the mean function can be taken into account by adding an extra term to the kernel. After accounting for the mean, the kind of structure that can be captured by a \gp{} model is entirely determined by its kernel. The kernel determines how the model generalizes, or extrapolates to new data. There are many possible choices of covariance function, and we can specify a wide range of models just by specifying the kernel of a \gp{}. For example, linear regression, splines, and Kalman filters are all examples of \gp{}s with particular kernels. However, these are just a few familiar examples out of a wide range of possibilities. %of the many possible models we can express through choosing a kernel. One of the main difficulties in using \gp{}s is constructing a kernel which represents the particular structure present in the data being modelled. \subsection{Model selection} The crucial property of \gp{}s that allows us to automatically construct models is that we can compute the \emph{marginal likelihood} of a dataset given a particular model, also known as the \emph{evidence} \citep{mackay1992bayesian}. The marginal likelihood allows one to compare models, balancing between the capacity of a model and its fit to the data~\citep{rasmussen2001occam,mackay2003information}. %discover the appropriate amount of detail to use, due to Bayesian Occam's razor % %Choosing a kernel, or kernel parameters, by maximizing the marginal likelihood will typically result in selecting the \emph{least} flexible model which still captures all the structure in the data. %For example, if a kernel has a parameter which controls the smoothness of the functions it models, the maximum-likelihood estimate of that parameter will usually correspond to the smoothest family of functions consistent with the observed function values. % The marginal likelihood under a \gp{} prior of a set of function values $\left[ f(\vx_1), f(\vx_2), \ldots f(\vx_N) \right] \defeq \vf(\vX)$ at locations $\vX$ is given by: % given by the rows of $\vX$: % \begin{align} p(\funcval | \vX, \mu(\cdot), k(\cdot, \cdot)) & = \N{\funcval}{\mu(\vX)}{k(\vX, \vX)} \label{eq:gp_marg_lik} \\ & = (2\pi)^{-\frac{N}{2}} \times \underwrite{ |k(\vX, \vX)|^{-\frac{1}{2}}\,}{\parbox{2.5cm}{\parbox{4cm}{\footnotesize controls model capacity}}} \nonumber \\ & \qquad \times \underwrite{\exp \left\{ -\frac{1}{2} \left( \funcval - \boldsymbol\mu(\vX) \right)\tra k(\vX, \vX)\inv \left(\funcval - \boldsymbol\mu(\vX) \right) \right\}}{\footnotesize encourages fit with data} \nonumber \end{align} % This multivariate Gaussian density is referred to as the \emph{marginal} likelihood because it implicitly integrates (marginalizes) over all possible functions values $\vf( \bar \vX)$, where $\bar \vX$ is the set of all locations where we have not observed the function. % Talk about the meaning of the different terms in the equation? \subsection{Prediction} %Even though we don't need to consider locations other than at the data when computing the marginal likelihood, We can ask the model which function values are likely to occur at any location, given the observations seen so far. By the formula for Gaussian conditionals (given in appendix~\ref{ch:appendix-gaussians}), the predictive distribution of a function value $f(\vx^\star)$ at a test point $\testpoint$ has the form: % \begin{align} p( f(\testpoint) | \funcval, \vX, \mu(\cdot), k(\cdot, \cdot)) = \mathcal{N} \big( f(\testpoint) \given & \underwrite{\mu(\testpoint) + k(\testpoint, \vX) k(\vX, \vX)\inv \left( \funcval - \mu(\vX) \right) }{\footnotesize predictive mean follows observations}, \nonumber \\ & \underwrite{k(\testpoint, \testpoint) - k(\testpoint, \vX) k(\vX, \vX)\inv k(\vX, \testpoint)}{\footnotesize predictive variance shrinks given more data} \big) \label{eq:predictive} \end{align} These expressions may look complex, but only require a few matrix operations to evaluate. Sampling a function from a \gp{} is also straightforward: a sample from a \gp{} at a finite set of locations is just a single sample from a single multivariate Gaussian distribution, given by \cref{eq:predictive}. \Cref{fig:gp-post} shows prior and posterior samples from a \gp{}, as well as contours of the predictive density. Our use of probabilities does not mean that we are assuming the function being learned is stochastic or random in any way; it is simply a consistent method of keeping track of uncertainty. \subsection{Useful properties of Gaussian processes} \label{sec:useful-properties} There are several reasons why \gp{}s in particular are well-suited for building a language of regression models: \begin{itemize} \item {\bf Analytic inference.} Given a kernel function and some observations, the predictive posterior distribution can be computed exactly in closed form. This is a rare property for nonparametric models to have. \item {\bf Expressivity.} Through the choice of covariance function, we can express a wide range of modeling assumptions. Some examples will be shown in \cref{ch:kernels}. \item {\bf Integration over hypotheses.} The fact that a \gp{} posterior, given a fixed kernel, lets us integrate exactly over a wide range of hypotheses means that overfitting is less of an issue than in comparable model classes. For example, compared to neural networks, relatively few parameters need to be estimated, which lessens the need for the complex optimization or regularization schemes. % %In contrast, much of the neural network literature is devoted to techniques for regularization and optimization. %TODO: add some citations. \item {\bf Model selection.} A side benefit of being able to integrate over all hypotheses is that we can compute the marginal likelihood of the data given a model. This gives us a principled way of comparing different models. \item {\bf Closed-form predictive distribution.} The predictive distribution of a \gp{} at a set of test points is simply a multivariate Gaussian distribution. This means that \gp{}s can easily be composed with other models or decision procedures. %For example, in reinforcement learning applications, \item {\bf Easy to analyze.} It may seem unsatisfying to restrict ourselves to a limited model class, as opposed to trying to do inference in the set of all computable functions. However, simple models can be used as well-understood building blocks for constructing more interesting models. %in diverse settings. For example, consider linear models. Although they form an extremely limited model class, they are simple, easy to analyze, and easy to incorporate into other models or procedures. Gaussian processes can be seen as an extension of linear models which retain these attractive properties~\citep[chapter 2]{rasmussen38gaussian}. %Linear models may seem like a hopelessly simple model class, but they're arguably the most useful modeling tools in existence. %\citet{rasmussen38gaussian} give a thorough introduction to \gp{}s, including many different types of analysis of the properties of this model. \end{itemize} \subsection{Limitations of Gaussian processes} There are several issues which make \gp{}s sometimes difficult to use: \begin{itemize} \item {\bf Slow inference.} Computing the matrix inverse in \cref{eq:gp_marg_lik,eq:predictive} takes $\mathcal{O}(N^3)$ time, making exact inference prohibitively slow for more than a few thousand datapoints. However, this problem can be addressed by approximate inference schemes~\citep{snelson2006sparse, quinonero2005unifying, hensman2013gaussian}. %Most \gp{} software packages implement several of these methods \item {\bf Light tails of the predictive distribution.} The predictive distribution of a standard \gp{} model is Gaussian. We may sometimes with to use non-Gaussian predictive likelihoods, for example in order to be robust to outliers, or to perform classification. Using non-Gaussian likelihoods requires approximate inference. Fortunately, mature software packages exist~\citep{GPy, GPML, VanRiiHarJylVeh14} which can automatically perform approximate inference for a wide variety of non-Gaussian likelihoods, and also implement sparse approximations. %\item {\bf Symmetric} %The predictive distribution of a \gp{} is always symmetric about its mean function. %This means that \gp{} models are inappropriate for modeling non-negative functions, such as likelihoods. %However, this problem can be addressed by simply exponentiating a \gp{} model, giving rise to a \emph{log-Gaussian process}. \item {\bf The need to choose a kernel.} The flexibility of \gp{} models raises the question of which kernel to use for a given problem. %means that we are also faced with the difficult task of choosing a kernel. Choosing a useful kernel is equivalent to learning a useful representation of the input. Kernel parameters can be set automatically by maximizing the marginal likelihood, but until recently, human experts were required to choose the parametric form of the kernel. %, usually from a small set of standard kernels. \Cref{ch:grammar} will show a way in which kernels can be automatically constructed for a given dataset. \end{itemize} \section{Outline and contributions of thesis} The main contribution of this thesis is to develop a method to automatically model, visualize, and describe a variety of statistical structures in data, by searching through an open-ended language of regression models. This thesis also includes a set of related results showing how Gaussian processes can be extended or composed with other models. %Furthermore, the fact that the marginal likelihood is available means that we can evaluate how much evidence the data provides for one structure over another, allowing an automatic search to construct models for us. Chapter~\ref{ch:kernels} is a tutorial showing how to build a wide variety of structured models of functions by constructing appropriate covariance functions. %TODO: Expand this description. We will also show how \gp{}s can produce nonparametric models of manifolds with diverse topological structures, such as cylinders, toruses and M\"obius strips. Chapter~\ref{ch:grammar} shows how to search over an open-ended language of models, built by adding and multiplying different kernels. Since we can evaluate each model by the marginal likelihood, we can automatically construct custom models for each dataset by a straightforward search procedure. We will show how the nature of \gp{}s allow the resulting models to be visualized by decomposing them into diverse, interpretable components, each capturing a different type of structure. Our experiments show that capturing such high-level structure sometimes allows one to extrapolate beyond the range of the data. One benefit of using a compositional model class is that the resulting models are relatively interpretable. Chapter~\ref{ch:description} demonstrates a system which automatically describes the structure implied by a given kernel on a given dataset, generating reports with graphs and English-language text describing the resulting model. %We will show several automatic analyses of time-series. Combined with the automatic model search developed in chapter~\ref{ch:grammar}, this system represents the beginnings of what could be called an ``automatic statistician'', capable of some aspects of model-building and explanation currently performed by experts. Chapter~\ref{ch:deep-limits} analyzes deep neural network models by characterizing the prior over functions obtained by composing \gp{} priors to form \emph{deep Gaussian processes}. %, and relates them to existing neep neural network architecures. We show that, as the number of layers increase, the amount of information retained about the original input diminishes to a single degree of freedom. A simple change to the network architecture fixes this pathology. We relate these models to neural networks, and as a side effect derive several forms of \emph{infinitely deep kernels}. Chapter~\ref{ch:additive} examines a more limited, but much faster way of discovering structure using \gp{}s. Specifying a kernel having many different types of structure, we use kernel parameters to discard whichever types of structure are \emph{not} found in the current dataset. The particular model class we examine is called \emph{additive Gaussian processes}, a model summing over exponentially-many \gp{}s, each depending on a different subset of the input variables. We give a polynomial-time inference algorithm for this model, and relate it to other model classes. For example, additive \gp{}s are shown to have the same covariance as a \gp{} that uses \emph{dropout}, a recently developed regularization technique for neural networks. Chapter~\ref{ch:warped} develops a Bayesian clustering model in which the clusters have nonparametric shapes, called the infinite warped mixture model. The density manifolds learned by this model follow the contours of the data density, and have interpretable, parametric forms in the latent space. The marginal likelihood lets us infer the effective dimension and shape of each cluster separately, as well as the number of clusters. \outbpdocument{ \bibliographystyle{plainnat} \bibliography{references.bib} } ================================================ FILE: kernels.tex ================================================ \input{common/header.tex} \inbpdocument \chapter{Expressing Structure with Kernels} \label{ch:kernels} This chapter shows how to use kernels to build models of functions with many different kinds of structure: additivity, symmetry, periodicity, interactions between variables, and changepoints. We also show several ways to encode group invariants into kernels. Combining a few simple kernels through addition and multiplication will give us a rich, open-ended language of models. The properties of kernels discussed in this chapter are mostly known in the literature. The original contribution of this chapter is to gather them into a coherent whole and to offer a tutorial showing the implications of different kernel choices, and some of the structures which can be obtained by combining them. %To the best of our knowledge, the different types of structure that could be obtained through sums and products of kernels had not been systematically explored before \citet{DuvLloGroetal13}. %TODO: Add more about contribution here. \section{Definition} %Since we'll be discussing kernels at length, we now give a precise definition. A kernel (also called a covariance function, kernel function, or covariance kernel), is a positive-definite function of two inputs $\vx, \vx'$. % in some space $\InputSpace$. %Formally, we write $\kernel(\vx, \vx'): \InputSpace \times \InputSpace \to \Reals$. In this chapter, $\vx$ and $\vx'$ are usually vectors in a Euclidean space, but kernels can also be defined on graphs, images, discrete or categorical inputs, or even text. Gaussian process models use a kernel to define the prior covariance between any two function values: % \begin{align} \textrm{Cov} \left[f(\vx), f(\vx') \right] = \kernel(\vx,\vx') \end{align} % Colloquially, kernels are often said to specify the similarity between two objects. This is slightly misleading in this context, since what is actually being specified is the similarity between two values of a \emph{function} evaluated on each object. The kernel specifies which functions are likely under the \gp{} prior, which in turn determines the generalization properties of the model. \section{A few basic kernels} \label{sec:basic-kernels} To begin understanding the types of structures expressible by \gp{}s, we will start by briefly examining the priors on functions encoded by some commonly used kernels: the squared-exponential (\kSE), periodic (\kPer), and linear (\kLin) kernels. These kernels are defined in \cref{fig:basic_kernels}. % \begin{figure}[h]% \centering \begin{tabular}{r|ccc} Kernel name: & Squared-exp (\kSE) & Periodic (\kPer) & Linear (\kLin) \\[10pt] $k(x, x') =$ & $\sigma_f^2 \exp\left(-\frac{(\inputVar - \inputVar')^2}{2\ell^2}\right)$ & $\sigma_f^2 \exp\left(-\frac{2}{\ell^2} \sin^2 \left( \pi \frac{\inputVar - \inputVar'}{p} \right)\right)$ & $\sigma_f^2 (\inputVar - c)(\inputVar' - c)$ \\[14pt] \raisebox{1cm}{Plot of $k(x,x')$:} & \kernpic{se_kernel} & \kernpic{per_kernel} & \kernpic{lin_kernel}\\ & $x -x'$ & $x -x'$ & \fixedx \\ %& & & \\ & \large $\downarrow$ & \large $\downarrow$ & \large $\downarrow$ \\ \raisebox{1cm}{\parbox{2.7cm}{\raggedleft Functions $f(x)$ sampled from \gp{} prior:}} & \kernpic{se_kernel_draws} & \kernpic{per_kernel_draws_s2} & \kernpic{lin_kernel_draws} \\ & $x$ & $x$ & $x$ \\ Type of structure: & local variation & repeating structure & linear functions \end{tabular} \vspace{6pt} \caption[Examples of structures expressible by some basic kernels] {Examples of structures expressible by some basic kernels. %Left and third columns: base kernels $k(\cdot,0)$. %Second and fourth columns: draws from a \sgp{} with each repective kernel. %The x-axis has the same range on all plots. } \label{fig:basic_kernels} \end{figure} % %There is nothing special about these kernels in particular, except that they represent a diverse set. Each covariance function corresponds to a different set of assumptions made about the function we wish to model. For example, using a squared-exp ($\kSE$) kernel implies that the function we are modeling has infinitely many derivatives. There exist many variants of ``local'' kernels similar to the $\kSE$ kernel, each encoding slightly different assumptions about the smoothness of the function being modeled. \paragraph{Kernel parameters} Each kernel has a number of parameters which specify the precise shape of the covariance function. These are sometimes referred to as \emph{hyper-parameters}, since they can be viewed as specifying a distribution over function parameters, instead of being parameters which specify a function directly. An example would be the lengthscale parameter $\ell$ of the $\kSE$ kernel, which specifies the width of the kernel and thereby the smoothness of the functions in the model. \paragraph{Stationary and Non-stationary} The $\kSE$ and $\kPer$ kernels are \emph{stationary}, meaning that their value only depends on the difference $x-x'$. This implies that the probability of observing a particular dataset remains the same even if we move all the $\vx$ values by the same amount. In contrast, the linear kernel ($\kLin$) is non-stationary, meaning that the corresponding \gp{} model will produce different predictions if the data were moved while the kernel parameters were kept fixed. \section{Combining kernels} What if the kind of structure we need is not expressed by any known kernel? For many types of structure, it is possible to build a ``made to order'' kernel with the desired properties. The next few sections of this chapter will explore ways in which kernels can be combined to create new ones with different properties. This will allow us to include as much high-level structure as necessary into our models. %For an overview, see \cite[Chapter~4]{rasmussen38gaussian}. \subsection{Notation} Below, we will focus on two ways of combining kernels: addition and multiplication. We will often write these operations in shorthand, without arguments: % \begin{align} k_a + k_b =& \,\, k_a(\vx, \vx') + k_b( \vx, \vx')\\ k_a \times k_b =& \,\, k_a(\vx, \vx') \times k_b(\vx, \vx') \end{align} All of the basic kernels we considered in \cref{sec:basic-kernels} are one-dimensional, but kernels over multi-dimensional inputs can be constructed by adding and multiplying between kernels on different dimensions. The dimension on which a kernel operates is denoted by a subscripted integer. For example, $\SE_2$ represents an \kSE{} kernel over the second dimension of vector $\vx$. To remove clutter, we will usually refer to kernels without specifying their parameters. \subsection{Combining properties through multiplication} %Multiplying kernels allows us to account for interactions between different input dimensions, or to combine different notions of similarity. %Loosely speaking, multip Multiplying two positive-definite kernels together always results in another positive-definite kernel. But what properties do these new kernels have? % \begin{figure} \centering \begin{tabular}{cccc} $\kLin \times \kLin$ & $\kSE \times \kPer$ & $\kLin \times \kSE$ & $\kLin \times \kPer$ \\ \kernpic{lin_times_lin} & \kernpic{longse_times_per} & \kernpic{se_times_lin} & \kernpic{lin_times_per}\\ \fixedx & $x -x'$ & \fixedx & \fixedx\\ %& & & \\ \large $\downarrow$ & \large $\downarrow$ & \large $\downarrow$ & \large $\downarrow$ \\ \kernpic{lin_times_lin_draws} & \kernpic{longse_times_per_draws_s2} & \kernpic{se_times_lin_draws_s2} & \kernpic{lin_times_per_draws_s2} \\ quadratic functions & locally \newline periodic & increasing variation & growing amplitude \\[10pt] \end{tabular} \caption[Examples of structures expressible by multiplying kernels] { Examples of one-dimensional structures expressible by multiplying kernels. %The x-axis has the same scale for all plots. Plots have same meaning as in figure \ref{fig:basic_kernels}.} \label{fig:kernels_times} \end{figure} % \Cref{fig:kernels_times} shows some kernels obtained by multiplying two basic kernels together. Working with kernels, rather than the parametric form of the function itself, allows us to express high-level properties of functions that do not necessarily have a simple parametric form. Here, we discuss a few examples: \begin{itemize} \item {\bf Polynomial Regression.} By multiplying together $T$ linear kernels, we obtain a prior on polynomials of degree $T$. %This class of functions also has a simple parametric form. The first column of \cref{fig:kernels_times} shows a quadratic kernel. \item {\bf Locally Periodic Functions.} In univariate data, multiplying a kernel by \kSE{} gives a way of converting global structure to local structure. For example, $\Per$ corresponds to exactly periodic structure, whereas $\Per \kerntimes \SE$ corresponds to locally periodic structure, as shown in the second column of \cref{fig:kernels_times}. \item {\bf Functions with Growing Amplitude.} Multiplying by a linear kernel means that the marginal standard deviation of the function being modeled grows linearly away from the location given by kernel parameter $c$. The third and fourth columns of \cref{fig:kernels_times} show two examples. \end{itemize} One can multiply any number of kernels together in this way to produce kernels combining several high-level properties. For example, the kernel $\kSE \kerntimes \kLin \kerntimes \kPer$ specifies a prior on functions which are locally periodic with linearly growing amplitude. We will see a real dataset having this kind of structure in \cref{ch:grammar}. \subsection{Building multi-dimensional models} %For instance, in multidimensional data, the multiplicative kernel $\SE_1 \kerntimes \SE_2$ represents a smoothly varying function of dimensions 1 and 2 which is not constrained to be additive. A flexible way to model functions having more than one input is to multiply together kernels defined on each individual input. For example, a product of $\kSE$ kernels over different dimensions, each having a different lengthscale parameter, is called the $\seard$ kernel: % \begin{align} \seard( \vx, \vx') = \prod_{d=1}^D \sigma_d^2 \exp \left( -\frac{1}{2} \frac{\left( x_d - x'_d \right)^2}{\ell_d^2} \right) = \sigma_f^2 \exp \left( -\frac{1}{2} \sum_{d=1}^D \frac{\left( x_d - x'_d \right)^2}{\ell_d^2} \right) \end{align} % \Cref{fig:product-of-se-kernels} illustrates the \seard{} kernel in two dimensions. % \begin{figure}[ht!] \centering \begin{tabular}{ccccccc} %$\kSE_1$ & \hspace{-0.4cm} $\kerntimes$ & $\kSE_2$ & $=$ & \hspace{-0.4cm} $\kSE_1 \kerntimes \kSE_2$ & & \hspace{-0.5cm} $f \sim \GPt{\vzero}{\kSE_1 \kerntimes \kSE_2}$ \\ \includegraphics[width=0.2\textwidth]{\additivefigsdir/2d-kernel/additive_kernel_sum_p1} \hspace{-0.3cm} & \hspace{-0.3cm} \raisebox{1cm}{$\times$} & \hspace{-0.3cm}\includegraphics[width=0.2\textwidth]{\additivefigsdir/2d-kernel/additive_kernel_sum_p2} \hspace{-0.3cm} & \hspace{-0.2cm} \raisebox{1cm}{=} \hspace{-0.2cm} & \hspace{-0.3cm} \includegraphics[width=0.2\textwidth]{\additivefigsdir/2d-kernel/sqexp_kernel} & \hspace{-0.3cm} \raisebox{1cm}{$\rightarrow$} \hspace{-0.3cm} & \hspace{-0.3cm} \includegraphics[width=0.2\textwidth]{\additivefigsdir/2d-kernel/sqexp_draw}\\ %$k_1(x_1, x_1')$ & & $k_2(x_2, x_2')$ & & \hspace{-0.4cm}\parbox{4cm}{$k_1(x_1,x_1') \kerntimes k_2(x_2,x_2')$} & & $f(x_1, x_2)$ \\[1em] $\kSE_1(x_1, x_1')$ & & $\kSE_2(x_2, x_2')$ & & \hspace{-0.4cm} $\kSE_1 \kerntimes \kSE_2$ & & \hspace{-0.5cm} \parbox{0.24\columnwidth}{$f(x_1, x_2)$ drawn from $\GPt{0}{\kSE_1 \kerntimes \kSE_2}$} \\ \end{tabular} \caption[A product of squared-exponential kernels across different dimensions]{ %\emph{Top left:} An additive kernel is a sum of kernels. %\emph{Bottom left:} A draw from an additive kernel corresponds to a sum of draws from independent \gp{} priors with the corresponding kernels. %\emph{Top right:} A product kernel. %\emph{Right:} A \gp{} prior with a product of kernels does not correspond to a product of draws from \gp{}s. %In this example, both kernels are composed of one dimensional squared-exponential kernels, but this need not be the case in general. A product of two one-dimensional kernels gives rise to a prior on functions which depend on both dimensions. } \label{fig:product-of-se-kernels} \end{figure} \ARD{} stands for automatic relevance determination, so named because estimating the lengthscale parameters $\ell_1, \ell_2, \dots, \ell_D$, implicitly determines the ``relevance'' of each dimension. Input dimensions with relatively large lengthscales imply relatively little variation along those dimensions in the function being modeled. $\seard$ kernels are the default kernel in most applications of \gp{}s. This may be partly because they have relatively few parameters to estimate, and because those parameters are relatively interpretable. In addition, there is a theoretical reason to use them: they are \emph{universal} kernels~\citep{micchelli2006universal}, capable of learning any continuous function given enough data, under some conditions. However, this flexibility means that they can sometimes be relatively slow to learn, due to the \emph{curse of dimensionality} \citep{bellman1956dynamic}. In general, the more structure we account for, the less data we need - the \emph{blessing of abstraction} \citep{goodman2011learning} counters the curse of dimensionality. Below, we will investigate ways to encode more structure into kernels. \section{Modeling sums of functions} \label{sec:modeling-sums-of-functions} An additive function is one which can be expressed as $f(\vx) = f_a(\vx) + f_b(\vx)$. Additivity is a useful modeling assumption in a wide variety of contexts, especially if it allows us to make strong assumptions about the individual components which make up the sum. Restricting the flexibility of component functions often aids in building interpretable models, and sometimes enables extrapolation in high dimensions. %TODO: make the se_long + se_short figure a bit more obvious \begin{figure}[h] \centering \begin{tabular}{cccc} %Composite & Draws from \gp{} & \gp{} posterior \\ \toprule $\kLin + \kPer$ & $\kSE + \kPer$ & $\kSE + \kLin$ & $\kSE^{(\textnormal{long})} + \kSE^{(\textnormal{short})}$ \\ \kernpic{lin_plus_per} & \kernpic{se_plus_per} & \kernpic{se_plus_lin} & \kernpic{shortse_plus_medse}\\ \fixedx & $x -x'$ & \fixedx & $x -x'$\\ %& & & \\ \large $\downarrow$ & \large $\downarrow$ & \large $\downarrow$ & \large $\downarrow$ \\ \kernpic{lin_plus_per_draws} & \kernpic{se_plus_per_draws_s7} & \kernpic{se_plus_lin_draws_s5} & \kernpic{shortse_plus_medse_draws_s5}\\ periodic plus trend & periodic plus noise & linear plus variation & slow \& fast variation \\[10pt] \end{tabular} \caption[Examples of structures expressible by adding kernels] { Examples of one-dimensional structures expressible by adding kernels. %The x-axis has the same scale for all plots. Rows have the same meaning as in \cref{fig:basic_kernels}. $\kSE^{(\textnormal{long})}$ denotes a $\kSE$ kernel whose lengthscale is long relative to that of $\kSE^{(\textnormal{short})}$ } \label{fig:kernels_plus} \end{figure} It is easy to encode additivity into \gp{} models. Suppose functions ${\function_a, \function_b}$ are drawn independently from \gp{} priors: % \begin{align} \function_a & \dist \GP(\mu_a, \kernel_a) \\ \function_b & \dist \GP(\mu_b, \kernel_b) \end{align} % Then the distribution of the sum of those functions is simply another \gp{}: % \begin{align} %\function := \function_a + \function_b \dist \GP(\mu_a + \mu_b, \kernel_a + \kernel_b). \end{align} % Kernels $\kernel_a$ and $\kernel_b$ can be of different types, allowing us to model the data as a sum of independent functions, each possibly representing a different type of structure. Any number of components can be summed this way. \subsection{Modeling noise} %We sometimes refer to a part of the signal that we don't care about, or don't expect to be able to extrapolate, as the noise. Additive noise can be modeled as an unknown, quickly-varying function added to the signal. This structure can be incorporated into a \gp{} model by adding a local kernel such as an \kSE{} with a short lengthscale, as in the fourth column of \cref{fig:kernels_plus}. The limit of the $\kSE$ kernel as its lengthscale goes to zero is a ``white noise'' (\kWN) kernel. Function values drawn from a \gp{} with a $\kWN$ kernel are independent draws from a Gaussian random variable. Given a kernel containing both signal and noise components, we may wish to isolate only the signal components. \Cref{sec:posterior-variance} shows how to decompose a \gp{} posterior into each of its additive components. In practice, there may not be a clear distinction between signal and noise. For example, \cref{sec:time_series} contains examples of models having long-term, medium-term, and short-term trends. Which parts we designate as the ``signal'' sometimes depends on the task at hand. \subsection{Additivity across multiple dimensions} \label{sec:additivity-multiple-dimensions} When modeling functions of multiple dimensions, summing kernels can give rise to additive structure across different dimensions. To be more precise, if the kernels being added together are each functions of only a subset of input dimensions, then the implied prior over functions decomposes in the same way. % \begin{figure} \centering \begin{tabular}{ccccc} \hspace{-0.2cm}\includegraphics[width=0.2\textwidth]{\additivefigsdir/2d-kernel/additive_kernel_sum_p2} & \hspace{-0.4cm} \raisebox{1cm}{+} \hspace{-0.4cm} & \includegraphics[width=0.2\textwidth]{\additivefigsdir/2d-kernel/additive_kernel_sum_p1} & \hspace{-0.4cm} \raisebox{1cm}{=} \hspace{-0.4cm} & \includegraphics[width=0.2\textwidth]{\additivefigsdir/2d-kernel/additive_kernel} \\ $k_1(x_1, x_1')$ & & $k_2(x_2, x_2')$ & & $k_1(x_1,x_1') + k_2(x_2,x_2')$ \\[1em] \large $\downarrow$ & & \large $\downarrow$ & & \large $\downarrow$ \\[-0.2em] \hspace{-0.2cm}\includegraphics[width=0.2\textwidth]{\additivefigsdir/2d-kernel/additive_kernel_draw_sum_p1} & \hspace{-0.4cm} \raisebox{1cm}{+} \hspace{-0.4cm} & \includegraphics[width=0.2\textwidth]{\additivefigsdir/2d-kernel/additive_kernel_draw_sum_p2} & \hspace{-0.4cm} \raisebox{1cm}{=} \hspace{-0.4cm} & \includegraphics[width=0.2\textwidth]{\additivefigsdir/2d-kernel/additive_kernel_draw_sum} \\ $f_1(x_1) \sim \GP\left(0, k_1\right)$ & & $f_2(x_2) \sim \GP\left(0, k_2\right)$ & & $f_1(x_1) + f_2(x_2)$ \\[1em] \end{tabular} \caption[Additive kernels correspond to additive functions]{ A sum of two orthogonal one-dimensional kernels. \emph{Top row:} An additive kernel is a sum of kernels. \emph{Bottom row:} A draw from an additive kernel corresponds to a sum of draws from independent \gp{} priors, each having the corresponding kernel. } \label{fig:sum-of-kernels} \end{figure} % For example, % \begin{align} %\function := f(x_1, x_2) \dist \GP(0, \kernel_1(x_1, x_1') + \kernel_2(x_2, x_2')) \label{eq:simple-additive-gp} \end{align} % is equivalent to the model % \begin{align} \function_1(x_1) & \dist \GP(0, \kernel_1(x_1, x_1'))\\ \function_2(x_2) & \dist \GP(0, \kernel_2(x_2, x_2'))\\ f(x_1, x_2) & = f_1(x_1) + f_2(x_2) \, . \end{align} % %Then we can model the sum of those functions through another \gp{}: % \Cref{fig:sum-of-kernels} illustrates a decomposition of this form. Note that the product of two kernels does not have an analogous interpretation as the product of two functions. \subsection{Extrapolation through additivity} \label{sec:additivity-extrapolation} Additive structure sometimes allows us to make predictions far from the training data. % \begin{figure} \centering \begin{tabular}{ccc} & \gp{} mean using & \gp{} mean using \\ True function: & sum of $\kSE$ kernels: & product of $\kSE$ kernels: \\ \parbox{0.33\columnwidth}{$f(x_1, x_2) = \sin(x_1) + \sin(x_2)$} & $k_1(x_1, x_1') \kernplus k_2(x_2, x_2')$ & $k_1(x_1, x_1') \kerntimes k_2(x_2, x_2')$ \\[0.2em] \hspace{-0.1in}\includegraphics[width=0.32\textwidth]{\additivefigsdir/1st_order_censored_truth} & \hspace{-0.1in}\includegraphics[width=0.32\textwidth]{\additivefigsdir/1st_order_censored_add} & \hspace{-0.1in}\includegraphics[width=0.32\textwidth]{\additivefigsdir/1st_order_censored_ard}\\[0em] %,clip,trim=16mm 17mm 0mm 0mm \end{tabular} \caption[Extrapolation in functions with additive structure] {\emph{Left:} A function with additive structure. %Inference in functions with additive structure. \emph{Center:} A \gp{} with an additive kernel can extrapolate away from the training data. \emph{Right:} A \gp{} with a product kernel allows a different function value for every combination of inputs, and so is uncertain about function values away from the training data. This causes the predictions to revert to the mean. % The additive GP is able to discover the additive pattern, and use it to fill in a distant mode. The ARD kernel can only interpolate, and thus predicts the mean in locations missing data. } \label{fig:synth2d} \end{figure} % \Cref{fig:synth2d} compares the extrapolations made by additive versus product-kernel \gp{} models, conditioned on data from a sum of two axis-aligned sine functions. The training points were evaluated in a small, L-shaped area. In this example, the additive model is able to correctly predict the height of the function at an unseen combinations of inputs. The product-kernel model is more flexible, and so remains uncertain about the function away from the data. %The ability of additive GPs to discover long-range structure suggests that this model may be well-suited to deal with covariate-shift problems. These types of additive models have been well-explored in the statistics literature. For example, generalized additive models~\citep{hastie1990generalized} have seen wide adoption. In high dimensions, we can also consider sums of functions of multiple input dimensions. \Cref{ch:additive} considers this model class in more detail. \subsection{Example: An additive model of concrete strength} \label{sec:concrete} To illustrate how additive kernels give rise to interpretable models, we built an additive model of the strength of concrete as a function of the amount of seven different ingredients (cement, slag, fly ash, water, plasticizer, coarse aggregate and fine aggregate), and the age of the concrete \citep{yeh1998modeling}. %We model measurements of the compressive strength of concrete, as a function of the concentration of 7 ingredients, plus the age of the concrete. Our simple model is a sum of 8 different one-dimensional functions, each depending on only one of these quantities: % \begin{align} f(\vx) & = f_1(\textnormal{cement}) + f_2(\textnormal{slag}) + f_3(\textnormal{fly ash}) + f_4(\textnormal{water}) \nonumber \\ & \quad + f_5(\textnormal{plasticizer}) + f_6(\textnormal{coarse}) + f_7(\textnormal{fine}) + f_8(\textnormal{age}) + \textnormal{noise} \label{eq:concrete} \end{align} % where $\textnormal{noise} \simiid \Nt{0}{\sigma^2_n}$. %The corresponding kernel has 9 additive components. Each of the functions $f_1, f_2, \ldots, f_8$ was modeled using a \gp{} with an $\kSE$ kernel. These eight $\kSE$ kernels plus a white noise kernel were added together as in \cref{eq:simple-additive-gp} to form a single \gp{} model whose kernel had 9 additive components. After learning the kernel parameters by maximizing the marginal likelihood of the data, one can visualize the predictive distribution of each component of the model. \newcommand{\concretepic}[1]{\includegraphics[width=0.3\columnwidth]{\decompfigsdir/concrete-component-#1}} \newcommand{\concretelegend}[0]{\raisebox{5mm}{\includegraphics[trim=56mm 1mm 2.1mm 34mm, clip, width=0.24\columnwidth]{\decompfigsdir/concrete-component-8-legend}}} % \begin{figure}[h!] \centering \begin{tabular}{cccc} %\includegraphics[width=0.3\textwidth]{\additivefigsdir/interpretable_1st_order1.pdf} & %\includegraphics[width=0.3\textwidth]{\additivefigsdir/interpretable_1st_order2.pdf}& \hspace{-6mm}\raisebox{0.7cm}{\rotatebox{90}{strength}}\hspace{-7mm} & \concretepic{1} & \concretepic{2} & \concretepic{3} \\ & cement (kg/m$^3$) & slag (kg/m$^3$) & fly ash (kg/m$^3$)\\[4mm] \hspace{-6mm}\raisebox{0.7cm}{\rotatebox{90}{strength}}\hspace{-7mm} & \concretepic{4} & \concretepic{5} & \concretepic{6} \\ & water (kg/m$^3$) & plasticizer (kg/m$^3$) & coarse (kg/m$^3$) \\[4mm] \hspace{-6mm}\raisebox{0.7cm}{\rotatebox{90}{strength}}\hspace{-7mm} & \concretepic{7} & \concretepic{8} & \concretelegend \\ & fine (kg/m$^3$) & age (days) \\ \end{tabular} \caption[Decomposition of posterior into interpretable one-dimensional functions] {The predictive distribution of each one-dimensional function in a multi-dimensional additive model. Blue crosses indicate the original data projected on to each dimension, red indicates the marginal posterior density of each function, and colored lines are samples from the marginal posterior distribution of each one-dimensional function. The vertical axis is the same for all plots. %The black line is the posterior mean of a \sgp{} with only one term in its kernel. %Right: the posterior mean of a \sgp{} with only one second-order term in its kernel. %rs = 1 - var(complete_mean - y)/ var(y) %R-squared = 0.9094 } \label{fig:interpretable functions} \end{figure} \Cref{fig:interpretable functions} shows the marginal posterior distribution of each of the eight one-dimensional functions in the model. %\cref{eq:concrete}. The parameters controlling the variance of two of the functions, $f_6(\textnormal{coarse})$ and $f_7(\textnormal{fine})$ were set to zero, meaning that the marginal likelihood preferred a parsimonious model which did not depend on these inputs. This is an example of the automatic sparsity that arises by maximizing marginal likelihood in \gp{} models, and is another example of automatic relevance determination (\ARD) \citep{neal1995bayesian}. The ability to learn kernel parameters in this way is much more difficult when using non-probabilistic methods such as Support Vector Machines \citep{cortes1995support}, for which cross-validation is often the best method to select kernel parameters. \subsection{Posterior variance of additive components} \label{sec:posterior-variance} %\footnote{Code is available at \url{github.com/duvenaud/phd-thesis/}} Here we derive the posterior variance and covariance of all of the additive components of a \gp{}. These formulas allow one to make plots such as \cref{fig:interpretable functions}. %These formulas let us plot the marginal variance of each component separately. These formulas can also be used to examine the posterior covariance between pairs of components. First, we write down the joint prior distribution over two functions drawn independently from \gp{} priors, and their sum. We distinguish between $\vecf(\vX)$ (the function values at training locations $[\vx_1, \vx_2, \ldots, \vx_N]\tra \defeq \vX$) and $\vecf(\vX^\star)$ (the function values at some set of query locations $[\vx_1^\star, \vx_2^\star, \ldots, \vx_N^\star]\tra \defeq \vX^\star$). % so that it's clear which matrices to use to extrapolate. Formally, if $f_1$ and $f_2$ are \emph{a priori} independent, and $f_1 \sim \gp( \mu_1, k_1)$ and ${f_2 \sim \gp( \mu_2, k_2)}$, then % \begin{align} \left[ \! \begin{array}{l} \vf_1(\vX) \\ \vf_1(\vX^\star) \\ \vf_2(\vX) \\ \vf_2(\vX^\star) \\ \vf_1(\vX) + \vf_2(\vX) \\ \vf_1(\vX^\star) + \vf_2(\vX^\star) \end{array} \! \right] \sim \Nt{ \left[ \! \begin{array}{c} \vmu_1 \\ \vmu_1^\star \\ \vmu_2 \\ \vmu_2^\star \\ \vmu_1 + \vmu_2 \\ \vmu_1^\star + \vmu_2^\star \end{array} \! \right] \! } {\! \left[ \! \begin{array}{llllll}%{cccccc} \vK_1 & \vK_1^\star & 0 & 0 & \vK_1 & \vK_1^\star \\ {\vK_1^\star}\tra & \vK_1^{\star\star} & 0 & 0 & \vK_1^\star & \vK_1^{\star\star} \\ 0 & 0 & \vK_2 & {\vK_2^\star} & \vK_2 & \vK_2^\star \\ 0 & 0 & {\vK_2^\star}\tra & \vK_2^{\star\star} & \vK_2^\star & \vK_2^{\star\star} \\ {\vK_1} & {\vK_1^\star}\tra & \vK_2 & {\vK_2^\star}\tra & \vK_1 + \vK_2 & \vK_1^\star + \vK_2^\star \\ {\vK_1^\star}\tra & \vK_1^{\star\star} & {\vK_2^\star}\tra & \vK_2^{\star\star} & {\vK_1^\star}\tra + {\vK_2^\star}\tra & \vK_1^{\star\star} + \vK_2^{\star\star}\\ \end{array} \! \right] } \end{align} % where we represent the Gram matrices, whose $i,j$th entry is given by $k(\vx_i, \vx_j)$ by % evaluated at all pairs of vectors in bold capitals as ${\vK_{i,j} = k(\vx_i, \vx_j)}$. % \begin{align} \vK_i & = k_i( \vX, \vX ) \\ \vK_i^\star & = k_i( \vX, \vX^\star ) \\ \vK_i^{\star\star} & = k_i( \vX^\star, \vX^\star ) \end{align} The formula for Gaussian conditionals \ref{eq:gauss_conditional} can be used to give the conditional distribution of a \gp{}-distributed function conditioned on its sum with another \gp{}-distributed function: % \begin{align} \vf_1(\vX^\star) \big| \vf_1(\vX) + \vf_2(\vX) \sim \Normal \Big( & \vmu_1^\star + \vK_1^{\star\tra} (\vK_1 + \vK_2)\inv \big[ \vf_1(\vX) + \vf_2(\vX) - \vmu_1 - \vmu_2 \big], \nonumber \\ & \vK_1^{\star\star} - \vK_1^{\star\tra} (\vK_1 + \vK_2)\inv \vK_1^{\star} \Big) %\vf_1(\vx^\star) | \vf(\vx) \sim \mathcal{N}\big( %& \vmu_1(\vx^\star) + \vk_1(\vx^{\star}, \vx) \left[ \vK_1(\vx, \vx) + \vK_2(\vx, \vx) \right]\inv \left( \vf(\vx) - \vmu_1(\vx) - \vmu_2(\vx) \right) , \nonumber \\ %& \vk_1(\vx^{\star}, \vx^{\star}) - \vk_1(\vx^{\star}, \vx) \left[ \vK_1(\vx, \vx) + \vK_2(\vx, \vx) \right] \inv \vk_1(\vx, \vx^{\star}) %\big). \end{align} % These formulas express the model's posterior uncertainty about the different components of the signal, integrating over the possible configurations of the other components. To extend these formulas to a sum of more than two functions, the term $\vK_1 + \vK_2$ can simply be replaced by $\sum_i \vK_i$ everywhere. \subsubsection{Posterior covariance of additive components} %One of the advantages of using a generative, model-based approach is that we can examine any aspect of the model we wish to. One can also compute the posterior covariance between the height of any two functions, conditioned on their sum: % \begin{align} \Cov \big[ \vf_1(\vX^\star), \vf_2(\vX^\star) \big| \vf(\vX) \big] & = - \vK_1^{\star\tra} (\vK_1 + \vK_2)\inv \vK_2^\star %\covargs{\vf_1(\vx^\star)}{\vf_2(\vx^\star) | \vf(\vx) } %& = - \vk_1(\vx^{\star}, \vx) \left[ \vK_1(\vx, \vx) + \vK_2(\vx, \vx) \right] \inv \vk_1(\vx, \vx^{\star}) \label{eq:post-component-cov} \end{align} % If this quantity is negative, it means that there is ambiguity about which of the two functions is high or low at that location. \begin{figure} \centering \renewcommand{\tabcolsep}{1mm} \def\incpic#1{\includegraphics[width=0.11\columnwidth]{../figures/decomp/concrete-#1}} \begin{tabular}{p{2mm}*{6}{c}} & {cement} & {slag} & {fly ash} & {water} & \parbox{0.1\columnwidth}{plasticizer} & {age} \\ \rotatebox{90}{$\;\;$ cement} & \incpic{Cement-Cement} & \incpic{Cement-Slag} & \incpic{Cement-Fly-Ash} & \incpic{Cement-Water} & \incpic{Cement-Plasticizer} & \incpic{Cement-Age} \\ \rotatebox{90}{$\;\;\;\;$ slag} & \incpic{Slag-Cement} & \incpic{Slag-Slag} & \incpic{Slag-Fly-Ash} & \incpic{Slag-Water} & \incpic{Slag-Plasticizer} & \incpic{Slag-Age} \\ \rotatebox{90}{{$\;\;$fly ash}} & \incpic{Fly-Ash-Cement} & \incpic{Fly-Ash-Slag} & \incpic{Fly-Ash-Fly-Ash} & \incpic{Fly-Ash-Water} & \incpic{Fly-Ash-Plasticizer} & \incpic{Fly-Ash-Age} \\ \rotatebox{90}{{$\quad$water}} & \incpic{Water-Cement} & \incpic{Water-Slag} & \incpic{Water-Fly-Ash} & \incpic{Water-Water} & \incpic{Water-Plasticizer} & \incpic{Water-Age} \\ \rotatebox{90}{{plasticizer}} & \incpic{Plasticizer-Cement} & \incpic{Plasticizer-Slag} & \incpic{Plasticizer-Fly-Ash} & \incpic{Plasticizer-Water} & \incpic{Plasticizer-Plasticizer} & \incpic{Plasticizer-Age}\\ \rotatebox{90}{$\;\;\;$ \phantom{t}age} & \incpic{Age-Cement} & \incpic{Age-Slag} & \incpic{Age-Fly-Ash} & \incpic{Age-Water} & \incpic{Age-Plasticizer} & \incpic{Age-Age} \\ \end{tabular} \fbox{ \begin{tabular}{c} Correlation \\[1ex] \includegraphics[width=0.1\columnwidth, clip, trim=6.2cm 0cm 0cm 0cm]{../figures/decomp/concrete-colorbar} \end{tabular} } \caption[Visualizing posterior correlations between components] {Posterior correlations between the heights of different one-dimensional functions in \cref{eq:concrete}, whose sum models concrete strength. %Each plot shows the posterior correlations between the height of two functions, evaluated across the range of the data upon which they depend. %Color indicates the amount of correlation between the function value of the two components. Red indicates high correlation, teal indicates no correlation, and blue indicates negative correlation. Plots on the diagonal show posterior correlations between different evaluations of the same function. Correlations are evaluated over the same input ranges as in \cref{fig:interpretable functions}. %Off-diagonal plots show posterior covariance between each pair of functions, as a function of both inputs. %Negative correlation means that one function is high and the other low, but which one is uncertain. Correlations with $f_6(\textnormal{coarse})$ and $f_7(\textnormal{fine})$ are not shown, because their estimated variance was zero. } \label{fig:interpretable interactions} \end{figure} % For example, \cref{fig:interpretable interactions} shows the posterior correlation between all non-zero components of the concrete model. This figure shows that most of the correlation occurs within components, but there is also negative correlation between the height of $f_1(\textnormal{cement})$ and $f_2(\textnormal{slag})$. %This reflects an ambiguity in the model about which one of these functions is high and the other low. \section{Changepoints} \label{sec:changepoint-definition} An example of how combining kernels can give rise to more structured priors is given by changepoint kernels, which can express a change between different types of structure. Changepoints kernels can be defined through addition and multiplication with sigmoidal functions such as $\sigma(x) = \nicefrac{1}{1 + \exp(-x)}$: % \begin{align} %\kCP(\kernel_1, \kernel_2) = \kernel_1 \kerntimes \boldsymbol\sigma \kernplus \kernel_2 \kerntimes \boldsymbol{\bar\sigma} %\kCP(\kernel_1, \kernel_2) & = \begin{array}{rl} (1-\sigma(x)) & k_2(x,x')(1-\sigma(x')) \\ % + \sigma(x) & k_1(x,x')\sigma(x') \end{array} \\ \kCP(\kernel_1, \kernel_2)(x,x') & = \sigma(x) k_1(x,x')\sigma(x') + (1-\sigma(x)) k_2(x,x')(1-\sigma(x')) \label{eq:cp} \end{align} % which can be written in shorthand as % \begin{align} \kCP(\kernel_1, \kernel_2)& = \kernel_1 \kerntimes \boldsymbol\sigma \kernplus \kernel_2 \kerntimes \boldsymbol{\bar\sigma} \end{align} where $\boldsymbol\sigma = \sigma(x)\sigma(x')$ and $\boldsymbol{\bar\sigma} = (1-\sigma(x))(1-\sigma(x'))$. This compound kernel expresses a change from one kernel to another. The parameters of the sigmoid determine where, and how rapidly, this change occurs. Figure~\ref{fig:changepoint_examples} shows some examples. % \newcommand{\cppic}[1]{\includegraphics[height=2.2cm,width=3.3cm]{\grammarfiguresdir/changepoint/#1}}% \begin{figure}[h] \centering \begin{tabular}{rcccc} & $\kCP(\kSE, \kPer)$ & $\kCP(\kSE, \kPer)$ & $\kCP(\kSE, \kSE)$ & $\kCP(\kPer, \kPer)$ \\ \raisebox{1cm}{$f(x)$} \hspace{-0.4cm} & \cppic{draw_1} & \cppic{draw_2} & \cppic{draw_3} & \cppic{draw_4} \\ & $x$ & $x$ & $x$ & $x$ \end{tabular} \caption[Draws from changepoint priors] {Draws from different priors on using changepoint kernels, constructed by adding and multiplying together base kernels with sigmoidal functions. } \label{fig:changepoint_examples} \end{figure} We can also build a model of functions whose structure changes only within some interval -- a \emph{change-window} -- by replacing $\sigma(x)$ with a product of two sigmoids, one increasing and one decreasing. \subsection{Multiplication by a known function} More generally, we can model an unknown function that's been multiplied by any fixed, known function $a(x)$, by multiplying the kernel by $a(\vx) a(\vx')$. Formally, % \begin{align} f(\iva) = a(\iva) g(\iva), \quad g \sim \GPdisttwo{0}{k(\iva, \iva')} \quad \iff \quad f \sim \GPdisttwo{0}{ a(\iva) k(\iva, \iva') a(\iva')} . \end{align} \section{Feature representation of kernels} \label{sec:mercer} % By Mercer's theorem \citep{mercer1909functions}, %Due to \citet{mercer1909functions}, we know that any positive-definite kernel can be represented as the inner product between a fixed set of features, evaluated at $\vx$ and at $\vx'$: % \begin{align} k(\vx, \vx') = \feat(\vx)\tra \feat(\vx') \end{align} For example, the squared-exponential kernel ($\kSE$) on the real line has a representation in terms of infinitely many radial-basis functions of the form ${h_i(x) \propto \exp( -\frac{1}{4\ell^2} (x - c_i)^2)}$. More generally, any stationary kernel can be represented by a set of sines and cosines - a Fourier representation \citep{bochner1959lectures}. In general, any particular feature representation of a kernel is not necessarily unique~\citep{minh2006mercer}. In some cases, the input to a kernel, $\vx$, can even be the implicit infinite-dimensional feature mapping of another kernel. Composing feature maps in this way leads to \emph{deep kernels}, which are explored in \cref{sec:deep_kernels}. %\cref{ch:deep-limits}. \subsection{Relation to linear regression} Surprisingly, \gp{} regression is equivalent to Bayesian linear regression on the implicit features $\feat(\vx)$ which give rise to the kernel: % \begin{align} f(\iva) = \vw\tra \feat(\iva), \quad \vw \sim \Nt{0}{\vI} \quad \iff \quad f \sim \GPdisttwo{0}{\feat(\iva) \tra \feat(\iva)} \end{align} % The link between Gaussian processes, linear regression, and neural networks is explored further in \cref{sec:relating}. %\cref{ch:deep-limits}. \subsection{Feature-space view of combining kernels} \def\feata{\va} \def\featb{\vb} %Many architectures for learning complex functions, such as convolutional networks \cite{lecun1989backpropagation} and sum-product networks \cite{poon2011sum}, include units which compute \texttt{and}-like and \texttt{or}-like operations. %Composite kernels can be viewed in this way too. %A sum of kernels can be understood as an \texttt{or}-like operation: two points are considered similar if either kernel has a high value. %Similarly, multiplying kernels is an \texttt{and}-like operation, since two points are considered similar only if both kernels have high values. We can also view kernel addition and multiplication as a combination of the features of the original kernels. %Viewing kernel addition from this point of view, if For example, given two kernels % \begin{align} k_a(\iva, \iva') & = \feata(\iva)\tra \feata(\iva')\\ k_b(\iva, \iva') & = \featb(\iva)\tra \featb(\iva') \end{align} % their addition has the form: % \begin{align} k_a(\iva, \iva') + k_b(\iva, \iva') & = \feata(\iva)\tra \feata(\iva') + \featb(\iva)\tra \featb(\iva') = \colvec{\feata(\iva)}{\featb(\iva)}\tra \colvec{\feata(\iva')}{\featb(\iva')} \end{align} % meaning that the features of $k_a + k_b$ are the concatenation of the features of each kernel. We can examine kernel multiplication in a similar way: % \begin{align} k_a(\iva, \iva') \times k_b(\iva, \iva') & = \left[ \feata(\iva)\tra \feata(\iva') \right] \times \left[ \featb(\iva)\tra \featb(\iva') \right] \\ %& = \left[ \begin{array}{c} \feat_1(\iva) \\ \feat_2(\iva) \end{array} \right]^\tra \left[ \begin{array}{c} \feat_1(\iva') \\ \feat_2(\iva') \end{array} \right] & = \sum_i a_i(\iva) a_i(\iva') \times \sum_j b_j(\iva) b_j(\iva') \\ %& = \sum_i \sum_j a_i(\iva) a_i(\iva') b_j(\iva) b_j(\iva') \\ & = \sum_{i,j} \big[ a_i(\iva) b_j(\iva) \big] \big[ a_i(\iva') b_j(\iva') \big] %& = \vecop{ \feata(\iva) \otimes \featb(\iva') } \tra \vecop{ \feata(\iva) \otimes \featb(\iva')} \end{align} % In words, the features of $k_a \kerntimes k_b$ are made of up all pairs of the original two sets of features. For example, the features of the product of two one-dimensional $\kSE$ kernels $(\kSE_1 \kerntimes \kSE_2)$ cover the plane with two-dimensional radial-basis functions of the form: % \begin{align} h_{ij}(x_1, x_2) \propto \exp \left( -\frac{1}{2} \frac{(x_1 - c_i)^2}{2\ell_1^2} \right) \exp \left( -\frac{1}{2} \frac{(x_2 - c_j)^2}{2\ell_2^2} \right) \end{align} \section{Expressing symmetries and invariances} \label{sec:expressing-symmetries} \def\gswitch{G_\textnormal{swap}} When modeling functions, encoding known symmetries can improve predictive accuracy. This section looks at different ways to encode symmetries into a prior on functions. Many types of symmetry can be enforced through operations on the kernel. We will demonstrate the properties of the resulting models by sampling functions from their priors. By using these functions to define smooth mappings from $\Reals^2 \to \Reals^3$, we will show how to build a nonparametric prior on an open-ended family of topological manifolds, such as cylinders, toruses, and M\"{o}bius strips. \subsection{Three recipes for invariant priors} Consider the scenario where we have a finite set of transformations of the input space $\{g_1, g_2, \ldots \}$ to which we wish our function to remain invariant: % \begin{align} f(\vx) = f(g(\vx)) \quad \forall \vx \in \mathcal{X}, \quad \forall g \in G \end{align} % As an example, imagine we wish to build a model of functions invariant to swapping their inputs: $f(x_1, x_2) = f(x_2, x_1)$, $\forall x_1,x_2$. Being invariant to a set of operations is equivalent to being invariant to all compositions of those operations, the set of which forms a group. \citep[chapter 21]{armstrong1988groups}. In our example, the elements of the group $\gswitch$ containing all operations the functions are invariant to has two elements:%, and is an example of the symmetric group $S2$: % \begin{align} g_1([x_1, x_2]) & = [x_2, x_1] \qquad \textnormal{(swap)} \\ g_2([x_1, x_2]) & = [x_1, x_2] \qquad \textnormal{(identity)} \end{align} How can we construct a prior on functions which respect these symmetries? \citet{ginsbourger2012argumentwise} and \citet{Invariances13} showed that the only way to construct a \gp{} prior on functions which respect a set of invariances is to construct a kernel which respects the same invariances with respect to each of its two inputs: % \begin{align} k(\vx, \vx') = k( g(\vx), g(\vx')), \quad \forall \vx, \vx' \in \InputSpace, \quad \forall g, g' \in G \end{align} % Formally, given a finite group $G$ whose elements are operations to which we wish our function to remain invariant, and $f \sim \GPt{0}{k(\vx,\vx')}$, then every $f$ is invariant under $G$ (up to a modification) if and only if $k(\cdot, \cdot)$ is argument-wise invariant under $G$. See \citet{Invariances13} for details. It might not always be clear how to construct a kernel respecting such argument-wise invariances. %Fortunately, for finite groups, there are a few simple ways to transform any kernel into one which is argument-wise invariant to actions under any finite group: Fortunately, there are a few simple ways to do this for any finite group: % % \begin{figure} \renewcommand{\tabcolsep}{1.5mm} \begin{tabular}{ccc} Additive method & Projection method & Product method \\[0.5ex] \includegraphics[width=0.3\columnwidth]{\symmetryfigsdir/symmetric-xy-naive-sample} & \includegraphics[width=0.3\columnwidth]{\symmetryfigsdir/symmetric-xy-projection-sample} & \includegraphics[width=0.3\columnwidth]{\symmetryfigsdir/symmetric-xy-prod-sample}\\ %$k(x, y, x', y') + k(x, y, y', x')$ & $k(x, y, x', y') \times k(x, y, y', x')$ & $k( \min(x, y), \max(x,y),$ \\ % $+ k(y, x, x', y') + k(y, x, y', x')$ & $\times k(y, x, x', y') \times k(y, x, y', x')$ & $\min(x', y'), \max(x',y') )$ %Draw from \gp{} with kernel: & Draw from \gp{} with kernel: & Draw from \gp{} with kernel: \\ $\begin{array}{r@{}l@{}} %& \kSE(x_1, x_2, x_1', x_2') \\ + \,\, & \kSE(x_1, x_2, x_2', x_1') & \kSE(x_1, x_1')\kerntimes \kSE(x_2, x_2') \\ + \,\, & \kSE(x_1, x_2')\kerntimes\kSE(x_2, x_1') \end{array}$ & $\begin{array}{r@{}l@{}} & \kSE( \min(x_1, x_2), \min(x_1', x_2')) \\ \kerntimes & \kSE( \max(x_1', x_2'), \max(x_1',x_2') ) \end{array}$ & $\begin{array}{r@{}l@{}} & \kSE(x_1, x_1')\kerntimes \kSE( x_2, x_2') \\ \times \,\, & \kSE(x_1, x_2')\kerntimes \kSE(x_2, x_1') \end{array}$ \end{tabular} \caption[Three ways to introduce symmetry]{ Functions drawn from three distinct \gp{} priors, each expressing symmetry about the line $x_1 = x_2$ using a different type of construction. %Three methods of introducing symmetry, illustrated through draws from the corresponding priors. %Left: The additive method. %Center: The product method. %Right: The projection method. %The additive method has half the marginal variance away from $y = x$, but the min method introduces a non-differentiable seam along $y = x$. All three methods introduce a different type of nonstationarity. } \label{fig:add_vs_min} \end{figure} % \begin{enumerate} \item {\bf Sum over the orbit.} The \emph{orbit} of $x$ with respect to a group $G$ is ${\{g(x) : g \in G\}}$, the set obtained by applying each element of $G$ to $x$. \citet{ginsbourger2012argumentwise} and \citet{kondor2008group} suggest enforcing invariances through a double sum over the orbits of $\vx$ and $\vx'$ with respect to G: % \begin{align} k_\textnormal{sum}(\vx, \vx') = \sum_{g, \in G} \sum_{g' \in G} k( g( \vx ), g'( \vx') ) \end{align} For the group $\gswitch$, this operation results in the kernel: % \begin{align} k_\textnormal{switch}(\vx, \vx') & = \sum_{g \in \gswitch} \sum_{g' \in \gswitch} k( g( \vx ), g'( \vx') ) \\ & = k(x_1, x_2, x_1', x_2') + k(x_1, x_2, x_2', x_1') \nonumber \\ & \quad + k(x_2, x_1, x_1', x_2') + k(x_2, x_1, x_2', x_1') \end{align} % For stationary kernels, some pairs of elements in this sum will be identical, and can be ignored. \Cref{fig:add_vs_min}(left) shows a draw from a \gp{} prior with a product of $\kSE$ kernels symmetrized in this way. This construction has the property that the marginal variance is doubled near $x_1 = x_2$, which may or may not be desirable. \item {\bf Project onto a fundamental domain.} \citet{Invariances13} also explored the possibility of projecting each datapoint into a fundamental domain of the group, using a mapping $A_G$: % \begin{align} k_\textnormal{proj}(\vx, \vx') = k( A_G(\vx), A_G( \vx') ) \end{align} % For example, a fundamental domain of the group $\gswitch$ is all ${\{x_1, x_2 : x_1 < x_2\}}$, a set which can be mapped to using $A_{\gswitch}( x_1, x_2 ) = \big[ \min(x_1, x_2), \max(x_1, x_2) \big]$. Constructing a kernel using this method introduces a non-differentiable ``seam'' along $x_1 = x_2$, as shown in \cref{fig:add_vs_min}(center). %The projection method also works for infinite groups, as we shall see below. \item {\bf Multiply over the orbit.} %\citet{adams2013product} Ryan P. Adams (personal communication) suggested a construction enforcing invariances through a double product over the orbits: % \begin{align} k_\textnormal{sum}(\vx, \vx') = \prod_{g \in G} \prod_{g' \in G} k( g( \vx ), g'( \vx') ) \end{align} % This method can sometimes produce \gp{} priors with zero variance in some regions, as in \cref{fig:add_vs_min}(right). %We include it here to show that each of these methods for enforcing symmetries modifies the resulting model in other ways as well. \end{enumerate} % There are often many possible ways to achieve a given symmetry, but we must be careful to do so without compromising other qualities of the model we are constructing. For example, simply setting $k(\vx, \vx') = 0$ gives rise to a \gp{} prior which obeys \emph{all possible} symmetries, but this is presumably not a model we wish to use. %In this section, we give recipes for expressing several classes of symmetries. Later, we will show how these can be combined to produce more interesting structures. \subsection{Example: Periodicity} %We can enforce periodicity on any subset of the dimensions: Periodicity in a one-dimensional function corresponds to the invariance % \begin{align} f(x) = f( x + \tau) \label{eq:periodic_invariance} \end{align} % where $\tau$ is the period. The most popular method for building a periodic kernel is due to \citet{mackay1998introduction}, who used the projection method in combination with an $\kSE$ kernel. A fundamental domain of the symmetry group is a circle, so the kernel % %The representer transformation for periodicity is simply $A(x) = [\sin(x), \cos(x)]$: % \begin{align} \kPer(x, x') = \kSE \left( \sin(x), \sin(x') \right) \kerntimes \kSE \left( \cos(x), \cos(x') \right) \end{align} % %We can also apply rotational symmetry repeatedy to a single dimension. achieves the invariance in \cref{eq:periodic_invariance}. Simple algebra reduces this kernel to the form given in \cref{fig:basic_kernels}. %We could also build a periodic kernel with period $\tau$ by the mapping $A(x) = \mod(x, \tau)$. %However, samples from this prior would be discontinuous at every integer multiple of $\tau$. \subsection{Example: Symmetry about zero} Another example of an easily-enforceable symmetry is symmetry about zero: % \begin{align} f(x) = f( -x) . \end{align} % This symmetry can be enforced using the sum over orbits method, by the transform % \begin{align} k_{\textnormal{reflect}}(x, x') & = k(x, x') + k(x, -x') + k(-x, x') + k(-x, -x'). \end{align} %This transformation can be applied to any subset of dimensions %\paragraph{Spherical Symmetry} %We can also enforce that a function expresses the symmetries obeyed by $n-spheres$ by simply transforming a set of $n - 1$ coordinates by: % %\begin{align} %x_1 & = \cos(\phi_1) \nonumber \\ %x_2 & = \sin(\phi_1) \cos(\phi_2) \nonumber \\ %x_3 & = \sin(\phi_1) \sin(\phi_2) \cos(\phi_3) \nonumber \\ %& \vdots \nonumber \\ %x_{n-1} & = \sin(\phi_1) \cdots \sin(\phi_{n-2}) \cos(\phi_{n-1}) \nonumber \\ %x_n & = \sin(\phi_1) \cdots \sin(\phi_{n-2}) \sin(\phi_{n-1}) %\end{align} %\cite{flanders1989} \subsection{Example: Translation invariance in images} Many models of images are invariant to spatial translations \citep{lecun1995convolutional}. Similarly, many models of sounds are also invariant to translation through time. Note that this sort of translation invariance is completely distinct from the stationarity of kernels such as $\kSE$ or $\kPer$. A stationary kernel implies that the prior is invariant to translations of the entire training and test set. In contrast, here we use translation invariance to refer to situations where the signal has been discretized, and each pixel (or the audio equivalent) corresponds to a different input dimension. We are interested in creating priors on functions that are invariant to swapping pixels in a manner that corresponds to shifting the signal in some direction: % \begin{align} f \Bigg( \raisebox{-2.5ex}{ \includegraphics[width=1cm]{\topologyfiguresdir/grid2} } \Bigg) = f \Bigg( \raisebox{-2.5ex}{ \includegraphics[width=1cm]{\topologyfiguresdir/grid3} } \Bigg) \end{align} % %In this setting, translation is equivalent to swapping dimensions of the input vector $\vx$. For example, in a one-dimensional image or audio signal, translation of an input vector by $i$ pixels can be defined as % \begin{align} \shift(\vx, i) = \big[ x_{\mod( i + 1, D )}, x_{\mod( i + 2, D )}, \dots, x_{\mod( i + D, D )} \big]\tra \end{align} % As above, translation invariance in one dimension can be achieved by a double sum over the orbit, given an initial translation-sensitive kernel between signals $k$: % \begin{align} %k \left( (x_1, x_2, \dots, x_D ), (x_1', x_2', \dots, x_D' ) \right) & = %\nonumber \\ %\sum_{i=1}^D \prod_{j=1}^D k( x_j, x_{ i + j \textnormal{mod $D$} }' ) k_\textnormal{invariant} \left( \vx, \vx' \right) = %\nonumber \\ \sum_{i=1}^D \sum_{j=1}^D k( \shift(\vx, i), \shift(\vx, j) ) \,. \end{align} % %This construction defines the kernel between two signals to be the sum of a kernel between all translations of those two signals. The extension to two dimensions, $\shift(\vx, i, j)$, is straightforward, but notationally cumbersome. \citet{kondor2008group} built a more elaborate kernel between images that was approximately invariant to both translation and rotation, using the projection method. %Is there a pathology of the additive construction that appears in the limit? %\subsection{Max-pooling} %What we'd really like to do is a max-pooling operation. However, in general, a kernel which is the max of other kernels is not PSD [put counterexample here?]. Is the max over co-ordinate switching PSD? \section{Generating topological manifolds} \label{sec:topological-manifolds} In this section we give a geometric illustration of the symmetries encoded by different compositions of kernels. The work presented in this section is based on a collaboration with David Reshef, Roger Grosse, Joshua B. Tenenbaum, and Zoubin Ghahramani. The derivation of the M\"obius kernel was my original contribution. Priors on functions obeying invariants can be used to create a prior on topological manifolds by using such functions to warp a simply-connected surface into a higher-dimensional space. For example, one can build a prior on 2-dimensional manifolds embedded in 3-dimensional space through a prior on mappings from $\mathbb{R}^2$ to $\mathbb{R}^3$. Such mappings can be constructed using three independent functions $[f_1(\vx), f_2(\vx), f_3(\vx)]$, each mapping from $\mathbb{R}^2$ to $\mathbb{R}$. Different \gp{} priors on these functions will implicitly give rise to different priors on warped surfaces. Symmetries in $[f_1, f_2, f_3]$ can connect different parts of the manifolds, giving rise to non-trivial topologies on the sampled surfaces. \begin{figure} \renewcommand{\tabcolsep}{1mm} \begin{tabular}{ccc} Euclidean $( \SE_1 \times \SE_2 )$ & Cylinder $( \SE_1 \times \Per_2 )$ & Toroid $( \Per_1 \times \Per_2 )$\\ \hspace{-0.5cm}\includegraphics[width=0.33\columnwidth,clip=true,trim=10mm 10mm 1mm 1mm]{\topologyfiguresdir/manifold} & \includegraphics[width=0.33\columnwidth,clip=true,trim=10mm 10mm 10mm 10mm]{\topologyfiguresdir/cylinder} & \includegraphics[width=0.33\columnwidth,clip=true,trim=1mm 1mm 1mm 1mm]{\topologyfiguresdir/torus} \\ \end{tabular} \caption[Generating 2D manifolds with different topological structures]{ Generating 2D manifolds with different topologies. By enforcing that the functions mapping from $\mathbb{R}^2$ to $\mathbb{R}^3$ obey certain symmetries, the surfaces created have corresponding topologies, ignoring self-intersections. } \label{fig:gen_surf} \end{figure} \Cref{fig:gen_surf} shows 2D meshes warped into 3D by functions drawn from \gp{} priors with various kernels, giving rise to a different topologies. % Higher-dimensional analogues of these shapes can be constructed by increasing the latent dimension and including corresponding terms in the kernel. For example, an $N$-dimensional latent space using kernel $\kPer_1 \kerntimes \kPer_2 \kerntimes \ldots \kerntimes \kPer_N$ will give rise to a prior on manifolds having the topology of $N$-dimensional toruses, ignoring self-intersections. This construction is similar in spirit to the \gp{} latent variable model (\gplvm{}) of \citet{lawrence2005probabilistic}, which learns a latent embedding of the data into a low-dimensional space, using a \gp{} prior on the mapping from the latent space to the observed space. \subsection{M\"{o}bius strips} %A prior on functions on M\"{o}bius strips can be constructed by enforcing the symmetries: % \begin{figure} \begin{tabular}[t]{ccc} %\begin{columns} \centering Draw from \gp{} with kernel: & & \\ %$( \Per_1 \times \Per_2 )$ and $f(x,y) = f(y,x)$ & generated parametrically\\ $\begin{array}{l@{}l@{}} & \Per(x_1, x_1') \kerntimes \Per(x_2, x_2') \\ \kernplus & \Per(x_1, x_2') \kerntimes \Per(x_2, x_1') \end{array}$ & $\begin{array}{c} \textnormal{ M\"{o}bius strip drawn from} \\ \textnormal{$\mathbb{R}^2 \to \mathbb{R}^3$ \gp{} prior} \end{array}$ & $\begin{array}{c} \textnormal{Sudanese M\"{o}bius strip} \\ \textnormal{generated parametrically} \end{array}$\\ %$ + \Per(x_1, x_2') \times \Per(x_2, x_1')$ & \\ %\includegraphics[width=0.45\columnwidth, height=0.45\columnwidth, clip=true,trim=2cm 2cm 2cm 1cm]{\topologyfiguresdir/mobius_regression} \\ \null\hspace{-5mm}\raisebox{2.75cm}{ \begin{tabular}{cc} \raisebox{1.65cm}{$x_2$\hspace{-1.5mm}} & \includegraphics[width=0.235\columnwidth, height=0.235\columnwidth, clip=true,trim=3cm 2cm 2cm 2cm]{\topologyfiguresdir/mobius_field} \\ & $x_1$ \\[-3em] \end{tabular}} & \raisebox{0.3cm}{\includegraphics[width=0.3\columnwidth,clip=true,trim=1cm 1.3cm 1mm 1mm]{\topologyfiguresdir/mobius}} & \raisebox{1cm}{\includegraphics[width=0.25\columnwidth,clip=true,trim=0cm 0cm 14.6cm 0cm]{\topologyfiguresdir/sudanese-wikipedia}} \\[-0.5em] \end{tabular} \caption[Generating M\"{o}bius strips]{Generating M\"{o}bius strips. \emph{Left:} A function drawn from a \sgp{} prior obeying the symmetries given by \cref{eq:mobius_symmetry_x,eq:mobius_symmetry_y,eq:mobius_symmetry_xy}. \emph{Center:} Simply-connected surfaces mapped from $\mathbb{R}^2$ to $\mathbb{R}^3$ by functions obeying those symmetries have a topology corresponding to a M\"{o}bius strip. Surfaces generated this way do not have the familiar shape of a flat surface connected to itself with a half-twist. Instead, they tend to look like \emph{Sudanese} M\"{o}bius strips \citep{sudanese1984}, whose edge has a circular shape. \emph{Right:} A Sudanese projection of a M\"{o}bius strip. Image adapted from \citet{sudanesepict}. } \label{fig:mobius} \end{figure} % A space having the topology of a M\"{o}bius strip can be constructed by enforcing invariance to the following operations~\citep[chapter 7]{reid2005geometry}: %A function defined on a M\"{o}bius strips can be constructed by enforcing the symmetries~\citep[chapter 7]{reid2005geometry}: % %\begin{align} %f(x_1, x_2) & = f( x_1 + \tau, x_2) \qquad \textnormal{(periodic in $x_1$)} \label%{eq:mobius_symmetry_x} \\ %f(x_1, x_2) & = f( x_1, x_2 + \tau) \qquad \textnormal{(periodic in $x_2$)} \label%{eq:mobius_symmetry_y} \\ %f(x_1, x_2) & = f( x_2, x_1) \quad \,\,\,\; \qquad \textnormal{(symmetric about $x_1 = x_2$)} %\label{eq:mobius_symmetry_xy} %\end{align} \begin{align} g_{p_1}([x_1, x_2]) & = [ x_1 + \tau, x_2] \qquad \textnormal{(periodic in $x_1$)} \label{eq:mobius_symmetry_x} \\ g_{p_2}([x_1, x_2]) & = [ x_1, x_2 + \tau] \qquad \textnormal{(periodic in $x_2$)} \label{eq:mobius_symmetry_y} \\ g_s([x_1, x_2]) & = [ x_2, x_1] \quad \,\,\,\; \qquad \textnormal{(symmetric about $x_1 = x_2$)} \label{eq:mobius_symmetry_xy} \end{align} % \Cref{sec:expressing-symmetries} already showed how to build \gp{} priors invariant to each of these types of transformations. We'll call a kernel which enforces these symmetries a \emph{M\"{o}bius kernel}. An example of such a kernel is: % \begin{align} k(x_1, x_2, x_1', x_2') = \Per(x_1, x_1') \kerntimes \Per(x_2, x_2') \kernplus \Per(x_1, x_2') \kerntimes \Per(x_2, x_1') \end{align} % Moving along the diagonal $x_1 = x_2$ of a function drawn from the corresponding \gp{} prior is equivalent to moving along the edge of a notional M\"{o}bius strip which has had that function mapped on to its surface. \Cref{fig:mobius}(left) shows an example of a function drawn from such a prior. \Cref{fig:mobius}(center) shows an example of a 2D mesh mapped to 3D by functions drawn from such a prior. This surface doesn't resemble the typical representation of a M\"{o}bius strip, %, because the edge of the M\"{o}bius strip is in roughly circular shape, as opposed to the double-loop that one obtains by gluing a strip of paper with a single twist. but instead resembles an embedding known as the Sudanese M\"{o}bius strip \citep{sudanese1984}, shown in \cref{fig:mobius}(right). %Another classic example of a function living on a Mobius strip is the auditory quality of 2-note intervals. The harmony of a pair of notes is periodic (over octaves) for each note, and the \section{Kernels on categorical variables} %Kernels can be defined over all types of data structures: Text, images, matrices, and even kernels . Coming up with a kernel on a new type of data used to be an easy way to get a NIPS paper. %\subsection{} %There is a simple way to do \gp{} regression over categorical variables: Categorical variables are variables which can take values only from a discrete, unordered set, such as $\{\texttt{blue}, \texttt{green}, \texttt{red}\}$. A simple way to construct a kernel over categorical variables is to represent that variable by a set of binary variables, using a one-of-k encoding. For example, if $\vx$ can take one of four values, $x \in \{ \texttt{A}, \texttt{B}, \texttt{C}, \texttt{D}\}$, then a one-of-k encoding of $x$ will correspond to four binary inputs, and $\oneofk(\texttt{C}) = [0, 0, 1, 0]$. Given a one-of-k encoding, we can place any multi-dimensional kernel on that space, such as the \seard{}: % \begin{align} k_{\textnormal{categorical}}( x, x') = \seard( \oneofk(x), \oneofk(x') ) \end{align} % Short lengthscales on any particular dimension of the $\seard$ kernel indicate that the function value corresponding to that category is uncorrelated with the others. More flexible parameterizations are also possible~\citep{pinheiro1996unconstrained}. %A more flexible parameterization suggested by %\citet{swersky2013categorical} %Kevin Swersky (personal communication) allows complete flexibility about which pairs of categories are similar to one another, replacing the $\seard$ kernel with a fully-parameterized kernel, $\sefull$: % %\begin{align} %\sefull( \vx, \vx') = \sigma^2_f \exp \left( -\frac{1}{2} \vx\tra \vL \vx' \right) %\end{align} % %where $\vL$ is a symmetric matrix, individually parameterizing the covariance between each pair of function values of categories. %Then, simply put a product of kernels on those dimensions. %This is the same as putting one SE ARD kernel on all of them. %Learning the lengthscale on each dimension of the $\kSE$ kernel will now encode how similar the value of the different categories are to one another. %The lengthscale hyperparameter will now encode whether, when that coding is active, the rest of the function changes. %If you notice that the estimated lengthscales for your categorical variables is short, your model is saying that it's not sharing any information between data of different categories. \section{Multiple outputs} Any \gp{} prior can easily be extended to the model multiple outputs: ${f_1(\vx), f_2(\vx), \dots, f_T(\vx)}$. This can be done by building a model of a single-output function which has had an extra input added that denotes the index of the output: $f_i(\vx) = f(\vx, i)$. This can be done by extending the original kernel $k(\vx, \vx')$ to have an extra discrete input dimension: $k(\vx, i, \vx', i')$. A simple and flexible construction of such a kernel multiplies the original kernel $k(\vx, \vx')$ with a categorical kernel on the output index~\citep{bonilla2007multi}: % \begin{align} k(\vx, i, \vx', i') = k_{\vx}(\vx, \vx') \kerntimes k_i(i,i') \end{align} \iffalse \section{Worked example: Building a structured kernel for a time-series} %\subsection{Modeling multiple periodicities} \begin{figure}[h] \begin{tabular}{ccc} Long-term trend & Weekly periodicity &Yearly periodicity \\ \includegraphics[width=0.31\columnwidth]{\examplefigsdir/births-component-1} & \includegraphics[width=0.31\columnwidth]{\examplefigsdir/births-component-3} & \includegraphics[width=0.31\columnwidth]{\examplefigsdir/births-component-2-zoom} \end{tabular} \caption[Composite model of births data]{A composite \gp{} model of births data. (blue)} \label{fig:quebec-decomp} \end{figure} \iffalse \begin{figure} \renewcommand{\tabcolsep}{1mm} \def \incpic#1{\includegraphics[width=0.200\columnwidth]{../figures/worked-example/births-#1}} \begin{tabular}{*{5}{c}} & {Long-term} & {Weekly} & {Yearly} & {Short-term} \\ \rotatebox{90}{{Long-term}} & \incpic{Long-term-Long-term} & \incpic{Long-term-Weekly} & \incpic{Long-term-Yearly} & \incpic{Long-term-Short-term} \\ \rotatebox{90}{{Weekly}} & \incpic{Weekly-Long-term} & \incpic{Weekly-Weekly} & \incpic{Weekly-Yearly} & \incpic{Weekly-Short-term} \\ \rotatebox{90}{{Yearly}} & \incpic{Yearly-Long-term} & \incpic{Yearly-Weekly} & \incpic{Yearly-Yearly} & \incpic{Yearly-Short-term} \\ \rotatebox{90}{{Short-term}} & \incpic{Short-term-Long-term} & \incpic{Short-term-Weekly} & \incpic{Short-term-Yearly} & \incpic{Short-term-Short-term} \\ \end{tabular} \caption[Two-way interactions in births data]{Two-way interactions in births data} \label{fig:quebec-decomp} \end{figure} \fi %\subsection{Incoportating discrete covariates} %\subsection{Breaking down the predictions, examining different parts of the model} \fi \section{Building a kernel in practice} %\subsection{Learning Kernel Parameters} This chapter outlined ways to choose the parametric form of a kernel in order to express different sorts of structure. Once the parametric form has been chosen, one still needs to choose, or integrate over, the kernel parameters. %Fortunately, typical kernels only have $\mathcal{O}(D)$ parameters, meaning that if $N$ is reasonably large, these parameters can be estimated by maximum marginal likelihood. If the kernel relatively few parameters, these parameters can be estimated by maximum marginal likelihood, using gradient-based optimizers. The kernel parameters estimated in \cref{sec:additivity-extrapolation,sec:concrete} were optimized using the \GPML{} toolbox~\citep{GPML}, available at \\\url{http://www.gaussianprocess.org/gpml/code}. %\subsection{Choosing the Kernel Form} %The marginal likelihood of a model is useful for choosing among parameters. %The marginal likelihood can also be used to selecting which type of kernel to use. % the form of the kernel. %For example, we might not know whether a particular structure or symmetry is present in the function we are trying to model. %Again, the fact that we can compare marginal likelihoods in \gp{}s means that we can %Because \gp{}s let us build models both with and without certain symmetries, %By building kernels with and without such structure, we can compute the marginal likelihoods of the corresponding \gp{} models. %The quantities represent the relative amount of evidence that the data provide for each of these possibilities, providing the assumptions of the model are correct. %To do so, we simple need to compare the marginal likelihood of the data %We demonstrate that marginal likeihood an be used to automatically search over such structures. A systematic search over kernel parameters is necessary when appropriate parameters are not known. Similarly, sometimes appropriate kernel structure is hard to guess. The next chapter will show how to perform an automatic search not just over kernel parameters, but also over an open-ended space of kernel expressions. \subsubsection{Source code} Source code to produce all figures and examples in this chapter is available at \\\url{http://www.github.com/duvenaud/phd-thesis}. %\section{Conclusion} %We've seen that kernels are a flexible and powerful language for building models of different types of functions. %However, for a given problem, it can difficult to specify an appropriate kernel, even after looking at the data. %A better procedure would be to compare the predictive performance, or marginal likelihood, of a few different kernels. %However, it might be difficult to enumerate all plausible kernels, and tedious to search over them. %In fact, choosing the kernel can be considered one of the main difficulties in doing inference. %Analogously, we usually don't expect to simply guess the best value of some parameter. %Rather, we specify a search space and an objective, and ask the computer to the search this space for us. \outbpdocument{ \bibliographystyle{plainnat} \bibliography{references.bib} } \iffalse \subsection{Example: Computing Molecular Energies} \begin{figure} \begin{center} \begin{tabular}{cc} Function on M\"{o}bius strip & \\ \includegraphics[width=0.3\columnwidth, height=0.3\columnwidth, clip=true,trim=3cm 2cm 2cm 2cm]{\topologyfiguresdir/mobius_field} & \begin{tikzpicture} %\pgfmathsetmacro{\r}{3cm} %\pgfmathsetmacro{\ho}{70} %\pgfmathsetmacro{\ht}{30} \newcommand{\radius}{3} \newcommand{\hone}{120} \newcommand{\htwo}{70} \newcommand{\hthree}{30} \coordinate (O) at (0, 0); \coordinate (left) at ({\radius*cos(\hone)}, {\radius*sin(\hone)}); \coordinate (right) at ({\radius*cos(\htwo)}, {\radius*sin(\htwo)}); \coordinate (zero) at ({\radius*cos(\hthree)}, {\radius*sin(\hthree)}); \draw[fill] (left) circle (2pt); \draw (left) node[below, left] {H}; \draw[fill] (right) circle (2pt); \draw (right) node[right] {H}; \draw[fill] (zero) circle (2pt); \draw (zero) node[right] {H}; \draw[fill] (O) circle (3pt); \draw (O) node[below] {C}; \draw (left) -- (O); \draw (right) -- (O); \draw (zero) -- (O); \begin{scope} \path[clip] (O) -- (right) -- (zero); \fill[red, opacity=0.5, draw=black] (O) circle (2); \node at ($(O)+(50:1.6)$) {$\theta_1$}; \end{scope} \begin{scope} \path[clip] (O) -- (left) -- (right); \fill[green, opacity=0.5, draw=black] (O) circle (1.8); \node at ($(O)+(90:1.4)$) {$\theta_2$}; \end{scope} \end{tikzpicture} \end{tabular} \end{center} \caption[The energy of a molecular configuration obeys the same symmetries as a M\"{o}bius strip]{An example of a function expressing the same symmetries as a M\"{o}bius strip in two of its arguments. The energy of a molecular configuration $f(\theta_1, \theta_2)$ depends only on the relative angles between atoms, and because each atom is indistinguishable, is invariant to permuting the atoms. } \label{fig:molecule} \end{figure} Figure \ref{fig:molecule} gives one example of a function which obeys the same symmetries as a M\"{o}bius strip, in some subsets of its arguments. \fi ================================================ FILE: misc/abstract.tex ================================================ % ************************** Thesis Abstract ***************************** % Use `abstract' as an option in the document class to print only the titlepage and the abstract. \begin{abstract} This thesis develops a method for automatically constructing, visualizing and describing a large class of models, useful for forecasting and finding structure in domains such as time series, geological formations, and physical dynamics. These models, based on Gaussian processes, can capture many types of statistical structure, such as periodicity, changepoints, additivity, and symmetries. Such structure can be encoded through \emph{kernels}, which have historically been hand-chosen by experts. We show how to automate this task, creating a system that explores an open-ended space of models and reports the structures discovered. To automatically construct Gaussian process models, we search over sums and products of kernels, maximizing the approximate marginal likelihood. We show how any model in this class can be automatically decomposed into qualitatively different parts, and how %every possible each component can be visualized and described through text. %We show how such models can be automatically decomposed, visualized, and described. %Combining these results, we present a procedure that takes in a dataset and outputs an automatically constructed model, along with a detailed report with plots and automatically generated text that illustrate the qualitatively different, and sometimes novel, types of structure discovered in the data. We combine these results into a procedure that, given a dataset, automatically constructs a model along with a detailed report containing plots and generated text that illustrate the structure discovered in the data. The introductory chapters contain a tutorial showing how to express many types of structure through kernels, and how adding and multiplying different kernels combines their properties. Examples also show how symmetric kernels can produce priors over topological manifolds such as cylinders, toruses, and M\"{o}bius strips, as well as their higher-dimensional generalizations. %Models can automatically be constructed by searching over sums and products of kernels, maximizing the approximate marginal likelihood of the data. %We show how such models can be automatically decomposed into qualitatively different types of structure, how these components can be automatically visualized, and finally how they can be automatically described using text. %Later chapters present a procedure which, given a dataset, outputs an automatically-constructed model, along with a detailed report. %These reports contain plots and automatically generated text that illustrate the qualitatively different, and sometimes novel, types of structure discovered in the data. %Models are automatically constructed by searching over sums and products of kernels, maximizing approximate marginal likelihood. %We show how such models can be automatically decomposed, visualized, and described. %Combining these results, we present a procedure which takes in a dataset and outputs an automatically-constructed model, along with a detailed report. %These reports contain plots and automatically generated text that illustrate the qualitatively different, and sometimes novel, types of structure discovered in the data. %To automatically construct Gaussian process models, we search over sums and products of kernels, maximizing approximate marginal likelihood. %This model-building procedure combines the properties of existing kernels to produce possibly novel statistical structures. %We show how such models can be decomposed automatically into qualitatively different parts, and how each component can automatically be visualized and described through text. %Combining these results, we present a procedure which takes in a dataset and outputs an automatically-constructed model and a detailed report. %These reports contain plots and text illustrating each of the types of structure discovered in the data. This thesis also explores several extensions to Gaussian process models. First, building on existing work that relates Gaussian processes and neural nets, we analyze natural extensions of these models to \emph{deep kernels} and \emph{deep Gaussian processes}. %Second, we examine models which efficiently sum over functions of all combinations of input variables, connecting this model class to the regularization method of \emph{dropout}. Second, we examine \emph{additive Gaussian processes}, showing their relation to the regularization method of \emph{dropout}. Third, we combine Gaussian processes with the Dirichlet process to produce the \emph{warped mixture model}: a Bayesian clustering model having nonparametric cluster shapes, and a corresponding latent space in which each cluster has an interpretable parametric form. \end{abstract} ================================================ FILE: misc/acknowledgement.tex ================================================ % ************************** Thesis Acknowledgements ***************************** \begin{acknowledgements} First, I would like to thank my supervisor, Carl Rasmussen, for so much advice and encouragement. It was wonderful working with someone who has spent many years thinking deeply about the business of modeling. Carl was patient while I spent the first few months of my PhD chasing half-baked ideas, and then gently suggested a series of ideas which actually worked. I'd also like to thank my advisor, Zoubin Ghahramani, for providing much encouragement, support, feedback and advice, and for being an example. %As well, one of the best things about the lab was that it was constantly populated with interesting visitors, thanks mainly to Zoubin. I'm also grateful for Zoubin's efforts to constantly populate the lab with interesting visitors. Thanks to Michael Osborne, whose thesis was an inspiration early on, and to Roman Garnett for being a sane third party. Sinead Williamson, Peter Orbanz and John Cunningham gave me valuable advice when I was still bewildered. Ferenc Husz\'ar and Dave Knowles made it clear that constantly asking questions was the right way to go. Tom Dean and Greg Corrado made me feel at home at Google. Philipp Hennig was a mentor to me during my time in T\"{u}bingen, and demonstrated an inspiring level of effectiveness. Who else finishes their conference papers with entire days to spare? I'd also like to thank Ryan Adams for hosting me during my visit to Harvard, for our enjoyable collaborations, and for building such an awesome group. Josh Tenenbaum's broad perspective and enthusiasm made the projects we worked on very rewarding. The time I got to spend with Roger Grosse was mind-expanding -- he constantly surprised me by pointing out basic unsolved questions about decades-old methods, and had extremely high standards for his own work. I'd like to thank Andrew McHutchon, Konstantina Palla, Alex Davies, Neil Houlsby, Koa Heaukulani, Miguel Hern\'{a}ndez-Lobato, Yue Wu, Ryan Turner, Roger Frigola, Sae Franklin, David Lopez-Paz, Mark van der Wilk and Rich Turner for making the lab feel like a family. I'd like to thank James Lloyd for many endless and enjoyable discussions, and for keeping a level head even during the depths of deadline death-marches. Christian Steinruecken showed me that amazing things are hidden all around. Thanks to Mel for reminding me that I needed to write a thesis, for supporting me during the final push, and for her love. My graduate study was supported by the National Sciences and Engineering Research Council of Canada, the Cambridge Commonwealth Trust, Pembroke College, a grant from the Engineering and Physical Sciences Research Council, and a grant from Google. \end{acknowledgements} ================================================ FILE: misc/declaration.tex ================================================ % ******************************* Thesis Declaration ******************************** \begin{declaration} I hereby declare that except where specific reference is made to the work of others, the contents of this dissertation are original and have not been submitted in whole or in part for consideration for any other degree or qualification in this, or any other University. This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration, except where specifically indicated in the text. This dissertation contains less than 60,000 words %including appendices, bibliography, footnotes, tables and equations and has less than 150 figures. % Author and date will be inserted automatically from thesis.tex \author \degreedate \end{declaration} ================================================ FILE: misc/dedication.tex ================================================ % ******************************* Thesis Dedidcation ******************************** \begin{dedication} I would like to dedicate this thesis to my loving parents ... \end{dedication} ================================================ FILE: misc/original-template/.gitignore ================================================ .DS_Store* *.aux* *.log* *.lof* *.lot* *.out* *.gz* *.toc* *.bbl* *.blg* *~* *.nlo* *make* *.d* *.bcf* *.blg* *~* *.nlo* *make *.d *.btmp* *.fls* *.glo* *.idx* *.ist* *.ilg* *.ind* *cookie* *#* *.nls* *.nlg* thesis.txt* thesis.bcf* thesis.run.xml* *run.xml* ================================================ FILE: misc/original-template/Abstract/abstract.tex ================================================ % ************************** Thesis Abstract ***************************** % Use `abstract' as an option in the document class to print only the titlepage and the abstract. \begin{abstract} This is where you write your abstract ... \end{abstract} ================================================ FILE: misc/original-template/Acknowledgement/acknowledgement.tex ================================================ % ************************** Thesis Acknowledgements ***************************** \begin{acknowledgements} And I would like to acknowledge ... \end{acknowledgements} ================================================ FILE: misc/original-template/Appendix1/appendix1.tex ================================================ % ******************************* Thesis Appendix A ******************************** \chapter{How to install \LaTeX} \section*{Windows OS} \subsection*{TeXLive package - full version} \begin{enumerate} \item Download the TeXLive ISO (2.2GB) from\\ \href{https://www.tug.org/texlive/}{https://www.tug.org/texlive/} \item Download WinCDEmu (if you don't have a virtual drive) from \\ \href{http://wincdemu.sysprogs.org/download/}{http://wincdemu.sysprogs.org/download/} \item To install Windows CD Emulator follow the instructions at\\ \href{http://wincdemu.sysprogs.org/tutorials/install/}{http://wincdemu.sysprogs.org/tutorials/install/} \item Right click the iso and mount it using the WinCDEmu as shown in \\ \href{http://wincdemu.sysprogs.org/tutorials/mount/}{http://wincdemu.sysprogs.org/tutorials/mount/} \item Open your virtual drive and run setup.pl \end{enumerate} or \subsection*{Basic MikTeX - TeX distribution} \begin{enumerate} \item Download Basic-MiK\TeX (32bit or 64bit) from\\ \href{http://miktex.org/download}{http://miktex.org/download} \item Run the installer \item To add a new package go to Start >> All Programs >> MikTex >> Maintenance (Admin) and choose Package Manager \item Select or search for packages to install \end{enumerate} \subsection*{TexStudio - Tex Editor} \begin{enumerate} \item Download TexStudio from\\ \href{http://texstudio.sourceforge.net/\#downloads}{http://texstudio.sourceforge.net/\#downloads} \item Run the installer \end{enumerate} \section*{Mac OS X} \subsection*{MacTeX - TeX distribution} \begin{enumerate} \item Download the file from\\ \href{https://www.tug.org/mactex/}{https://www.tug.org/mactex/} \item Extract and double click to run the installer. It does the entire configuration, sit back and relax. \end{enumerate} \subsection*{TexStudio - Tex Editor} \begin{enumerate} \item Download TexStudio from\\ \href{http://texstudio.sourceforge.net/\#downloads}{http://texstudio.sourceforge.net/\#downloads} \item Extract and Start \end{enumerate} \section*{Unix/Linux} \subsection*{TeXLive - TeX distribution} \subsubsection*{Getting the distribution:} \begin{enumerate} \item TexLive can be downloaded from\\ \href{http://www.tug.org/texlive/acquire-netinstall.html}{http://www.tug.org/texlive/acquire-netinstall.html}. \item TexLive is provided by most operating system you can use (rpm,apt-get or yum) to get TexLive distributions \end{enumerate} \subsubsection*{Installation} \begin{enumerate} \item Mount the ISO file in the mnt directory \begin{verbatim} mount -t iso9660 -o ro,loop,noauto /your/texlive####.iso /mnt \end{verbatim} \item Install wget on your OS (use rpm, apt-get or yum install) \item Run the installer script install-tl. \begin{verbatim} cd /your/download/directory ./install-tl \end{verbatim} \item Enter command `i' for installation \item Post-Installation configuration:\\ \href{http://www.tug.org/texlive/doc/texlive-en/texlive-en.html\#x1-320003.4.1}{http://www.tug.org/texlive/doc/texlive-en/texlive-en.html\#x1-320003.4.1} \item Set the path for the directory of TexLive binaries in your .bashrc file \end{enumerate} \subsubsection*{For 32Bit OS} For Bourne-compatible shells such as bash, and using Intel x86 GNU/Linux and a default directory setup as an example, the file to edit might be \begin{verbatim} edit $~/.bashrc file and add following lines PATH=/usr/local/texlive/2011/bin/i386-linux:$PATH; export PATH MANPATH=/usr/local/texlive/2011/texmf/doc/man:$MANPATH; export MANPATH INFOPATH=/usr/local/texlive/2011/texmf/doc/info:$INFOPATH; export INFOPATH \end{verbatim} \subsubsection*{For 64Bit} \begin{verbatim} edit $~/.bashrc file and add following lines PATH=/usr/local/texlive/2011/bin/x86_64-linux:$PATH; export PATH MANPATH=/usr/local/texlive/2011/texmf/doc/man:$MANPATH; export MANPATH INFOPATH=/usr/local/texlive/2011/texmf/doc/info:$INFOPATH; export INFOPATH \end{verbatim} %\subsection{Installing directly using Linux packages} \subsubsection*{Fedora/RedHat/CENTOS:} \begin{verbatim} sudo yum install texlive sudo yum install psutils \end{verbatim} \subsubsection*{SUSE:} \begin{verbatim} sudo zypper install texlive \end{verbatim} \subsubsection*{Debian/Ubuntu:} \begin{verbatim} sudo apt-get install texlive texlive-latex-extra sudo apt-get install psutils \end{verbatim} ================================================ FILE: misc/original-template/Appendix2/appendix2.tex ================================================ % ******************************* Thesis Appendix B ******************************** \chapter{Installing the CUED Class file} \LaTeX.cls files can be accessed system-wide when they are placed in the /tex/latex directory, where is the root directory of the user’s \TeX installation. On systems that have a local texmf tree (), which may be named ``texmf-local'' or ``localtexmf'', it may be advisable to install packages in , rather than as the contents of the former, unlike that of the latter, are preserved after the \LaTeX system is reinstalled and/or upgraded. It is recommended that the user create a subdirectory /tex/latex/CUED for all CUED related \LaTeX class and package files. On some \LaTeX systems, the directory look-up tables will need to be refreshed after making additions or deletions to the system files. For \TeX Live systems this is accomplished via executing ``texhash'' as root. MIK\TeX users can run ``initexmf -u'' to accomplish the same thing. Users not willing or able to install the files system-wide can install them in their personal directories, but will then have to provide the path (full or relative) in addition to the filename when referring to them in \LaTeX. ================================================ FILE: misc/original-template/Chapter1/chapter1.tex ================================================ %***************************************************************************************** %*********************************** First Chapter *************************************** %***************************************************************************************** \chapter{Getting Started} %Title of the First Chapter \ifpdf \graphicspath{{Chapter1/Figs/Raster/}{Chapter1/Figs/PDF/}{Chapter1/Figs/}} \else \graphicspath{{Chapter1/Figs/Vector/}{Chapter1/Figs/}} \fi %********************************** %First Section ************************************** \section{What is Loren Ipsum? Title with Math \texorpdfstring{$\sigma$}{[sigma]}} %Section - 1.1 Lorem Ipsum is simply dummy text of the printing and typesetting industry (see Section~\ref{section1.3}). Lorem Ipsum~\citep{Aup91} has been the industry's standard dummy text ever since the 1500s, when an unknown printer took a galley of type and scrambled it to make a type specimen book. It has survived not only five centuries, but also the leap into electronic typesetting, remaining essentially unchanged. It was popularised in the 1960s with the release of Letraset sheets containing Lorem Ipsum passages, and more recently with desktop publishing software like Aldus PageMaker including versions of Lorem Ipsum~\citep{AAB95,Con90,LM65}. The most famous equation in the world: $E^2 = (m_0c^2)^2 + (pc)^2$, which is known as the \textbf{energy-mass-momentum} relation as an in-line equation. A {\em \LaTeX{} class file}\index{\LaTeX{} class file@LaTeX class file} is a file, which holds style information for a particular \LaTeX{}. \begin{eqnarray} CIF: \hspace*{5mm}F_0^j(a) &=& \frac{1}{2\pi \iota} \oint_{\gamma} \frac{F_0^j(z)}{z - a} dz \end{eqnarray} \nomenclature[z-cif]{$CIF$}{Cauchy's Integral Formula} % first letter Z is for Acronyms \nomenclature[a-F]{$F$}{complex function} % first letter A is for Roman symbols \nomenclature[g-p]{$\pi$}{ $\simeq 3.14\ldots$} % first letter G is for Greek Symbols \nomenclature[g-i]{$\iota$}{unit imaginary number $\sqrt{-1}$} % first letter G is for Greek Symbols \nomenclature[g-g]{$\gamma$}{a simply closed curve on a complex plane} % first letter G is for Greek Symbols \nomenclature[x-i]{$\oint_\gamma$}{integration around a curve $\gamma$} % first letter X is for Other Symbols \nomenclature[r-j]{$j$}{superscript index} % first letter R is for superscripts \nomenclature[s-0]{$0$}{subscript index} % first letter S is for subscripts %********************************** %Second Section ************************************* \section{Why do we use Loren Ipsum?} %Section - 1.2 It is a long established fact that a reader will be distracted by the readable content of a page when looking at its layout. The point of using Lorem Ipsum is that it has a more-or-less normal distribution of letters, as opposed to using `Content here, content here', making it look like readable English. Many desktop publishing packages and web page editors now use Lorem Ipsum as their default model text, and a search for `lorem ipsum' will uncover many web sites still in their infancy. Various versions have evolved over the years, sometimes by accident, sometimes on purpose (injected humour and the like). %********************************** % Third Section ************************************* \section{Where does it come from?} %Section - 1.3 \label{section1.3} Contrary to popular belief, Lorem Ipsum is not simply random text. It has roots in a piece of classical Latin literature from 45 BC, making it over 2000 years old. Richard McClintock, a Latin professor at Hampden-Sydney College in Virginia, looked up one of the more obscure Latin words, consectetur, from a Lorem Ipsum passage, and going through the cites of the word in classical literature, discovered the undoubtable source. Lorem Ipsum comes from sections 1.10.32 and 1.10.33 of "de Finibus Bonorum et Malorum" (The Extremes of Good and Evil) by Cicero, written in 45 BC. This book is a treatise on the theory of ethics, very popular during the Renaissance. The first line of Lorem Ipsum, "Lorem ipsum dolor sit amet..", comes from a line in section 1.10.32. The standard chunk of Lorem Ipsum used since the 1500s is reproduced below for those interested. Sections 1.10.32 and 1.10.33 from ``de Finibus Bonorum et Malorum" by Cicero are also reproduced in their exact original form, accompanied by English versions from the 1914 translation by H. Rackham ``Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum." Section 1.10.32 of ``de Finibus Bonorum et Malorum", written by Cicero in 45 BC: ``Sed ut perspiciatis unde omnis iste natus error sit voluptatem accusantium doloremque laudantium, totam rem aperiam, eaque ipsa quae ab illo inventore veritatis et quasi architecto beatae vitae dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit aut fugit, sed quia consequuntur magni dolores eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci velit, sed quia non numquam eius modi tempora incidunt ut labore et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure reprehenderit qui in ea voluptate velit esse quam nihil molestiae consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla pariatur?" 1914 translation by H. Rackham: ``But I must explain to you how all this mistaken idea of denouncing pleasure and praising pain was born and I will give you a complete account of the system, and expound the actual teachings of the great explorer of the truth, the master-builder of human happiness. No one rejects, dislikes, or avoids pleasure itself, because it is pleasure, but because those who do not know how to pursue pleasure rationally encounter consequences that are extremely painful. Nor again is there anyone who loves or pursues or desires to obtain pain of itself, because it is pain, but because occasionally circumstances occur in which toil and pain can procure him some great pleasure. To take a trivial example, which of us ever undertakes laborious physical exercise, except to obtain some advantage from it? But who has any right to find fault with a man who chooses to enjoy a pleasure that has no annoying consequences, or one who avoids a pain that produces no resultant pleasure?" Section 1.10.33 of ``de Finibus Bonorum et Malorum", written by Cicero in 45 BC: ``At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Temporibus autem quibusdam et aut officiis debitis aut rerum necessitatibus saepe eveniet ut et voluptates repudiandae sint et molestiae non recusandae. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat." 1914 translation by H. Rackham: ``On the other hand, we denounce with righteous indignation and dislike men who are so beguiled and demoralized by the charms of pleasure of the moment, so blinded by desire, that they cannot foresee the pain and trouble that are bound to ensue; and equal blame belongs to those who fail in their duty through weakness of will, which is the same as saying through shrinking from toil and pain. These cases are perfectly simple and easy to distinguish. In a free hour, when our power of choice is untrammelled and when nothing prevents our being able to do what we like best, every pleasure is to be welcomed and every pain avoided. But in certain circumstances and owing to the claims of duty or the obligations of business it will frequently occur that pleasures have to be repudiated and annoyances accepted. 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64.092 279.105 66.315 274.926 63.831 c 278.203 67.85 278.574 72.346 278.574 72.346 c 280.961 72.139 l 280.961 72.139 267.805 73.292 267.363 76.932 c 266.961 80.296 271.375 84.003 271.375 84.003 c 271.375 84.003 269.738 83.233 267.535 80.737 c 267.535 80.737 266.934 81.132 266.406 81.553 c 271.75 90.811 276.035 94.042 275.012 97.428 c 274.395 100.229 277.898 103.378 277.898 103.378 c 277.898 103.378 273.18 100.651 274.059 98.382 c 271.273 100.6 273.855 105.483 273.855 105.483 c 273.855 105.483 269.961 101.194 272.227 98.327 c 269.211 97.374 266.648 94.675 264.848 91.479 c 262.809 88.421 261.855 85.428 261.992 82.471 c 257.574 81.893 l 257.574 81.893 257.051 82.483 256.621 83.05 c 259.633 92.878 264.555 97.69 262.945 100.893 c 262.297 102.151 262.195 103.378 262.57 104.53 c 262.57 104.53 263.832 105.272 264.789 105.682 c 266.246 106.471 265.93 103.51 265.93 103.51 c 265.93 103.51 266.723 105.307 266.301 106.21 c 271.75 108.803 283.477 107.999 283.477 107.999 c 274.434 111.225 l 274.434 111.225 278.047 111.499 285.379 108.542 c 284.816 109.385 282.691 111.432 282.691 111.432 c 282.691 111.432 290.375 106.526 293.809 103.378 c 292.488 103.405 289.594 102.8 289.594 102.8 c 289.594 102.8 295.93 102.979 296.617 102.979 c 308.734 108.749 317.305 109.065 319.688 106.878 c h 275.59 103.581 m 278.918 104.839 281.094 104.839 282.113 103.581 c 283.031 102.288 282.691 100.76 281.164 98.96 c 281.672 101.268 281.625 102.288 280.789 103.378 c 279.949 104.464 275.59 103.581 275.59 103.581 c h 277.727 109.065 m 273.312 108.948 l 273.48 110.479 l h 261.414 101.846 m 260.633 103.206 l 261.414 103.581 l h 251.625 75.37 m 251.625 75.37 251.484 73.948 252 72.139 c 251.523 72.639 248.965 75.581 247.988 76.729 c 243.285 82.214 244.711 89.991 253.531 87.467 c 253.602 88.589 253.707 98.538 260.055 102.628 c 260.055 102.628 260.27 101.815 260.836 101.065 c 259.898 100.6 253.918 98.01 254.109 85.155 c 251.961 87.796 242.316 85.456 251.625 75.37 c h 261.008 60.245 m 262.672 59.596 264.746 60.042 267.16 61.569 c 267.16 61.569 271.008 61.503 273.18 62.768 c 271.75 61.553 270.422 58.885 270.422 58.885 c 270.422 58.885 274.449 64.464 278.047 63.776 c 279.527 63.46 280.008 62.35 280.789 59.667 c 282.184 55.452 282.895 51.374 282.895 47.393 c 282.523 43.69 281.469 41.382 279.805 40.495 c 278 39.987 276.301 40.428 274.637 41.823 c 270.219 45.866 l 271 49.3 l 269.844 47.022 268.824 45.729 267.941 45.46 c 265.258 44.71 l 264.102 42.026 l 262.367 40.292 l 261.348 40.53 260.973 41.55 261.211 43.35 c 262.164 47.971 l 260.633 45.288 l 260.262 41.448 l 258.73 42.229 258.219 43.35 258.73 44.882 c 260.055 48.721 l 257.949 45.663 l 254.684 45.085 l 252.406 45.288 l 252.137 46.175 252.578 46.987 253.734 47.768 c 257.371 49.706 l 260.836 51.608 l 262.164 52.936 l 262.809 53.581 263.898 54.128 265.426 54.499 c 268.867 54.78 276.168 50.659 276.168 50.659 c 276.168 50.659 268.707 56.604 263.785 55.178 c 261.484 56.686 258.73 55.823 258.73 55.823 c 257.066 55.042 255.707 54.092 254.684 52.936 c 253.562 51.815 252.852 51.374 252.578 51.608 c 252.988 55.249 l 253.496 56.776 253.09 57.932 251.832 58.714 c 252.953 59.971 254.312 60.925 255.84 61.569 c 252.883 60.925 250.098 58.952 247.41 55.62 c 244.59 52.186 242.551 48.721 241.262 45.288 c 234.77 41.448 l 237.047 44.71 l 237.965 45.729 238.133 46.885 237.625 48.143 c 237.25 48.925 236.469 49.163 235.348 48.925 c 234.191 48.553 l 232.082 46.444 l 231.676 47.393 l 231.676 47.393 244.184 64.624 257.777 65.206 c 258.152 62.655 259.242 61.022 261.008 60.245 c h 230.145 47.971 m 232.457 45.288 l 234.191 47.393 l 236.469 47.768 l 236.977 46.885 236.875 45.932 236.094 44.882 c 234.191 42.397 l 233.273 40.596 231.676 39.475 229.398 38.967 c 225.559 39.917 l 224.027 41.073 l 223.246 41.073 l 224.773 46.038 l 226.711 48.925 l 225.352 47.971 l 223.688 48.346 222.43 48.925 221.512 49.706 c 220.629 50.725 220.492 51.542 221.137 52.186 c 222.16 52.698 223.383 52.698 224.773 52.186 c 228.039 50.081 l h 228.617 44.132 m 227.359 43.624 227.086 42.772 227.836 41.651 c 228.48 40.495 229.465 40.053 230.723 40.292 c 228.617 42.026 l h 234.191 21.327 m 234.191 21.327 237.535 24.546 239.152 25.339 c 242.516 16.526 248.309 13.831 253.152 13.987 c 255.137 14.225 272.199 18.749 261.961 38.987 c 262.914 39.303 264.68 40.089 264.68 40.089 c 265.629 43.553 l 269.266 44.71 l 268.484 41.514 268.555 39.139 269.473 37.604 c 269.641 37.233 l 272.324 33.971 l 273.348 32.952 273.992 31.217 274.262 28.803 c 272.598 28.53 270.902 28.19 269.844 27.241 c 279.262 29.385 283.477 23.229 283.477 23.229 c 282.691 20.921 l 282.895 18.061 l 284.02 22.073 l 286.16 24.553 l 287.043 21.87 287.109 19.592 286.328 17.655 c 285.547 15.991 285.309 13.882 285.582 11.335 c 277.527 13.065 l 269.062 14.596 l 264.848 13.44 l 259.684 12.288 l 255.094 12.116 l 251.422 11.913 l 249.145 11.163 l 246.969 10.382 244.422 9.159 241.465 7.495 c 234.191 3.854 l 230.352 2.124 l 230.352 3.514 230.145 5.01 229.77 6.542 c 227.461 10.757 l 226.578 12.147 226.203 13.749 226.305 15.55 c 226.883 16.534 l 229.566 14.018 l 231.504 10.178 l 231.504 12.624 230.859 14.561 229.566 15.956 c 233.984 21.124 l h 298.805 47.393 m 294.316 46.104 291.191 44.233 289.387 41.823 c 287.484 39.546 285.957 37.096 284.801 34.55 c 283.543 32.135 281.742 29.96 279.43 28.022 c 276.371 28.803 l 276.168 29.553 l 278.988 30.842 281.164 32.815 282.691 35.499 c 284.086 38.182 285.887 40.803 288.062 43.35 c 290.238 45.8 293.809 47.159 298.805 47.393 c h 217.672 43.928 m 217.809 44.846 218.656 45.8 220.188 46.815 c 224.605 47.393 l 223.449 46.038 l 222.871 44.507 l 221.578 43.487 220.426 42.975 219.406 42.975 c 218.387 42.842 217.809 43.147 217.672 43.928 c h 222.668 43.553 m 222.871 41.073 l 219.984 42.026 l 222.668 43.553 l f* 0.968627 0.596078 0.572549 rg 321.031 96.272 m 321.031 92.565 319.637 88.284 316.816 83.425 c 313.621 78.698 309.645 75.061 304.922 72.514 c 300.195 70.1 296.117 69.014 292.652 69.253 c 295.199 72.444 297.371 76.6 297.441 79.21 c 301.75 77.796 305.5 82.096 305.5 82.096 c 305.5 82.096 298.363 77.585 295.742 81.518 c 294.352 83.46 294.215 85.94 295.336 88.999 c 296.355 92.057 298.293 94.948 301.113 97.632 c 303.938 100.178 308.223 102.639 310.699 103.581 c 322.754 109.331 321.031 96.272 321.031 96.272 c f* 1 g 266.004 81.721 m 265.242 81.815 262.945 82.303 262.945 82.303 c 264.848 85.733 l 265.121 87.128 264.828 88.35 263.898 88.046 c 267.359 96.315 273.602 98.432 273.867 97.108 c 274.605 92.979 269.582 88.483 266.004 81.721 c h 256.215 83.257 m 255.824 83.671 255.094 84.581 255.094 84.581 c 256.621 87.842 l 256.863 89.374 256.488 90.292 255.469 90.526 c 255.84 93.62 256.828 96.034 258.355 97.835 c 259.887 99.635 261.645 101.233 261.992 99.94 c 263.285 96.421 259.793 94.303 256.215 83.257 c h 252.781 82.303 m 255.094 82.096 l 256.621 80.737 l 254.516 80.94 l 252.781 82.303 l f* 0.988235 0.909804 0.682353 rg 266.582 76.151 m 266.582 76.151 263.152 73.592 258.527 73.671 c 255.891 73.749 253.734 76.729 252.578 77.1 c 254.312 75.37 l 244.223 81.132 251.047 89.702 256.215 82.471 c 255.332 83.073 253.531 83.257 253.531 83.257 c 252.648 83.491 252.07 83.323 251.832 82.678 c 251.457 81.655 251.695 80.772 252.578 79.991 c 254.109 78.26 l 255.84 79.042 l 257.574 79.616 l 257.574 79.616 257.957 80.475 257.949 81.147 c 262.543 82.296 265.703 81.237 267.363 80.569 c 265.73 78.112 266.582 76.151 266.582 76.151 c h 271.953 73.092 m 271.953 73.092 274.926 72.667 278.152 72.346 c 278.152 68.749 274.773 64.425 273.688 63.678 c 271.645 62.401 268.109 62.147 268.109 62.147 c 270.625 65.038 l 268.52 63.096 l 267.227 62.214 266.07 61.569 265.051 61.194 c 263.012 60.311 260.77 60.893 259.477 62.928 c 257.598 65.893 257.574 69.456 257.574 69.456 c 257.371 66.565 l 255.094 68.878 l 255.094 68.878 253.285 70.792 252.781 71.358 c 252.039 73.87 252.238 75.057 252.238 75.057 c 253.43 74.425 255.469 74.249 255.469 74.249 c 256.488 73.194 257.812 72.616 259.477 72.514 c 260.465 67.143 l 260.699 64.967 261.855 63.745 263.898 63.507 c 265.426 63.507 267.09 64.389 268.895 66.19 c 270.695 67.858 271.715 70.135 271.953 73.092 c f* 0.929412 0.117647 0.172549 rg 266.379 66.768 m 267.672 69.315 269.129 70.577 270.797 70.577 c 270.016 68.913 269.062 67.448 267.941 66.19 c 266.785 65.038 265.215 63.948 263.32 64.257 c 260.27 64.827 267.043 70.042 267.16 69.456 c 267.277 68.87 266.379 66.768 266.379 66.768 c f* 0.984314 0.803922 0.376471 rg 262.57 54.671 m 260.262 52.764 l 258.527 50.827 l 255.57 50.081 253.531 49.132 252.406 47.971 c 251.016 46.815 250.676 45.526 251.422 44.132 c 254.109 43.553 l 257.574 44.71 l 258.152 41.073 l 258.152 41.073 259.715 39.182 260.836 38.967 c 266.008 32.8 266.562 16.686 252.578 14.971 c 245.215 14.065 227.754 31.21 251.051 58.135 c 252.07 57.628 252.578 56.846 252.578 55.823 c 251.832 52.936 l 251.32 51.917 251.492 51.1 252.406 50.456 c 254.684 51.815 l 258.355 54.671 l 259.613 55.589 261.043 55.589 262.57 54.671 c h 292.48 24.964 m 290.953 29.925 l 292.074 30.436 293.094 30.132 294.012 28.971 c 294.793 27.819 295.168 25.917 295.168 23.229 c 294.656 19.116 293.363 15.921 291.324 13.643 c 289.152 11.467 287.688 10.655 286.906 11.163 c 286.023 12.182 285.887 13.643 286.535 15.55 c 287.859 21.698 l 286.906 26.698 l 286.668 28.464 287.043 29.553 288.062 29.925 c 289.082 30.061 289.965 29.483 290.746 28.225 c h 226.137 20.171 m 226.371 19.151 226.238 17.995 225.727 16.706 c 225.082 15.682 225.082 14.155 225.727 12.116 c 228.242 7.323 l 229.398 3.854 l 229.5 2.839 229.262 2.053 228.617 1.546 c 224.98 2.327 l 223.18 3.245 221.578 5.01 220.188 7.698 c 219.168 10.01 218.793 12.182 219.031 14.221 c 219.406 16.397 220.051 17.858 220.934 18.639 c 222.293 15.55 l 221.715 18.233 l 222.871 20.921 l 224.773 21.124 l 226.137 20.171 l f* 0.0627451 0.0588235 0.0509804 rg 248.922 84.108 m 246.723 86.315 238.891 91.077 238.891 91.077 c 238.891 91.077 246.301 87.69 249.5 84.604 c 249.637 84.448 249.617 84.21 249.457 84.069 c 249.387 84.007 249.297 83.975 249.211 83.975 c 249.102 83.975 248.996 84.022 248.922 84.108 c f 248.586 82.038 m 243.762 83.51 238.152 83.51 238.152 83.51 c 238.152 83.51 243.441 84.303 248.828 82.753 c 249.023 82.686 249.129 82.471 249.062 82.276 c 249.035 82.186 248.973 82.116 248.898 82.069 c 248.805 82.018 248.691 82.003 248.586 82.038 c f 262.953 78.772 m 262.879 78.971 262.977 79.19 263.176 79.264 c 271.801 81.288 279.34 80.178 279.34 80.178 c 279.34 80.178 267.953 79.897 263.441 78.553 c 263.445 78.55 l 263.398 78.534 263.355 78.526 263.312 78.526 c 263.156 78.526 263.012 78.62 262.953 78.772 c f 281.602 74.303 m 281.594 74.303 273.113 76.846 266.219 76.995 c 264.891 76.995 263.578 76.905 262.336 76.71 c 262.34 76.706 l 262.129 76.675 261.934 76.815 261.902 77.022 c 261.867 77.229 262.008 77.428 262.219 77.46 c 263.508 77.667 264.855 77.757 266.219 77.757 c 273.91 77.753 281.602 74.303 281.602 74.303 c f 113.98 180.616 m 113.98 180.616 106.926 177.385 100.598 172.639 c 106.004 175.21 112.859 177.78 112.859 177.78 c h 121.371 31.284 m 81.219 36.245 73.832 68.811 47.797 66.151 c 48.781 66.835 51.438 69.514 51.438 69.514 c 51.438 69.514 43.641 71.448 33.145 63.612 c 32.184 65.956 27.422 79.757 39.02 93.538 c 6.723 111.397 21.059 137.483 20.566 160.971 c 20.566 170.991 15.488 185.76 0 184.643 c 33.754 199.866 65.66 179.69 83.312 166.01 c 90.852 172.081 98.953 177.104 107.68 181.143 c 107.68 181.143 113.676 183.616 114.848 184.19 c 129.672 213.577 157.535 216.659 157.535 216.659 c 157.535 216.659 135.934 201.909 141.754 181.143 c 150.699 181.342 157.195 178.124 157.195 178.124 c 157.195 178.124 149.578 179.034 142.402 178.292 c 142.523 176.991 142.883 172.561 142.883 172.561 c 142.883 172.561 148.789 171.385 154.969 165.698 c 157.754 167.561 159.766 164.596 159.766 164.596 c 167.246 168.62 175.379 148.987 175.379 148.987 c 190.383 133.811 180.77 102.8 180.77 102.8 c 180.461 100.456 180.141 99.178 181.109 97.378 c 183.879 93.686 191.789 96.233 194.973 86.225 c 193.418 87.007 192.828 87.608 192.402 87.495 c 200.402 79.096 195.723 70.788 189.383 68.694 c 189.383 68.694 190.438 70.569 190.652 71.401 c 187.438 67.229 178.543 61.69 178.543 61.69 c 179.672 65.362 l 166.125 48.573 156.961 49.1 146.598 37.178 c 146.598 37.178 139.559 17.983 121.371 31.284 c f* 0.470588 0.505882 0.509804 rg 154.969 215.725 m 151.465 212.37 l 144.633 204.885 l 141.867 201.807 139.41 198.307 137.293 194.385 c 135.062 190.233 133.652 185.772 133.004 181.003 c 133.004 181.003 125.387 180.616 114.059 177.811 c 115.469 181.143 l 123.434 184.19 l 132.832 187.046 l 129.137 187.467 124.082 186.788 117.699 184.983 c 122.078 192.405 126.961 198.366 132.352 202.823 c 137.578 207.061 142.348 210.194 146.695 212.198 c h 93.336 88.143 m 91.953 88.995 90.43 89.585 88.734 89.897 c 90.316 89.135 91.473 88.456 92.234 87.807 c 93.984 86.706 l 93.984 86.706 86.93 72.151 92.027 67.581 c 91.676 58.792 93.961 61.167 93.961 61.167 c 95.23 55.971 97.43 51.018 100.508 46.249 c 103.473 41.592 106.832 38.034 110.527 35.577 c 100.453 39.925 92.18 44.639 85.684 49.749 c 79.219 54.858 73.32 59.151 68.012 62.651 c 62.703 66.264 56.805 67.76 50.34 67.112 c 53.527 69.514 l 56.238 69.342 l 58.129 68.694 l 50.496 74.346 36.258 71.315 33.625 66.323 c 33.227 93.596 42.453 90.565 61.18 111.069 c 61.18 111.069 65.133 116.01 62.449 118.862 c 60.953 119.936 59.684 120.217 58.609 119.678 c 57.34 119.37 56.297 118.663 55.422 117.592 c 54.008 114.401 l 53.586 117.167 54.23 122.69 61.488 123.01 c 54.98 127.35 48.898 121.092 48.898 121.092 c 48.898 121.092 52.605 128.671 59.262 129.874 c 60.051 128.432 l 60.25 128.01 60.84 127.725 61.801 127.643 c 60.305 129.987 59.938 132.245 60.699 134.475 c 61.547 136.706 62.703 138.37 64.199 139.417 c 62.59 137.413 61.914 135.55 62.109 133.854 c 62.109 132.132 62.59 130.803 63.551 129.874 c 64.312 129.112 65.301 128.8 66.57 128.913 c 73.305 130.339 72.305 138.456 72.305 138.456 c 73.262 134.475 l 73.574 137.241 73.121 140.319 71.852 143.733 c 70.355 147.237 68.719 150.37 66.91 153.108 c 62.984 159.065 58.102 164.26 52.258 168.721 c 47.375 172.448 42.152 175.635 36.645 178.292 c 26.227 182.948 15.188 185.292 3.5 185.292 c 12.961 187.635 21.539 188.428 29.305 187.69 c 36.957 186.928 44.016 185.237 50.508 182.581 c 56.746 180.128 62.422 177.276 67.531 173.971 c 72.418 170.893 76.934 168.214 81.082 165.87 c 85.121 163.639 88.762 162.339 92.066 162.03 c 88.254 163.78 l 86.758 164.315 85.488 164.967 84.414 165.698 c 86.531 167.62 89.891 170.018 94.465 172.87 c 99.012 175.835 104.062 178.546 109.598 181.003 c 115.102 183.542 120.523 185.178 125.832 185.94 c 117.871 183.6 111.121 180.717 105.617 177.331 c 105.617 177.331 94.457 170.01 92.715 167.928 c 120.109 181.405 142.883 180.577 148.926 179.729 c 148.926 179.729 119.848 178.635 112.449 173.042 c 112.449 173.042 139.492 177.186 154.348 164.741 c 153.922 163.78 153.863 162.706 154.176 161.55 c 154.402 160.507 154.77 159.6 155.305 158.839 c 155.617 163.467 l 156.039 164.854 157.227 164.995 159.117 163.948 c 158.156 162.03 157.848 159.913 158.156 157.569 c 158.469 155.225 159.117 153.053 160.078 151.046 c 161.035 149.155 161.941 147.831 162.789 147.065 c 162.957 149.467 l 163.266 153.448 l 163.352 154.831 163.773 155.678 164.539 155.987 c 165.387 156.3 166.711 155.819 168.52 154.55 c 173.77 149.296 l 173.77 149.296 189.066 135.26 179.02 101.671 c 178.781 100.717 178.668 94.964 184.133 94.018 c 189.594 93.069 192.262 89.249 192.262 89.249 c 189.723 89.643 l 189.723 89.643 201.984 78.964 190.965 70.303 c 191.176 70.612 192.668 72.94 191.781 73.971 c 186.297 68.065 180.363 64.065 180.363 64.065 c 180.363 64.065 182.477 68.327 183.004 71.401 c 172.719 57.87 164.848 54.042 164.848 54.042 c 167.418 58.077 169.363 62.284 170.75 66.632 c 170.75 66.632 179.836 71.581 182.859 80.635 c 183.352 81.553 l 183.891 83.788 184.133 85.944 183.82 87.327 c 183.66 88.128 l 183.348 89.624 182.926 91.37 180.941 92.749 c 178.078 94.831 173.629 94.018 173.629 94.018 c 176.902 103.788 178.43 111.917 178.23 118.409 c 178.031 125.1 177.016 130.182 175.211 133.682 c 173.402 137.1 171.848 138.964 170.578 139.272 c 168.688 139.585 167.191 139.19 166.117 138.143 c 164.961 137.186 164.199 135.686 163.887 133.682 c 163.152 136.026 162.25 137.495 161.18 138.143 c 160.02 138.796 159.059 138.456 158.328 137.186 c 156.098 144.354 l 153.863 147.858 l 152.059 149.975 150.309 150.792 148.613 150.257 c 147.344 149.835 146.582 148.76 146.387 147.065 c 146.387 145.147 l 146.695 141.026 l 147.004 138.905 147.484 136.819 148.137 134.811 c 146.133 137.553 144.465 138.764 143.195 138.456 c 141.809 138.143 141.078 136.706 140.965 134.163 c 137.773 138.456 l 133.004 142.604 l 131.195 143.651 129.5 143.987 127.891 143.565 c 125.551 142.917 123.801 140.374 122.641 135.913 c 121.598 131.565 121.598 125.721 122.641 118.409 c 123.801 111.069 126.68 102.741 131.254 93.397 c 131.254 93.397 128.234 93.889 126.48 93.878 c 125.516 99.573 122.5 102.46 122.5 102.46 c 122.5 102.46 129.473 91.225 121.852 83.827 c 120.891 83.038 l 117.391 80.69 113.832 79.76 110.219 80.182 c 106.688 80.596 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228.422 470.124 228.324 471.324 228.121 472.624 c 227.723 472.523 226.723 472.425 226.324 472.425 c 225.824 471.124 225.422 469.824 224.922 468.523 c 226.121 468.425 227.324 468.222 228.523 468.124 c 228.621 465.222 225.723 464.722 223.523 464.023 c 223.824 467.624 223.422 471.722 225.922 474.722 c f q 1 0 0 1 0 515.085388 cm 225.922 -40.363 m 227.723 -38.961 229.824 -40.461 231.824 -40.762 c 232.223 -43.363 232.422 -45.863 232.422 -48.461 c 231.121 -47.762 229.824 -46.961 228.523 -46.262 c 228.422 -44.961 228.324 -43.762 228.121 -42.461 c 227.723 -42.562 226.723 -42.66 226.324 -42.66 c 225.824 -43.961 225.422 -45.262 224.922 -46.562 c 226.121 -46.66 227.324 -46.863 228.523 -46.961 c 228.621 -49.863 225.723 -50.363 223.523 -51.062 c 223.824 -47.461 223.422 -43.363 225.922 -40.363 c h S Q 0.380392 0.360784 0.345098 rg 94.125 473.124 m 95.223 473.222 96.324 473.425 97.422 473.523 c 97.523 473.324 97.625 472.824 97.723 472.624 c 95.625 468.925 93.723 465.124 92.922 460.925 c 93.023 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c 100.824 291.023 101.023 292.222 101.223 292.824 c 101.723 292.921 102.922 292.921 103.523 292.921 c 103.223 292.421 102.824 291.324 102.621 290.824 c 103.922 290.624 105.223 290.324 106.523 290.222 c 106.824 289.023 107.121 287.921 107.422 286.722 c 105.621 287.324 103.922 287.921 102.121 288.624 c 100.824 288.722 99.625 288.324 98.422 288.023 c 96.125 288.921 93.824 290.023 91.625 291.023 c 91.523 291.824 91.223 293.421 91.125 294.222 c f q 1 0 0 1 0 515.085388 cm 91.125 -220.863 m 92.223 -221.262 93.324 -221.664 94.422 -222.164 c 94.324 -222.664 94.223 -223.664 94.125 -224.062 c 94.922 -223.762 95.723 -223.461 96.523 -223.062 c 97.922 -223.562 99.324 -224.164 100.723 -224.664 c 100.824 -224.062 101.023 -222.863 101.223 -222.262 c 101.723 -222.164 102.922 -222.164 103.523 -222.164 c 103.223 -222.664 102.824 -223.762 102.621 -224.262 c 103.922 -224.461 105.223 -224.762 106.523 -224.863 c 106.824 -226.062 107.121 -227.164 107.422 -228.363 c 105.621 -227.762 103.922 -227.164 102.121 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185.523 358.718 185.523 357.922 c 185.523 356.136 184.012 354.687 182.148 354.687 c 181.246 354.687 180.422 355.031 179.816 355.586 c 170.418 347.964 158.766 343.035 146.02 341.976 c 146.113 341.683 146.16 341.371 146.16 341.05 c 146.16 339.265 144.652 337.82 142.789 337.82 c 140.922 337.82 139.414 339.265 139.414 341.05 c 139.414 341.3 139.441 341.539 139.496 341.773 c 126.363 342.047 114.23 346.422 104.312 353.668 c 103.715 352.761 102.66 352.16 101.457 352.16 c 99.594 352.16 98.082 353.605 98.082 355.39 c 98.082 356.355 98.523 357.218 99.223 357.812 c 86.609 369.238 78.684 385.777 78.684 404.172 c 78.684 419.992 84.547 434.433 94.203 445.433 c 93.48 446.027 93.023 446.91 93.023 447.894 c 93.023 449.679 94.531 451.125 96.395 451.125 c 97.34 451.125 98.195 450.754 98.809 450.156 c 108.711 459.289 121.57 465.238 135.777 466.386 c 135.766 466.5 135.758 466.613 135.758 466.73 c 135.758 468.515 137.27 469.964 139.133 469.964 c 140.996 469.964 142.504 468.515 142.504 466.73 c 142.504 466.675 142.504 466.621 142.5 466.566 c f* 0.211765 0.447059 0.756863 rg 109.484 180.246 m 127.949 173.164 149.637 172.187 171.406 180.246 c 178.078 154.027 173.238 126.39 142.59 124.789 c 117.105 124.093 103.488 137.894 109.484 180.246 c f* 0.137255 0.121569 0.12549 rg 1.50009 w q 1 0 0 1 0 579.929321 cm 109.484 -399.684 m 127.949 -406.766 149.637 -407.742 171.406 -399.684 c 178.078 -425.902 173.238 -453.539 142.59 -455.141 c 117.105 -455.836 103.488 -442.035 109.484 -399.684 c h 109.484 -399.684 m S Q 1 g [ 3.00018 1.50009] 0 d q 1 0 0 1 0 579.929321 cm 112.227 -403.902 m 128.746 -410.062 149.613 -411.008 169.086 -404 c 172.812 -426.605 169.848 -450.828 142.43 -452.223 c 119.629 -452.824 108.035 -440.629 112.227 -403.902 c h 112.227 -403.902 m S Q 0.137255 0.121569 0.12549 rg 157.16 151.722 m 141.645 151.722 l 141.645 150.765 l 157.16 150.765 l h 141.086 155.816 m 143.965 155.816 146.324 153.589 146.539 150.765 c 141.645 150.765 l 141.645 149.082 l 146.41 149.082 l 145.84 146.672 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96.304 206.758 121.027 c f* 1 g 1.50009 w [ 3.00018 1.50009] 0 d q 1 0 0 1 0 579.929321 cm 35.863 -443.172 m 46.129 -454.078 60.781 -458.156 74.723 -462.141 c 250.98 -443.172 m 240.715 -454.078 226.062 -458.156 212.121 -462.141 c 236.176 -488.047 m 236.176 -488.047 226.461 -491.285 215.82 -482.957 c 205.18 -474.629 204.719 -462.602 203.793 -457.051 c 202.867 -451.5 207.031 -371.93 207.031 -371.93 c 207.031 -371.93 186.676 -385.809 146.891 -387.656 c 107.105 -389.508 79.352 -373.059 79.352 -373.059 c 82.125 -457.512 l 82.125 -457.512 84.902 -470.004 74.723 -479.254 c 64.547 -488.508 51.594 -488.047 51.594 -488.047 c S Q 0.211765 0.447059 0.756863 rg 245.523 206.98 m 244.949 229.914 l 230.16 215.39 204.664 201.777 182.43 193.91 c 187.695 174.593 l 224.281 188.23 234.559 194.824 245.523 206.98 c f* 33.145 210.129 m 33.719 233.058 l 48.508 218.539 74.004 204.925 96.242 197.058 c 90.973 177.738 l 54.387 191.379 44.109 197.968 33.145 210.129 c f* 1 g [ 3.00018 1.50009] 0 d q 1 0 0 1 0 579.929321 cm 35.402 -352.035 m 35.402 -352.035 46.969 -362.676 62.234 -371.004 c 77.5 -379.332 92.305 -384.418 92.305 -384.418 c 87.676 -397.836 l 87.676 -397.836 61.77 -387.195 50.207 -380.719 c 38.641 -374.242 36.328 -368.691 36.328 -368.691 c 244.039 -356.664 m 244.039 -356.664 232.477 -365.914 223.684 -371.465 c 214.895 -377.02 187.602 -387.656 187.602 -387.656 c 190.84 -400.609 l 190.84 -400.609 210.73 -395.008 226.461 -384.828 c 242.188 -374.652 243.578 -371.004 243.578 -371.004 c S Q 0.137255 0.121569 0.12549 rg 83.648 198.5 m 87.234 198.312 89.719 195.312 89.191 191.793 c 88.668 188.277 85.328 185.578 81.738 185.765 c 78.148 185.953 75.664 188.957 76.191 192.472 c 76.719 195.988 80.059 198.687 83.648 198.5 c f* 0.50003 w [] 0.0 d q 1 0 0 1 0 579.929321 cm 83.648 -381.43 m 87.234 -381.617 89.719 -384.617 89.191 -388.137 c 88.668 -391.652 85.328 -394.352 81.738 -394.164 c 78.148 -393.977 75.664 -390.973 76.191 -387.457 c 76.719 -383.941 80.059 -381.242 83.648 -381.43 c h 83.648 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470.968 l 532.312 470.968 532.113 491.488 556.625 499.187 c 556.625 499.187 534.41 496.968 530.109 470.722 c f* 461.523 468.918 m 459.293 469.117 l 459.293 469.117 455.43 489.992 429.633 497.277 c 429.633 497.277 452.07 495.515 461.523 468.918 c f* 0.980392 0.878431 0.0941176 rg 393.676 225.199 m 393.676 225.199 376.949 192.898 372.105 175.898 c 367.262 158.898 362.422 144.875 363.742 128.726 c 365.062 112.574 365.062 101.527 365.062 101.527 c 386.191 100.675 l 386.191 100.675 385.754 114.277 385.312 124.476 c 384.871 134.675 385.754 141.898 388.832 152.101 c 391.914 162.3 396.316 177.175 398.957 182.273 c 401.602 187.375 404.68 195.875 404.68 195.875 c 393.676 225.199 l f* 603.039 237.492 m 603.039 237.492 615.262 201.039 618 178.527 c 620.738 156.015 623.504 132.429 623.504 132.429 c 603.32 125.461 l 603.32 125.461 595.473 158.695 593.879 178.527 c 592.281 198.359 577.723 201.578 603.039 237.492 c f* 498 478.238 m 498 478.238 l 556.23 478.238 603.875 429.312 603.875 369.515 c 603.875 247.191 l 608.66 131.105 630.414 91.863 511.012 88.269 c 388.082 84.57 389.203 129.019 392.242 239.726 c 392.129 369.515 l 392.078 429.312 439.77 478.238 498 478.238 c f* 0.137255 0.121569 0.12549 rg 361.137 102.554 m 361.137 102.554 360.125 100.535 361.137 96.672 c 362.145 92.804 363.32 90.789 362.984 87.425 c 362.648 84.062 361.977 85.406 362.312 83.726 c 362.648 82.047 362.145 80.027 362.145 78.347 c 362.145 76.668 366.008 62.211 368.027 58.679 c 370.043 55.152 l 370.043 55.152 370.883 44.73 378.953 43.89 c 387.02 43.047 388.199 47.25 388.199 47.25 c 388.199 47.25 397.781 30.945 408.371 40.191 c 418.961 49.437 414.59 56.496 414.59 56.496 c 414.59 56.496 426.188 55.824 426.355 69.773 c 426.523 83.726 407.527 81.879 407.527 81.879 c 407.527 81.879 399.797 86.922 395.594 88.937 c 391.391 90.957 389.879 89.609 389.879 89.609 c 388.703 90.957 l 388.703 90.957 391.223 93.308 391.223 96.839 c 391.223 100.367 391.391 102.218 391.391 102.218 c 391.391 102.218 394.922 104.57 390.047 105.074 c 385.172 105.578 381.641 106.418 371.219 106.757 c 360.797 107.093 356.262 105.074 361.137 102.554 c f* q 1 0 0 1 0 579.929321 cm 361.137 -477.375 m 361.137 -477.375 360.125 -479.395 361.137 -483.258 c 362.145 -487.125 363.32 -489.141 362.984 -492.504 c 362.648 -495.867 361.977 -494.523 362.312 -496.203 c 362.648 -497.883 362.145 -499.902 362.145 -501.582 c 362.145 -503.262 366.008 -517.719 368.027 -521.25 c 370.043 -524.777 l 370.043 -524.777 370.883 -535.199 378.953 -536.039 c 387.02 -536.883 388.199 -532.68 388.199 -532.68 c 388.199 -532.68 397.781 -548.984 408.371 -539.738 c 418.961 -530.492 414.59 -523.434 414.59 -523.434 c 414.59 -523.434 426.188 -524.105 426.355 -510.156 c 426.523 -496.203 407.527 -498.051 407.527 -498.051 c 407.527 -498.051 399.797 -493.008 395.594 -490.992 c 391.391 -488.973 389.879 -490.32 389.879 -490.32 c 388.703 -488.973 l 388.703 -488.973 391.223 -486.621 391.223 -483.09 c 391.223 -479.562 391.391 -477.711 391.391 -477.711 c 391.391 -477.711 394.922 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607.203 112.308 606.477 113.218 c 605.75 114.129 602.297 117.214 601.023 124.3 c 599.754 131.39 599.934 131.57 599.934 131.57 c 599.934 131.57 617.195 133.023 625.555 139.019 c f* q 1 0 0 1 0 579.929321 cm 625.555 -440.91 m 625.555 -440.91 628.461 -440 628.461 -440.91 c 628.461 -441.816 626.098 -442 626.828 -444.363 c 627.555 -446.723 626.828 -449.449 627.008 -453.445 c 627.188 -457.445 627.734 -464.711 629.371 -468.711 c 631.004 -472.707 635.184 -487.426 633.73 -494.148 c 632.277 -500.871 632.277 -502.324 630.098 -504.688 c 627.918 -507.047 625.555 -506.867 625.555 -506.867 c 625.555 -506.867 624.465 -514.133 621.195 -515.406 c 617.922 -516.68 617.379 -516.316 617.379 -516.316 c 617.379 -516.316 613.199 -522.312 607.02 -519.766 c 600.844 -517.223 600.844 -516.133 600.844 -516.133 c 600.844 -516.133 590.484 -513.77 587.941 -505.777 c 585.398 -497.781 590.848 -492.695 594.121 -492.512 c 597.391 -492.332 l 597.391 -492.332 597.754 -484.516 601.57 -477.066 c 605.387 -469.617 607.203 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47.519 m 506.781 47.519 504.484 39.691 505.453 37.418 c 506.418 35.144 508.906 36.375 513.676 35.781 c 518.445 35.191 522.992 33.64 528.324 31.906 c 533.656 30.175 536.336 29.472 552.234 29.91 c 568.137 30.351 567.852 30.839 568.75 32.265 c 569.645 33.691 570.598 40.586 563.59 43.64 c 556.582 46.695 552.945 47.425 548.484 47.847 c 544.023 48.273 545.02 51.144 545.02 51.144 c 545.02 51.144 529.734 47.957 506.781 47.519 c f* q 1 0 0 1 0 579.929321 cm 506.781 -532.41 m 506.781 -532.41 504.484 -540.238 505.453 -542.512 c 506.418 -544.785 508.906 -543.555 513.676 -544.148 c 518.445 -544.738 522.992 -546.289 528.324 -548.023 c 533.656 -549.754 536.336 -550.457 552.234 -550.02 c 568.137 -549.578 567.852 -549.09 568.75 -547.664 c 569.645 -546.238 570.598 -539.344 563.59 -536.289 c 556.582 -533.234 552.945 -532.504 548.484 -532.082 c 544.023 -531.656 545.02 -528.785 545.02 -528.785 c 545.02 -528.785 529.734 -531.973 506.781 -532.41 c h 506.781 -532.41 m S Q 0.129412 0.12549 0.121569 rg 446.195 25.968 m 446.895 22.062 l 446.895 22.062 452.77 18.605 468.355 20.761 c 483.938 22.914 490.488 26.972 490.488 26.972 c 492.535 26.347 l 492.535 26.347 496.078 26.429 497.277 27.472 c 498.473 28.515 498.422 30.129 498.242 32.23 c 498.066 34.336 l 498.066 34.336 497.383 34.324 496.207 34.3 c 495.434 33.695 494.156 33.488 492.043 33.297 c 488 32.929 488.574 31.925 485.508 29.402 c 482.441 26.875 478.352 25.867 465.469 25.621 c 452.59 25.371 448.02 26.808 448.02 26.808 c 448.02 26.808 447.824 26.964 447.598 27.347 c 447.152 26.941 446.688 26.488 446.195 25.968 c f* 0.137255 0.121569 0.12549 rg q 1 0 0 1 0 579.929321 cm 446.195 -553.961 m 446.895 -557.867 l 446.895 -557.867 452.77 -561.324 468.355 -559.168 c 483.938 -557.016 490.488 -552.957 490.488 -552.957 c 492.535 -553.582 l 492.535 -553.582 496.078 -553.5 497.277 -552.457 c 498.473 -551.414 498.422 -549.801 498.242 -547.699 c 498.066 -545.594 l 498.066 -545.594 497.383 -545.605 496.207 -545.629 c 495.434 -546.234 494.156 -546.441 492.043 -546.633 c 488 -547 488.574 -548.004 485.508 -550.527 c 482.441 -553.055 478.352 -554.062 465.469 -554.309 c 452.59 -554.559 448.02 -553.121 448.02 -553.121 c 448.02 -553.121 447.824 -552.965 447.598 -552.582 c 447.152 -552.988 446.688 -553.441 446.195 -553.961 c h 446.195 -553.961 m S Q 0.129412 0.12549 0.121569 rg 504.156 34.875 m 504.156 34.875 502.551 30.089 505.375 29.218 c 508.203 28.351 516.219 28.328 516.219 28.328 c 517.766 29.898 l 517.766 29.898 523.52 29.605 526.91 27.597 c 530.301 25.589 531.801 24.109 545.762 24.269 c 559.719 24.433 564.727 24.961 567.191 25.871 c 569.66 26.777 569.211 27.269 569.309 28.714 c 569.406 30.16 569.781 30.957 569.781 30.957 c 568.09 31.406 l 567.098 30.613 564.41 30.246 552.234 29.91 c 536.336 29.472 533.656 30.175 528.324 31.906 c 522.992 33.64 518.445 35.191 513.676 35.781 c 508.906 36.375 506.418 35.144 505.453 37.418 c 505.422 37.5 l 504.68 36.82 504.23 35.957 504.156 34.875 c f* 0.137255 0.121569 0.12549 rg q 1 0 0 1 0 579.929321 cm 504.156 -545.055 m 504.156 -545.055 502.551 -549.84 505.375 -550.711 c 508.203 -551.578 516.219 -551.602 516.219 -551.602 c 517.766 -550.031 l 517.766 -550.031 523.52 -550.324 526.91 -552.332 c 530.301 -554.34 531.801 -555.82 545.762 -555.66 c 559.719 -555.496 564.727 -554.969 567.191 -554.059 c 569.66 -553.152 569.211 -552.66 569.309 -551.215 c 569.406 -549.77 569.781 -548.973 569.781 -548.973 c 568.09 -548.523 l 567.098 -549.316 564.41 -549.684 552.234 -550.02 c 536.336 -550.457 533.656 -549.754 528.324 -548.023 c 522.992 -546.289 518.445 -544.738 513.676 -544.148 c 508.906 -543.555 506.418 -544.785 505.453 -542.512 c 505.422 -542.43 l 504.68 -543.109 504.23 -543.973 504.156 -545.055 c h 504.156 -545.055 m S Q 0.211765 0.447059 0.756863 rg 433.281 235.547 m 438.121 141.773 l 417.27 143.347 401.109 150.496 392.488 163.66 c 392.488 163.66 390.895 104.582 419.785 91.047 c 448.672 77.511 455.035 69.394 457.52 58.414 c 460 47.437 457.934 47.437 462.066 46.121 c 466.203 44.804 483.723 44.714 489.516 44.714 c 495.305 44.714 498.199 46.035 499.441 52.621 c 500.68 59.207 499.895 68.781 499.191 72.664 c 498.242 77.898 504.641 77.8 503.875 72.562 c 503.305 68.66 502.906 59.207 504.148 52.621 c 505.391 46.035 508.285 44.714 514.074 44.714 c 519.867 44.714 538.062 44.804 542.199 46.121 c 546.336 47.437 544.27 47.437 546.75 58.414 c 549.23 69.394 553.059 77.328 584.484 91.047 c 615.91 104.765 611.777 163.66 611.777 163.66 c 603.445 152.804 589.883 145.656 565.859 140.332 c 570.699 235.836 l 528.812 211.379 475.168 211.089 433.281 235.547 c f* 468.406 200.05 m 486.875 192.964 508.562 191.988 530.332 200.05 c 537.004 173.828 532.164 146.191 501.516 144.589 c 476.031 143.894 462.414 157.699 468.406 200.05 c f* 0.137255 0.121569 0.12549 rg 1.50009 w q 1 0 0 1 0 579.929321 cm 468.406 -379.879 m 486.875 -386.965 508.562 -387.941 530.332 -379.879 c 537.004 -406.102 532.164 -433.738 501.516 -435.34 c 476.031 -436.035 462.414 -422.23 468.406 -379.879 c h 468.406 -379.879 m S Q 1 g [ 3.00018 1.50009] 0 d q 1 0 0 1 0 579.929321 cm 471.148 -384.102 m 487.668 -390.258 508.535 -391.207 528.012 -384.199 c 531.738 -406.801 528.77 -431.027 501.352 -432.418 c 478.555 -433.023 466.961 -420.824 471.148 -384.102 c h 471.148 -384.102 m S Q 0.137255 0.121569 0.12549 rg 516.086 171.527 m 500.57 171.527 l 500.57 170.57 l 516.086 170.57 l h 500.008 175.621 m 502.891 175.621 505.25 173.394 505.465 170.57 c 500.57 170.57 l 500.57 168.886 l 505.336 168.886 l 504.766 166.472 502.598 164.675 500.008 164.675 c 496.988 164.675 494.535 167.129 494.535 170.148 c 494.535 173.172 496.988 175.621 500.008 175.621 c h 500.008 186.285 m 508.457 186.285 515.387 179.793 516.086 171.527 c 512.246 171.527 l 500.219 183.554 l 487.023 170.359 l 500.219 157.164 l 511.824 168.769 l 516.086 168.769 l 515.387 160.504 508.457 154.011 500.008 154.011 c 491.098 154.011 483.871 161.238 483.871 170.148 c 483.871 179.058 491.098 186.285 500.008 186.285 c f* 0.50003 w [] 0.0 d q 1 0 0 1 0 579.929321 cm 516.086 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48.011 529.184 51.004 c 529.184 51.004 542.379 49.304 545.812 53.64 c f* 458.418 53.64 m 458.734 51.507 l 458.84 50.781 466.262 48.011 475.051 51.004 c 475.051 51.004 461.855 49.304 458.418 53.64 c f* 439.895 140.832 m 439.895 140.832 439.895 124.984 427.867 118.433 c 415.836 111.882 407.973 112.398 407.973 112.398 c 407.973 112.398 433.879 116.105 439.895 140.832 c f* 565.684 140.832 m 565.684 140.832 565.684 124.984 577.711 118.433 c 589.738 111.882 597.602 112.398 597.602 112.398 c 597.602 112.398 571.695 116.105 565.684 140.832 c f* 1 g 1.50009 w [ 3.00018 1.50009] 0 d q 1 0 0 1 0 579.929321 cm 394.789 -423.367 m 405.055 -434.277 419.707 -438.352 433.648 -442.336 c 609.906 -423.367 m 599.637 -434.277 584.988 -438.352 571.047 -442.336 c 595.102 -468.242 m 595.102 -468.242 585.387 -471.48 574.746 -463.152 c 564.105 -454.828 563.645 -442.797 562.719 -437.246 c 561.793 -431.695 565.957 -352.125 565.957 -352.125 c 565.957 -352.125 545.602 -366.004 505.816 -367.855 c 466.031 -369.707 438.273 -353.258 438.273 -353.258 c 441.051 -437.711 l 441.051 -437.711 443.824 -450.199 433.648 -459.453 c 423.473 -468.703 410.52 -468.242 410.52 -468.242 c S Q 0.211765 0.447059 0.756863 rg 604.449 226.781 m 603.875 249.714 l 589.086 235.195 563.59 221.582 541.352 213.714 c 546.621 194.394 l 583.207 208.035 593.484 214.625 604.449 226.781 c f* 392.07 229.929 m 392.645 252.863 l 407.434 238.343 432.93 224.73 455.164 216.863 c 449.895 197.543 l 413.312 211.179 403.035 217.773 392.07 229.929 c f* 1 g [ 3.00018 1.50009] 0 d q 1 0 0 1 0 579.929321 cm 394.328 -332.234 m 394.328 -332.234 405.891 -342.875 421.156 -351.199 c 436.426 -359.527 451.227 -364.617 451.227 -364.617 c 446.602 -378.031 l 446.602 -378.031 420.695 -367.391 409.129 -360.914 c 397.562 -354.438 395.25 -348.887 395.25 -348.887 c 602.965 -336.859 m 602.965 -336.859 591.398 -346.113 582.609 -351.664 c 573.82 -357.215 546.527 -367.855 546.527 -367.855 c 549.766 -380.809 l 549.766 -380.809 569.656 -375.203 585.387 -365.027 c 601.113 -354.848 602.504 -351.199 602.504 -351.199 c S Q 0.137255 0.121569 0.12549 rg 442.57 218.304 m 446.16 218.117 448.645 215.113 448.117 211.597 c 447.59 208.082 444.254 205.382 440.664 205.57 c 437.074 205.757 434.59 208.761 435.117 212.277 c 435.645 215.793 438.98 218.492 442.57 218.304 c f* 0.50003 w [] 0.0 d q 1 0 0 1 0 579.929321 cm 442.57 -361.625 m 446.16 -361.812 448.645 -364.816 448.117 -368.332 c 447.59 -371.848 444.254 -374.547 440.664 -374.359 c 437.074 -374.172 434.59 -371.168 435.117 -367.652 c 435.645 -364.137 438.98 -361.438 442.57 -361.625 c h 442.57 -361.625 m S Q 0.223529 0.207843 0.211765 rg 442.211 215.906 m 444.449 215.789 446 213.918 445.672 211.726 c 445.344 209.531 443.262 207.847 441.023 207.964 c 438.785 208.082 437.234 209.957 437.562 212.148 c 437.891 214.339 439.973 216.023 442.211 215.906 c f* 0.137255 0.121569 0.12549 rg q 1 0 0 1 0 579.929321 cm 442.211 -364.023 m 444.449 -364.141 446 -366.012 445.672 -368.203 c 445.344 -370.398 443.262 -372.082 441.023 -371.965 c 438.785 -371.848 437.234 -369.973 437.562 -367.781 c 437.891 -365.59 439.973 -363.906 442.211 -364.023 c h 442.211 -364.023 m S Q 0 0.678431 0.937255 rg 440.891 214.617 m 442.535 214.617 l 442.383 212.726 l 444.379 212.644 l 443.953 211.007 l 442.25 211.082 l 442.105 209.257 l 440.461 209.257 l 440.613 211.148 l 438.617 211.234 l 439.043 212.867 l 440.746 212.797 l 440.891 214.617 l f* 0.137255 0.121569 0.12549 rg q 1 0 0 1 0 579.929321 cm 440.891 -365.312 m 442.535 -365.312 l 442.383 -367.203 l 444.379 -367.285 l 443.953 -368.922 l 442.25 -368.848 l 442.105 -370.672 l 440.461 -370.672 l 440.613 -368.781 l 438.617 -368.695 l 439.043 -367.062 l 440.746 -367.133 l h 440.891 -365.312 m S Q 553.637 215.035 m 550.324 213.57 548.652 209.718 549.906 206.433 c 551.16 203.148 554.863 201.672 558.176 203.136 c 561.488 204.601 563.16 208.453 561.906 211.738 c 560.652 215.023 556.949 216.5 553.637 215.035 c f* q 1 0 0 1 0 579.929321 cm 553.637 -364.895 m 550.324 -366.359 548.652 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misc/original-template/Chapter2/chapter2.tex ================================================ %***************************************************************************************** %*********************************** Second Chapter ************************************** %***************************************************************************************** \chapter{My Second Chapter} \ifpdf \graphicspath{{Chapter2/Figs/Raster/}{Chapter2/Figs/PDF/}{Chapter2/Figs/}} \else \graphicspath{{Chapter2/Figs/Vector/}{Chapter2/Figs/}} \fi \section[Short title]{Reasonably Long Section Title} % Uncomment this line, when you have siunitx package loaded. %The SI Units for dynamic viscosity is \si{\newton\second\per\metre\squared}. I'm going to randomly include a picture Figure~\ref{fig:minion}. If you have trouble viewing this document contact Krishna \href{mailto:kks32@cam.ac.uk}{kks32@cam.ac.uk}. \begin{figure}[htbp!] \centering \includegraphics[width=1.0\textwidth]{minion} \caption[Minion]{This is just a long figure caption for the minion in Despicable Me from Pixar} \label{fig:minion} \end{figure} \section*{Enumeration} \begin{enumerate} \item The first topic is dull \item The second topic is duller \begin{enumerate} \item The first subtopic is silly \item The second subtopic is stupid \end{enumerate} \item The third topic is dullest \end{enumerate} \section*{itemize} \begin{itemize} \item The first topic is dull \item The second topic is duller \begin{itemize} \item The first subtopic is silly \item The second subtopic is stupid \end{itemize} \item The third topic is dullest \end{itemize} \section*{description} \begin{description} \item[The first topic] is dull \item[The second topic] is duller \begin{description} \item[The first subtopic] is silly \item[The second subtopic] is stupid \end{description} \item[The third topic] is dullest \end{description} \clearpage \tochide\section{Hidden Section} \textbf{Lorem ipsum dolor sit amet}, \textit{consectetur adipiscing elit}. In magna nisi, aliquam id blandit id, congue ac est. Fusce porta consequat leo. Proin feugiat at felis vel consectetur. Ut tempus ipsum sit amet congue posuere. Nulla varius rutrum quam. Donec sed purus luctus, faucibus velit id, ultrices sapien. Cras diam purus, tincidunt eget tristique ut, egestas quis nulla. Curabitur vel iaculis lectus. Nunc nulla urna, ultrices et eleifend in, accumsan ut erat. In ut ante leo. Aenean a lacinia nisl, sit amet ullamcorper dolor. Maecenas blandit, tortor ut scelerisque congue, velit diam volutpat metus, sed vestibulum eros justo ut nulla. Etiam nec ipsum non enim luctus porta in in massa. Cras arcu urna, malesuada ut tellus ut, pellentesque mollis risus.Morbi vel tortor imperdiet arcu auctor mattis sit amet eu nisi. Nulla gravida urna vel nisl egestas varius. Aliquam posuere ante quis malesuada dignissim. Mauris ultrices tristique eros, a dignissim nisl iaculis nec. Praesent dapibus tincidunt mauris nec tempor. Curabitur et consequat nisi. Quisque viverra egestas risus, ut sodales enim blandit at. Mauris quis odio nulla. Cras euismod turpis magna, in facilisis diam congue non. Mauris faucibus nisl a orci dictum, et tempus mi cursus. Etiam elementum tristique lacus, sit amet eleifend nibh eleifend sed \footnote{My footnote goes blah blah blah! \dots}. Maecenas dapibu augue ut urna malesuada, non tempor nibh mollis. Donec sed sem sollicitudin, convallis velit aliquam, tincidunt diam. In eu venenatis lorem. Aliquam non augue porttitor tellus faucibus porta et nec ante. Proin sodales, libero vitae commodo sodales, dolor nisi cursus magna, non tincidunt ipsum nibh eget purus. Nam rutrum tincidunt arcu, tincidunt vulputate mi sagittis id. Proin et nisi nec orci tincidunt auctor et porta elit. Praesent eu dolor ac magna cursus euismod. Integer non dictum nunc. \begin{landscape} \section*{Subplots} I can cite Wall-E (see Fig.~\ref{fig:WallE}) and Minions in despicable me (Fig.~\ref{fig:Minnion}) or I can cite the whole figure as Fig.~\ref{fig:animations} \begin{figure} \centering \subfloat[A Tom and Jerry]{\label{fig:TomJerry}\includegraphics[width=0.3\textwidth]{TomandJerry}} \subfloat[A Wall-E]{\label{fig:WallE}\includegraphics[width=0.3\textwidth]{WallE}} \subfloat[A Minion]{\label{fig:Minnion}\includegraphics[width=0.3\textwidth]{minion}} \caption{Best Animations} \label{fig:animations} \end{figure} \end{landscape} ================================================ FILE: misc/original-template/Chapter3/chapter3.tex ================================================ \chapter{My Third Chapter} % **************************** Define Graphics Path ************************** \ifpdf \graphicspath{{Chapter3/Figs/Raster/}{Chapter3/Figs/PDF/}{Chapter3/Figs/}} \else \graphicspath{{Chapter3/Figs/Vector/}{Chapter3/Figs/}} \fi \section{First Section of the Third Chapter} And now I begin my third chapter here \dots And now to cite some more people~\citet{Rea85,Ancey1996} \subsection{First Subsection in the First Section} \dots and some more \subsection{Second Subsection in the First Section} \dots and some more \dots \subsubsection{First subsub section in the second subsection} \dots and some more in the first subsub section otherwise it all looks the same doesn't it? well we can add some text to it \dots \subsection{Third Subsection in the First Section} \dots and some more \dots \subsubsection{First subsub section in the third subsection} \dots and some more in the first subsub section otherwise it all looks the same doesn't it? well we can add some text to it and some more and some more and some more and some more and some more and some more and some more \dots \subsubsection{Second subsub section in the third subsection} \dots and some more in the first subsub section otherwise it all looks the same doesn't it? well we can add some text to it \dots \section{Second Section of the Third Chapter} and here I write more \dots Now we can refer to the table using Table.~\ref{t:borders}. \begin{table}[h] \caption{Table with Borders} \centering \label{t:borders} \begin{tabular}{|l|c| r|} \hline 1 & 2 & 3 \\ \hline 4 & 5 & 6 \\ \hline 7 & 8 & 9 \\ \hline \end{tabular} \end{table} ================================================ FILE: misc/original-template/Classes/PhDThesisPSnPDF.cls ================================================ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% Class ``PhD Thesis PSnPDF'' %% %% %% %% A PhD thesis LaTeX template for Cambridge University Engineering Department%% %% %% %% Version: v1.0 %% %% Authors: Krishna Kumar %% %% Date: 2013/11/16 (inception) %% %% License: MIT License (c) 2013 Krishna Kumar %% %% GitHub Repo: https://github.com/kks32/phd-thesis-template/ %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % ************************** Class Identification ****************************** \NeedsTeXFormat{LaTeX2e} \ProvidesClass{PhDThesisPSnPDF}[2013/12/10 version 1.0 by Krishna Kumar] \typeout{https://github.com/kks32/phd-thesis-template/} % ****************************************************************************** % **************************** Class Definition ******************************** % ****************************************************************************** % *********************** Define a Print/Online Version ************************ \newif\if@print\@printfalse \DeclareOption{print}{\@printtrue} % ****************************** Define index ********************************** \newif\ifPHD@index\PHD@indexfalse \DeclareOption{index}{\PHD@indextrue} % ******************************* Font Option ********************************** \newif\ifsetFont\setFontfalse % Font is not set \newif\ifPHD@times\PHD@timesfalse % Times with Math Support \DeclareOption{times}{\PHD@timestrue} \newif\ifPHD@fourier\PHD@fourierfalse % Fourier with Math Support \DeclareOption{fourier}{\PHD@fouriertrue} \newif\ifPHD@customfont\PHD@customfontfalse % Custom Font with Math Support \DeclareOption{customfont}{\PHD@customfonttrue} % ******************************* Bibliography ********************************* \newif\ifsetBib\setBibfalse % Custom Bibliography = true/false \newif\ifsetBiBLaTeX\setBiBLaTeXfalse % BiBLaTeX = True / False \newif\ifPHD@biblatex\PHD@biblatexfalse % BiBLaTeX \DeclareOption{biblatex}{\PHD@biblatextrue} \newif\ifPHD@authoryear\PHD@authoryearfalse % Author-Year citation \DeclareOption{authoryear}{\PHD@authoryeartrue} \newif\ifPHD@numbered\PHD@numberedfalse % Numbered citiation \DeclareOption{numbered}{\PHD@numberedtrue} \newif\ifPHD@custombib\PHD@custombibfalse % Custom Bibliography \DeclareOption{custombib}{\PHD@custombibtrue} % ************************* Header / Footer Styling **************************** \newif\ifPHD@pageStyleI\PHD@pageStyleIfalse % Set Page StyleI \DeclareOption{PageStyleI}{\PHD@pageStyleItrue} \newif\ifPHD@pageStyleII\PHD@pageStyleIIfalse % Set Page StyleII \DeclareOption{PageStyleII}{\PHD@pageStyleIItrue} % ***************************** Custom Margins ******************************** \newif\ifsetMargin\setMarginfalse % Margins are not set \newif\ifPHD@custommargin\PHD@custommarginfalse % Custom margin \DeclareOption{custommargin}{\PHD@custommargintrue} % **************************** Separate Abstract ****************************** \newif \ifdefineAbstract\defineAbstractfalse %To enable Separate abstract \newif\ifPHD@abstract\PHD@abstractfalse % Enable Separate Abstract \DeclareOption{abstract}{ \PHD@abstracttrue \ClassWarning{PhDThesisPSnPDF}{You have chosen an option that generates only the Title page and an abstract with PhD title and author name, if this was intentional, ignore this warning. Congratulations on submitting your thesis!! If not, please remove the option `abstract' from the document class and recompile. Good luck with your writing!} \PassOptionsToClass{oneside}{book} } % ****************** Chapter Mode - To print only selected chapters ************ \newif \ifdefineChapter\defineChapterfalse %To enable Separate abstract \newif\ifPHD@chapter\PHD@chapterfalse % Enable Separate Abstract \DeclareOption{chapter}{ \PHD@chaptertrue \ClassWarning{PhDThesisPSnPDF}{You have chosen an option that generates only selected chapters with references, if this was intentional, ignore this warning. If not, please remove the option `chapter' from the document class and recompile. Good luck with your writing!} } \ProcessOptions\relax% % *************************** Pre-defined Options ****************************** % Font Size \newcommand\PHD@ptsize{12pt} %Set Default Size as 12 \DeclareOption{10pt}{ \ClassWarning{PhDThesisPSnPDF}{The Cambridge University PhD thesis guidelines recommend using a minimum font size of 11pt (12pt is preferred) and 10pt for footnotes.} \renewcommand\PHD@ptsize{10pt} } \DeclareOption{11pt}{\renewcommand\PHD@ptsize{11pt}}% \DeclareOption{12pt}{\renewcommand\PHD@ptsize{12pt}}% \PassOptionsToClass{\PHD@ptsize}{book}% % Page Size \newcommand\PHD@papersize{a4paper} % Set Default as a4paper \DeclareOption{a4paper}{\renewcommand\PHD@papersize{a4paper}} \DeclareOption{a5paper}{\renewcommand\PHD@papersize{a5paper}} \DeclareOption{letterpaper}{ \ClassWarning{PhDThesisPSnPDF}{The Cambridge University Engineering Deparment PhD thesis guidelines recommend using A4 or A5paper} \renewcommand\PHD@papersize{letterpaper} } \PassOptionsToClass{\PHD@papersize}{book}% % Column layout \DeclareOption{oneside}{\PassOptionsToClass{\CurrentOption}{book}}% \DeclareOption{twoside}{\PassOptionsToClass{\CurrentOption}{book}}% % Draft Mode \DeclareOption{draft}{\PassOptionsToClass{\CurrentOption}{book}}% % Generates Warning for unknown options \DeclareOption*{ \ClassWarning{PhDThesisPSnPDF}{Unknown or non-standard option '\CurrentOption'. I'll see if I can load it from the book class. If you get a warning unused global option(s): `\CurrentOption` then the option is not supported!} \PassOptionsToClass{\CurrentOption}{book} } % Determine whether to run pdftex or dvips \ProcessOptions\relax% \newif\ifsetDVI\setDVIfalse \ifx\pdfoutput\undefined % we are not running PDFLaTeX \setDVItrue \LoadClass[dvips,fleqn,openright]{book}% \else % we are running PDFLaTeX \ifnum \pdfoutput>0 %PDF-Output \setDVIfalse \LoadClass[pdftex,fleqn,openright]{book}% \else %DVI-output \setDVItrue \LoadClass[fleqn,openright]{book}% \fi \fi %* ***************************** Print / Online ******************************** % Defines a print / online version to define page-layout and hyperrefering \ifsetDVI \special{papersize=\the\paperwidth,\the\paperheight} \RequirePackage[dvips,unicode=true]{hyperref} \else \RequirePackage[unicode=true]{hyperref} \pdfpagewidth=\the\paperwidth \pdfpageheight=\the\paperheight \fi \if@print % For Print version \hypersetup{ plainpages=false, pdfstartview=FitV, pdftoolbar=true, pdfmenubar=true, bookmarksopen=true, bookmarksnumbered=true, breaklinks=true, linktocpage, colorlinks=true, linkcolor=black, urlcolor=black, citecolor=black, anchorcolor=black } \ifPHD@custommargin \setMarginfalse \else \ifsetDVI % Odd and Even side Margin for binding and set viewmode for PDF \RequirePackage[dvips,paper=\PHD@papersize,hmarginratio=1:1,vmarginratio=1:1,scale=0.75,bindingoffset=5mm]{geometry} \else \RequirePackage[pdftex,paper=\PHD@papersize,hmarginratio=1:1,vmarginratio=1:1,scale=0.75,bindingoffset=5mm]{geometry} \fi \setMargintrue \fi \if@twoside \hypersetup{pdfpagelayout=TwoPageRight} \else \hypersetup{pdfpagelayout=OneColumn} \fi \else % For PDF Online version \hypersetup{ plainpages=false, pdfstartview=FitV, pdftoolbar=true, pdfmenubar=true, bookmarksopen=true, bookmarksnumbered=true, breaklinks=true, linktocpage, colorlinks=true, linkcolor=blue, urlcolor=blue, citecolor=blue, anchorcolor=green } \ifPHD@custommargin \setMarginfalse \else % No Margin staggering on Odd and Even side \ifsetDVI \RequirePackage[dvips,paper=\PHD@papersize,hmarginratio=1:1,vmarginratio=1:1,scale=0.75]{geometry} \else \RequirePackage[pdftex,paper=\PHD@papersize,hmarginratio=1:1,vmarginratio=1:1,scale=0.75]{geometry} \fi \setMargintrue \fi \hypersetup{pdfpagelayout=OneColumn} \fi % ****************************************************************************** % ******************************** Packages ************************************ % ****************************************************************************** % ************************** Layout and Formatting ***************************** \def\pdfshellescape{1} \RequirePackage{lscape} % Supports Landscape Layout \RequirePackage{setspace} % Define line spacing in paragraph \RequirePackage{calc} % To calculate vertical spacing % ************************* Conditional Statements ***************************** \RequirePackage{ifthen} % Used in LaTeX Class files for conditional statements \RequirePackage{ifpdf} % Check for pdfLaTeX % *********************** Table of Contents & Appendices *********************** % add Bibliography, List of figures and tables to contents \RequirePackage{tocbibind} % Add appendices \RequirePackage[title,titletoc]{appendix} % *************************** Graphics and Figures ***************************** \RequirePackage[usenames, dvipsnames]{color} \ifpdf % Convert eps figures to pdf \RequirePackage{epstopdf} \RequirePackage[pdftex]{graphicx} \DeclareGraphicsExtensions{.png, .jpg, .pdf} \pdfcompresslevel=9 \graphicspath{{Figs/Raster/}{Figs/}} \else \RequirePackage{graphicx} \DeclareGraphicsExtensions{.eps, .ps} \graphicspath{{Figs/Vector/}{Figs/}} \fi % ************************ URL Package and Definition ************************** \RequirePackage{url} % Redefining urlstyle to use smaller fontsize in References with URLs \newcommand{\url@leostyle}{% \@ifundefined{selectfont}{\renewcommand{\UrlFont}{\sf}} {\renewcommand{\UrlFont}{\small\ttfamily}}} \urlstyle{leo} % ******************************* Bibliography ********************************* \ifPHD@authoryear \ifPHD@biblatex \RequirePackage[backend=biber, style=authoryear, citestyle=alphabetic, sorting=nty, natbib=true]{biblatex} \setBiBLaTeXtrue \else \RequirePackage[round, sort, numbers, authoryear]{natbib} %author year \fi \setBibtrue \else \ifPHD@numbered \ifPHD@biblatex \RequirePackage[backend=biber, style=numeric-comp, citestyle=numeric, sorting=none, natbib=true]{biblatex} \setBiBLaTeXtrue \else \RequirePackage[numbers,sort&compress]{natbib} % numbered citation \fi \setBibtrue \else \ifPHD@custombib \setBibfalse \ifPHD@biblatex \setBiBLaTeXtrue \fi \else \ifPHD@biblatex \RequirePackage[backend=biber, style=numeric-comp, citestyle=numeric, sorting=none, natbib=true]{biblatex} \setBiBLaTeXtrue \else \RequirePackage[numbers,sort&compress]{natbib} % Default - numbered \fi \setBibtrue \ClassWarning{PhDThesisPSnPDF}{No bibliography style was specified. Default numbered style is used. If you would like to use a different style, use `authoryear' or `numbered' in the options in documentclass or use `custombib` and define the natbib package or biblatex package with required style in the Preamble.tex file} \fi \fi \fi % *********************** To copy ligatures and Fonts ************************** \RequirePackage{textcomp} % Font Selection \ifPHD@times \RequirePackage{mathptmx} % times roman, including math (where possible) \setFonttrue \else \ifPHD@fourier \RequirePackage{fourier} % Fourier \setFonttrue \else \ifPHD@customfont \setFontfalse \else \ClassWarning{PhDThesisPSnPDf}{Using default font Latin Modern. If you would like to use other pre-defined fonts use `times' (The Cambridge University PhD thesis guidelines recommend using Times font) or `fourier' or load a custom font in the preamble.tex file by specifying `customfont' in the class options} \RequirePackage{lmodern} \setFonttrue \fi \fi \fi \RequirePackage[utf8]{inputenc} \RequirePackage[T1]{fontenc} \input{glyphtounicode} \pdfglyphtounicode{f_f}{FB00} \pdfglyphtounicode{f_i}{FB01} \pdfglyphtounicode{f_l}{FB02} \pdfglyphtounicode{f_f_i}{FB03} \pdfglyphtounicode{f_f_l}{FB04} \pdfgentounicode=1 % ****************************************************************************** % **************************** Pre-defined Settings **************************** % ****************************************************************************** % *************************** Setting PDF Meta-Data **************************** \ifpdf \AtBeginDocument{ \hypersetup{ pdftitle = {\@title}, pdfauthor = {\@author}, pdfsubject={\@subject}, pdfkeywords={\@keywords} } } \fi % ************************** TOC and Hide Sections ***************************** \newcommand{\nocontentsline}[3]{} \newcommand{\tochide}[2]{ \bgroup\let \addcontentsline=\nocontentsline#1{#2} \egroup} % Removes pagenumber appearing from TOC \addtocontents{toc}{\protect\thispagestyle{empty}} % ***************************** Header Formatting ****************************** % Custom Header with Chapter Number, Page Number and Section Numbering \RequirePackage{fancyhdr} % Define custom header % Set Fancy Header Command is defined to Load FancyHdr after Geometry is defined \newcommand{\setFancyHdr}{ \pagestyle{fancy} \ifPHD@pageStyleI % Style 1: Sets Page Number at the Top and Chapter/Section Name on LE/RO \renewcommand{\chaptermark}[1]{\markboth{##1}{}} \renewcommand{\sectionmark}[1]{\markright{\thesection\ ##1\ }} \fancyhf{} \fancyhead[RO]{\nouppercase \rightmark\hspace{0.25em} | \hspace{0.25em} \bfseries{\thepage}} \fancyhead[LE]{ {\bfseries\thepage} \hspace{0.25em} | \hspace{0.25em} \nouppercase \leftmark} \else \ifPHD@pageStyleII % Style 2: Sets Page Number at the Bottom with Chapter/Section Name on LO/RE \renewcommand{\chaptermark}[1]{\markboth{##1}{}} \renewcommand{\sectionmark}[1]{\markright{\thesection\ ##1}} \fancyhf{} \fancyhead[RO]{\bfseries\nouppercase \rightmark} \fancyhead[LE]{\bfseries \nouppercase \leftmark} \fancyfoot[C]{\thepage} \else % Default Style: Sets Page Number at the Top (LE/RO) with Chapter/Section Name % on LO/RE and an empty footer \renewcommand{\chaptermark}[1]{\markboth {##1}{}} \renewcommand{\sectionmark}[1]{\markright{\thesection\ ##1}} \fancyhf{} \fancyhead[LO]{\nouppercase \rightmark} \fancyhead[LE,RO]{\bfseries\thepage} \fancyhead[RE]{\nouppercase \leftmark} \fi \fi } \setlength{\headheight}{14.5pt} %\renewcommand{\headrulewidth}{0.5pt} %\renewcommand{\footrulewidth}{0pt} \fancypagestyle{plain}{ \fancyhead{} \renewcommand{\headrulewidth}{0pt} } % If Margin has been set (default margin print/online version) \ifsetMargin \setFancyHdr % Apply fancy header settings otherwise apply it in preamble \fi % **************** Clear Header Style on the Last Empty Odd pages ************** \renewcommand{\cleardoublepage}{\clearpage\if@twoside \ifodd\c@page\else% \hbox{}% \thispagestyle{empty} % Empty header styles \newpage% \if@twocolumn\hbox{}\newpage\fi\fi\fi} % ******************************** Roman Pages ********************************* % The romanpages environment set the page numbering to lowercase roman one % for the contents and figures lists. It also resets % page-numbering for the remainder of the dissertation (arabic, starting at 1). \newenvironment{romanpages}{ \setcounter{page}{1} \renewcommand{\thepage}{\roman{page}}} {\newpage\renewcommand{\thepage}{\arabic{page}}} % ******************************* Appendices ********************************** % Appending the appendix name to the letter in TOC % Especially when backmatter is enabled %\renewcommand\backmatter{ % \def\chaptermark##1{\markboth{% % \ifnum \c@secnumdepth > \m@ne \@chapapp\ \thechapter: \fi ##1}{% % \ifnum \c@secnumdepth > \m@ne \@chapapp\ \thechapter: \fi ##1}}% % \def\sectionmark##1{\relax}} % ****************************************************************************** % **************************** Macro Definitions ******************************* % ****************************************************************************** % These macros are used to declare arguments needed for the % construction of the title page and other preamble. % The year and term the degree will be officially conferred \newcommand{\@degreedate}{} \newcommand{\degreedate}[1]{\renewcommand{\@degreedate}{#1}} % The full (unabbreviated) name of the degree \newcommand{\@degree}{} \newcommand{\degree}[1]{\renewcommand{\@degree}{#1}} % The name of your department(eg. Engineering, Maths, Physics) \newcommand{\@dept}{} \newcommand{\dept}[1]{\renewcommand{\@dept}{#1}} % The name of your college (eg. King's) \newcommand{\@college}{} \newcommand{\college}[1]{\renewcommand{\@college}{#1}} % The name of your University \newcommand{\@university}{} \newcommand{\university}[1]{\renewcommand{\@university}{#1}} % Defining the crest \newcommand{\@crest}{} \newcommand{\crest}[1]{\renewcommand{\@crest}{#1}} % Submission Text \newcommand{\submissiontext}{This dissertation is submitted for the degree of } % keywords (These keywords will appear in the PDF meta-information % called `pdfkeywords`.) \newcommand{\@keywords}{} \newcommand{\keywords}[1]{\renewcommand{\@keywords}{#1}} % subjectline (This subject will appear in the PDF meta-information % called `pdfsubject`.) \newcommand{\@subject}{} \newcommand{\subject}[1]{\renewcommand{\@subject}{#1}} % These macros define an environment for front matter that is always % single column even in a double-column document. \newenvironment{alwayssingle}{% \@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn \else\newpage\fi} {\if@restonecol\twocolumn\else\newpage\fi} % Set single column even under two column layout \newcommand{\setsinglecolumn}{ \if@twocolumn \onecolumn \else \fi } % ****************************************************************************** % *************************** Front Matter Layout ****************************** % ****************************************************************************** % ******************************** Title Page ********************************** \renewcommand{\maketitle}{ % To compute the free vertical space in Title page \computeTitlePageSpacing \begin{singlespace} \begin{center} % Title {\Huge \bfseries{\@title} \par} \vspace{.25\PHD@titlepagespacing} % Crest {\@crest \par} \vspace{.2\PHD@titlepagespacing} % Author {\Large \bfseries{\@author} \par} \vspace*{1ex} % Department and University {\large \@dept \par} \vspace*{1ex} {\large \@university \par} \vspace{.2\PHD@titlepagespacing} % Submission Text {\large \submissiontext \par} \vspace*{1ex} {\large \it {\@degree} \par} \end{center} \vfill \large \begin{minipage}{0.49\textwidth} \flushleft\hspace*{\oddsidemargin}\@college \end{minipage} \begin{minipage}{0.49\textwidth} \flushright \@degreedate \end{minipage} \end{singlespace} } % ********************************* Dedication ********************************* % The dedication environment makes sure the dedication gets its % own page, centered \newenvironment{dedication} { \cleardoublepage \setsinglecolumn \vspace*{0.2\textheight} \thispagestyle{empty} \centering } % ******************************* Declaration ********************************** % The declaration environment puts a large, bold, centered % "Declaration" label at the top of the page. \newenvironment{declaration}{ \cleardoublepage \setsinglecolumn \chapter*{\centering \Large Declaration} \thispagestyle{empty} }{ \flushright \@author{}\\ \@degreedate{} \vfill } % ****************************** Acknowledgements ******************************** % The acknowledgements environment puts a large, bold, centered % "Acknowledgements" label at the top of the page. \newenvironment{acknowledgements}{ \cleardoublepage \setsinglecolumn \chapter*{\centering \Large Acknowledgements} \thispagestyle{empty} } % ******************************* Nomenclature ********************************* \usepackage{nomencl} \makenomenclature \renewcommand{\nomgroup}[1]{% \ifthenelse{\equal{#1}{A}}{\item[\textbf{Roman Symbols}]}{% \ifthenelse{\equal{#1}{G}}{\item[\textbf{Greek Symbols}]}{% \ifthenelse{\equal{#1}{Z}}{\item[\textbf{Acronyms / Abbreviations}]}{% \ifthenelse{\equal{#1}{R}}{\item[\textbf{Superscripts}]}{% \ifthenelse{\equal{#1}{S}}{\item[\textbf{Subscripts}]}{% \ifthenelse{\equal{#1}{X}}{\item[\textbf{Other Symbols}]} {} }% matches mathematical symbols > X }% matches Subscripts > S }% matches Superscripts > R }% matches Abbreviations > Z }% matches Greek Symbols > G }% matches Roman Symbols > A % To add nomenclature in the header \renewcommand{\nompreamble}{\markboth{\nomname}{\nomname}} % Add nomenclature to contents and print out nomenclature \newcommand{\printnomencl}[1][]{ \ifthenelse{\equal {#1}{}} {\printnomenclature} {\printnomenclature[#1]} \addcontentsline{toc}{chapter}{\nomname} } % ***************************** Create the index ******************************* \ifPHD@index \RequirePackage{makeidx} \makeindex \newcommand{\printthesisindex}{ \cleardoublepage \phantomsection \printindex} \else \newcommand{\printthesisindex}{} \fi % ***************************** Chapter Mode *********************************** % The chapter mode allows user to only print particular chapters with references % All other options are disabled by default % To include only specific chapters without TOC, LOF, Title and Front Matter % To send it to supervisior for changes \ifPHD@chapter \defineChaptertrue % Disable the table of contents, figures, tables, index and nomenclature \renewcommand{\maketitle}{} \renewcommand{\tableofcontents}{} \renewcommand{\listoffigures}{} \renewcommand{\listoftables}{} \renewcommand{\printnomencl}{} \renewcommand{\printthesisindex}{} \else \defineChapterfalse \fi % ******************************** Abstract ************************************ % The abstract environment puts a large, bold, centered "Abstract" label at % the top of the page. Defines both abstract and separate abstract environment % To include only the Title and the abstract pages for submission to BoGS \ifPHD@abstract \defineAbstracttrue % Disable the table of contents, figures, tables, index and nomenclature \renewcommand{\tableofcontents}{} \renewcommand{\listoffigures}{} \renewcommand{\listoftables}{} \renewcommand{\printnomencl}{} \renewcommand{\printthesisindex}{} \renewcommand{\bibname}{} \renewcommand{\bibliography}[1]{\thispagestyle{empty}} \else \defineAbstractfalse \fi \newenvironment{abstract} { \ifPHD@abstract % Separate abstract as per Student Registry guidelines \thispagestyle{empty} \setsinglecolumn \begin{center} { \Large {\bfseries {\@title}} \par} {{\large \vspace*{1em} \@author} \par} \end{center} \else % Normal abstract in the thesis \cleardoublepage \setsinglecolumn \chapter*{\centering \Large Abstract} \thispagestyle{empty} \fi } % ******************************** Line Spacing ******************************** % Set spacing as 1.5 line spacing for the PhD Thesis \onehalfspace % ******************** To compute empty space in title page ******************** % Boxes below are used to space differt contents on the title page \newcommand{\computeTitlePageSpacing}{ % Title Box \newsavebox{\PHD@Title} \begin{lrbox}{\PHD@Title} \begin{minipage}[c]{0.98\textwidth} \centering \Huge \bfseries{\@title} \end{minipage} \end{lrbox} % University Crest Box \newsavebox{\PHD@crest} \begin{lrbox}{\PHD@crest} \@crest \end{lrbox} % Author Box \newsavebox{\PHD@author} \begin{lrbox}{\PHD@author} \begin{minipage}[c]{\textwidth} \centering \Large \bfseries{\@author} \end{minipage} \end{lrbox} % Department Box \newsavebox{\PHD@dept} \begin{lrbox}{\PHD@dept} \begin{minipage}[c]{\textwidth} \centering {\large \@dept \par} \vspace*{1ex} {\large \@university \par} \end{minipage} \end{lrbox} % Submission Box \newsavebox{\PHD@submission} \begin{lrbox}{\PHD@submission} \begin{minipage}[c]{\textwidth} \begin{center} \large \submissiontext \par \vspace*{1ex} \large \it {\@degree} \par \end{center} \end{minipage} \end{lrbox} % College and Date Box \newsavebox{\PHD@collegedate} \begin{lrbox}{\PHD@collegedate} \begin{minipage}[c]{\textwidth} \large \begin{minipage}{0.45\textwidth} \flushleft\@college \end{minipage} \begin{minipage}{0.45\textwidth} \flushright \@degreedate \end{minipage} \end{minipage} \end{lrbox} % Now to compute the free vertical space \newlength{\PHD@titlepagespacing} \setlength{\PHD@titlepagespacing}{ \textheight % - \totalheightof{\usebox{\PHD@Title}} - \totalheightof{\usebox{\PHD@crest}} - \totalheightof{\usebox{\PHD@author}} - \totalheightof{\usebox{\PHD@dept}} - \totalheightof{\usebox{\PHD@submission}} - \totalheightof{\usebox{\PHD@collegedate}} } } ================================================ FILE: misc/original-template/Classes/glyphtounicode.tex ================================================ % this file was converted from the following files: % - glyphlist.txt (Adobe Glyph List v2.0) % - zapfdingbats.txt (ITC Zapf Dingbats Glyph List) % - texglyphlist.txt (lcdf-typetools texglyphlist.txt, v2.33) % - additional.tex (additional entries) % % Notes: % - entries containing duplicates in glyphlist.txt like % 'dalethatafpatah;05D3 05B2' are ignored (commented out) % % - entries containing duplicates in texglyphlist.txt like % 'angbracketleft;27E8,2329' are changed so that only the first % choice remains, ie 'angbracketleft;27E8' % % - a few entries in texglyphlist.txt like Delta, Omega, etc. are % commented out (they are already in glyphlist.txt) % entries from glyphlist.txt: \pdfglyphtounicode{A}{0041} \pdfglyphtounicode{AE}{00C6} \pdfglyphtounicode{AEacute}{01FC} \pdfglyphtounicode{AEmacron}{01E2} \pdfglyphtounicode{AEsmall}{F7E6} \pdfglyphtounicode{Aacute}{00C1} \pdfglyphtounicode{Aacutesmall}{F7E1} \pdfglyphtounicode{Abreve}{0102} \pdfglyphtounicode{Abreveacute}{1EAE} \pdfglyphtounicode{Abrevecyrillic}{04D0} \pdfglyphtounicode{Abrevedotbelow}{1EB6} \pdfglyphtounicode{Abrevegrave}{1EB0} \pdfglyphtounicode{Abrevehookabove}{1EB2} \pdfglyphtounicode{Abrevetilde}{1EB4} \pdfglyphtounicode{Acaron}{01CD} \pdfglyphtounicode{Acircle}{24B6} \pdfglyphtounicode{Acircumflex}{00C2} \pdfglyphtounicode{Acircumflexacute}{1EA4} \pdfglyphtounicode{Acircumflexdotbelow}{1EAC} \pdfglyphtounicode{Acircumflexgrave}{1EA6} \pdfglyphtounicode{Acircumflexhookabove}{1EA8} \pdfglyphtounicode{Acircumflexsmall}{F7E2} \pdfglyphtounicode{Acircumflextilde}{1EAA} \pdfglyphtounicode{Acute}{F6C9} \pdfglyphtounicode{Acutesmall}{F7B4} \pdfglyphtounicode{Acyrillic}{0410} \pdfglyphtounicode{Adblgrave}{0200} \pdfglyphtounicode{Adieresis}{00C4} \pdfglyphtounicode{Adieresiscyrillic}{04D2} \pdfglyphtounicode{Adieresismacron}{01DE} \pdfglyphtounicode{Adieresissmall}{F7E4} \pdfglyphtounicode{Adotbelow}{1EA0} \pdfglyphtounicode{Adotmacron}{01E0} \pdfglyphtounicode{Agrave}{00C0} \pdfglyphtounicode{Agravesmall}{F7E0} \pdfglyphtounicode{Ahookabove}{1EA2} \pdfglyphtounicode{Aiecyrillic}{04D4} \pdfglyphtounicode{Ainvertedbreve}{0202} \pdfglyphtounicode{Alpha}{0391} \pdfglyphtounicode{Alphatonos}{0386} \pdfglyphtounicode{Amacron}{0100} \pdfglyphtounicode{Amonospace}{FF21} \pdfglyphtounicode{Aogonek}{0104} \pdfglyphtounicode{Aring}{00C5} \pdfglyphtounicode{Aringacute}{01FA} \pdfglyphtounicode{Aringbelow}{1E00} \pdfglyphtounicode{Aringsmall}{F7E5} \pdfglyphtounicode{Asmall}{F761} \pdfglyphtounicode{Atilde}{00C3} \pdfglyphtounicode{Atildesmall}{F7E3} \pdfglyphtounicode{Aybarmenian}{0531} \pdfglyphtounicode{B}{0042} \pdfglyphtounicode{Bcircle}{24B7} \pdfglyphtounicode{Bdotaccent}{1E02} \pdfglyphtounicode{Bdotbelow}{1E04} \pdfglyphtounicode{Becyrillic}{0411} \pdfglyphtounicode{Benarmenian}{0532} \pdfglyphtounicode{Beta}{0392} \pdfglyphtounicode{Bhook}{0181} \pdfglyphtounicode{Blinebelow}{1E06} \pdfglyphtounicode{Bmonospace}{FF22} \pdfglyphtounicode{Brevesmall}{F6F4} \pdfglyphtounicode{Bsmall}{F762} \pdfglyphtounicode{Btopbar}{0182} \pdfglyphtounicode{C}{0043} \pdfglyphtounicode{Caarmenian}{053E} \pdfglyphtounicode{Cacute}{0106} \pdfglyphtounicode{Caron}{F6CA} \pdfglyphtounicode{Caronsmall}{F6F5} \pdfglyphtounicode{Ccaron}{010C} \pdfglyphtounicode{Ccedilla}{00C7} \pdfglyphtounicode{Ccedillaacute}{1E08} \pdfglyphtounicode{Ccedillasmall}{F7E7} \pdfglyphtounicode{Ccircle}{24B8} \pdfglyphtounicode{Ccircumflex}{0108} \pdfglyphtounicode{Cdot}{010A} \pdfglyphtounicode{Cdotaccent}{010A} \pdfglyphtounicode{Cedillasmall}{F7B8} \pdfglyphtounicode{Chaarmenian}{0549} \pdfglyphtounicode{Cheabkhasiancyrillic}{04BC} \pdfglyphtounicode{Checyrillic}{0427} \pdfglyphtounicode{Chedescenderabkhasiancyrillic}{04BE} \pdfglyphtounicode{Chedescendercyrillic}{04B6} \pdfglyphtounicode{Chedieresiscyrillic}{04F4} \pdfglyphtounicode{Cheharmenian}{0543} \pdfglyphtounicode{Chekhakassiancyrillic}{04CB} \pdfglyphtounicode{Cheverticalstrokecyrillic}{04B8} \pdfglyphtounicode{Chi}{03A7} \pdfglyphtounicode{Chook}{0187} \pdfglyphtounicode{Circumflexsmall}{F6F6} \pdfglyphtounicode{Cmonospace}{FF23} \pdfglyphtounicode{Coarmenian}{0551} \pdfglyphtounicode{Csmall}{F763} \pdfglyphtounicode{D}{0044} \pdfglyphtounicode{DZ}{01F1} \pdfglyphtounicode{DZcaron}{01C4} \pdfglyphtounicode{Daarmenian}{0534} \pdfglyphtounicode{Dafrican}{0189} \pdfglyphtounicode{Dcaron}{010E} \pdfglyphtounicode{Dcedilla}{1E10} \pdfglyphtounicode{Dcircle}{24B9} \pdfglyphtounicode{Dcircumflexbelow}{1E12} \pdfglyphtounicode{Dcroat}{0110} \pdfglyphtounicode{Ddotaccent}{1E0A} \pdfglyphtounicode{Ddotbelow}{1E0C} \pdfglyphtounicode{Decyrillic}{0414} \pdfglyphtounicode{Deicoptic}{03EE} \pdfglyphtounicode{Delta}{2206} \pdfglyphtounicode{Deltagreek}{0394} \pdfglyphtounicode{Dhook}{018A} \pdfglyphtounicode{Dieresis}{F6CB} \pdfglyphtounicode{DieresisAcute}{F6CC} \pdfglyphtounicode{DieresisGrave}{F6CD} \pdfglyphtounicode{Dieresissmall}{F7A8} \pdfglyphtounicode{Digammagreek}{03DC} \pdfglyphtounicode{Djecyrillic}{0402} \pdfglyphtounicode{Dlinebelow}{1E0E} \pdfglyphtounicode{Dmonospace}{FF24} \pdfglyphtounicode{Dotaccentsmall}{F6F7} \pdfglyphtounicode{Dslash}{0110} \pdfglyphtounicode{Dsmall}{F764} \pdfglyphtounicode{Dtopbar}{018B} \pdfglyphtounicode{Dz}{01F2} \pdfglyphtounicode{Dzcaron}{01C5} \pdfglyphtounicode{Dzeabkhasiancyrillic}{04E0} \pdfglyphtounicode{Dzecyrillic}{0405} \pdfglyphtounicode{Dzhecyrillic}{040F} \pdfglyphtounicode{E}{0045} \pdfglyphtounicode{Eacute}{00C9} \pdfglyphtounicode{Eacutesmall}{F7E9} \pdfglyphtounicode{Ebreve}{0114} \pdfglyphtounicode{Ecaron}{011A} \pdfglyphtounicode{Ecedillabreve}{1E1C} \pdfglyphtounicode{Echarmenian}{0535} \pdfglyphtounicode{Ecircle}{24BA} \pdfglyphtounicode{Ecircumflex}{00CA} \pdfglyphtounicode{Ecircumflexacute}{1EBE} \pdfglyphtounicode{Ecircumflexbelow}{1E18} \pdfglyphtounicode{Ecircumflexdotbelow}{1EC6} \pdfglyphtounicode{Ecircumflexgrave}{1EC0} \pdfglyphtounicode{Ecircumflexhookabove}{1EC2} \pdfglyphtounicode{Ecircumflexsmall}{F7EA} \pdfglyphtounicode{Ecircumflextilde}{1EC4} \pdfglyphtounicode{Ecyrillic}{0404} \pdfglyphtounicode{Edblgrave}{0204} \pdfglyphtounicode{Edieresis}{00CB} \pdfglyphtounicode{Edieresissmall}{F7EB} \pdfglyphtounicode{Edot}{0116} \pdfglyphtounicode{Edotaccent}{0116} \pdfglyphtounicode{Edotbelow}{1EB8} \pdfglyphtounicode{Efcyrillic}{0424} \pdfglyphtounicode{Egrave}{00C8} \pdfglyphtounicode{Egravesmall}{F7E8} \pdfglyphtounicode{Eharmenian}{0537} \pdfglyphtounicode{Ehookabove}{1EBA} \pdfglyphtounicode{Eightroman}{2167} \pdfglyphtounicode{Einvertedbreve}{0206} \pdfglyphtounicode{Eiotifiedcyrillic}{0464} \pdfglyphtounicode{Elcyrillic}{041B} \pdfglyphtounicode{Elevenroman}{216A} \pdfglyphtounicode{Emacron}{0112} \pdfglyphtounicode{Emacronacute}{1E16} \pdfglyphtounicode{Emacrongrave}{1E14} \pdfglyphtounicode{Emcyrillic}{041C} \pdfglyphtounicode{Emonospace}{FF25} \pdfglyphtounicode{Encyrillic}{041D} \pdfglyphtounicode{Endescendercyrillic}{04A2} \pdfglyphtounicode{Eng}{014A} \pdfglyphtounicode{Enghecyrillic}{04A4} \pdfglyphtounicode{Enhookcyrillic}{04C7} \pdfglyphtounicode{Eogonek}{0118} \pdfglyphtounicode{Eopen}{0190} \pdfglyphtounicode{Epsilon}{0395} \pdfglyphtounicode{Epsilontonos}{0388} \pdfglyphtounicode{Ercyrillic}{0420} \pdfglyphtounicode{Ereversed}{018E} \pdfglyphtounicode{Ereversedcyrillic}{042D} \pdfglyphtounicode{Escyrillic}{0421} \pdfglyphtounicode{Esdescendercyrillic}{04AA} \pdfglyphtounicode{Esh}{01A9} \pdfglyphtounicode{Esmall}{F765} \pdfglyphtounicode{Eta}{0397} \pdfglyphtounicode{Etarmenian}{0538} \pdfglyphtounicode{Etatonos}{0389} \pdfglyphtounicode{Eth}{00D0} \pdfglyphtounicode{Ethsmall}{F7F0} \pdfglyphtounicode{Etilde}{1EBC} \pdfglyphtounicode{Etildebelow}{1E1A} \pdfglyphtounicode{Euro}{20AC} \pdfglyphtounicode{Ezh}{01B7} \pdfglyphtounicode{Ezhcaron}{01EE} \pdfglyphtounicode{Ezhreversed}{01B8} \pdfglyphtounicode{F}{0046} \pdfglyphtounicode{Fcircle}{24BB} \pdfglyphtounicode{Fdotaccent}{1E1E} \pdfglyphtounicode{Feharmenian}{0556} \pdfglyphtounicode{Feicoptic}{03E4} \pdfglyphtounicode{Fhook}{0191} \pdfglyphtounicode{Fitacyrillic}{0472} \pdfglyphtounicode{Fiveroman}{2164} \pdfglyphtounicode{Fmonospace}{FF26} \pdfglyphtounicode{Fourroman}{2163} \pdfglyphtounicode{Fsmall}{F766} \pdfglyphtounicode{G}{0047} \pdfglyphtounicode{GBsquare}{3387} \pdfglyphtounicode{Gacute}{01F4} \pdfglyphtounicode{Gamma}{0393} \pdfglyphtounicode{Gammaafrican}{0194} \pdfglyphtounicode{Gangiacoptic}{03EA} \pdfglyphtounicode{Gbreve}{011E} \pdfglyphtounicode{Gcaron}{01E6} \pdfglyphtounicode{Gcedilla}{0122} \pdfglyphtounicode{Gcircle}{24BC} \pdfglyphtounicode{Gcircumflex}{011C} \pdfglyphtounicode{Gcommaaccent}{0122} \pdfglyphtounicode{Gdot}{0120} \pdfglyphtounicode{Gdotaccent}{0120} \pdfglyphtounicode{Gecyrillic}{0413} \pdfglyphtounicode{Ghadarmenian}{0542} \pdfglyphtounicode{Ghemiddlehookcyrillic}{0494} \pdfglyphtounicode{Ghestrokecyrillic}{0492} \pdfglyphtounicode{Gheupturncyrillic}{0490} \pdfglyphtounicode{Ghook}{0193} \pdfglyphtounicode{Gimarmenian}{0533} \pdfglyphtounicode{Gjecyrillic}{0403} \pdfglyphtounicode{Gmacron}{1E20} \pdfglyphtounicode{Gmonospace}{FF27} \pdfglyphtounicode{Grave}{F6CE} \pdfglyphtounicode{Gravesmall}{F760} \pdfglyphtounicode{Gsmall}{F767} \pdfglyphtounicode{Gsmallhook}{029B} \pdfglyphtounicode{Gstroke}{01E4} \pdfglyphtounicode{H}{0048} \pdfglyphtounicode{H18533}{25CF} \pdfglyphtounicode{H18543}{25AA} \pdfglyphtounicode{H18551}{25AB} \pdfglyphtounicode{H22073}{25A1} \pdfglyphtounicode{HPsquare}{33CB} \pdfglyphtounicode{Haabkhasiancyrillic}{04A8} \pdfglyphtounicode{Hadescendercyrillic}{04B2} \pdfglyphtounicode{Hardsigncyrillic}{042A} \pdfglyphtounicode{Hbar}{0126} \pdfglyphtounicode{Hbrevebelow}{1E2A} \pdfglyphtounicode{Hcedilla}{1E28} \pdfglyphtounicode{Hcircle}{24BD} \pdfglyphtounicode{Hcircumflex}{0124} \pdfglyphtounicode{Hdieresis}{1E26} \pdfglyphtounicode{Hdotaccent}{1E22} \pdfglyphtounicode{Hdotbelow}{1E24} \pdfglyphtounicode{Hmonospace}{FF28} \pdfglyphtounicode{Hoarmenian}{0540} \pdfglyphtounicode{Horicoptic}{03E8} \pdfglyphtounicode{Hsmall}{F768} \pdfglyphtounicode{Hungarumlaut}{F6CF} \pdfglyphtounicode{Hungarumlautsmall}{F6F8} \pdfglyphtounicode{Hzsquare}{3390} \pdfglyphtounicode{I}{0049} \pdfglyphtounicode{IAcyrillic}{042F} \pdfglyphtounicode{IJ}{0132} \pdfglyphtounicode{IUcyrillic}{042E} \pdfglyphtounicode{Iacute}{00CD} \pdfglyphtounicode{Iacutesmall}{F7ED} \pdfglyphtounicode{Ibreve}{012C} \pdfglyphtounicode{Icaron}{01CF} \pdfglyphtounicode{Icircle}{24BE} \pdfglyphtounicode{Icircumflex}{00CE} \pdfglyphtounicode{Icircumflexsmall}{F7EE} \pdfglyphtounicode{Icyrillic}{0406} \pdfglyphtounicode{Idblgrave}{0208} \pdfglyphtounicode{Idieresis}{00CF} \pdfglyphtounicode{Idieresisacute}{1E2E} \pdfglyphtounicode{Idieresiscyrillic}{04E4} \pdfglyphtounicode{Idieresissmall}{F7EF} \pdfglyphtounicode{Idot}{0130} \pdfglyphtounicode{Idotaccent}{0130} \pdfglyphtounicode{Idotbelow}{1ECA} \pdfglyphtounicode{Iebrevecyrillic}{04D6} \pdfglyphtounicode{Iecyrillic}{0415} \pdfglyphtounicode{Ifraktur}{2111} \pdfglyphtounicode{Igrave}{00CC} \pdfglyphtounicode{Igravesmall}{F7EC} \pdfglyphtounicode{Ihookabove}{1EC8} \pdfglyphtounicode{Iicyrillic}{0418} \pdfglyphtounicode{Iinvertedbreve}{020A} \pdfglyphtounicode{Iishortcyrillic}{0419} \pdfglyphtounicode{Imacron}{012A} \pdfglyphtounicode{Imacroncyrillic}{04E2} \pdfglyphtounicode{Imonospace}{FF29} \pdfglyphtounicode{Iniarmenian}{053B} \pdfglyphtounicode{Iocyrillic}{0401} \pdfglyphtounicode{Iogonek}{012E} \pdfglyphtounicode{Iota}{0399} \pdfglyphtounicode{Iotaafrican}{0196} \pdfglyphtounicode{Iotadieresis}{03AA} \pdfglyphtounicode{Iotatonos}{038A} \pdfglyphtounicode{Ismall}{F769} \pdfglyphtounicode{Istroke}{0197} \pdfglyphtounicode{Itilde}{0128} \pdfglyphtounicode{Itildebelow}{1E2C} \pdfglyphtounicode{Izhitsacyrillic}{0474} \pdfglyphtounicode{Izhitsadblgravecyrillic}{0476} \pdfglyphtounicode{J}{004A} \pdfglyphtounicode{Jaarmenian}{0541} \pdfglyphtounicode{Jcircle}{24BF} \pdfglyphtounicode{Jcircumflex}{0134} \pdfglyphtounicode{Jecyrillic}{0408} \pdfglyphtounicode{Jheharmenian}{054B} \pdfglyphtounicode{Jmonospace}{FF2A} \pdfglyphtounicode{Jsmall}{F76A} \pdfglyphtounicode{K}{004B} \pdfglyphtounicode{KBsquare}{3385} \pdfglyphtounicode{KKsquare}{33CD} \pdfglyphtounicode{Kabashkircyrillic}{04A0} \pdfglyphtounicode{Kacute}{1E30} \pdfglyphtounicode{Kacyrillic}{041A} \pdfglyphtounicode{Kadescendercyrillic}{049A} \pdfglyphtounicode{Kahookcyrillic}{04C3} \pdfglyphtounicode{Kappa}{039A} \pdfglyphtounicode{Kastrokecyrillic}{049E} \pdfglyphtounicode{Kaverticalstrokecyrillic}{049C} \pdfglyphtounicode{Kcaron}{01E8} \pdfglyphtounicode{Kcedilla}{0136} \pdfglyphtounicode{Kcircle}{24C0} \pdfglyphtounicode{Kcommaaccent}{0136} \pdfglyphtounicode{Kdotbelow}{1E32} \pdfglyphtounicode{Keharmenian}{0554} \pdfglyphtounicode{Kenarmenian}{053F} \pdfglyphtounicode{Khacyrillic}{0425} \pdfglyphtounicode{Kheicoptic}{03E6} \pdfglyphtounicode{Khook}{0198} \pdfglyphtounicode{Kjecyrillic}{040C} \pdfglyphtounicode{Klinebelow}{1E34} \pdfglyphtounicode{Kmonospace}{FF2B} \pdfglyphtounicode{Koppacyrillic}{0480} \pdfglyphtounicode{Koppagreek}{03DE} \pdfglyphtounicode{Ksicyrillic}{046E} \pdfglyphtounicode{Ksmall}{F76B} \pdfglyphtounicode{L}{004C} \pdfglyphtounicode{LJ}{01C7} \pdfglyphtounicode{LL}{F6BF} \pdfglyphtounicode{Lacute}{0139} \pdfglyphtounicode{Lambda}{039B} \pdfglyphtounicode{Lcaron}{013D} \pdfglyphtounicode{Lcedilla}{013B} \pdfglyphtounicode{Lcircle}{24C1} \pdfglyphtounicode{Lcircumflexbelow}{1E3C} \pdfglyphtounicode{Lcommaaccent}{013B} \pdfglyphtounicode{Ldot}{013F} \pdfglyphtounicode{Ldotaccent}{013F} \pdfglyphtounicode{Ldotbelow}{1E36} \pdfglyphtounicode{Ldotbelowmacron}{1E38} \pdfglyphtounicode{Liwnarmenian}{053C} \pdfglyphtounicode{Lj}{01C8} \pdfglyphtounicode{Ljecyrillic}{0409} \pdfglyphtounicode{Llinebelow}{1E3A} \pdfglyphtounicode{Lmonospace}{FF2C} \pdfglyphtounicode{Lslash}{0141} \pdfglyphtounicode{Lslashsmall}{F6F9} \pdfglyphtounicode{Lsmall}{F76C} \pdfglyphtounicode{M}{004D} \pdfglyphtounicode{MBsquare}{3386} \pdfglyphtounicode{Macron}{F6D0} \pdfglyphtounicode{Macronsmall}{F7AF} \pdfglyphtounicode{Macute}{1E3E} \pdfglyphtounicode{Mcircle}{24C2} \pdfglyphtounicode{Mdotaccent}{1E40} \pdfglyphtounicode{Mdotbelow}{1E42} \pdfglyphtounicode{Menarmenian}{0544} \pdfglyphtounicode{Mmonospace}{FF2D} \pdfglyphtounicode{Msmall}{F76D} \pdfglyphtounicode{Mturned}{019C} \pdfglyphtounicode{Mu}{039C} \pdfglyphtounicode{N}{004E} \pdfglyphtounicode{NJ}{01CA} \pdfglyphtounicode{Nacute}{0143} \pdfglyphtounicode{Ncaron}{0147} \pdfglyphtounicode{Ncedilla}{0145} \pdfglyphtounicode{Ncircle}{24C3} \pdfglyphtounicode{Ncircumflexbelow}{1E4A} \pdfglyphtounicode{Ncommaaccent}{0145} \pdfglyphtounicode{Ndotaccent}{1E44} \pdfglyphtounicode{Ndotbelow}{1E46} \pdfglyphtounicode{Nhookleft}{019D} \pdfglyphtounicode{Nineroman}{2168} \pdfglyphtounicode{Nj}{01CB} \pdfglyphtounicode{Njecyrillic}{040A} \pdfglyphtounicode{Nlinebelow}{1E48} \pdfglyphtounicode{Nmonospace}{FF2E} \pdfglyphtounicode{Nowarmenian}{0546} \pdfglyphtounicode{Nsmall}{F76E} \pdfglyphtounicode{Ntilde}{00D1} \pdfglyphtounicode{Ntildesmall}{F7F1} \pdfglyphtounicode{Nu}{039D} \pdfglyphtounicode{O}{004F} \pdfglyphtounicode{OE}{0152} \pdfglyphtounicode{OEsmall}{F6FA} \pdfglyphtounicode{Oacute}{00D3} \pdfglyphtounicode{Oacutesmall}{F7F3} \pdfglyphtounicode{Obarredcyrillic}{04E8} \pdfglyphtounicode{Obarreddieresiscyrillic}{04EA} \pdfglyphtounicode{Obreve}{014E} \pdfglyphtounicode{Ocaron}{01D1} \pdfglyphtounicode{Ocenteredtilde}{019F} \pdfglyphtounicode{Ocircle}{24C4} \pdfglyphtounicode{Ocircumflex}{00D4} \pdfglyphtounicode{Ocircumflexacute}{1ED0} \pdfglyphtounicode{Ocircumflexdotbelow}{1ED8} \pdfglyphtounicode{Ocircumflexgrave}{1ED2} \pdfglyphtounicode{Ocircumflexhookabove}{1ED4} \pdfglyphtounicode{Ocircumflexsmall}{F7F4} \pdfglyphtounicode{Ocircumflextilde}{1ED6} \pdfglyphtounicode{Ocyrillic}{041E} \pdfglyphtounicode{Odblacute}{0150} \pdfglyphtounicode{Odblgrave}{020C} \pdfglyphtounicode{Odieresis}{00D6} \pdfglyphtounicode{Odieresiscyrillic}{04E6} \pdfglyphtounicode{Odieresissmall}{F7F6} \pdfglyphtounicode{Odotbelow}{1ECC} \pdfglyphtounicode{Ogoneksmall}{F6FB} \pdfglyphtounicode{Ograve}{00D2} \pdfglyphtounicode{Ogravesmall}{F7F2} \pdfglyphtounicode{Oharmenian}{0555} \pdfglyphtounicode{Ohm}{2126} \pdfglyphtounicode{Ohookabove}{1ECE} \pdfglyphtounicode{Ohorn}{01A0} \pdfglyphtounicode{Ohornacute}{1EDA} \pdfglyphtounicode{Ohorndotbelow}{1EE2} \pdfglyphtounicode{Ohorngrave}{1EDC} \pdfglyphtounicode{Ohornhookabove}{1EDE} \pdfglyphtounicode{Ohorntilde}{1EE0} \pdfglyphtounicode{Ohungarumlaut}{0150} \pdfglyphtounicode{Oi}{01A2} \pdfglyphtounicode{Oinvertedbreve}{020E} \pdfglyphtounicode{Omacron}{014C} \pdfglyphtounicode{Omacronacute}{1E52} \pdfglyphtounicode{Omacrongrave}{1E50} \pdfglyphtounicode{Omega}{2126} \pdfglyphtounicode{Omegacyrillic}{0460} \pdfglyphtounicode{Omegagreek}{03A9} \pdfglyphtounicode{Omegaroundcyrillic}{047A} \pdfglyphtounicode{Omegatitlocyrillic}{047C} \pdfglyphtounicode{Omegatonos}{038F} \pdfglyphtounicode{Omicron}{039F} \pdfglyphtounicode{Omicrontonos}{038C} \pdfglyphtounicode{Omonospace}{FF2F} \pdfglyphtounicode{Oneroman}{2160} \pdfglyphtounicode{Oogonek}{01EA} \pdfglyphtounicode{Oogonekmacron}{01EC} \pdfglyphtounicode{Oopen}{0186} \pdfglyphtounicode{Oslash}{00D8} \pdfglyphtounicode{Oslashacute}{01FE} \pdfglyphtounicode{Oslashsmall}{F7F8} \pdfglyphtounicode{Osmall}{F76F} \pdfglyphtounicode{Ostrokeacute}{01FE} \pdfglyphtounicode{Otcyrillic}{047E} \pdfglyphtounicode{Otilde}{00D5} \pdfglyphtounicode{Otildeacute}{1E4C} \pdfglyphtounicode{Otildedieresis}{1E4E} \pdfglyphtounicode{Otildesmall}{F7F5} \pdfglyphtounicode{P}{0050} \pdfglyphtounicode{Pacute}{1E54} \pdfglyphtounicode{Pcircle}{24C5} \pdfglyphtounicode{Pdotaccent}{1E56} \pdfglyphtounicode{Pecyrillic}{041F} \pdfglyphtounicode{Peharmenian}{054A} \pdfglyphtounicode{Pemiddlehookcyrillic}{04A6} \pdfglyphtounicode{Phi}{03A6} \pdfglyphtounicode{Phook}{01A4} \pdfglyphtounicode{Pi}{03A0} \pdfglyphtounicode{Piwrarmenian}{0553} \pdfglyphtounicode{Pmonospace}{FF30} \pdfglyphtounicode{Psi}{03A8} \pdfglyphtounicode{Psicyrillic}{0470} \pdfglyphtounicode{Psmall}{F770} \pdfglyphtounicode{Q}{0051} \pdfglyphtounicode{Qcircle}{24C6} \pdfglyphtounicode{Qmonospace}{FF31} \pdfglyphtounicode{Qsmall}{F771} \pdfglyphtounicode{R}{0052} \pdfglyphtounicode{Raarmenian}{054C} \pdfglyphtounicode{Racute}{0154} \pdfglyphtounicode{Rcaron}{0158} \pdfglyphtounicode{Rcedilla}{0156} \pdfglyphtounicode{Rcircle}{24C7} \pdfglyphtounicode{Rcommaaccent}{0156} \pdfglyphtounicode{Rdblgrave}{0210} \pdfglyphtounicode{Rdotaccent}{1E58} \pdfglyphtounicode{Rdotbelow}{1E5A} \pdfglyphtounicode{Rdotbelowmacron}{1E5C} \pdfglyphtounicode{Reharmenian}{0550} \pdfglyphtounicode{Rfraktur}{211C} \pdfglyphtounicode{Rho}{03A1} \pdfglyphtounicode{Ringsmall}{F6FC} \pdfglyphtounicode{Rinvertedbreve}{0212} \pdfglyphtounicode{Rlinebelow}{1E5E} \pdfglyphtounicode{Rmonospace}{FF32} \pdfglyphtounicode{Rsmall}{F772} \pdfglyphtounicode{Rsmallinverted}{0281} \pdfglyphtounicode{Rsmallinvertedsuperior}{02B6} \pdfglyphtounicode{S}{0053} \pdfglyphtounicode{SF010000}{250C} \pdfglyphtounicode{SF020000}{2514} \pdfglyphtounicode{SF030000}{2510} \pdfglyphtounicode{SF040000}{2518} \pdfglyphtounicode{SF050000}{253C} \pdfglyphtounicode{SF060000}{252C} \pdfglyphtounicode{SF070000}{2534} \pdfglyphtounicode{SF080000}{251C} \pdfglyphtounicode{SF090000}{2524} \pdfglyphtounicode{SF100000}{2500} \pdfglyphtounicode{SF110000}{2502} \pdfglyphtounicode{SF190000}{2561} \pdfglyphtounicode{SF200000}{2562} \pdfglyphtounicode{SF210000}{2556} \pdfglyphtounicode{SF220000}{2555} \pdfglyphtounicode{SF230000}{2563} \pdfglyphtounicode{SF240000}{2551} \pdfglyphtounicode{SF250000}{2557} \pdfglyphtounicode{SF260000}{255D} \pdfglyphtounicode{SF270000}{255C} \pdfglyphtounicode{SF280000}{255B} \pdfglyphtounicode{SF360000}{255E} \pdfglyphtounicode{SF370000}{255F} \pdfglyphtounicode{SF380000}{255A} \pdfglyphtounicode{SF390000}{2554} \pdfglyphtounicode{SF400000}{2569} \pdfglyphtounicode{SF410000}{2566} \pdfglyphtounicode{SF420000}{2560} \pdfglyphtounicode{SF430000}{2550} \pdfglyphtounicode{SF440000}{256C} \pdfglyphtounicode{SF450000}{2567} \pdfglyphtounicode{SF460000}{2568} \pdfglyphtounicode{SF470000}{2564} \pdfglyphtounicode{SF480000}{2565} \pdfglyphtounicode{SF490000}{2559} \pdfglyphtounicode{SF500000}{2558} \pdfglyphtounicode{SF510000}{2552} \pdfglyphtounicode{SF520000}{2553} \pdfglyphtounicode{SF530000}{256B} \pdfglyphtounicode{SF540000}{256A} \pdfglyphtounicode{Sacute}{015A} \pdfglyphtounicode{Sacutedotaccent}{1E64} \pdfglyphtounicode{Sampigreek}{03E0} \pdfglyphtounicode{Scaron}{0160} \pdfglyphtounicode{Scarondotaccent}{1E66} \pdfglyphtounicode{Scaronsmall}{F6FD} \pdfglyphtounicode{Scedilla}{015E} \pdfglyphtounicode{Schwa}{018F} \pdfglyphtounicode{Schwacyrillic}{04D8} \pdfglyphtounicode{Schwadieresiscyrillic}{04DA} \pdfglyphtounicode{Scircle}{24C8} \pdfglyphtounicode{Scircumflex}{015C} \pdfglyphtounicode{Scommaaccent}{0218} \pdfglyphtounicode{Sdotaccent}{1E60} \pdfglyphtounicode{Sdotbelow}{1E62} \pdfglyphtounicode{Sdotbelowdotaccent}{1E68} \pdfglyphtounicode{Seharmenian}{054D} \pdfglyphtounicode{Sevenroman}{2166} \pdfglyphtounicode{Shaarmenian}{0547} \pdfglyphtounicode{Shacyrillic}{0428} \pdfglyphtounicode{Shchacyrillic}{0429} \pdfglyphtounicode{Sheicoptic}{03E2} \pdfglyphtounicode{Shhacyrillic}{04BA} \pdfglyphtounicode{Shimacoptic}{03EC} \pdfglyphtounicode{Sigma}{03A3} \pdfglyphtounicode{Sixroman}{2165} \pdfglyphtounicode{Smonospace}{FF33} \pdfglyphtounicode{Softsigncyrillic}{042C} \pdfglyphtounicode{Ssmall}{F773} \pdfglyphtounicode{Stigmagreek}{03DA} \pdfglyphtounicode{T}{0054} \pdfglyphtounicode{Tau}{03A4} \pdfglyphtounicode{Tbar}{0166} \pdfglyphtounicode{Tcaron}{0164} \pdfglyphtounicode{Tcedilla}{0162} \pdfglyphtounicode{Tcircle}{24C9} \pdfglyphtounicode{Tcircumflexbelow}{1E70} \pdfglyphtounicode{Tcommaaccent}{0162} \pdfglyphtounicode{Tdotaccent}{1E6A} \pdfglyphtounicode{Tdotbelow}{1E6C} \pdfglyphtounicode{Tecyrillic}{0422} \pdfglyphtounicode{Tedescendercyrillic}{04AC} \pdfglyphtounicode{Tenroman}{2169} \pdfglyphtounicode{Tetsecyrillic}{04B4} \pdfglyphtounicode{Theta}{0398} \pdfglyphtounicode{Thook}{01AC} \pdfglyphtounicode{Thorn}{00DE} \pdfglyphtounicode{Thornsmall}{F7FE} \pdfglyphtounicode{Threeroman}{2162} \pdfglyphtounicode{Tildesmall}{F6FE} \pdfglyphtounicode{Tiwnarmenian}{054F} \pdfglyphtounicode{Tlinebelow}{1E6E} \pdfglyphtounicode{Tmonospace}{FF34} \pdfglyphtounicode{Toarmenian}{0539} \pdfglyphtounicode{Tonefive}{01BC} \pdfglyphtounicode{Tonesix}{0184} \pdfglyphtounicode{Tonetwo}{01A7} \pdfglyphtounicode{Tretroflexhook}{01AE} \pdfglyphtounicode{Tsecyrillic}{0426} \pdfglyphtounicode{Tshecyrillic}{040B} \pdfglyphtounicode{Tsmall}{F774} \pdfglyphtounicode{Twelveroman}{216B} \pdfglyphtounicode{Tworoman}{2161} \pdfglyphtounicode{U}{0055} \pdfglyphtounicode{Uacute}{00DA} \pdfglyphtounicode{Uacutesmall}{F7FA} \pdfglyphtounicode{Ubreve}{016C} \pdfglyphtounicode{Ucaron}{01D3} \pdfglyphtounicode{Ucircle}{24CA} \pdfglyphtounicode{Ucircumflex}{00DB} \pdfglyphtounicode{Ucircumflexbelow}{1E76} \pdfglyphtounicode{Ucircumflexsmall}{F7FB} \pdfglyphtounicode{Ucyrillic}{0423} \pdfglyphtounicode{Udblacute}{0170} \pdfglyphtounicode{Udblgrave}{0214} \pdfglyphtounicode{Udieresis}{00DC} \pdfglyphtounicode{Udieresisacute}{01D7} \pdfglyphtounicode{Udieresisbelow}{1E72} \pdfglyphtounicode{Udieresiscaron}{01D9} \pdfglyphtounicode{Udieresiscyrillic}{04F0} \pdfglyphtounicode{Udieresisgrave}{01DB} \pdfglyphtounicode{Udieresismacron}{01D5} \pdfglyphtounicode{Udieresissmall}{F7FC} \pdfglyphtounicode{Udotbelow}{1EE4} \pdfglyphtounicode{Ugrave}{00D9} \pdfglyphtounicode{Ugravesmall}{F7F9} \pdfglyphtounicode{Uhookabove}{1EE6} \pdfglyphtounicode{Uhorn}{01AF} \pdfglyphtounicode{Uhornacute}{1EE8} \pdfglyphtounicode{Uhorndotbelow}{1EF0} \pdfglyphtounicode{Uhorngrave}{1EEA} \pdfglyphtounicode{Uhornhookabove}{1EEC} \pdfglyphtounicode{Uhorntilde}{1EEE} \pdfglyphtounicode{Uhungarumlaut}{0170} \pdfglyphtounicode{Uhungarumlautcyrillic}{04F2} \pdfglyphtounicode{Uinvertedbreve}{0216} \pdfglyphtounicode{Ukcyrillic}{0478} \pdfglyphtounicode{Umacron}{016A} \pdfglyphtounicode{Umacroncyrillic}{04EE} \pdfglyphtounicode{Umacrondieresis}{1E7A} \pdfglyphtounicode{Umonospace}{FF35} \pdfglyphtounicode{Uogonek}{0172} \pdfglyphtounicode{Upsilon}{03A5} \pdfglyphtounicode{Upsilon1}{03D2} \pdfglyphtounicode{Upsilonacutehooksymbolgreek}{03D3} \pdfglyphtounicode{Upsilonafrican}{01B1} \pdfglyphtounicode{Upsilondieresis}{03AB} \pdfglyphtounicode{Upsilondieresishooksymbolgreek}{03D4} \pdfglyphtounicode{Upsilonhooksymbol}{03D2} \pdfglyphtounicode{Upsilontonos}{038E} \pdfglyphtounicode{Uring}{016E} \pdfglyphtounicode{Ushortcyrillic}{040E} \pdfglyphtounicode{Usmall}{F775} \pdfglyphtounicode{Ustraightcyrillic}{04AE} \pdfglyphtounicode{Ustraightstrokecyrillic}{04B0} \pdfglyphtounicode{Utilde}{0168} \pdfglyphtounicode{Utildeacute}{1E78} \pdfglyphtounicode{Utildebelow}{1E74} \pdfglyphtounicode{V}{0056} \pdfglyphtounicode{Vcircle}{24CB} \pdfglyphtounicode{Vdotbelow}{1E7E} \pdfglyphtounicode{Vecyrillic}{0412} \pdfglyphtounicode{Vewarmenian}{054E} \pdfglyphtounicode{Vhook}{01B2} \pdfglyphtounicode{Vmonospace}{FF36} \pdfglyphtounicode{Voarmenian}{0548} \pdfglyphtounicode{Vsmall}{F776} \pdfglyphtounicode{Vtilde}{1E7C} \pdfglyphtounicode{W}{0057} \pdfglyphtounicode{Wacute}{1E82} \pdfglyphtounicode{Wcircle}{24CC} \pdfglyphtounicode{Wcircumflex}{0174} \pdfglyphtounicode{Wdieresis}{1E84} \pdfglyphtounicode{Wdotaccent}{1E86} \pdfglyphtounicode{Wdotbelow}{1E88} \pdfglyphtounicode{Wgrave}{1E80} \pdfglyphtounicode{Wmonospace}{FF37} \pdfglyphtounicode{Wsmall}{F777} \pdfglyphtounicode{X}{0058} \pdfglyphtounicode{Xcircle}{24CD} \pdfglyphtounicode{Xdieresis}{1E8C} \pdfglyphtounicode{Xdotaccent}{1E8A} \pdfglyphtounicode{Xeharmenian}{053D} \pdfglyphtounicode{Xi}{039E} \pdfglyphtounicode{Xmonospace}{FF38} \pdfglyphtounicode{Xsmall}{F778} \pdfglyphtounicode{Y}{0059} \pdfglyphtounicode{Yacute}{00DD} \pdfglyphtounicode{Yacutesmall}{F7FD} \pdfglyphtounicode{Yatcyrillic}{0462} \pdfglyphtounicode{Ycircle}{24CE} \pdfglyphtounicode{Ycircumflex}{0176} \pdfglyphtounicode{Ydieresis}{0178} \pdfglyphtounicode{Ydieresissmall}{F7FF} \pdfglyphtounicode{Ydotaccent}{1E8E} \pdfglyphtounicode{Ydotbelow}{1EF4} \pdfglyphtounicode{Yericyrillic}{042B} \pdfglyphtounicode{Yerudieresiscyrillic}{04F8} \pdfglyphtounicode{Ygrave}{1EF2} \pdfglyphtounicode{Yhook}{01B3} \pdfglyphtounicode{Yhookabove}{1EF6} \pdfglyphtounicode{Yiarmenian}{0545} \pdfglyphtounicode{Yicyrillic}{0407} \pdfglyphtounicode{Yiwnarmenian}{0552} \pdfglyphtounicode{Ymonospace}{FF39} \pdfglyphtounicode{Ysmall}{F779} \pdfglyphtounicode{Ytilde}{1EF8} \pdfglyphtounicode{Yusbigcyrillic}{046A} \pdfglyphtounicode{Yusbigiotifiedcyrillic}{046C} \pdfglyphtounicode{Yuslittlecyrillic}{0466} \pdfglyphtounicode{Yuslittleiotifiedcyrillic}{0468} \pdfglyphtounicode{Z}{005A} \pdfglyphtounicode{Zaarmenian}{0536} \pdfglyphtounicode{Zacute}{0179} \pdfglyphtounicode{Zcaron}{017D} \pdfglyphtounicode{Zcaronsmall}{F6FF} \pdfglyphtounicode{Zcircle}{24CF} \pdfglyphtounicode{Zcircumflex}{1E90} \pdfglyphtounicode{Zdot}{017B} \pdfglyphtounicode{Zdotaccent}{017B} \pdfglyphtounicode{Zdotbelow}{1E92} \pdfglyphtounicode{Zecyrillic}{0417} \pdfglyphtounicode{Zedescendercyrillic}{0498} \pdfglyphtounicode{Zedieresiscyrillic}{04DE} \pdfglyphtounicode{Zeta}{0396} \pdfglyphtounicode{Zhearmenian}{053A} \pdfglyphtounicode{Zhebrevecyrillic}{04C1} \pdfglyphtounicode{Zhecyrillic}{0416} \pdfglyphtounicode{Zhedescendercyrillic}{0496} \pdfglyphtounicode{Zhedieresiscyrillic}{04DC} \pdfglyphtounicode{Zlinebelow}{1E94} \pdfglyphtounicode{Zmonospace}{FF3A} \pdfglyphtounicode{Zsmall}{F77A} \pdfglyphtounicode{Zstroke}{01B5} \pdfglyphtounicode{a}{0061} \pdfglyphtounicode{aabengali}{0986} \pdfglyphtounicode{aacute}{00E1} \pdfglyphtounicode{aadeva}{0906} \pdfglyphtounicode{aagujarati}{0A86} \pdfglyphtounicode{aagurmukhi}{0A06} \pdfglyphtounicode{aamatragurmukhi}{0A3E} \pdfglyphtounicode{aarusquare}{3303} \pdfglyphtounicode{aavowelsignbengali}{09BE} \pdfglyphtounicode{aavowelsigndeva}{093E} \pdfglyphtounicode{aavowelsigngujarati}{0ABE} \pdfglyphtounicode{abbreviationmarkarmenian}{055F} \pdfglyphtounicode{abbreviationsigndeva}{0970} \pdfglyphtounicode{abengali}{0985} \pdfglyphtounicode{abopomofo}{311A} \pdfglyphtounicode{abreve}{0103} \pdfglyphtounicode{abreveacute}{1EAF} \pdfglyphtounicode{abrevecyrillic}{04D1} \pdfglyphtounicode{abrevedotbelow}{1EB7} \pdfglyphtounicode{abrevegrave}{1EB1} \pdfglyphtounicode{abrevehookabove}{1EB3} \pdfglyphtounicode{abrevetilde}{1EB5} \pdfglyphtounicode{acaron}{01CE} \pdfglyphtounicode{acircle}{24D0} \pdfglyphtounicode{acircumflex}{00E2} \pdfglyphtounicode{acircumflexacute}{1EA5} \pdfglyphtounicode{acircumflexdotbelow}{1EAD} \pdfglyphtounicode{acircumflexgrave}{1EA7} \pdfglyphtounicode{acircumflexhookabove}{1EA9} \pdfglyphtounicode{acircumflextilde}{1EAB} \pdfglyphtounicode{acute}{00B4} \pdfglyphtounicode{acutebelowcmb}{0317} \pdfglyphtounicode{acutecmb}{0301} \pdfglyphtounicode{acutecomb}{0301} \pdfglyphtounicode{acutedeva}{0954} \pdfglyphtounicode{acutelowmod}{02CF} \pdfglyphtounicode{acutetonecmb}{0341} \pdfglyphtounicode{acyrillic}{0430} \pdfglyphtounicode{adblgrave}{0201} \pdfglyphtounicode{addakgurmukhi}{0A71} \pdfglyphtounicode{adeva}{0905} \pdfglyphtounicode{adieresis}{00E4} \pdfglyphtounicode{adieresiscyrillic}{04D3} \pdfglyphtounicode{adieresismacron}{01DF} \pdfglyphtounicode{adotbelow}{1EA1} \pdfglyphtounicode{adotmacron}{01E1} \pdfglyphtounicode{ae}{00E6} \pdfglyphtounicode{aeacute}{01FD} \pdfglyphtounicode{aekorean}{3150} \pdfglyphtounicode{aemacron}{01E3} \pdfglyphtounicode{afii00208}{2015} \pdfglyphtounicode{afii08941}{20A4} \pdfglyphtounicode{afii10017}{0410} \pdfglyphtounicode{afii10018}{0411} \pdfglyphtounicode{afii10019}{0412} \pdfglyphtounicode{afii10020}{0413} \pdfglyphtounicode{afii10021}{0414} \pdfglyphtounicode{afii10022}{0415} \pdfglyphtounicode{afii10023}{0401} \pdfglyphtounicode{afii10024}{0416} \pdfglyphtounicode{afii10025}{0417} \pdfglyphtounicode{afii10026}{0418} \pdfglyphtounicode{afii10027}{0419} \pdfglyphtounicode{afii10028}{041A} \pdfglyphtounicode{afii10029}{041B} \pdfglyphtounicode{afii10030}{041C} \pdfglyphtounicode{afii10031}{041D} \pdfglyphtounicode{afii10032}{041E} \pdfglyphtounicode{afii10033}{041F} \pdfglyphtounicode{afii10034}{0420} \pdfglyphtounicode{afii10035}{0421} \pdfglyphtounicode{afii10036}{0422} \pdfglyphtounicode{afii10037}{0423} \pdfglyphtounicode{afii10038}{0424} \pdfglyphtounicode{afii10039}{0425} \pdfglyphtounicode{afii10040}{0426} \pdfglyphtounicode{afii10041}{0427} \pdfglyphtounicode{afii10042}{0428} 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\pdfglyphtounicode{afii10070}{0435} \pdfglyphtounicode{afii10071}{0451} \pdfglyphtounicode{afii10072}{0436} \pdfglyphtounicode{afii10073}{0437} \pdfglyphtounicode{afii10074}{0438} \pdfglyphtounicode{afii10075}{0439} \pdfglyphtounicode{afii10076}{043A} \pdfglyphtounicode{afii10077}{043B} \pdfglyphtounicode{afii10078}{043C} \pdfglyphtounicode{afii10079}{043D} \pdfglyphtounicode{afii10080}{043E} \pdfglyphtounicode{afii10081}{043F} \pdfglyphtounicode{afii10082}{0440} \pdfglyphtounicode{afii10083}{0441} \pdfglyphtounicode{afii10084}{0442} \pdfglyphtounicode{afii10085}{0443} \pdfglyphtounicode{afii10086}{0444} \pdfglyphtounicode{afii10087}{0445} \pdfglyphtounicode{afii10088}{0446} \pdfglyphtounicode{afii10089}{0447} \pdfglyphtounicode{afii10090}{0448} \pdfglyphtounicode{afii10091}{0449} \pdfglyphtounicode{afii10092}{044A} \pdfglyphtounicode{afii10093}{044B} \pdfglyphtounicode{afii10094}{044C} \pdfglyphtounicode{afii10095}{044D} \pdfglyphtounicode{afii10096}{044E} 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\pdfglyphtounicode{afii57670}{05D6} \pdfglyphtounicode{afii57671}{05D7} \pdfglyphtounicode{afii57672}{05D8} \pdfglyphtounicode{afii57673}{05D9} \pdfglyphtounicode{afii57674}{05DA} \pdfglyphtounicode{afii57675}{05DB} \pdfglyphtounicode{afii57676}{05DC} \pdfglyphtounicode{afii57677}{05DD} \pdfglyphtounicode{afii57678}{05DE} \pdfglyphtounicode{afii57679}{05DF} \pdfglyphtounicode{afii57680}{05E0} \pdfglyphtounicode{afii57681}{05E1} \pdfglyphtounicode{afii57682}{05E2} \pdfglyphtounicode{afii57683}{05E3} \pdfglyphtounicode{afii57684}{05E4} \pdfglyphtounicode{afii57685}{05E5} \pdfglyphtounicode{afii57686}{05E6} \pdfglyphtounicode{afii57687}{05E7} \pdfglyphtounicode{afii57688}{05E8} \pdfglyphtounicode{afii57689}{05E9} \pdfglyphtounicode{afii57690}{05EA} \pdfglyphtounicode{afii57694}{FB2A} \pdfglyphtounicode{afii57695}{FB2B} \pdfglyphtounicode{afii57700}{FB4B} \pdfglyphtounicode{afii57705}{FB1F} \pdfglyphtounicode{afii57716}{05F0} \pdfglyphtounicode{afii57717}{05F1} \pdfglyphtounicode{afii57718}{05F2} \pdfglyphtounicode{afii57723}{FB35} \pdfglyphtounicode{afii57793}{05B4} \pdfglyphtounicode{afii57794}{05B5} \pdfglyphtounicode{afii57795}{05B6} \pdfglyphtounicode{afii57796}{05BB} \pdfglyphtounicode{afii57797}{05B8} \pdfglyphtounicode{afii57798}{05B7} \pdfglyphtounicode{afii57799}{05B0} \pdfglyphtounicode{afii57800}{05B2} \pdfglyphtounicode{afii57801}{05B1} \pdfglyphtounicode{afii57802}{05B3} \pdfglyphtounicode{afii57803}{05C2} \pdfglyphtounicode{afii57804}{05C1} \pdfglyphtounicode{afii57806}{05B9} \pdfglyphtounicode{afii57807}{05BC} \pdfglyphtounicode{afii57839}{05BD} \pdfglyphtounicode{afii57841}{05BF} \pdfglyphtounicode{afii57842}{05C0} \pdfglyphtounicode{afii57929}{02BC} \pdfglyphtounicode{afii61248}{2105} \pdfglyphtounicode{afii61289}{2113} \pdfglyphtounicode{afii61352}{2116} \pdfglyphtounicode{afii61573}{202C} \pdfglyphtounicode{afii61574}{202D} \pdfglyphtounicode{afii61575}{202E} \pdfglyphtounicode{afii61664}{200C} \pdfglyphtounicode{afii63167}{066D} \pdfglyphtounicode{afii64937}{02BD} \pdfglyphtounicode{agrave}{00E0} \pdfglyphtounicode{agujarati}{0A85} \pdfglyphtounicode{agurmukhi}{0A05} \pdfglyphtounicode{ahiragana}{3042} \pdfglyphtounicode{ahookabove}{1EA3} \pdfglyphtounicode{aibengali}{0990} \pdfglyphtounicode{aibopomofo}{311E} \pdfglyphtounicode{aideva}{0910} \pdfglyphtounicode{aiecyrillic}{04D5} \pdfglyphtounicode{aigujarati}{0A90} \pdfglyphtounicode{aigurmukhi}{0A10} \pdfglyphtounicode{aimatragurmukhi}{0A48} \pdfglyphtounicode{ainarabic}{0639} \pdfglyphtounicode{ainfinalarabic}{FECA} \pdfglyphtounicode{aininitialarabic}{FECB} \pdfglyphtounicode{ainmedialarabic}{FECC} \pdfglyphtounicode{ainvertedbreve}{0203} \pdfglyphtounicode{aivowelsignbengali}{09C8} \pdfglyphtounicode{aivowelsigndeva}{0948} \pdfglyphtounicode{aivowelsigngujarati}{0AC8} \pdfglyphtounicode{akatakana}{30A2} \pdfglyphtounicode{akatakanahalfwidth}{FF71} \pdfglyphtounicode{akorean}{314F} \pdfglyphtounicode{alef}{05D0} \pdfglyphtounicode{alefarabic}{0627} \pdfglyphtounicode{alefdageshhebrew}{FB30} \pdfglyphtounicode{aleffinalarabic}{FE8E} \pdfglyphtounicode{alefhamzaabovearabic}{0623} \pdfglyphtounicode{alefhamzaabovefinalarabic}{FE84} \pdfglyphtounicode{alefhamzabelowarabic}{0625} \pdfglyphtounicode{alefhamzabelowfinalarabic}{FE88} \pdfglyphtounicode{alefhebrew}{05D0} \pdfglyphtounicode{aleflamedhebrew}{FB4F} \pdfglyphtounicode{alefmaddaabovearabic}{0622} \pdfglyphtounicode{alefmaddaabovefinalarabic}{FE82} \pdfglyphtounicode{alefmaksuraarabic}{0649} \pdfglyphtounicode{alefmaksurafinalarabic}{FEF0} \pdfglyphtounicode{alefmaksurainitialarabic}{FEF3} \pdfglyphtounicode{alefmaksuramedialarabic}{FEF4} \pdfglyphtounicode{alefpatahhebrew}{FB2E} \pdfglyphtounicode{alefqamatshebrew}{FB2F} \pdfglyphtounicode{aleph}{2135} \pdfglyphtounicode{allequal}{224C} \pdfglyphtounicode{alpha}{03B1} \pdfglyphtounicode{alphatonos}{03AC} \pdfglyphtounicode{amacron}{0101} \pdfglyphtounicode{amonospace}{FF41} \pdfglyphtounicode{ampersand}{0026} \pdfglyphtounicode{ampersandmonospace}{FF06} \pdfglyphtounicode{ampersandsmall}{F726} \pdfglyphtounicode{amsquare}{33C2} \pdfglyphtounicode{anbopomofo}{3122} \pdfglyphtounicode{angbopomofo}{3124} \pdfglyphtounicode{angkhankhuthai}{0E5A} \pdfglyphtounicode{angle}{2220} \pdfglyphtounicode{anglebracketleft}{3008} \pdfglyphtounicode{anglebracketleftvertical}{FE3F} \pdfglyphtounicode{anglebracketright}{3009} \pdfglyphtounicode{anglebracketrightvertical}{FE40} \pdfglyphtounicode{angleleft}{2329} \pdfglyphtounicode{angleright}{232A} \pdfglyphtounicode{angstrom}{212B} \pdfglyphtounicode{anoteleia}{0387} \pdfglyphtounicode{anudattadeva}{0952} \pdfglyphtounicode{anusvarabengali}{0982} \pdfglyphtounicode{anusvaradeva}{0902} \pdfglyphtounicode{anusvaragujarati}{0A82} \pdfglyphtounicode{aogonek}{0105} \pdfglyphtounicode{apaatosquare}{3300} \pdfglyphtounicode{aparen}{249C} \pdfglyphtounicode{apostrophearmenian}{055A} \pdfglyphtounicode{apostrophemod}{02BC} \pdfglyphtounicode{apple}{F8FF} \pdfglyphtounicode{approaches}{2250} \pdfglyphtounicode{approxequal}{2248} \pdfglyphtounicode{approxequalorimage}{2252} \pdfglyphtounicode{approximatelyequal}{2245} \pdfglyphtounicode{araeaekorean}{318E} \pdfglyphtounicode{araeakorean}{318D} \pdfglyphtounicode{arc}{2312} \pdfglyphtounicode{arighthalfring}{1E9A} \pdfglyphtounicode{aring}{00E5} \pdfglyphtounicode{aringacute}{01FB} \pdfglyphtounicode{aringbelow}{1E01} \pdfglyphtounicode{arrowboth}{2194} \pdfglyphtounicode{arrowdashdown}{21E3} \pdfglyphtounicode{arrowdashleft}{21E0} \pdfglyphtounicode{arrowdashright}{21E2} \pdfglyphtounicode{arrowdashup}{21E1} \pdfglyphtounicode{arrowdblboth}{21D4} \pdfglyphtounicode{arrowdbldown}{21D3} \pdfglyphtounicode{arrowdblleft}{21D0} \pdfglyphtounicode{arrowdblright}{21D2} \pdfglyphtounicode{arrowdblup}{21D1} \pdfglyphtounicode{arrowdown}{2193} \pdfglyphtounicode{arrowdownleft}{2199} \pdfglyphtounicode{arrowdownright}{2198} \pdfglyphtounicode{arrowdownwhite}{21E9} \pdfglyphtounicode{arrowheaddownmod}{02C5} \pdfglyphtounicode{arrowheadleftmod}{02C2} \pdfglyphtounicode{arrowheadrightmod}{02C3} \pdfglyphtounicode{arrowheadupmod}{02C4} \pdfglyphtounicode{arrowhorizex}{F8E7} \pdfglyphtounicode{arrowleft}{2190} \pdfglyphtounicode{arrowleftdbl}{21D0} \pdfglyphtounicode{arrowleftdblstroke}{21CD} \pdfglyphtounicode{arrowleftoverright}{21C6} \pdfglyphtounicode{arrowleftwhite}{21E6} \pdfglyphtounicode{arrowright}{2192} \pdfglyphtounicode{arrowrightdblstroke}{21CF} \pdfglyphtounicode{arrowrightheavy}{279E} \pdfglyphtounicode{arrowrightoverleft}{21C4} \pdfglyphtounicode{arrowrightwhite}{21E8} \pdfglyphtounicode{arrowtableft}{21E4} \pdfglyphtounicode{arrowtabright}{21E5} \pdfglyphtounicode{arrowup}{2191} \pdfglyphtounicode{arrowupdn}{2195} \pdfglyphtounicode{arrowupdnbse}{21A8} \pdfglyphtounicode{arrowupdownbase}{21A8} \pdfglyphtounicode{arrowupleft}{2196} \pdfglyphtounicode{arrowupleftofdown}{21C5} \pdfglyphtounicode{arrowupright}{2197} \pdfglyphtounicode{arrowupwhite}{21E7} \pdfglyphtounicode{arrowvertex}{F8E6} \pdfglyphtounicode{asciicircum}{005E} \pdfglyphtounicode{asciicircummonospace}{FF3E} \pdfglyphtounicode{asciitilde}{007E} \pdfglyphtounicode{asciitildemonospace}{FF5E} \pdfglyphtounicode{ascript}{0251} \pdfglyphtounicode{ascriptturned}{0252} \pdfglyphtounicode{asmallhiragana}{3041} \pdfglyphtounicode{asmallkatakana}{30A1} \pdfglyphtounicode{asmallkatakanahalfwidth}{FF67} \pdfglyphtounicode{asterisk}{002A} \pdfglyphtounicode{asteriskaltonearabic}{066D} \pdfglyphtounicode{asteriskarabic}{066D} \pdfglyphtounicode{asteriskmath}{2217} \pdfglyphtounicode{asteriskmonospace}{FF0A} \pdfglyphtounicode{asterisksmall}{FE61} \pdfglyphtounicode{asterism}{2042} \pdfglyphtounicode{asuperior}{F6E9} \pdfglyphtounicode{asymptoticallyequal}{2243} \pdfglyphtounicode{at}{0040} \pdfglyphtounicode{atilde}{00E3} \pdfglyphtounicode{atmonospace}{FF20} \pdfglyphtounicode{atsmall}{FE6B} \pdfglyphtounicode{aturned}{0250} \pdfglyphtounicode{aubengali}{0994} \pdfglyphtounicode{aubopomofo}{3120} \pdfglyphtounicode{audeva}{0914} \pdfglyphtounicode{augujarati}{0A94} \pdfglyphtounicode{augurmukhi}{0A14} \pdfglyphtounicode{aulengthmarkbengali}{09D7} \pdfglyphtounicode{aumatragurmukhi}{0A4C} \pdfglyphtounicode{auvowelsignbengali}{09CC} \pdfglyphtounicode{auvowelsigndeva}{094C} \pdfglyphtounicode{auvowelsigngujarati}{0ACC} \pdfglyphtounicode{avagrahadeva}{093D} \pdfglyphtounicode{aybarmenian}{0561} \pdfglyphtounicode{ayin}{05E2} \pdfglyphtounicode{ayinaltonehebrew}{FB20} \pdfglyphtounicode{ayinhebrew}{05E2} \pdfglyphtounicode{b}{0062} \pdfglyphtounicode{babengali}{09AC} \pdfglyphtounicode{backslash}{005C} \pdfglyphtounicode{backslashmonospace}{FF3C} \pdfglyphtounicode{badeva}{092C} \pdfglyphtounicode{bagujarati}{0AAC} \pdfglyphtounicode{bagurmukhi}{0A2C} \pdfglyphtounicode{bahiragana}{3070} \pdfglyphtounicode{bahtthai}{0E3F} \pdfglyphtounicode{bakatakana}{30D0} \pdfglyphtounicode{bar}{007C} \pdfglyphtounicode{barmonospace}{FF5C} \pdfglyphtounicode{bbopomofo}{3105} \pdfglyphtounicode{bcircle}{24D1} \pdfglyphtounicode{bdotaccent}{1E03} \pdfglyphtounicode{bdotbelow}{1E05} \pdfglyphtounicode{beamedsixteenthnotes}{266C} \pdfglyphtounicode{because}{2235} \pdfglyphtounicode{becyrillic}{0431} \pdfglyphtounicode{beharabic}{0628} \pdfglyphtounicode{behfinalarabic}{FE90} \pdfglyphtounicode{behinitialarabic}{FE91} \pdfglyphtounicode{behiragana}{3079} \pdfglyphtounicode{behmedialarabic}{FE92} \pdfglyphtounicode{behmeeminitialarabic}{FC9F} \pdfglyphtounicode{behmeemisolatedarabic}{FC08} \pdfglyphtounicode{behnoonfinalarabic}{FC6D} \pdfglyphtounicode{bekatakana}{30D9} \pdfglyphtounicode{benarmenian}{0562} \pdfglyphtounicode{bet}{05D1} \pdfglyphtounicode{beta}{03B2} \pdfglyphtounicode{betasymbolgreek}{03D0} \pdfglyphtounicode{betdagesh}{FB31} \pdfglyphtounicode{betdageshhebrew}{FB31} \pdfglyphtounicode{bethebrew}{05D1} \pdfglyphtounicode{betrafehebrew}{FB4C} \pdfglyphtounicode{bhabengali}{09AD} \pdfglyphtounicode{bhadeva}{092D} \pdfglyphtounicode{bhagujarati}{0AAD} \pdfglyphtounicode{bhagurmukhi}{0A2D} \pdfglyphtounicode{bhook}{0253} \pdfglyphtounicode{bihiragana}{3073} \pdfglyphtounicode{bikatakana}{30D3} \pdfglyphtounicode{bilabialclick}{0298} \pdfglyphtounicode{bindigurmukhi}{0A02} \pdfglyphtounicode{birusquare}{3331} \pdfglyphtounicode{blackcircle}{25CF} \pdfglyphtounicode{blackdiamond}{25C6} \pdfglyphtounicode{blackdownpointingtriangle}{25BC} \pdfglyphtounicode{blackleftpointingpointer}{25C4} \pdfglyphtounicode{blackleftpointingtriangle}{25C0} \pdfglyphtounicode{blacklenticularbracketleft}{3010} \pdfglyphtounicode{blacklenticularbracketleftvertical}{FE3B} \pdfglyphtounicode{blacklenticularbracketright}{3011} \pdfglyphtounicode{blacklenticularbracketrightvertical}{FE3C} \pdfglyphtounicode{blacklowerlefttriangle}{25E3} \pdfglyphtounicode{blacklowerrighttriangle}{25E2} \pdfglyphtounicode{blackrectangle}{25AC} \pdfglyphtounicode{blackrightpointingpointer}{25BA} \pdfglyphtounicode{blackrightpointingtriangle}{25B6} \pdfglyphtounicode{blacksmallsquare}{25AA} \pdfglyphtounicode{blacksmilingface}{263B} \pdfglyphtounicode{blacksquare}{25A0} \pdfglyphtounicode{blackstar}{2605} \pdfglyphtounicode{blackupperlefttriangle}{25E4} \pdfglyphtounicode{blackupperrighttriangle}{25E5} \pdfglyphtounicode{blackuppointingsmalltriangle}{25B4} \pdfglyphtounicode{blackuppointingtriangle}{25B2} \pdfglyphtounicode{blank}{2423} \pdfglyphtounicode{blinebelow}{1E07} \pdfglyphtounicode{block}{2588} \pdfglyphtounicode{bmonospace}{FF42} \pdfglyphtounicode{bobaimaithai}{0E1A} \pdfglyphtounicode{bohiragana}{307C} \pdfglyphtounicode{bokatakana}{30DC} \pdfglyphtounicode{bparen}{249D} \pdfglyphtounicode{bqsquare}{33C3} \pdfglyphtounicode{braceex}{F8F4} \pdfglyphtounicode{braceleft}{007B} \pdfglyphtounicode{braceleftbt}{F8F3} \pdfglyphtounicode{braceleftmid}{F8F2} \pdfglyphtounicode{braceleftmonospace}{FF5B} \pdfglyphtounicode{braceleftsmall}{FE5B} \pdfglyphtounicode{bracelefttp}{F8F1} \pdfglyphtounicode{braceleftvertical}{FE37} \pdfglyphtounicode{braceright}{007D} \pdfglyphtounicode{bracerightbt}{F8FE} \pdfglyphtounicode{bracerightmid}{F8FD} \pdfglyphtounicode{bracerightmonospace}{FF5D} \pdfglyphtounicode{bracerightsmall}{FE5C} \pdfglyphtounicode{bracerighttp}{F8FC} \pdfglyphtounicode{bracerightvertical}{FE38} \pdfglyphtounicode{bracketleft}{005B} \pdfglyphtounicode{bracketleftbt}{F8F0} \pdfglyphtounicode{bracketleftex}{F8EF} \pdfglyphtounicode{bracketleftmonospace}{FF3B} \pdfglyphtounicode{bracketlefttp}{F8EE} \pdfglyphtounicode{bracketright}{005D} \pdfglyphtounicode{bracketrightbt}{F8FB} \pdfglyphtounicode{bracketrightex}{F8FA} \pdfglyphtounicode{bracketrightmonospace}{FF3D} \pdfglyphtounicode{bracketrighttp}{F8F9} \pdfglyphtounicode{breve}{02D8} \pdfglyphtounicode{brevebelowcmb}{032E} \pdfglyphtounicode{brevecmb}{0306} \pdfglyphtounicode{breveinvertedbelowcmb}{032F} \pdfglyphtounicode{breveinvertedcmb}{0311} \pdfglyphtounicode{breveinverteddoublecmb}{0361} \pdfglyphtounicode{bridgebelowcmb}{032A} \pdfglyphtounicode{bridgeinvertedbelowcmb}{033A} \pdfglyphtounicode{brokenbar}{00A6} \pdfglyphtounicode{bstroke}{0180} \pdfglyphtounicode{bsuperior}{F6EA} \pdfglyphtounicode{btopbar}{0183} \pdfglyphtounicode{buhiragana}{3076} \pdfglyphtounicode{bukatakana}{30D6} \pdfglyphtounicode{bullet}{2022} \pdfglyphtounicode{bulletinverse}{25D8} \pdfglyphtounicode{bulletoperator}{2219} \pdfglyphtounicode{bullseye}{25CE} \pdfglyphtounicode{c}{0063} \pdfglyphtounicode{caarmenian}{056E} \pdfglyphtounicode{cabengali}{099A} \pdfglyphtounicode{cacute}{0107} \pdfglyphtounicode{cadeva}{091A} \pdfglyphtounicode{cagujarati}{0A9A} \pdfglyphtounicode{cagurmukhi}{0A1A} \pdfglyphtounicode{calsquare}{3388} \pdfglyphtounicode{candrabindubengali}{0981} \pdfglyphtounicode{candrabinducmb}{0310} \pdfglyphtounicode{candrabindudeva}{0901} \pdfglyphtounicode{candrabindugujarati}{0A81} \pdfglyphtounicode{capslock}{21EA} \pdfglyphtounicode{careof}{2105} \pdfglyphtounicode{caron}{02C7} \pdfglyphtounicode{caronbelowcmb}{032C} \pdfglyphtounicode{caroncmb}{030C} \pdfglyphtounicode{carriagereturn}{21B5} \pdfglyphtounicode{cbopomofo}{3118} \pdfglyphtounicode{ccaron}{010D} \pdfglyphtounicode{ccedilla}{00E7} \pdfglyphtounicode{ccedillaacute}{1E09} \pdfglyphtounicode{ccircle}{24D2} \pdfglyphtounicode{ccircumflex}{0109} \pdfglyphtounicode{ccurl}{0255} \pdfglyphtounicode{cdot}{010B} \pdfglyphtounicode{cdotaccent}{010B} \pdfglyphtounicode{cdsquare}{33C5} \pdfglyphtounicode{cedilla}{00B8} \pdfglyphtounicode{cedillacmb}{0327} \pdfglyphtounicode{cent}{00A2} \pdfglyphtounicode{centigrade}{2103} \pdfglyphtounicode{centinferior}{F6DF} \pdfglyphtounicode{centmonospace}{FFE0} \pdfglyphtounicode{centoldstyle}{F7A2} \pdfglyphtounicode{centsuperior}{F6E0} \pdfglyphtounicode{chaarmenian}{0579} \pdfglyphtounicode{chabengali}{099B} \pdfglyphtounicode{chadeva}{091B} \pdfglyphtounicode{chagujarati}{0A9B} \pdfglyphtounicode{chagurmukhi}{0A1B} \pdfglyphtounicode{chbopomofo}{3114} \pdfglyphtounicode{cheabkhasiancyrillic}{04BD} \pdfglyphtounicode{checkmark}{2713} \pdfglyphtounicode{checyrillic}{0447} \pdfglyphtounicode{chedescenderabkhasiancyrillic}{04BF} \pdfglyphtounicode{chedescendercyrillic}{04B7} \pdfglyphtounicode{chedieresiscyrillic}{04F5} \pdfglyphtounicode{cheharmenian}{0573} \pdfglyphtounicode{chekhakassiancyrillic}{04CC} \pdfglyphtounicode{cheverticalstrokecyrillic}{04B9} \pdfglyphtounicode{chi}{03C7} \pdfglyphtounicode{chieuchacirclekorean}{3277} \pdfglyphtounicode{chieuchaparenkorean}{3217} \pdfglyphtounicode{chieuchcirclekorean}{3269} \pdfglyphtounicode{chieuchkorean}{314A} \pdfglyphtounicode{chieuchparenkorean}{3209} \pdfglyphtounicode{chochangthai}{0E0A} \pdfglyphtounicode{chochanthai}{0E08} \pdfglyphtounicode{chochingthai}{0E09} \pdfglyphtounicode{chochoethai}{0E0C} \pdfglyphtounicode{chook}{0188} \pdfglyphtounicode{cieucacirclekorean}{3276} \pdfglyphtounicode{cieucaparenkorean}{3216} \pdfglyphtounicode{cieuccirclekorean}{3268} \pdfglyphtounicode{cieuckorean}{3148} \pdfglyphtounicode{cieucparenkorean}{3208} \pdfglyphtounicode{cieucuparenkorean}{321C} \pdfglyphtounicode{circle}{25CB} \pdfglyphtounicode{circlemultiply}{2297} \pdfglyphtounicode{circleot}{2299} \pdfglyphtounicode{circleplus}{2295} \pdfglyphtounicode{circlepostalmark}{3036} \pdfglyphtounicode{circlewithlefthalfblack}{25D0} \pdfglyphtounicode{circlewithrighthalfblack}{25D1} \pdfglyphtounicode{circumflex}{02C6} \pdfglyphtounicode{circumflexbelowcmb}{032D} \pdfglyphtounicode{circumflexcmb}{0302} \pdfglyphtounicode{clear}{2327} \pdfglyphtounicode{clickalveolar}{01C2} \pdfglyphtounicode{clickdental}{01C0} \pdfglyphtounicode{clicklateral}{01C1} \pdfglyphtounicode{clickretroflex}{01C3} \pdfglyphtounicode{club}{2663} \pdfglyphtounicode{clubsuitblack}{2663} \pdfglyphtounicode{clubsuitwhite}{2667} \pdfglyphtounicode{cmcubedsquare}{33A4} \pdfglyphtounicode{cmonospace}{FF43} \pdfglyphtounicode{cmsquaredsquare}{33A0} \pdfglyphtounicode{coarmenian}{0581} \pdfglyphtounicode{colon}{003A} \pdfglyphtounicode{colonmonetary}{20A1} \pdfglyphtounicode{colonmonospace}{FF1A} \pdfglyphtounicode{colonsign}{20A1} \pdfglyphtounicode{colonsmall}{FE55} \pdfglyphtounicode{colontriangularhalfmod}{02D1} \pdfglyphtounicode{colontriangularmod}{02D0} \pdfglyphtounicode{comma}{002C} \pdfglyphtounicode{commaabovecmb}{0313} \pdfglyphtounicode{commaaboverightcmb}{0315} \pdfglyphtounicode{commaaccent}{F6C3} \pdfglyphtounicode{commaarabic}{060C} \pdfglyphtounicode{commaarmenian}{055D} \pdfglyphtounicode{commainferior}{F6E1} \pdfglyphtounicode{commamonospace}{FF0C} \pdfglyphtounicode{commareversedabovecmb}{0314} \pdfglyphtounicode{commareversedmod}{02BD} \pdfglyphtounicode{commasmall}{FE50} \pdfglyphtounicode{commasuperior}{F6E2} \pdfglyphtounicode{commaturnedabovecmb}{0312} \pdfglyphtounicode{commaturnedmod}{02BB} \pdfglyphtounicode{compass}{263C} \pdfglyphtounicode{congruent}{2245} \pdfglyphtounicode{contourintegral}{222E} \pdfglyphtounicode{control}{2303} \pdfglyphtounicode{controlACK}{0006} \pdfglyphtounicode{controlBEL}{0007} \pdfglyphtounicode{controlBS}{0008} \pdfglyphtounicode{controlCAN}{0018} \pdfglyphtounicode{controlCR}{000D} \pdfglyphtounicode{controlDC1}{0011} \pdfglyphtounicode{controlDC2}{0012} \pdfglyphtounicode{controlDC3}{0013} \pdfglyphtounicode{controlDC4}{0014} \pdfglyphtounicode{controlDEL}{007F} \pdfglyphtounicode{controlDLE}{0010} \pdfglyphtounicode{controlEM}{0019} \pdfglyphtounicode{controlENQ}{0005} \pdfglyphtounicode{controlEOT}{0004} \pdfglyphtounicode{controlESC}{001B} \pdfglyphtounicode{controlETB}{0017} \pdfglyphtounicode{controlETX}{0003} \pdfglyphtounicode{controlFF}{000C} \pdfglyphtounicode{controlFS}{001C} \pdfglyphtounicode{controlGS}{001D} \pdfglyphtounicode{controlHT}{0009} \pdfglyphtounicode{controlLF}{000A} \pdfglyphtounicode{controlNAK}{0015} \pdfglyphtounicode{controlRS}{001E} \pdfglyphtounicode{controlSI}{000F} \pdfglyphtounicode{controlSO}{000E} \pdfglyphtounicode{controlSOT}{0002} \pdfglyphtounicode{controlSTX}{0001} \pdfglyphtounicode{controlSUB}{001A} \pdfglyphtounicode{controlSYN}{0016} \pdfglyphtounicode{controlUS}{001F} \pdfglyphtounicode{controlVT}{000B} \pdfglyphtounicode{copyright}{00A9} \pdfglyphtounicode{copyrightsans}{F8E9} \pdfglyphtounicode{copyrightserif}{F6D9} \pdfglyphtounicode{cornerbracketleft}{300C} \pdfglyphtounicode{cornerbracketlefthalfwidth}{FF62} \pdfglyphtounicode{cornerbracketleftvertical}{FE41} \pdfglyphtounicode{cornerbracketright}{300D} \pdfglyphtounicode{cornerbracketrighthalfwidth}{FF63} \pdfglyphtounicode{cornerbracketrightvertical}{FE42} \pdfglyphtounicode{corporationsquare}{337F} \pdfglyphtounicode{cosquare}{33C7} \pdfglyphtounicode{coverkgsquare}{33C6} \pdfglyphtounicode{cparen}{249E} \pdfglyphtounicode{cruzeiro}{20A2} \pdfglyphtounicode{cstretched}{0297} \pdfglyphtounicode{curlyand}{22CF} \pdfglyphtounicode{curlyor}{22CE} \pdfglyphtounicode{currency}{00A4} \pdfglyphtounicode{cyrBreve}{F6D1} \pdfglyphtounicode{cyrFlex}{F6D2} \pdfglyphtounicode{cyrbreve}{F6D4} \pdfglyphtounicode{cyrflex}{F6D5} \pdfglyphtounicode{d}{0064} \pdfglyphtounicode{daarmenian}{0564} \pdfglyphtounicode{dabengali}{09A6} \pdfglyphtounicode{dadarabic}{0636} \pdfglyphtounicode{dadeva}{0926} \pdfglyphtounicode{dadfinalarabic}{FEBE} \pdfglyphtounicode{dadinitialarabic}{FEBF} \pdfglyphtounicode{dadmedialarabic}{FEC0} \pdfglyphtounicode{dagesh}{05BC} \pdfglyphtounicode{dageshhebrew}{05BC} \pdfglyphtounicode{dagger}{2020} \pdfglyphtounicode{daggerdbl}{2021} \pdfglyphtounicode{dagujarati}{0AA6} \pdfglyphtounicode{dagurmukhi}{0A26} \pdfglyphtounicode{dahiragana}{3060} \pdfglyphtounicode{dakatakana}{30C0} \pdfglyphtounicode{dalarabic}{062F} \pdfglyphtounicode{dalet}{05D3} \pdfglyphtounicode{daletdagesh}{FB33} \pdfglyphtounicode{daletdageshhebrew}{FB33} % dalethatafpatah;05D3 05B2 % dalethatafpatahhebrew;05D3 05B2 % dalethatafsegol;05D3 05B1 % dalethatafsegolhebrew;05D3 05B1 \pdfglyphtounicode{dalethebrew}{05D3} % dalethiriq;05D3 05B4 % dalethiriqhebrew;05D3 05B4 % daletholam;05D3 05B9 % daletholamhebrew;05D3 05B9 % daletpatah;05D3 05B7 % daletpatahhebrew;05D3 05B7 % daletqamats;05D3 05B8 % daletqamatshebrew;05D3 05B8 % daletqubuts;05D3 05BB % daletqubutshebrew;05D3 05BB % daletsegol;05D3 05B6 % daletsegolhebrew;05D3 05B6 % daletsheva;05D3 05B0 % daletshevahebrew;05D3 05B0 % dalettsere;05D3 05B5 % dalettserehebrew;05D3 05B5 \pdfglyphtounicode{dalfinalarabic}{FEAA} \pdfglyphtounicode{dammaarabic}{064F} \pdfglyphtounicode{dammalowarabic}{064F} \pdfglyphtounicode{dammatanaltonearabic}{064C} \pdfglyphtounicode{dammatanarabic}{064C} \pdfglyphtounicode{danda}{0964} \pdfglyphtounicode{dargahebrew}{05A7} \pdfglyphtounicode{dargalefthebrew}{05A7} \pdfglyphtounicode{dasiapneumatacyrilliccmb}{0485} \pdfglyphtounicode{dblGrave}{F6D3} \pdfglyphtounicode{dblanglebracketleft}{300A} \pdfglyphtounicode{dblanglebracketleftvertical}{FE3D} \pdfglyphtounicode{dblanglebracketright}{300B} \pdfglyphtounicode{dblanglebracketrightvertical}{FE3E} \pdfglyphtounicode{dblarchinvertedbelowcmb}{032B} \pdfglyphtounicode{dblarrowleft}{21D4} \pdfglyphtounicode{dblarrowright}{21D2} \pdfglyphtounicode{dbldanda}{0965} \pdfglyphtounicode{dblgrave}{F6D6} \pdfglyphtounicode{dblgravecmb}{030F} \pdfglyphtounicode{dblintegral}{222C} \pdfglyphtounicode{dbllowline}{2017} \pdfglyphtounicode{dbllowlinecmb}{0333} \pdfglyphtounicode{dbloverlinecmb}{033F} \pdfglyphtounicode{dblprimemod}{02BA} \pdfglyphtounicode{dblverticalbar}{2016} \pdfglyphtounicode{dblverticallineabovecmb}{030E} \pdfglyphtounicode{dbopomofo}{3109} \pdfglyphtounicode{dbsquare}{33C8} \pdfglyphtounicode{dcaron}{010F} \pdfglyphtounicode{dcedilla}{1E11} \pdfglyphtounicode{dcircle}{24D3} \pdfglyphtounicode{dcircumflexbelow}{1E13} \pdfglyphtounicode{dcroat}{0111} \pdfglyphtounicode{ddabengali}{09A1} \pdfglyphtounicode{ddadeva}{0921} \pdfglyphtounicode{ddagujarati}{0AA1} \pdfglyphtounicode{ddagurmukhi}{0A21} \pdfglyphtounicode{ddalarabic}{0688} \pdfglyphtounicode{ddalfinalarabic}{FB89} \pdfglyphtounicode{dddhadeva}{095C} \pdfglyphtounicode{ddhabengali}{09A2} \pdfglyphtounicode{ddhadeva}{0922} \pdfglyphtounicode{ddhagujarati}{0AA2} \pdfglyphtounicode{ddhagurmukhi}{0A22} \pdfglyphtounicode{ddotaccent}{1E0B} \pdfglyphtounicode{ddotbelow}{1E0D} \pdfglyphtounicode{decimalseparatorarabic}{066B} \pdfglyphtounicode{decimalseparatorpersian}{066B} \pdfglyphtounicode{decyrillic}{0434} \pdfglyphtounicode{degree}{00B0} \pdfglyphtounicode{dehihebrew}{05AD} \pdfglyphtounicode{dehiragana}{3067} \pdfglyphtounicode{deicoptic}{03EF} \pdfglyphtounicode{dekatakana}{30C7} \pdfglyphtounicode{deleteleft}{232B} \pdfglyphtounicode{deleteright}{2326} \pdfglyphtounicode{delta}{03B4} \pdfglyphtounicode{deltaturned}{018D} \pdfglyphtounicode{denominatorminusonenumeratorbengali}{09F8} \pdfglyphtounicode{dezh}{02A4} \pdfglyphtounicode{dhabengali}{09A7} \pdfglyphtounicode{dhadeva}{0927} \pdfglyphtounicode{dhagujarati}{0AA7} \pdfglyphtounicode{dhagurmukhi}{0A27} \pdfglyphtounicode{dhook}{0257} \pdfglyphtounicode{dialytikatonos}{0385} \pdfglyphtounicode{dialytikatonoscmb}{0344} \pdfglyphtounicode{diamond}{2666} \pdfglyphtounicode{diamondsuitwhite}{2662} \pdfglyphtounicode{dieresis}{00A8} \pdfglyphtounicode{dieresisacute}{F6D7} \pdfglyphtounicode{dieresisbelowcmb}{0324} \pdfglyphtounicode{dieresiscmb}{0308} \pdfglyphtounicode{dieresisgrave}{F6D8} \pdfglyphtounicode{dieresistonos}{0385} \pdfglyphtounicode{dihiragana}{3062} \pdfglyphtounicode{dikatakana}{30C2} \pdfglyphtounicode{dittomark}{3003} \pdfglyphtounicode{divide}{00F7} \pdfglyphtounicode{divides}{2223} \pdfglyphtounicode{divisionslash}{2215} \pdfglyphtounicode{djecyrillic}{0452} \pdfglyphtounicode{dkshade}{2593} \pdfglyphtounicode{dlinebelow}{1E0F} \pdfglyphtounicode{dlsquare}{3397} \pdfglyphtounicode{dmacron}{0111} \pdfglyphtounicode{dmonospace}{FF44} \pdfglyphtounicode{dnblock}{2584} \pdfglyphtounicode{dochadathai}{0E0E} \pdfglyphtounicode{dodekthai}{0E14} \pdfglyphtounicode{dohiragana}{3069} \pdfglyphtounicode{dokatakana}{30C9} \pdfglyphtounicode{dollar}{0024} \pdfglyphtounicode{dollarinferior}{F6E3} \pdfglyphtounicode{dollarmonospace}{FF04} \pdfglyphtounicode{dollaroldstyle}{F724} \pdfglyphtounicode{dollarsmall}{FE69} \pdfglyphtounicode{dollarsuperior}{F6E4} \pdfglyphtounicode{dong}{20AB} \pdfglyphtounicode{dorusquare}{3326} \pdfglyphtounicode{dotaccent}{02D9} \pdfglyphtounicode{dotaccentcmb}{0307} \pdfglyphtounicode{dotbelowcmb}{0323} \pdfglyphtounicode{dotbelowcomb}{0323} \pdfglyphtounicode{dotkatakana}{30FB} \pdfglyphtounicode{dotlessi}{0131} \pdfglyphtounicode{dotlessj}{F6BE} \pdfglyphtounicode{dotlessjstrokehook}{0284} \pdfglyphtounicode{dotmath}{22C5} \pdfglyphtounicode{dottedcircle}{25CC} \pdfglyphtounicode{doubleyodpatah}{FB1F} \pdfglyphtounicode{doubleyodpatahhebrew}{FB1F} \pdfglyphtounicode{downtackbelowcmb}{031E} \pdfglyphtounicode{downtackmod}{02D5} \pdfglyphtounicode{dparen}{249F} \pdfglyphtounicode{dsuperior}{F6EB} \pdfglyphtounicode{dtail}{0256} \pdfglyphtounicode{dtopbar}{018C} \pdfglyphtounicode{duhiragana}{3065} \pdfglyphtounicode{dukatakana}{30C5} \pdfglyphtounicode{dz}{01F3} \pdfglyphtounicode{dzaltone}{02A3} \pdfglyphtounicode{dzcaron}{01C6} \pdfglyphtounicode{dzcurl}{02A5} \pdfglyphtounicode{dzeabkhasiancyrillic}{04E1} \pdfglyphtounicode{dzecyrillic}{0455} \pdfglyphtounicode{dzhecyrillic}{045F} \pdfglyphtounicode{e}{0065} \pdfglyphtounicode{eacute}{00E9} \pdfglyphtounicode{earth}{2641} \pdfglyphtounicode{ebengali}{098F} \pdfglyphtounicode{ebopomofo}{311C} \pdfglyphtounicode{ebreve}{0115} \pdfglyphtounicode{ecandradeva}{090D} \pdfglyphtounicode{ecandragujarati}{0A8D} \pdfglyphtounicode{ecandravowelsigndeva}{0945} \pdfglyphtounicode{ecandravowelsigngujarati}{0AC5} \pdfglyphtounicode{ecaron}{011B} \pdfglyphtounicode{ecedillabreve}{1E1D} \pdfglyphtounicode{echarmenian}{0565} \pdfglyphtounicode{echyiwnarmenian}{0587} \pdfglyphtounicode{ecircle}{24D4} \pdfglyphtounicode{ecircumflex}{00EA} \pdfglyphtounicode{ecircumflexacute}{1EBF} \pdfglyphtounicode{ecircumflexbelow}{1E19} \pdfglyphtounicode{ecircumflexdotbelow}{1EC7} \pdfglyphtounicode{ecircumflexgrave}{1EC1} \pdfglyphtounicode{ecircumflexhookabove}{1EC3} \pdfglyphtounicode{ecircumflextilde}{1EC5} \pdfglyphtounicode{ecyrillic}{0454} \pdfglyphtounicode{edblgrave}{0205} \pdfglyphtounicode{edeva}{090F} \pdfglyphtounicode{edieresis}{00EB} \pdfglyphtounicode{edot}{0117} \pdfglyphtounicode{edotaccent}{0117} \pdfglyphtounicode{edotbelow}{1EB9} \pdfglyphtounicode{eegurmukhi}{0A0F} \pdfglyphtounicode{eematragurmukhi}{0A47} \pdfglyphtounicode{efcyrillic}{0444} \pdfglyphtounicode{egrave}{00E8} \pdfglyphtounicode{egujarati}{0A8F} \pdfglyphtounicode{eharmenian}{0567} \pdfglyphtounicode{ehbopomofo}{311D} \pdfglyphtounicode{ehiragana}{3048} \pdfglyphtounicode{ehookabove}{1EBB} \pdfglyphtounicode{eibopomofo}{311F} \pdfglyphtounicode{eight}{0038} \pdfglyphtounicode{eightarabic}{0668} \pdfglyphtounicode{eightbengali}{09EE} \pdfglyphtounicode{eightcircle}{2467} \pdfglyphtounicode{eightcircleinversesansserif}{2791} \pdfglyphtounicode{eightdeva}{096E} \pdfglyphtounicode{eighteencircle}{2471} \pdfglyphtounicode{eighteenparen}{2485} \pdfglyphtounicode{eighteenperiod}{2499} \pdfglyphtounicode{eightgujarati}{0AEE} \pdfglyphtounicode{eightgurmukhi}{0A6E} \pdfglyphtounicode{eighthackarabic}{0668} \pdfglyphtounicode{eighthangzhou}{3028} \pdfglyphtounicode{eighthnotebeamed}{266B} \pdfglyphtounicode{eightideographicparen}{3227} \pdfglyphtounicode{eightinferior}{2088} \pdfglyphtounicode{eightmonospace}{FF18} \pdfglyphtounicode{eightoldstyle}{F738} \pdfglyphtounicode{eightparen}{247B} \pdfglyphtounicode{eightperiod}{248F} \pdfglyphtounicode{eightpersian}{06F8} \pdfglyphtounicode{eightroman}{2177} \pdfglyphtounicode{eightsuperior}{2078} \pdfglyphtounicode{eightthai}{0E58} \pdfglyphtounicode{einvertedbreve}{0207} \pdfglyphtounicode{eiotifiedcyrillic}{0465} \pdfglyphtounicode{ekatakana}{30A8} \pdfglyphtounicode{ekatakanahalfwidth}{FF74} \pdfglyphtounicode{ekonkargurmukhi}{0A74} \pdfglyphtounicode{ekorean}{3154} \pdfglyphtounicode{elcyrillic}{043B} \pdfglyphtounicode{element}{2208} \pdfglyphtounicode{elevencircle}{246A} \pdfglyphtounicode{elevenparen}{247E} \pdfglyphtounicode{elevenperiod}{2492} \pdfglyphtounicode{elevenroman}{217A} \pdfglyphtounicode{ellipsis}{2026} \pdfglyphtounicode{ellipsisvertical}{22EE} \pdfglyphtounicode{emacron}{0113} \pdfglyphtounicode{emacronacute}{1E17} \pdfglyphtounicode{emacrongrave}{1E15} \pdfglyphtounicode{emcyrillic}{043C} \pdfglyphtounicode{emdash}{2014} \pdfglyphtounicode{emdashvertical}{FE31} \pdfglyphtounicode{emonospace}{FF45} \pdfglyphtounicode{emphasismarkarmenian}{055B} \pdfglyphtounicode{emptyset}{2205} \pdfglyphtounicode{enbopomofo}{3123} \pdfglyphtounicode{encyrillic}{043D} \pdfglyphtounicode{endash}{2013} \pdfglyphtounicode{endashvertical}{FE32} \pdfglyphtounicode{endescendercyrillic}{04A3} \pdfglyphtounicode{eng}{014B} \pdfglyphtounicode{engbopomofo}{3125} \pdfglyphtounicode{enghecyrillic}{04A5} \pdfglyphtounicode{enhookcyrillic}{04C8} \pdfglyphtounicode{enspace}{2002} \pdfglyphtounicode{eogonek}{0119} \pdfglyphtounicode{eokorean}{3153} \pdfglyphtounicode{eopen}{025B} \pdfglyphtounicode{eopenclosed}{029A} \pdfglyphtounicode{eopenreversed}{025C} \pdfglyphtounicode{eopenreversedclosed}{025E} \pdfglyphtounicode{eopenreversedhook}{025D} \pdfglyphtounicode{eparen}{24A0} \pdfglyphtounicode{epsilon}{03B5} \pdfglyphtounicode{epsilontonos}{03AD} \pdfglyphtounicode{equal}{003D} \pdfglyphtounicode{equalmonospace}{FF1D} \pdfglyphtounicode{equalsmall}{FE66} \pdfglyphtounicode{equalsuperior}{207C} \pdfglyphtounicode{equivalence}{2261} \pdfglyphtounicode{erbopomofo}{3126} \pdfglyphtounicode{ercyrillic}{0440} \pdfglyphtounicode{ereversed}{0258} \pdfglyphtounicode{ereversedcyrillic}{044D} \pdfglyphtounicode{escyrillic}{0441} \pdfglyphtounicode{esdescendercyrillic}{04AB} \pdfglyphtounicode{esh}{0283} \pdfglyphtounicode{eshcurl}{0286} \pdfglyphtounicode{eshortdeva}{090E} \pdfglyphtounicode{eshortvowelsigndeva}{0946} \pdfglyphtounicode{eshreversedloop}{01AA} \pdfglyphtounicode{eshsquatreversed}{0285} \pdfglyphtounicode{esmallhiragana}{3047} \pdfglyphtounicode{esmallkatakana}{30A7} \pdfglyphtounicode{esmallkatakanahalfwidth}{FF6A} \pdfglyphtounicode{estimated}{212E} \pdfglyphtounicode{esuperior}{F6EC} \pdfglyphtounicode{eta}{03B7} \pdfglyphtounicode{etarmenian}{0568} \pdfglyphtounicode{etatonos}{03AE} \pdfglyphtounicode{eth}{00F0} \pdfglyphtounicode{etilde}{1EBD} \pdfglyphtounicode{etildebelow}{1E1B} \pdfglyphtounicode{etnahtafoukhhebrew}{0591} \pdfglyphtounicode{etnahtafoukhlefthebrew}{0591} \pdfglyphtounicode{etnahtahebrew}{0591} \pdfglyphtounicode{etnahtalefthebrew}{0591} \pdfglyphtounicode{eturned}{01DD} \pdfglyphtounicode{eukorean}{3161} \pdfglyphtounicode{euro}{20AC} \pdfglyphtounicode{evowelsignbengali}{09C7} \pdfglyphtounicode{evowelsigndeva}{0947} \pdfglyphtounicode{evowelsigngujarati}{0AC7} \pdfglyphtounicode{exclam}{0021} \pdfglyphtounicode{exclamarmenian}{055C} \pdfglyphtounicode{exclamdbl}{203C} \pdfglyphtounicode{exclamdown}{00A1} \pdfglyphtounicode{exclamdownsmall}{F7A1} \pdfglyphtounicode{exclammonospace}{FF01} \pdfglyphtounicode{exclamsmall}{F721} \pdfglyphtounicode{existential}{2203} \pdfglyphtounicode{ezh}{0292} \pdfglyphtounicode{ezhcaron}{01EF} \pdfglyphtounicode{ezhcurl}{0293} \pdfglyphtounicode{ezhreversed}{01B9} \pdfglyphtounicode{ezhtail}{01BA} \pdfglyphtounicode{f}{0066} \pdfglyphtounicode{fadeva}{095E} \pdfglyphtounicode{fagurmukhi}{0A5E} \pdfglyphtounicode{fahrenheit}{2109} \pdfglyphtounicode{fathaarabic}{064E} \pdfglyphtounicode{fathalowarabic}{064E} \pdfglyphtounicode{fathatanarabic}{064B} \pdfglyphtounicode{fbopomofo}{3108} \pdfglyphtounicode{fcircle}{24D5} \pdfglyphtounicode{fdotaccent}{1E1F} \pdfglyphtounicode{feharabic}{0641} \pdfglyphtounicode{feharmenian}{0586} \pdfglyphtounicode{fehfinalarabic}{FED2} \pdfglyphtounicode{fehinitialarabic}{FED3} \pdfglyphtounicode{fehmedialarabic}{FED4} \pdfglyphtounicode{feicoptic}{03E5} \pdfglyphtounicode{female}{2640} \pdfglyphtounicode{ff}{FB00} \pdfglyphtounicode{ffi}{FB03} \pdfglyphtounicode{ffl}{FB04} \pdfglyphtounicode{fi}{FB01} \pdfglyphtounicode{fifteencircle}{246E} \pdfglyphtounicode{fifteenparen}{2482} \pdfglyphtounicode{fifteenperiod}{2496} \pdfglyphtounicode{figuredash}{2012} \pdfglyphtounicode{filledbox}{25A0} \pdfglyphtounicode{filledrect}{25AC} \pdfglyphtounicode{finalkaf}{05DA} \pdfglyphtounicode{finalkafdagesh}{FB3A} \pdfglyphtounicode{finalkafdageshhebrew}{FB3A} \pdfglyphtounicode{finalkafhebrew}{05DA} % finalkafqamats;05DA 05B8 % finalkafqamatshebrew;05DA 05B8 % finalkafsheva;05DA 05B0 % finalkafshevahebrew;05DA 05B0 \pdfglyphtounicode{finalmem}{05DD} \pdfglyphtounicode{finalmemhebrew}{05DD} \pdfglyphtounicode{finalnun}{05DF} \pdfglyphtounicode{finalnunhebrew}{05DF} \pdfglyphtounicode{finalpe}{05E3} \pdfglyphtounicode{finalpehebrew}{05E3} \pdfglyphtounicode{finaltsadi}{05E5} \pdfglyphtounicode{finaltsadihebrew}{05E5} \pdfglyphtounicode{firsttonechinese}{02C9} \pdfglyphtounicode{fisheye}{25C9} \pdfglyphtounicode{fitacyrillic}{0473} \pdfglyphtounicode{five}{0035} \pdfglyphtounicode{fivearabic}{0665} \pdfglyphtounicode{fivebengali}{09EB} \pdfglyphtounicode{fivecircle}{2464} \pdfglyphtounicode{fivecircleinversesansserif}{278E} \pdfglyphtounicode{fivedeva}{096B} \pdfglyphtounicode{fiveeighths}{215D} \pdfglyphtounicode{fivegujarati}{0AEB} \pdfglyphtounicode{fivegurmukhi}{0A6B} \pdfglyphtounicode{fivehackarabic}{0665} \pdfglyphtounicode{fivehangzhou}{3025} \pdfglyphtounicode{fiveideographicparen}{3224} \pdfglyphtounicode{fiveinferior}{2085} \pdfglyphtounicode{fivemonospace}{FF15} \pdfglyphtounicode{fiveoldstyle}{F735} \pdfglyphtounicode{fiveparen}{2478} \pdfglyphtounicode{fiveperiod}{248C} \pdfglyphtounicode{fivepersian}{06F5} \pdfglyphtounicode{fiveroman}{2174} \pdfglyphtounicode{fivesuperior}{2075} \pdfglyphtounicode{fivethai}{0E55} \pdfglyphtounicode{fl}{FB02} \pdfglyphtounicode{florin}{0192} \pdfglyphtounicode{fmonospace}{FF46} \pdfglyphtounicode{fmsquare}{3399} \pdfglyphtounicode{fofanthai}{0E1F} \pdfglyphtounicode{fofathai}{0E1D} \pdfglyphtounicode{fongmanthai}{0E4F} \pdfglyphtounicode{forall}{2200} \pdfglyphtounicode{four}{0034} \pdfglyphtounicode{fourarabic}{0664} \pdfglyphtounicode{fourbengali}{09EA} \pdfglyphtounicode{fourcircle}{2463} \pdfglyphtounicode{fourcircleinversesansserif}{278D} \pdfglyphtounicode{fourdeva}{096A} \pdfglyphtounicode{fourgujarati}{0AEA} \pdfglyphtounicode{fourgurmukhi}{0A6A} \pdfglyphtounicode{fourhackarabic}{0664} \pdfglyphtounicode{fourhangzhou}{3024} \pdfglyphtounicode{fourideographicparen}{3223} \pdfglyphtounicode{fourinferior}{2084} \pdfglyphtounicode{fourmonospace}{FF14} \pdfglyphtounicode{fournumeratorbengali}{09F7} \pdfglyphtounicode{fouroldstyle}{F734} \pdfglyphtounicode{fourparen}{2477} \pdfglyphtounicode{fourperiod}{248B} \pdfglyphtounicode{fourpersian}{06F4} \pdfglyphtounicode{fourroman}{2173} \pdfglyphtounicode{foursuperior}{2074} \pdfglyphtounicode{fourteencircle}{246D} \pdfglyphtounicode{fourteenparen}{2481} \pdfglyphtounicode{fourteenperiod}{2495} \pdfglyphtounicode{fourthai}{0E54} \pdfglyphtounicode{fourthtonechinese}{02CB} \pdfglyphtounicode{fparen}{24A1} \pdfglyphtounicode{fraction}{2044} \pdfglyphtounicode{franc}{20A3} \pdfglyphtounicode{g}{0067} \pdfglyphtounicode{gabengali}{0997} \pdfglyphtounicode{gacute}{01F5} \pdfglyphtounicode{gadeva}{0917} \pdfglyphtounicode{gafarabic}{06AF} \pdfglyphtounicode{gaffinalarabic}{FB93} \pdfglyphtounicode{gafinitialarabic}{FB94} \pdfglyphtounicode{gafmedialarabic}{FB95} \pdfglyphtounicode{gagujarati}{0A97} \pdfglyphtounicode{gagurmukhi}{0A17} \pdfglyphtounicode{gahiragana}{304C} \pdfglyphtounicode{gakatakana}{30AC} \pdfglyphtounicode{gamma}{03B3} \pdfglyphtounicode{gammalatinsmall}{0263} \pdfglyphtounicode{gammasuperior}{02E0} \pdfglyphtounicode{gangiacoptic}{03EB} \pdfglyphtounicode{gbopomofo}{310D} \pdfglyphtounicode{gbreve}{011F} \pdfglyphtounicode{gcaron}{01E7} \pdfglyphtounicode{gcedilla}{0123} \pdfglyphtounicode{gcircle}{24D6} \pdfglyphtounicode{gcircumflex}{011D} \pdfglyphtounicode{gcommaaccent}{0123} \pdfglyphtounicode{gdot}{0121} \pdfglyphtounicode{gdotaccent}{0121} \pdfglyphtounicode{gecyrillic}{0433} \pdfglyphtounicode{gehiragana}{3052} \pdfglyphtounicode{gekatakana}{30B2} \pdfglyphtounicode{geometricallyequal}{2251} \pdfglyphtounicode{gereshaccenthebrew}{059C} \pdfglyphtounicode{gereshhebrew}{05F3} \pdfglyphtounicode{gereshmuqdamhebrew}{059D} \pdfglyphtounicode{germandbls}{00DF} \pdfglyphtounicode{gershayimaccenthebrew}{059E} \pdfglyphtounicode{gershayimhebrew}{05F4} \pdfglyphtounicode{getamark}{3013} \pdfglyphtounicode{ghabengali}{0998} \pdfglyphtounicode{ghadarmenian}{0572} \pdfglyphtounicode{ghadeva}{0918} \pdfglyphtounicode{ghagujarati}{0A98} \pdfglyphtounicode{ghagurmukhi}{0A18} \pdfglyphtounicode{ghainarabic}{063A} \pdfglyphtounicode{ghainfinalarabic}{FECE} \pdfglyphtounicode{ghaininitialarabic}{FECF} \pdfglyphtounicode{ghainmedialarabic}{FED0} \pdfglyphtounicode{ghemiddlehookcyrillic}{0495} \pdfglyphtounicode{ghestrokecyrillic}{0493} \pdfglyphtounicode{gheupturncyrillic}{0491} \pdfglyphtounicode{ghhadeva}{095A} \pdfglyphtounicode{ghhagurmukhi}{0A5A} \pdfglyphtounicode{ghook}{0260} \pdfglyphtounicode{ghzsquare}{3393} \pdfglyphtounicode{gihiragana}{304E} \pdfglyphtounicode{gikatakana}{30AE} \pdfglyphtounicode{gimarmenian}{0563} \pdfglyphtounicode{gimel}{05D2} \pdfglyphtounicode{gimeldagesh}{FB32} \pdfglyphtounicode{gimeldageshhebrew}{FB32} \pdfglyphtounicode{gimelhebrew}{05D2} \pdfglyphtounicode{gjecyrillic}{0453} \pdfglyphtounicode{glottalinvertedstroke}{01BE} \pdfglyphtounicode{glottalstop}{0294} \pdfglyphtounicode{glottalstopinverted}{0296} \pdfglyphtounicode{glottalstopmod}{02C0} \pdfglyphtounicode{glottalstopreversed}{0295} \pdfglyphtounicode{glottalstopreversedmod}{02C1} \pdfglyphtounicode{glottalstopreversedsuperior}{02E4} \pdfglyphtounicode{glottalstopstroke}{02A1} \pdfglyphtounicode{glottalstopstrokereversed}{02A2} \pdfglyphtounicode{gmacron}{1E21} \pdfglyphtounicode{gmonospace}{FF47} \pdfglyphtounicode{gohiragana}{3054} \pdfglyphtounicode{gokatakana}{30B4} \pdfglyphtounicode{gparen}{24A2} \pdfglyphtounicode{gpasquare}{33AC} \pdfglyphtounicode{gradient}{2207} \pdfglyphtounicode{grave}{0060} \pdfglyphtounicode{gravebelowcmb}{0316} \pdfglyphtounicode{gravecmb}{0300} \pdfglyphtounicode{gravecomb}{0300} \pdfglyphtounicode{gravedeva}{0953} \pdfglyphtounicode{gravelowmod}{02CE} \pdfglyphtounicode{gravemonospace}{FF40} \pdfglyphtounicode{gravetonecmb}{0340} \pdfglyphtounicode{greater}{003E} \pdfglyphtounicode{greaterequal}{2265} \pdfglyphtounicode{greaterequalorless}{22DB} \pdfglyphtounicode{greatermonospace}{FF1E} \pdfglyphtounicode{greaterorequivalent}{2273} \pdfglyphtounicode{greaterorless}{2277} \pdfglyphtounicode{greateroverequal}{2267} \pdfglyphtounicode{greatersmall}{FE65} \pdfglyphtounicode{gscript}{0261} \pdfglyphtounicode{gstroke}{01E5} \pdfglyphtounicode{guhiragana}{3050} \pdfglyphtounicode{guillemotleft}{00AB} \pdfglyphtounicode{guillemotright}{00BB} \pdfglyphtounicode{guilsinglleft}{2039} \pdfglyphtounicode{guilsinglright}{203A} \pdfglyphtounicode{gukatakana}{30B0} \pdfglyphtounicode{guramusquare}{3318} \pdfglyphtounicode{gysquare}{33C9} \pdfglyphtounicode{h}{0068} \pdfglyphtounicode{haabkhasiancyrillic}{04A9} \pdfglyphtounicode{haaltonearabic}{06C1} \pdfglyphtounicode{habengali}{09B9} \pdfglyphtounicode{hadescendercyrillic}{04B3} \pdfglyphtounicode{hadeva}{0939} \pdfglyphtounicode{hagujarati}{0AB9} \pdfglyphtounicode{hagurmukhi}{0A39} \pdfglyphtounicode{haharabic}{062D} \pdfglyphtounicode{hahfinalarabic}{FEA2} \pdfglyphtounicode{hahinitialarabic}{FEA3} \pdfglyphtounicode{hahiragana}{306F} \pdfglyphtounicode{hahmedialarabic}{FEA4} \pdfglyphtounicode{haitusquare}{332A} \pdfglyphtounicode{hakatakana}{30CF} \pdfglyphtounicode{hakatakanahalfwidth}{FF8A} \pdfglyphtounicode{halantgurmukhi}{0A4D} \pdfglyphtounicode{hamzaarabic}{0621} % hamzadammaarabic;0621 064F % hamzadammatanarabic;0621 064C % hamzafathaarabic;0621 064E % hamzafathatanarabic;0621 064B \pdfglyphtounicode{hamzalowarabic}{0621} % hamzalowkasraarabic;0621 0650 % hamzalowkasratanarabic;0621 064D % hamzasukunarabic;0621 0652 \pdfglyphtounicode{hangulfiller}{3164} \pdfglyphtounicode{hardsigncyrillic}{044A} \pdfglyphtounicode{harpoonleftbarbup}{21BC} \pdfglyphtounicode{harpoonrightbarbup}{21C0} \pdfglyphtounicode{hasquare}{33CA} \pdfglyphtounicode{hatafpatah}{05B2} \pdfglyphtounicode{hatafpatah16}{05B2} \pdfglyphtounicode{hatafpatah23}{05B2} \pdfglyphtounicode{hatafpatah2f}{05B2} \pdfglyphtounicode{hatafpatahhebrew}{05B2} \pdfglyphtounicode{hatafpatahnarrowhebrew}{05B2} \pdfglyphtounicode{hatafpatahquarterhebrew}{05B2} \pdfglyphtounicode{hatafpatahwidehebrew}{05B2} \pdfglyphtounicode{hatafqamats}{05B3} \pdfglyphtounicode{hatafqamats1b}{05B3} \pdfglyphtounicode{hatafqamats28}{05B3} \pdfglyphtounicode{hatafqamats34}{05B3} \pdfglyphtounicode{hatafqamatshebrew}{05B3} \pdfglyphtounicode{hatafqamatsnarrowhebrew}{05B3} \pdfglyphtounicode{hatafqamatsquarterhebrew}{05B3} \pdfglyphtounicode{hatafqamatswidehebrew}{05B3} \pdfglyphtounicode{hatafsegol}{05B1} \pdfglyphtounicode{hatafsegol17}{05B1} \pdfglyphtounicode{hatafsegol24}{05B1} \pdfglyphtounicode{hatafsegol30}{05B1} \pdfglyphtounicode{hatafsegolhebrew}{05B1} \pdfglyphtounicode{hatafsegolnarrowhebrew}{05B1} \pdfglyphtounicode{hatafsegolquarterhebrew}{05B1} \pdfglyphtounicode{hatafsegolwidehebrew}{05B1} \pdfglyphtounicode{hbar}{0127} \pdfglyphtounicode{hbopomofo}{310F} \pdfglyphtounicode{hbrevebelow}{1E2B} \pdfglyphtounicode{hcedilla}{1E29} \pdfglyphtounicode{hcircle}{24D7} \pdfglyphtounicode{hcircumflex}{0125} \pdfglyphtounicode{hdieresis}{1E27} \pdfglyphtounicode{hdotaccent}{1E23} \pdfglyphtounicode{hdotbelow}{1E25} \pdfglyphtounicode{he}{05D4} \pdfglyphtounicode{heart}{2665} \pdfglyphtounicode{heartsuitblack}{2665} \pdfglyphtounicode{heartsuitwhite}{2661} \pdfglyphtounicode{hedagesh}{FB34} \pdfglyphtounicode{hedageshhebrew}{FB34} \pdfglyphtounicode{hehaltonearabic}{06C1} \pdfglyphtounicode{heharabic}{0647} \pdfglyphtounicode{hehebrew}{05D4} \pdfglyphtounicode{hehfinalaltonearabic}{FBA7} \pdfglyphtounicode{hehfinalalttwoarabic}{FEEA} \pdfglyphtounicode{hehfinalarabic}{FEEA} \pdfglyphtounicode{hehhamzaabovefinalarabic}{FBA5} \pdfglyphtounicode{hehhamzaaboveisolatedarabic}{FBA4} \pdfglyphtounicode{hehinitialaltonearabic}{FBA8} \pdfglyphtounicode{hehinitialarabic}{FEEB} \pdfglyphtounicode{hehiragana}{3078} \pdfglyphtounicode{hehmedialaltonearabic}{FBA9} \pdfglyphtounicode{hehmedialarabic}{FEEC} \pdfglyphtounicode{heiseierasquare}{337B} \pdfglyphtounicode{hekatakana}{30D8} \pdfglyphtounicode{hekatakanahalfwidth}{FF8D} \pdfglyphtounicode{hekutaarusquare}{3336} \pdfglyphtounicode{henghook}{0267} \pdfglyphtounicode{herutusquare}{3339} \pdfglyphtounicode{het}{05D7} \pdfglyphtounicode{hethebrew}{05D7} \pdfglyphtounicode{hhook}{0266} \pdfglyphtounicode{hhooksuperior}{02B1} \pdfglyphtounicode{hieuhacirclekorean}{327B} \pdfglyphtounicode{hieuhaparenkorean}{321B} \pdfglyphtounicode{hieuhcirclekorean}{326D} \pdfglyphtounicode{hieuhkorean}{314E} \pdfglyphtounicode{hieuhparenkorean}{320D} \pdfglyphtounicode{hihiragana}{3072} \pdfglyphtounicode{hikatakana}{30D2} \pdfglyphtounicode{hikatakanahalfwidth}{FF8B} \pdfglyphtounicode{hiriq}{05B4} \pdfglyphtounicode{hiriq14}{05B4} \pdfglyphtounicode{hiriq21}{05B4} \pdfglyphtounicode{hiriq2d}{05B4} \pdfglyphtounicode{hiriqhebrew}{05B4} \pdfglyphtounicode{hiriqnarrowhebrew}{05B4} \pdfglyphtounicode{hiriqquarterhebrew}{05B4} \pdfglyphtounicode{hiriqwidehebrew}{05B4} \pdfglyphtounicode{hlinebelow}{1E96} \pdfglyphtounicode{hmonospace}{FF48} \pdfglyphtounicode{hoarmenian}{0570} \pdfglyphtounicode{hohipthai}{0E2B} \pdfglyphtounicode{hohiragana}{307B} \pdfglyphtounicode{hokatakana}{30DB} \pdfglyphtounicode{hokatakanahalfwidth}{FF8E} \pdfglyphtounicode{holam}{05B9} \pdfglyphtounicode{holam19}{05B9} \pdfglyphtounicode{holam26}{05B9} \pdfglyphtounicode{holam32}{05B9} \pdfglyphtounicode{holamhebrew}{05B9} \pdfglyphtounicode{holamnarrowhebrew}{05B9} \pdfglyphtounicode{holamquarterhebrew}{05B9} \pdfglyphtounicode{holamwidehebrew}{05B9} \pdfglyphtounicode{honokhukthai}{0E2E} \pdfglyphtounicode{hookabovecomb}{0309} \pdfglyphtounicode{hookcmb}{0309} \pdfglyphtounicode{hookpalatalizedbelowcmb}{0321} \pdfglyphtounicode{hookretroflexbelowcmb}{0322} \pdfglyphtounicode{hoonsquare}{3342} \pdfglyphtounicode{horicoptic}{03E9} \pdfglyphtounicode{horizontalbar}{2015} \pdfglyphtounicode{horncmb}{031B} \pdfglyphtounicode{hotsprings}{2668} \pdfglyphtounicode{house}{2302} \pdfglyphtounicode{hparen}{24A3} \pdfglyphtounicode{hsuperior}{02B0} \pdfglyphtounicode{hturned}{0265} \pdfglyphtounicode{huhiragana}{3075} \pdfglyphtounicode{huiitosquare}{3333} \pdfglyphtounicode{hukatakana}{30D5} \pdfglyphtounicode{hukatakanahalfwidth}{FF8C} \pdfglyphtounicode{hungarumlaut}{02DD} \pdfglyphtounicode{hungarumlautcmb}{030B} \pdfglyphtounicode{hv}{0195} \pdfglyphtounicode{hyphen}{002D} \pdfglyphtounicode{hypheninferior}{F6E5} \pdfglyphtounicode{hyphenmonospace}{FF0D} \pdfglyphtounicode{hyphensmall}{FE63} \pdfglyphtounicode{hyphensuperior}{F6E6} \pdfglyphtounicode{hyphentwo}{2010} \pdfglyphtounicode{i}{0069} \pdfglyphtounicode{iacute}{00ED} \pdfglyphtounicode{iacyrillic}{044F} \pdfglyphtounicode{ibengali}{0987} \pdfglyphtounicode{ibopomofo}{3127} \pdfglyphtounicode{ibreve}{012D} \pdfglyphtounicode{icaron}{01D0} \pdfglyphtounicode{icircle}{24D8} \pdfglyphtounicode{icircumflex}{00EE} \pdfglyphtounicode{icyrillic}{0456} \pdfglyphtounicode{idblgrave}{0209} \pdfglyphtounicode{ideographearthcircle}{328F} \pdfglyphtounicode{ideographfirecircle}{328B} \pdfglyphtounicode{ideographicallianceparen}{323F} \pdfglyphtounicode{ideographiccallparen}{323A} \pdfglyphtounicode{ideographiccentrecircle}{32A5} \pdfglyphtounicode{ideographicclose}{3006} \pdfglyphtounicode{ideographiccomma}{3001} \pdfglyphtounicode{ideographiccommaleft}{FF64} \pdfglyphtounicode{ideographiccongratulationparen}{3237} \pdfglyphtounicode{ideographiccorrectcircle}{32A3} \pdfglyphtounicode{ideographicearthparen}{322F} \pdfglyphtounicode{ideographicenterpriseparen}{323D} \pdfglyphtounicode{ideographicexcellentcircle}{329D} \pdfglyphtounicode{ideographicfestivalparen}{3240} \pdfglyphtounicode{ideographicfinancialcircle}{3296} \pdfglyphtounicode{ideographicfinancialparen}{3236} \pdfglyphtounicode{ideographicfireparen}{322B} \pdfglyphtounicode{ideographichaveparen}{3232} \pdfglyphtounicode{ideographichighcircle}{32A4} \pdfglyphtounicode{ideographiciterationmark}{3005} \pdfglyphtounicode{ideographiclaborcircle}{3298} \pdfglyphtounicode{ideographiclaborparen}{3238} \pdfglyphtounicode{ideographicleftcircle}{32A7} \pdfglyphtounicode{ideographiclowcircle}{32A6} \pdfglyphtounicode{ideographicmedicinecircle}{32A9} \pdfglyphtounicode{ideographicmetalparen}{322E} \pdfglyphtounicode{ideographicmoonparen}{322A} \pdfglyphtounicode{ideographicnameparen}{3234} \pdfglyphtounicode{ideographicperiod}{3002} \pdfglyphtounicode{ideographicprintcircle}{329E} \pdfglyphtounicode{ideographicreachparen}{3243} \pdfglyphtounicode{ideographicrepresentparen}{3239} \pdfglyphtounicode{ideographicresourceparen}{323E} \pdfglyphtounicode{ideographicrightcircle}{32A8} \pdfglyphtounicode{ideographicsecretcircle}{3299} \pdfglyphtounicode{ideographicselfparen}{3242} \pdfglyphtounicode{ideographicsocietyparen}{3233} \pdfglyphtounicode{ideographicspace}{3000} \pdfglyphtounicode{ideographicspecialparen}{3235} \pdfglyphtounicode{ideographicstockparen}{3231} \pdfglyphtounicode{ideographicstudyparen}{323B} \pdfglyphtounicode{ideographicsunparen}{3230} \pdfglyphtounicode{ideographicsuperviseparen}{323C} \pdfglyphtounicode{ideographicwaterparen}{322C} \pdfglyphtounicode{ideographicwoodparen}{322D} \pdfglyphtounicode{ideographiczero}{3007} \pdfglyphtounicode{ideographmetalcircle}{328E} \pdfglyphtounicode{ideographmooncircle}{328A} \pdfglyphtounicode{ideographnamecircle}{3294} \pdfglyphtounicode{ideographsuncircle}{3290} \pdfglyphtounicode{ideographwatercircle}{328C} \pdfglyphtounicode{ideographwoodcircle}{328D} \pdfglyphtounicode{ideva}{0907} \pdfglyphtounicode{idieresis}{00EF} \pdfglyphtounicode{idieresisacute}{1E2F} \pdfglyphtounicode{idieresiscyrillic}{04E5} \pdfglyphtounicode{idotbelow}{1ECB} \pdfglyphtounicode{iebrevecyrillic}{04D7} \pdfglyphtounicode{iecyrillic}{0435} \pdfglyphtounicode{ieungacirclekorean}{3275} \pdfglyphtounicode{ieungaparenkorean}{3215} \pdfglyphtounicode{ieungcirclekorean}{3267} \pdfglyphtounicode{ieungkorean}{3147} \pdfglyphtounicode{ieungparenkorean}{3207} \pdfglyphtounicode{igrave}{00EC} \pdfglyphtounicode{igujarati}{0A87} \pdfglyphtounicode{igurmukhi}{0A07} \pdfglyphtounicode{ihiragana}{3044} \pdfglyphtounicode{ihookabove}{1EC9} \pdfglyphtounicode{iibengali}{0988} \pdfglyphtounicode{iicyrillic}{0438} \pdfglyphtounicode{iideva}{0908} \pdfglyphtounicode{iigujarati}{0A88} \pdfglyphtounicode{iigurmukhi}{0A08} \pdfglyphtounicode{iimatragurmukhi}{0A40} \pdfglyphtounicode{iinvertedbreve}{020B} \pdfglyphtounicode{iishortcyrillic}{0439} \pdfglyphtounicode{iivowelsignbengali}{09C0} \pdfglyphtounicode{iivowelsigndeva}{0940} \pdfglyphtounicode{iivowelsigngujarati}{0AC0} \pdfglyphtounicode{ij}{0133} \pdfglyphtounicode{ikatakana}{30A4} \pdfglyphtounicode{ikatakanahalfwidth}{FF72} \pdfglyphtounicode{ikorean}{3163} \pdfglyphtounicode{ilde}{02DC} \pdfglyphtounicode{iluyhebrew}{05AC} \pdfglyphtounicode{imacron}{012B} \pdfglyphtounicode{imacroncyrillic}{04E3} \pdfglyphtounicode{imageorapproximatelyequal}{2253} \pdfglyphtounicode{imatragurmukhi}{0A3F} \pdfglyphtounicode{imonospace}{FF49} \pdfglyphtounicode{increment}{2206} \pdfglyphtounicode{infinity}{221E} \pdfglyphtounicode{iniarmenian}{056B} \pdfglyphtounicode{integral}{222B} \pdfglyphtounicode{integralbottom}{2321} \pdfglyphtounicode{integralbt}{2321} \pdfglyphtounicode{integralex}{F8F5} \pdfglyphtounicode{integraltop}{2320} \pdfglyphtounicode{integraltp}{2320} \pdfglyphtounicode{intersection}{2229} \pdfglyphtounicode{intisquare}{3305} \pdfglyphtounicode{invbullet}{25D8} \pdfglyphtounicode{invcircle}{25D9} \pdfglyphtounicode{invsmileface}{263B} \pdfglyphtounicode{iocyrillic}{0451} \pdfglyphtounicode{iogonek}{012F} \pdfglyphtounicode{iota}{03B9} \pdfglyphtounicode{iotadieresis}{03CA} \pdfglyphtounicode{iotadieresistonos}{0390} \pdfglyphtounicode{iotalatin}{0269} \pdfglyphtounicode{iotatonos}{03AF} \pdfglyphtounicode{iparen}{24A4} \pdfglyphtounicode{irigurmukhi}{0A72} \pdfglyphtounicode{ismallhiragana}{3043} \pdfglyphtounicode{ismallkatakana}{30A3} \pdfglyphtounicode{ismallkatakanahalfwidth}{FF68} \pdfglyphtounicode{issharbengali}{09FA} \pdfglyphtounicode{istroke}{0268} \pdfglyphtounicode{isuperior}{F6ED} \pdfglyphtounicode{iterationhiragana}{309D} \pdfglyphtounicode{iterationkatakana}{30FD} \pdfglyphtounicode{itilde}{0129} \pdfglyphtounicode{itildebelow}{1E2D} \pdfglyphtounicode{iubopomofo}{3129} \pdfglyphtounicode{iucyrillic}{044E} \pdfglyphtounicode{ivowelsignbengali}{09BF} \pdfglyphtounicode{ivowelsigndeva}{093F} \pdfglyphtounicode{ivowelsigngujarati}{0ABF} \pdfglyphtounicode{izhitsacyrillic}{0475} \pdfglyphtounicode{izhitsadblgravecyrillic}{0477} \pdfglyphtounicode{j}{006A} \pdfglyphtounicode{jaarmenian}{0571} \pdfglyphtounicode{jabengali}{099C} \pdfglyphtounicode{jadeva}{091C} \pdfglyphtounicode{jagujarati}{0A9C} \pdfglyphtounicode{jagurmukhi}{0A1C} \pdfglyphtounicode{jbopomofo}{3110} \pdfglyphtounicode{jcaron}{01F0} \pdfglyphtounicode{jcircle}{24D9} \pdfglyphtounicode{jcircumflex}{0135} \pdfglyphtounicode{jcrossedtail}{029D} \pdfglyphtounicode{jdotlessstroke}{025F} \pdfglyphtounicode{jecyrillic}{0458} \pdfglyphtounicode{jeemarabic}{062C} \pdfglyphtounicode{jeemfinalarabic}{FE9E} \pdfglyphtounicode{jeeminitialarabic}{FE9F} \pdfglyphtounicode{jeemmedialarabic}{FEA0} \pdfglyphtounicode{jeharabic}{0698} \pdfglyphtounicode{jehfinalarabic}{FB8B} \pdfglyphtounicode{jhabengali}{099D} \pdfglyphtounicode{jhadeva}{091D} \pdfglyphtounicode{jhagujarati}{0A9D} \pdfglyphtounicode{jhagurmukhi}{0A1D} \pdfglyphtounicode{jheharmenian}{057B} \pdfglyphtounicode{jis}{3004} \pdfglyphtounicode{jmonospace}{FF4A} \pdfglyphtounicode{jparen}{24A5} \pdfglyphtounicode{jsuperior}{02B2} \pdfglyphtounicode{k}{006B} \pdfglyphtounicode{kabashkircyrillic}{04A1} \pdfglyphtounicode{kabengali}{0995} \pdfglyphtounicode{kacute}{1E31} \pdfglyphtounicode{kacyrillic}{043A} \pdfglyphtounicode{kadescendercyrillic}{049B} \pdfglyphtounicode{kadeva}{0915} \pdfglyphtounicode{kaf}{05DB} \pdfglyphtounicode{kafarabic}{0643} \pdfglyphtounicode{kafdagesh}{FB3B} \pdfglyphtounicode{kafdageshhebrew}{FB3B} \pdfglyphtounicode{kaffinalarabic}{FEDA} \pdfglyphtounicode{kafhebrew}{05DB} \pdfglyphtounicode{kafinitialarabic}{FEDB} \pdfglyphtounicode{kafmedialarabic}{FEDC} \pdfglyphtounicode{kafrafehebrew}{FB4D} \pdfglyphtounicode{kagujarati}{0A95} \pdfglyphtounicode{kagurmukhi}{0A15} \pdfglyphtounicode{kahiragana}{304B} \pdfglyphtounicode{kahookcyrillic}{04C4} \pdfglyphtounicode{kakatakana}{30AB} \pdfglyphtounicode{kakatakanahalfwidth}{FF76} \pdfglyphtounicode{kappa}{03BA} \pdfglyphtounicode{kappasymbolgreek}{03F0} \pdfglyphtounicode{kapyeounmieumkorean}{3171} \pdfglyphtounicode{kapyeounphieuphkorean}{3184} \pdfglyphtounicode{kapyeounpieupkorean}{3178} \pdfglyphtounicode{kapyeounssangpieupkorean}{3179} \pdfglyphtounicode{karoriisquare}{330D} \pdfglyphtounicode{kashidaautoarabic}{0640} \pdfglyphtounicode{kashidaautonosidebearingarabic}{0640} \pdfglyphtounicode{kasmallkatakana}{30F5} \pdfglyphtounicode{kasquare}{3384} \pdfglyphtounicode{kasraarabic}{0650} \pdfglyphtounicode{kasratanarabic}{064D} \pdfglyphtounicode{kastrokecyrillic}{049F} \pdfglyphtounicode{katahiraprolongmarkhalfwidth}{FF70} \pdfglyphtounicode{kaverticalstrokecyrillic}{049D} \pdfglyphtounicode{kbopomofo}{310E} \pdfglyphtounicode{kcalsquare}{3389} \pdfglyphtounicode{kcaron}{01E9} \pdfglyphtounicode{kcedilla}{0137} \pdfglyphtounicode{kcircle}{24DA} \pdfglyphtounicode{kcommaaccent}{0137} \pdfglyphtounicode{kdotbelow}{1E33} \pdfglyphtounicode{keharmenian}{0584} \pdfglyphtounicode{kehiragana}{3051} \pdfglyphtounicode{kekatakana}{30B1} \pdfglyphtounicode{kekatakanahalfwidth}{FF79} \pdfglyphtounicode{kenarmenian}{056F} \pdfglyphtounicode{kesmallkatakana}{30F6} \pdfglyphtounicode{kgreenlandic}{0138} \pdfglyphtounicode{khabengali}{0996} \pdfglyphtounicode{khacyrillic}{0445} \pdfglyphtounicode{khadeva}{0916} \pdfglyphtounicode{khagujarati}{0A96} \pdfglyphtounicode{khagurmukhi}{0A16} \pdfglyphtounicode{khaharabic}{062E} \pdfglyphtounicode{khahfinalarabic}{FEA6} \pdfglyphtounicode{khahinitialarabic}{FEA7} \pdfglyphtounicode{khahmedialarabic}{FEA8} \pdfglyphtounicode{kheicoptic}{03E7} \pdfglyphtounicode{khhadeva}{0959} \pdfglyphtounicode{khhagurmukhi}{0A59} \pdfglyphtounicode{khieukhacirclekorean}{3278} \pdfglyphtounicode{khieukhaparenkorean}{3218} \pdfglyphtounicode{khieukhcirclekorean}{326A} \pdfglyphtounicode{khieukhkorean}{314B} \pdfglyphtounicode{khieukhparenkorean}{320A} \pdfglyphtounicode{khokhaithai}{0E02} \pdfglyphtounicode{khokhonthai}{0E05} \pdfglyphtounicode{khokhuatthai}{0E03} \pdfglyphtounicode{khokhwaithai}{0E04} \pdfglyphtounicode{khomutthai}{0E5B} \pdfglyphtounicode{khook}{0199} \pdfglyphtounicode{khorakhangthai}{0E06} \pdfglyphtounicode{khzsquare}{3391} \pdfglyphtounicode{kihiragana}{304D} \pdfglyphtounicode{kikatakana}{30AD} \pdfglyphtounicode{kikatakanahalfwidth}{FF77} \pdfglyphtounicode{kiroguramusquare}{3315} \pdfglyphtounicode{kiromeetorusquare}{3316} \pdfglyphtounicode{kirosquare}{3314} \pdfglyphtounicode{kiyeokacirclekorean}{326E} \pdfglyphtounicode{kiyeokaparenkorean}{320E} \pdfglyphtounicode{kiyeokcirclekorean}{3260} \pdfglyphtounicode{kiyeokkorean}{3131} \pdfglyphtounicode{kiyeokparenkorean}{3200} \pdfglyphtounicode{kiyeoksioskorean}{3133} \pdfglyphtounicode{kjecyrillic}{045C} \pdfglyphtounicode{klinebelow}{1E35} \pdfglyphtounicode{klsquare}{3398} \pdfglyphtounicode{kmcubedsquare}{33A6} \pdfglyphtounicode{kmonospace}{FF4B} \pdfglyphtounicode{kmsquaredsquare}{33A2} \pdfglyphtounicode{kohiragana}{3053} \pdfglyphtounicode{kohmsquare}{33C0} \pdfglyphtounicode{kokaithai}{0E01} \pdfglyphtounicode{kokatakana}{30B3} \pdfglyphtounicode{kokatakanahalfwidth}{FF7A} \pdfglyphtounicode{kooposquare}{331E} \pdfglyphtounicode{koppacyrillic}{0481} \pdfglyphtounicode{koreanstandardsymbol}{327F} \pdfglyphtounicode{koroniscmb}{0343} \pdfglyphtounicode{kparen}{24A6} \pdfglyphtounicode{kpasquare}{33AA} \pdfglyphtounicode{ksicyrillic}{046F} \pdfglyphtounicode{ktsquare}{33CF} \pdfglyphtounicode{kturned}{029E} \pdfglyphtounicode{kuhiragana}{304F} \pdfglyphtounicode{kukatakana}{30AF} \pdfglyphtounicode{kukatakanahalfwidth}{FF78} \pdfglyphtounicode{kvsquare}{33B8} \pdfglyphtounicode{kwsquare}{33BE} \pdfglyphtounicode{l}{006C} \pdfglyphtounicode{labengali}{09B2} \pdfglyphtounicode{lacute}{013A} \pdfglyphtounicode{ladeva}{0932} \pdfglyphtounicode{lagujarati}{0AB2} \pdfglyphtounicode{lagurmukhi}{0A32} \pdfglyphtounicode{lakkhangyaothai}{0E45} \pdfglyphtounicode{lamaleffinalarabic}{FEFC} \pdfglyphtounicode{lamalefhamzaabovefinalarabic}{FEF8} \pdfglyphtounicode{lamalefhamzaaboveisolatedarabic}{FEF7} \pdfglyphtounicode{lamalefhamzabelowfinalarabic}{FEFA} \pdfglyphtounicode{lamalefhamzabelowisolatedarabic}{FEF9} \pdfglyphtounicode{lamalefisolatedarabic}{FEFB} \pdfglyphtounicode{lamalefmaddaabovefinalarabic}{FEF6} \pdfglyphtounicode{lamalefmaddaaboveisolatedarabic}{FEF5} \pdfglyphtounicode{lamarabic}{0644} \pdfglyphtounicode{lambda}{03BB} \pdfglyphtounicode{lambdastroke}{019B} \pdfglyphtounicode{lamed}{05DC} \pdfglyphtounicode{lameddagesh}{FB3C} \pdfglyphtounicode{lameddageshhebrew}{FB3C} \pdfglyphtounicode{lamedhebrew}{05DC} % lamedholam;05DC 05B9 % lamedholamdagesh;05DC 05B9 05BC % lamedholamdageshhebrew;05DC 05B9 05BC % lamedholamhebrew;05DC 05B9 \pdfglyphtounicode{lamfinalarabic}{FEDE} \pdfglyphtounicode{lamhahinitialarabic}{FCCA} \pdfglyphtounicode{laminitialarabic}{FEDF} \pdfglyphtounicode{lamjeeminitialarabic}{FCC9} \pdfglyphtounicode{lamkhahinitialarabic}{FCCB} \pdfglyphtounicode{lamlamhehisolatedarabic}{FDF2} \pdfglyphtounicode{lammedialarabic}{FEE0} \pdfglyphtounicode{lammeemhahinitialarabic}{FD88} \pdfglyphtounicode{lammeeminitialarabic}{FCCC} % lammeemjeeminitialarabic;FEDF FEE4 FEA0 % lammeemkhahinitialarabic;FEDF FEE4 FEA8 \pdfglyphtounicode{largecircle}{25EF} \pdfglyphtounicode{lbar}{019A} \pdfglyphtounicode{lbelt}{026C} \pdfglyphtounicode{lbopomofo}{310C} \pdfglyphtounicode{lcaron}{013E} \pdfglyphtounicode{lcedilla}{013C} \pdfglyphtounicode{lcircle}{24DB} \pdfglyphtounicode{lcircumflexbelow}{1E3D} \pdfglyphtounicode{lcommaaccent}{013C} \pdfglyphtounicode{ldot}{0140} \pdfglyphtounicode{ldotaccent}{0140} \pdfglyphtounicode{ldotbelow}{1E37} \pdfglyphtounicode{ldotbelowmacron}{1E39} \pdfglyphtounicode{leftangleabovecmb}{031A} \pdfglyphtounicode{lefttackbelowcmb}{0318} \pdfglyphtounicode{less}{003C} \pdfglyphtounicode{lessequal}{2264} \pdfglyphtounicode{lessequalorgreater}{22DA} \pdfglyphtounicode{lessmonospace}{FF1C} \pdfglyphtounicode{lessorequivalent}{2272} \pdfglyphtounicode{lessorgreater}{2276} \pdfglyphtounicode{lessoverequal}{2266} \pdfglyphtounicode{lesssmall}{FE64} \pdfglyphtounicode{lezh}{026E} \pdfglyphtounicode{lfblock}{258C} \pdfglyphtounicode{lhookretroflex}{026D} \pdfglyphtounicode{lira}{20A4} \pdfglyphtounicode{liwnarmenian}{056C} \pdfglyphtounicode{lj}{01C9} \pdfglyphtounicode{ljecyrillic}{0459} \pdfglyphtounicode{ll}{F6C0} \pdfglyphtounicode{lladeva}{0933} \pdfglyphtounicode{llagujarati}{0AB3} \pdfglyphtounicode{llinebelow}{1E3B} \pdfglyphtounicode{llladeva}{0934} \pdfglyphtounicode{llvocalicbengali}{09E1} \pdfglyphtounicode{llvocalicdeva}{0961} \pdfglyphtounicode{llvocalicvowelsignbengali}{09E3} \pdfglyphtounicode{llvocalicvowelsigndeva}{0963} \pdfglyphtounicode{lmiddletilde}{026B} \pdfglyphtounicode{lmonospace}{FF4C} \pdfglyphtounicode{lmsquare}{33D0} \pdfglyphtounicode{lochulathai}{0E2C} \pdfglyphtounicode{logicaland}{2227} \pdfglyphtounicode{logicalnot}{00AC} \pdfglyphtounicode{logicalnotreversed}{2310} \pdfglyphtounicode{logicalor}{2228} \pdfglyphtounicode{lolingthai}{0E25} \pdfglyphtounicode{longs}{017F} \pdfglyphtounicode{lowlinecenterline}{FE4E} \pdfglyphtounicode{lowlinecmb}{0332} \pdfglyphtounicode{lowlinedashed}{FE4D} \pdfglyphtounicode{lozenge}{25CA} \pdfglyphtounicode{lparen}{24A7} \pdfglyphtounicode{lslash}{0142} \pdfglyphtounicode{lsquare}{2113} \pdfglyphtounicode{lsuperior}{F6EE} \pdfglyphtounicode{ltshade}{2591} \pdfglyphtounicode{luthai}{0E26} \pdfglyphtounicode{lvocalicbengali}{098C} \pdfglyphtounicode{lvocalicdeva}{090C} \pdfglyphtounicode{lvocalicvowelsignbengali}{09E2} \pdfglyphtounicode{lvocalicvowelsigndeva}{0962} \pdfglyphtounicode{lxsquare}{33D3} \pdfglyphtounicode{m}{006D} \pdfglyphtounicode{mabengali}{09AE} \pdfglyphtounicode{macron}{00AF} \pdfglyphtounicode{macronbelowcmb}{0331} \pdfglyphtounicode{macroncmb}{0304} \pdfglyphtounicode{macronlowmod}{02CD} \pdfglyphtounicode{macronmonospace}{FFE3} \pdfglyphtounicode{macute}{1E3F} \pdfglyphtounicode{madeva}{092E} \pdfglyphtounicode{magujarati}{0AAE} \pdfglyphtounicode{magurmukhi}{0A2E} \pdfglyphtounicode{mahapakhhebrew}{05A4} \pdfglyphtounicode{mahapakhlefthebrew}{05A4} \pdfglyphtounicode{mahiragana}{307E} \pdfglyphtounicode{maichattawalowleftthai}{F895} \pdfglyphtounicode{maichattawalowrightthai}{F894} \pdfglyphtounicode{maichattawathai}{0E4B} \pdfglyphtounicode{maichattawaupperleftthai}{F893} \pdfglyphtounicode{maieklowleftthai}{F88C} \pdfglyphtounicode{maieklowrightthai}{F88B} \pdfglyphtounicode{maiekthai}{0E48} \pdfglyphtounicode{maiekupperleftthai}{F88A} \pdfglyphtounicode{maihanakatleftthai}{F884} \pdfglyphtounicode{maihanakatthai}{0E31} \pdfglyphtounicode{maitaikhuleftthai}{F889} \pdfglyphtounicode{maitaikhuthai}{0E47} \pdfglyphtounicode{maitholowleftthai}{F88F} \pdfglyphtounicode{maitholowrightthai}{F88E} \pdfglyphtounicode{maithothai}{0E49} \pdfglyphtounicode{maithoupperleftthai}{F88D} \pdfglyphtounicode{maitrilowleftthai}{F892} \pdfglyphtounicode{maitrilowrightthai}{F891} \pdfglyphtounicode{maitrithai}{0E4A} \pdfglyphtounicode{maitriupperleftthai}{F890} \pdfglyphtounicode{maiyamokthai}{0E46} \pdfglyphtounicode{makatakana}{30DE} \pdfglyphtounicode{makatakanahalfwidth}{FF8F} \pdfglyphtounicode{male}{2642} \pdfglyphtounicode{mansyonsquare}{3347} \pdfglyphtounicode{maqafhebrew}{05BE} \pdfglyphtounicode{mars}{2642} \pdfglyphtounicode{masoracirclehebrew}{05AF} \pdfglyphtounicode{masquare}{3383} \pdfglyphtounicode{mbopomofo}{3107} \pdfglyphtounicode{mbsquare}{33D4} \pdfglyphtounicode{mcircle}{24DC} \pdfglyphtounicode{mcubedsquare}{33A5} \pdfglyphtounicode{mdotaccent}{1E41} \pdfglyphtounicode{mdotbelow}{1E43} \pdfglyphtounicode{meemarabic}{0645} \pdfglyphtounicode{meemfinalarabic}{FEE2} \pdfglyphtounicode{meeminitialarabic}{FEE3} \pdfglyphtounicode{meemmedialarabic}{FEE4} \pdfglyphtounicode{meemmeeminitialarabic}{FCD1} \pdfglyphtounicode{meemmeemisolatedarabic}{FC48} \pdfglyphtounicode{meetorusquare}{334D} \pdfglyphtounicode{mehiragana}{3081} \pdfglyphtounicode{meizierasquare}{337E} \pdfglyphtounicode{mekatakana}{30E1} \pdfglyphtounicode{mekatakanahalfwidth}{FF92} \pdfglyphtounicode{mem}{05DE} \pdfglyphtounicode{memdagesh}{FB3E} \pdfglyphtounicode{memdageshhebrew}{FB3E} \pdfglyphtounicode{memhebrew}{05DE} \pdfglyphtounicode{menarmenian}{0574} \pdfglyphtounicode{merkhahebrew}{05A5} \pdfglyphtounicode{merkhakefulahebrew}{05A6} \pdfglyphtounicode{merkhakefulalefthebrew}{05A6} \pdfglyphtounicode{merkhalefthebrew}{05A5} \pdfglyphtounicode{mhook}{0271} \pdfglyphtounicode{mhzsquare}{3392} \pdfglyphtounicode{middledotkatakanahalfwidth}{FF65} \pdfglyphtounicode{middot}{00B7} \pdfglyphtounicode{mieumacirclekorean}{3272} \pdfglyphtounicode{mieumaparenkorean}{3212} \pdfglyphtounicode{mieumcirclekorean}{3264} \pdfglyphtounicode{mieumkorean}{3141} \pdfglyphtounicode{mieumpansioskorean}{3170} \pdfglyphtounicode{mieumparenkorean}{3204} \pdfglyphtounicode{mieumpieupkorean}{316E} \pdfglyphtounicode{mieumsioskorean}{316F} \pdfglyphtounicode{mihiragana}{307F} \pdfglyphtounicode{mikatakana}{30DF} \pdfglyphtounicode{mikatakanahalfwidth}{FF90} \pdfglyphtounicode{minus}{2212} \pdfglyphtounicode{minusbelowcmb}{0320} \pdfglyphtounicode{minuscircle}{2296} \pdfglyphtounicode{minusmod}{02D7} \pdfglyphtounicode{minusplus}{2213} \pdfglyphtounicode{minute}{2032} \pdfglyphtounicode{miribaarusquare}{334A} \pdfglyphtounicode{mirisquare}{3349} \pdfglyphtounicode{mlonglegturned}{0270} \pdfglyphtounicode{mlsquare}{3396} \pdfglyphtounicode{mmcubedsquare}{33A3} \pdfglyphtounicode{mmonospace}{FF4D} \pdfglyphtounicode{mmsquaredsquare}{339F} \pdfglyphtounicode{mohiragana}{3082} \pdfglyphtounicode{mohmsquare}{33C1} \pdfglyphtounicode{mokatakana}{30E2} \pdfglyphtounicode{mokatakanahalfwidth}{FF93} \pdfglyphtounicode{molsquare}{33D6} \pdfglyphtounicode{momathai}{0E21} \pdfglyphtounicode{moverssquare}{33A7} \pdfglyphtounicode{moverssquaredsquare}{33A8} \pdfglyphtounicode{mparen}{24A8} \pdfglyphtounicode{mpasquare}{33AB} \pdfglyphtounicode{mssquare}{33B3} \pdfglyphtounicode{msuperior}{F6EF} \pdfglyphtounicode{mturned}{026F} \pdfglyphtounicode{mu}{00B5} \pdfglyphtounicode{mu1}{00B5} \pdfglyphtounicode{muasquare}{3382} \pdfglyphtounicode{muchgreater}{226B} \pdfglyphtounicode{muchless}{226A} \pdfglyphtounicode{mufsquare}{338C} \pdfglyphtounicode{mugreek}{03BC} \pdfglyphtounicode{mugsquare}{338D} \pdfglyphtounicode{muhiragana}{3080} \pdfglyphtounicode{mukatakana}{30E0} \pdfglyphtounicode{mukatakanahalfwidth}{FF91} \pdfglyphtounicode{mulsquare}{3395} \pdfglyphtounicode{multiply}{00D7} \pdfglyphtounicode{mumsquare}{339B} \pdfglyphtounicode{munahhebrew}{05A3} \pdfglyphtounicode{munahlefthebrew}{05A3} \pdfglyphtounicode{musicalnote}{266A} \pdfglyphtounicode{musicalnotedbl}{266B} \pdfglyphtounicode{musicflatsign}{266D} \pdfglyphtounicode{musicsharpsign}{266F} \pdfglyphtounicode{mussquare}{33B2} \pdfglyphtounicode{muvsquare}{33B6} \pdfglyphtounicode{muwsquare}{33BC} \pdfglyphtounicode{mvmegasquare}{33B9} \pdfglyphtounicode{mvsquare}{33B7} \pdfglyphtounicode{mwmegasquare}{33BF} \pdfglyphtounicode{mwsquare}{33BD} \pdfglyphtounicode{n}{006E} \pdfglyphtounicode{nabengali}{09A8} \pdfglyphtounicode{nabla}{2207} \pdfglyphtounicode{nacute}{0144} \pdfglyphtounicode{nadeva}{0928} \pdfglyphtounicode{nagujarati}{0AA8} \pdfglyphtounicode{nagurmukhi}{0A28} \pdfglyphtounicode{nahiragana}{306A} \pdfglyphtounicode{nakatakana}{30CA} \pdfglyphtounicode{nakatakanahalfwidth}{FF85} \pdfglyphtounicode{napostrophe}{0149} \pdfglyphtounicode{nasquare}{3381} \pdfglyphtounicode{nbopomofo}{310B} \pdfglyphtounicode{nbspace}{00A0} \pdfglyphtounicode{ncaron}{0148} \pdfglyphtounicode{ncedilla}{0146} \pdfglyphtounicode{ncircle}{24DD} \pdfglyphtounicode{ncircumflexbelow}{1E4B} \pdfglyphtounicode{ncommaaccent}{0146} \pdfglyphtounicode{ndotaccent}{1E45} \pdfglyphtounicode{ndotbelow}{1E47} \pdfglyphtounicode{nehiragana}{306D} \pdfglyphtounicode{nekatakana}{30CD} \pdfglyphtounicode{nekatakanahalfwidth}{FF88} \pdfglyphtounicode{newsheqelsign}{20AA} \pdfglyphtounicode{nfsquare}{338B} \pdfglyphtounicode{ngabengali}{0999} \pdfglyphtounicode{ngadeva}{0919} \pdfglyphtounicode{ngagujarati}{0A99} \pdfglyphtounicode{ngagurmukhi}{0A19} \pdfglyphtounicode{ngonguthai}{0E07} \pdfglyphtounicode{nhiragana}{3093} \pdfglyphtounicode{nhookleft}{0272} \pdfglyphtounicode{nhookretroflex}{0273} \pdfglyphtounicode{nieunacirclekorean}{326F} \pdfglyphtounicode{nieunaparenkorean}{320F} \pdfglyphtounicode{nieuncieuckorean}{3135} \pdfglyphtounicode{nieuncirclekorean}{3261} \pdfglyphtounicode{nieunhieuhkorean}{3136} \pdfglyphtounicode{nieunkorean}{3134} \pdfglyphtounicode{nieunpansioskorean}{3168} \pdfglyphtounicode{nieunparenkorean}{3201} \pdfglyphtounicode{nieunsioskorean}{3167} \pdfglyphtounicode{nieuntikeutkorean}{3166} \pdfglyphtounicode{nihiragana}{306B} \pdfglyphtounicode{nikatakana}{30CB} \pdfglyphtounicode{nikatakanahalfwidth}{FF86} \pdfglyphtounicode{nikhahitleftthai}{F899} \pdfglyphtounicode{nikhahitthai}{0E4D} \pdfglyphtounicode{nine}{0039} \pdfglyphtounicode{ninearabic}{0669} \pdfglyphtounicode{ninebengali}{09EF} \pdfglyphtounicode{ninecircle}{2468} \pdfglyphtounicode{ninecircleinversesansserif}{2792} \pdfglyphtounicode{ninedeva}{096F} \pdfglyphtounicode{ninegujarati}{0AEF} \pdfglyphtounicode{ninegurmukhi}{0A6F} \pdfglyphtounicode{ninehackarabic}{0669} \pdfglyphtounicode{ninehangzhou}{3029} \pdfglyphtounicode{nineideographicparen}{3228} \pdfglyphtounicode{nineinferior}{2089} \pdfglyphtounicode{ninemonospace}{FF19} \pdfglyphtounicode{nineoldstyle}{F739} \pdfglyphtounicode{nineparen}{247C} \pdfglyphtounicode{nineperiod}{2490} \pdfglyphtounicode{ninepersian}{06F9} \pdfglyphtounicode{nineroman}{2178} \pdfglyphtounicode{ninesuperior}{2079} \pdfglyphtounicode{nineteencircle}{2472} \pdfglyphtounicode{nineteenparen}{2486} \pdfglyphtounicode{nineteenperiod}{249A} \pdfglyphtounicode{ninethai}{0E59} \pdfglyphtounicode{nj}{01CC} \pdfglyphtounicode{njecyrillic}{045A} \pdfglyphtounicode{nkatakana}{30F3} \pdfglyphtounicode{nkatakanahalfwidth}{FF9D} \pdfglyphtounicode{nlegrightlong}{019E} \pdfglyphtounicode{nlinebelow}{1E49} \pdfglyphtounicode{nmonospace}{FF4E} \pdfglyphtounicode{nmsquare}{339A} \pdfglyphtounicode{nnabengali}{09A3} \pdfglyphtounicode{nnadeva}{0923} \pdfglyphtounicode{nnagujarati}{0AA3} \pdfglyphtounicode{nnagurmukhi}{0A23} \pdfglyphtounicode{nnnadeva}{0929} \pdfglyphtounicode{nohiragana}{306E} \pdfglyphtounicode{nokatakana}{30CE} \pdfglyphtounicode{nokatakanahalfwidth}{FF89} \pdfglyphtounicode{nonbreakingspace}{00A0} \pdfglyphtounicode{nonenthai}{0E13} \pdfglyphtounicode{nonuthai}{0E19} \pdfglyphtounicode{noonarabic}{0646} \pdfglyphtounicode{noonfinalarabic}{FEE6} \pdfglyphtounicode{noonghunnaarabic}{06BA} \pdfglyphtounicode{noonghunnafinalarabic}{FB9F} % noonhehinitialarabic;FEE7 FEEC \pdfglyphtounicode{nooninitialarabic}{FEE7} \pdfglyphtounicode{noonjeeminitialarabic}{FCD2} \pdfglyphtounicode{noonjeemisolatedarabic}{FC4B} \pdfglyphtounicode{noonmedialarabic}{FEE8} \pdfglyphtounicode{noonmeeminitialarabic}{FCD5} \pdfglyphtounicode{noonmeemisolatedarabic}{FC4E} \pdfglyphtounicode{noonnoonfinalarabic}{FC8D} \pdfglyphtounicode{notcontains}{220C} \pdfglyphtounicode{notelement}{2209} \pdfglyphtounicode{notelementof}{2209} \pdfglyphtounicode{notequal}{2260} \pdfglyphtounicode{notgreater}{226F} \pdfglyphtounicode{notgreaternorequal}{2271} \pdfglyphtounicode{notgreaternorless}{2279} \pdfglyphtounicode{notidentical}{2262} \pdfglyphtounicode{notless}{226E} \pdfglyphtounicode{notlessnorequal}{2270} \pdfglyphtounicode{notparallel}{2226} \pdfglyphtounicode{notprecedes}{2280} \pdfglyphtounicode{notsubset}{2284} \pdfglyphtounicode{notsucceeds}{2281} \pdfglyphtounicode{notsuperset}{2285} \pdfglyphtounicode{nowarmenian}{0576} \pdfglyphtounicode{nparen}{24A9} \pdfglyphtounicode{nssquare}{33B1} \pdfglyphtounicode{nsuperior}{207F} \pdfglyphtounicode{ntilde}{00F1} \pdfglyphtounicode{nu}{03BD} \pdfglyphtounicode{nuhiragana}{306C} \pdfglyphtounicode{nukatakana}{30CC} \pdfglyphtounicode{nukatakanahalfwidth}{FF87} \pdfglyphtounicode{nuktabengali}{09BC} \pdfglyphtounicode{nuktadeva}{093C} \pdfglyphtounicode{nuktagujarati}{0ABC} \pdfglyphtounicode{nuktagurmukhi}{0A3C} \pdfglyphtounicode{numbersign}{0023} \pdfglyphtounicode{numbersignmonospace}{FF03} \pdfglyphtounicode{numbersignsmall}{FE5F} \pdfglyphtounicode{numeralsigngreek}{0374} \pdfglyphtounicode{numeralsignlowergreek}{0375} \pdfglyphtounicode{numero}{2116} \pdfglyphtounicode{nun}{05E0} \pdfglyphtounicode{nundagesh}{FB40} \pdfglyphtounicode{nundageshhebrew}{FB40} \pdfglyphtounicode{nunhebrew}{05E0} \pdfglyphtounicode{nvsquare}{33B5} \pdfglyphtounicode{nwsquare}{33BB} \pdfglyphtounicode{nyabengali}{099E} \pdfglyphtounicode{nyadeva}{091E} \pdfglyphtounicode{nyagujarati}{0A9E} \pdfglyphtounicode{nyagurmukhi}{0A1E} \pdfglyphtounicode{o}{006F} \pdfglyphtounicode{oacute}{00F3} \pdfglyphtounicode{oangthai}{0E2D} \pdfglyphtounicode{obarred}{0275} \pdfglyphtounicode{obarredcyrillic}{04E9} \pdfglyphtounicode{obarreddieresiscyrillic}{04EB} \pdfglyphtounicode{obengali}{0993} \pdfglyphtounicode{obopomofo}{311B} \pdfglyphtounicode{obreve}{014F} \pdfglyphtounicode{ocandradeva}{0911} \pdfglyphtounicode{ocandragujarati}{0A91} \pdfglyphtounicode{ocandravowelsigndeva}{0949} \pdfglyphtounicode{ocandravowelsigngujarati}{0AC9} \pdfglyphtounicode{ocaron}{01D2} \pdfglyphtounicode{ocircle}{24DE} \pdfglyphtounicode{ocircumflex}{00F4} \pdfglyphtounicode{ocircumflexacute}{1ED1} \pdfglyphtounicode{ocircumflexdotbelow}{1ED9} \pdfglyphtounicode{ocircumflexgrave}{1ED3} \pdfglyphtounicode{ocircumflexhookabove}{1ED5} \pdfglyphtounicode{ocircumflextilde}{1ED7} \pdfglyphtounicode{ocyrillic}{043E} \pdfglyphtounicode{odblacute}{0151} \pdfglyphtounicode{odblgrave}{020D} \pdfglyphtounicode{odeva}{0913} \pdfglyphtounicode{odieresis}{00F6} \pdfglyphtounicode{odieresiscyrillic}{04E7} \pdfglyphtounicode{odotbelow}{1ECD} \pdfglyphtounicode{oe}{0153} \pdfglyphtounicode{oekorean}{315A} \pdfglyphtounicode{ogonek}{02DB} \pdfglyphtounicode{ogonekcmb}{0328} \pdfglyphtounicode{ograve}{00F2} \pdfglyphtounicode{ogujarati}{0A93} \pdfglyphtounicode{oharmenian}{0585} \pdfglyphtounicode{ohiragana}{304A} \pdfglyphtounicode{ohookabove}{1ECF} \pdfglyphtounicode{ohorn}{01A1} \pdfglyphtounicode{ohornacute}{1EDB} \pdfglyphtounicode{ohorndotbelow}{1EE3} \pdfglyphtounicode{ohorngrave}{1EDD} \pdfglyphtounicode{ohornhookabove}{1EDF} \pdfglyphtounicode{ohorntilde}{1EE1} \pdfglyphtounicode{ohungarumlaut}{0151} \pdfglyphtounicode{oi}{01A3} \pdfglyphtounicode{oinvertedbreve}{020F} \pdfglyphtounicode{okatakana}{30AA} \pdfglyphtounicode{okatakanahalfwidth}{FF75} \pdfglyphtounicode{okorean}{3157} \pdfglyphtounicode{olehebrew}{05AB} \pdfglyphtounicode{omacron}{014D} \pdfglyphtounicode{omacronacute}{1E53} \pdfglyphtounicode{omacrongrave}{1E51} \pdfglyphtounicode{omdeva}{0950} \pdfglyphtounicode{omega}{03C9} \pdfglyphtounicode{omega1}{03D6} \pdfglyphtounicode{omegacyrillic}{0461} \pdfglyphtounicode{omegalatinclosed}{0277} \pdfglyphtounicode{omegaroundcyrillic}{047B} \pdfglyphtounicode{omegatitlocyrillic}{047D} \pdfglyphtounicode{omegatonos}{03CE} \pdfglyphtounicode{omgujarati}{0AD0} \pdfglyphtounicode{omicron}{03BF} \pdfglyphtounicode{omicrontonos}{03CC} \pdfglyphtounicode{omonospace}{FF4F} \pdfglyphtounicode{one}{0031} \pdfglyphtounicode{onearabic}{0661} \pdfglyphtounicode{onebengali}{09E7} \pdfglyphtounicode{onecircle}{2460} \pdfglyphtounicode{onecircleinversesansserif}{278A} \pdfglyphtounicode{onedeva}{0967} \pdfglyphtounicode{onedotenleader}{2024} \pdfglyphtounicode{oneeighth}{215B} \pdfglyphtounicode{onefitted}{F6DC} \pdfglyphtounicode{onegujarati}{0AE7} \pdfglyphtounicode{onegurmukhi}{0A67} \pdfglyphtounicode{onehackarabic}{0661} \pdfglyphtounicode{onehalf}{00BD} \pdfglyphtounicode{onehangzhou}{3021} \pdfglyphtounicode{oneideographicparen}{3220} \pdfglyphtounicode{oneinferior}{2081} \pdfglyphtounicode{onemonospace}{FF11} \pdfglyphtounicode{onenumeratorbengali}{09F4} \pdfglyphtounicode{oneoldstyle}{F731} \pdfglyphtounicode{oneparen}{2474} \pdfglyphtounicode{oneperiod}{2488} \pdfglyphtounicode{onepersian}{06F1} \pdfglyphtounicode{onequarter}{00BC} \pdfglyphtounicode{oneroman}{2170} \pdfglyphtounicode{onesuperior}{00B9} \pdfglyphtounicode{onethai}{0E51} \pdfglyphtounicode{onethird}{2153} \pdfglyphtounicode{oogonek}{01EB} \pdfglyphtounicode{oogonekmacron}{01ED} \pdfglyphtounicode{oogurmukhi}{0A13} \pdfglyphtounicode{oomatragurmukhi}{0A4B} \pdfglyphtounicode{oopen}{0254} \pdfglyphtounicode{oparen}{24AA} \pdfglyphtounicode{openbullet}{25E6} \pdfglyphtounicode{option}{2325} \pdfglyphtounicode{ordfeminine}{00AA} \pdfglyphtounicode{ordmasculine}{00BA} \pdfglyphtounicode{orthogonal}{221F} \pdfglyphtounicode{oshortdeva}{0912} \pdfglyphtounicode{oshortvowelsigndeva}{094A} \pdfglyphtounicode{oslash}{00F8} \pdfglyphtounicode{oslashacute}{01FF} \pdfglyphtounicode{osmallhiragana}{3049} \pdfglyphtounicode{osmallkatakana}{30A9} \pdfglyphtounicode{osmallkatakanahalfwidth}{FF6B} \pdfglyphtounicode{ostrokeacute}{01FF} \pdfglyphtounicode{osuperior}{F6F0} \pdfglyphtounicode{otcyrillic}{047F} \pdfglyphtounicode{otilde}{00F5} \pdfglyphtounicode{otildeacute}{1E4D} \pdfglyphtounicode{otildedieresis}{1E4F} \pdfglyphtounicode{oubopomofo}{3121} \pdfglyphtounicode{overline}{203E} \pdfglyphtounicode{overlinecenterline}{FE4A} \pdfglyphtounicode{overlinecmb}{0305} \pdfglyphtounicode{overlinedashed}{FE49} \pdfglyphtounicode{overlinedblwavy}{FE4C} \pdfglyphtounicode{overlinewavy}{FE4B} \pdfglyphtounicode{overscore}{00AF} \pdfglyphtounicode{ovowelsignbengali}{09CB} \pdfglyphtounicode{ovowelsigndeva}{094B} \pdfglyphtounicode{ovowelsigngujarati}{0ACB} \pdfglyphtounicode{p}{0070} \pdfglyphtounicode{paampssquare}{3380} \pdfglyphtounicode{paasentosquare}{332B} \pdfglyphtounicode{pabengali}{09AA} \pdfglyphtounicode{pacute}{1E55} \pdfglyphtounicode{padeva}{092A} \pdfglyphtounicode{pagedown}{21DF} \pdfglyphtounicode{pageup}{21DE} \pdfglyphtounicode{pagujarati}{0AAA} \pdfglyphtounicode{pagurmukhi}{0A2A} \pdfglyphtounicode{pahiragana}{3071} \pdfglyphtounicode{paiyannoithai}{0E2F} \pdfglyphtounicode{pakatakana}{30D1} \pdfglyphtounicode{palatalizationcyrilliccmb}{0484} \pdfglyphtounicode{palochkacyrillic}{04C0} \pdfglyphtounicode{pansioskorean}{317F} \pdfglyphtounicode{paragraph}{00B6} \pdfglyphtounicode{parallel}{2225} \pdfglyphtounicode{parenleft}{0028} \pdfglyphtounicode{parenleftaltonearabic}{FD3E} \pdfglyphtounicode{parenleftbt}{F8ED} \pdfglyphtounicode{parenleftex}{F8EC} \pdfglyphtounicode{parenleftinferior}{208D} \pdfglyphtounicode{parenleftmonospace}{FF08} \pdfglyphtounicode{parenleftsmall}{FE59} \pdfglyphtounicode{parenleftsuperior}{207D} \pdfglyphtounicode{parenlefttp}{F8EB} \pdfglyphtounicode{parenleftvertical}{FE35} \pdfglyphtounicode{parenright}{0029} \pdfglyphtounicode{parenrightaltonearabic}{FD3F} \pdfglyphtounicode{parenrightbt}{F8F8} \pdfglyphtounicode{parenrightex}{F8F7} \pdfglyphtounicode{parenrightinferior}{208E} \pdfglyphtounicode{parenrightmonospace}{FF09} \pdfglyphtounicode{parenrightsmall}{FE5A} \pdfglyphtounicode{parenrightsuperior}{207E} \pdfglyphtounicode{parenrighttp}{F8F6} \pdfglyphtounicode{parenrightvertical}{FE36} \pdfglyphtounicode{partialdiff}{2202} \pdfglyphtounicode{paseqhebrew}{05C0} \pdfglyphtounicode{pashtahebrew}{0599} \pdfglyphtounicode{pasquare}{33A9} \pdfglyphtounicode{patah}{05B7} \pdfglyphtounicode{patah11}{05B7} \pdfglyphtounicode{patah1d}{05B7} \pdfglyphtounicode{patah2a}{05B7} \pdfglyphtounicode{patahhebrew}{05B7} \pdfglyphtounicode{patahnarrowhebrew}{05B7} \pdfglyphtounicode{patahquarterhebrew}{05B7} \pdfglyphtounicode{patahwidehebrew}{05B7} \pdfglyphtounicode{pazerhebrew}{05A1} \pdfglyphtounicode{pbopomofo}{3106} \pdfglyphtounicode{pcircle}{24DF} \pdfglyphtounicode{pdotaccent}{1E57} \pdfglyphtounicode{pe}{05E4} \pdfglyphtounicode{pecyrillic}{043F} \pdfglyphtounicode{pedagesh}{FB44} \pdfglyphtounicode{pedageshhebrew}{FB44} \pdfglyphtounicode{peezisquare}{333B} \pdfglyphtounicode{pefinaldageshhebrew}{FB43} \pdfglyphtounicode{peharabic}{067E} \pdfglyphtounicode{peharmenian}{057A} \pdfglyphtounicode{pehebrew}{05E4} \pdfglyphtounicode{pehfinalarabic}{FB57} \pdfglyphtounicode{pehinitialarabic}{FB58} \pdfglyphtounicode{pehiragana}{307A} \pdfglyphtounicode{pehmedialarabic}{FB59} \pdfglyphtounicode{pekatakana}{30DA} \pdfglyphtounicode{pemiddlehookcyrillic}{04A7} \pdfglyphtounicode{perafehebrew}{FB4E} \pdfglyphtounicode{percent}{0025} \pdfglyphtounicode{percentarabic}{066A} \pdfglyphtounicode{percentmonospace}{FF05} \pdfglyphtounicode{percentsmall}{FE6A} \pdfglyphtounicode{period}{002E} \pdfglyphtounicode{periodarmenian}{0589} \pdfglyphtounicode{periodcentered}{00B7} \pdfglyphtounicode{periodhalfwidth}{FF61} \pdfglyphtounicode{periodinferior}{F6E7} \pdfglyphtounicode{periodmonospace}{FF0E} \pdfglyphtounicode{periodsmall}{FE52} \pdfglyphtounicode{periodsuperior}{F6E8} \pdfglyphtounicode{perispomenigreekcmb}{0342} \pdfglyphtounicode{perpendicular}{22A5} \pdfglyphtounicode{perthousand}{2030} \pdfglyphtounicode{peseta}{20A7} \pdfglyphtounicode{pfsquare}{338A} \pdfglyphtounicode{phabengali}{09AB} \pdfglyphtounicode{phadeva}{092B} \pdfglyphtounicode{phagujarati}{0AAB} \pdfglyphtounicode{phagurmukhi}{0A2B} \pdfglyphtounicode{phi}{03C6} \pdfglyphtounicode{phi1}{03D5} \pdfglyphtounicode{phieuphacirclekorean}{327A} \pdfglyphtounicode{phieuphaparenkorean}{321A} \pdfglyphtounicode{phieuphcirclekorean}{326C} \pdfglyphtounicode{phieuphkorean}{314D} \pdfglyphtounicode{phieuphparenkorean}{320C} \pdfglyphtounicode{philatin}{0278} \pdfglyphtounicode{phinthuthai}{0E3A} \pdfglyphtounicode{phisymbolgreek}{03D5} \pdfglyphtounicode{phook}{01A5} \pdfglyphtounicode{phophanthai}{0E1E} \pdfglyphtounicode{phophungthai}{0E1C} \pdfglyphtounicode{phosamphaothai}{0E20} \pdfglyphtounicode{pi}{03C0} \pdfglyphtounicode{pieupacirclekorean}{3273} \pdfglyphtounicode{pieupaparenkorean}{3213} \pdfglyphtounicode{pieupcieuckorean}{3176} \pdfglyphtounicode{pieupcirclekorean}{3265} \pdfglyphtounicode{pieupkiyeokkorean}{3172} \pdfglyphtounicode{pieupkorean}{3142} \pdfglyphtounicode{pieupparenkorean}{3205} \pdfglyphtounicode{pieupsioskiyeokkorean}{3174} \pdfglyphtounicode{pieupsioskorean}{3144} \pdfglyphtounicode{pieupsiostikeutkorean}{3175} \pdfglyphtounicode{pieupthieuthkorean}{3177} \pdfglyphtounicode{pieuptikeutkorean}{3173} \pdfglyphtounicode{pihiragana}{3074} \pdfglyphtounicode{pikatakana}{30D4} \pdfglyphtounicode{pisymbolgreek}{03D6} \pdfglyphtounicode{piwrarmenian}{0583} \pdfglyphtounicode{plus}{002B} \pdfglyphtounicode{plusbelowcmb}{031F} \pdfglyphtounicode{pluscircle}{2295} \pdfglyphtounicode{plusminus}{00B1} \pdfglyphtounicode{plusmod}{02D6} \pdfglyphtounicode{plusmonospace}{FF0B} \pdfglyphtounicode{plussmall}{FE62} \pdfglyphtounicode{plussuperior}{207A} \pdfglyphtounicode{pmonospace}{FF50} \pdfglyphtounicode{pmsquare}{33D8} \pdfglyphtounicode{pohiragana}{307D} \pdfglyphtounicode{pointingindexdownwhite}{261F} \pdfglyphtounicode{pointingindexleftwhite}{261C} \pdfglyphtounicode{pointingindexrightwhite}{261E} \pdfglyphtounicode{pointingindexupwhite}{261D} \pdfglyphtounicode{pokatakana}{30DD} \pdfglyphtounicode{poplathai}{0E1B} \pdfglyphtounicode{postalmark}{3012} \pdfglyphtounicode{postalmarkface}{3020} \pdfglyphtounicode{pparen}{24AB} \pdfglyphtounicode{precedes}{227A} \pdfglyphtounicode{prescription}{211E} \pdfglyphtounicode{primemod}{02B9} \pdfglyphtounicode{primereversed}{2035} \pdfglyphtounicode{product}{220F} \pdfglyphtounicode{projective}{2305} \pdfglyphtounicode{prolongedkana}{30FC} \pdfglyphtounicode{propellor}{2318} \pdfglyphtounicode{propersubset}{2282} \pdfglyphtounicode{propersuperset}{2283} \pdfglyphtounicode{proportion}{2237} \pdfglyphtounicode{proportional}{221D} \pdfglyphtounicode{psi}{03C8} \pdfglyphtounicode{psicyrillic}{0471} \pdfglyphtounicode{psilipneumatacyrilliccmb}{0486} \pdfglyphtounicode{pssquare}{33B0} \pdfglyphtounicode{puhiragana}{3077} \pdfglyphtounicode{pukatakana}{30D7} \pdfglyphtounicode{pvsquare}{33B4} \pdfglyphtounicode{pwsquare}{33BA} \pdfglyphtounicode{q}{0071} \pdfglyphtounicode{qadeva}{0958} \pdfglyphtounicode{qadmahebrew}{05A8} \pdfglyphtounicode{qafarabic}{0642} \pdfglyphtounicode{qaffinalarabic}{FED6} \pdfglyphtounicode{qafinitialarabic}{FED7} \pdfglyphtounicode{qafmedialarabic}{FED8} \pdfglyphtounicode{qamats}{05B8} \pdfglyphtounicode{qamats10}{05B8} \pdfglyphtounicode{qamats1a}{05B8} \pdfglyphtounicode{qamats1c}{05B8} \pdfglyphtounicode{qamats27}{05B8} \pdfglyphtounicode{qamats29}{05B8} \pdfglyphtounicode{qamats33}{05B8} \pdfglyphtounicode{qamatsde}{05B8} \pdfglyphtounicode{qamatshebrew}{05B8} \pdfglyphtounicode{qamatsnarrowhebrew}{05B8} \pdfglyphtounicode{qamatsqatanhebrew}{05B8} \pdfglyphtounicode{qamatsqatannarrowhebrew}{05B8} \pdfglyphtounicode{qamatsqatanquarterhebrew}{05B8} \pdfglyphtounicode{qamatsqatanwidehebrew}{05B8} \pdfglyphtounicode{qamatsquarterhebrew}{05B8} \pdfglyphtounicode{qamatswidehebrew}{05B8} \pdfglyphtounicode{qarneyparahebrew}{059F} \pdfglyphtounicode{qbopomofo}{3111} \pdfglyphtounicode{qcircle}{24E0} \pdfglyphtounicode{qhook}{02A0} \pdfglyphtounicode{qmonospace}{FF51} \pdfglyphtounicode{qof}{05E7} \pdfglyphtounicode{qofdagesh}{FB47} \pdfglyphtounicode{qofdageshhebrew}{FB47} % qofhatafpatah;05E7 05B2 % qofhatafpatahhebrew;05E7 05B2 % qofhatafsegol;05E7 05B1 % qofhatafsegolhebrew;05E7 05B1 \pdfglyphtounicode{qofhebrew}{05E7} % qofhiriq;05E7 05B4 % qofhiriqhebrew;05E7 05B4 % qofholam;05E7 05B9 % qofholamhebrew;05E7 05B9 % qofpatah;05E7 05B7 % qofpatahhebrew;05E7 05B7 % qofqamats;05E7 05B8 % qofqamatshebrew;05E7 05B8 % qofqubuts;05E7 05BB % qofqubutshebrew;05E7 05BB % qofsegol;05E7 05B6 % qofsegolhebrew;05E7 05B6 % qofsheva;05E7 05B0 % qofshevahebrew;05E7 05B0 % qoftsere;05E7 05B5 % qoftserehebrew;05E7 05B5 \pdfglyphtounicode{qparen}{24AC} \pdfglyphtounicode{quarternote}{2669} \pdfglyphtounicode{qubuts}{05BB} \pdfglyphtounicode{qubuts18}{05BB} \pdfglyphtounicode{qubuts25}{05BB} \pdfglyphtounicode{qubuts31}{05BB} \pdfglyphtounicode{qubutshebrew}{05BB} \pdfglyphtounicode{qubutsnarrowhebrew}{05BB} \pdfglyphtounicode{qubutsquarterhebrew}{05BB} \pdfglyphtounicode{qubutswidehebrew}{05BB} \pdfglyphtounicode{question}{003F} \pdfglyphtounicode{questionarabic}{061F} \pdfglyphtounicode{questionarmenian}{055E} \pdfglyphtounicode{questiondown}{00BF} \pdfglyphtounicode{questiondownsmall}{F7BF} \pdfglyphtounicode{questiongreek}{037E} \pdfglyphtounicode{questionmonospace}{FF1F} \pdfglyphtounicode{questionsmall}{F73F} \pdfglyphtounicode{quotedbl}{0022} \pdfglyphtounicode{quotedblbase}{201E} \pdfglyphtounicode{quotedblleft}{201C} \pdfglyphtounicode{quotedblmonospace}{FF02} \pdfglyphtounicode{quotedblprime}{301E} \pdfglyphtounicode{quotedblprimereversed}{301D} \pdfglyphtounicode{quotedblright}{201D} \pdfglyphtounicode{quoteleft}{2018} \pdfglyphtounicode{quoteleftreversed}{201B} \pdfglyphtounicode{quotereversed}{201B} \pdfglyphtounicode{quoteright}{2019} \pdfglyphtounicode{quoterightn}{0149} \pdfglyphtounicode{quotesinglbase}{201A} \pdfglyphtounicode{quotesingle}{0027} \pdfglyphtounicode{quotesinglemonospace}{FF07} \pdfglyphtounicode{r}{0072} \pdfglyphtounicode{raarmenian}{057C} \pdfglyphtounicode{rabengali}{09B0} \pdfglyphtounicode{racute}{0155} \pdfglyphtounicode{radeva}{0930} \pdfglyphtounicode{radical}{221A} \pdfglyphtounicode{radicalex}{F8E5} \pdfglyphtounicode{radoverssquare}{33AE} \pdfglyphtounicode{radoverssquaredsquare}{33AF} \pdfglyphtounicode{radsquare}{33AD} \pdfglyphtounicode{rafe}{05BF} \pdfglyphtounicode{rafehebrew}{05BF} \pdfglyphtounicode{ragujarati}{0AB0} \pdfglyphtounicode{ragurmukhi}{0A30} \pdfglyphtounicode{rahiragana}{3089} \pdfglyphtounicode{rakatakana}{30E9} \pdfglyphtounicode{rakatakanahalfwidth}{FF97} \pdfglyphtounicode{ralowerdiagonalbengali}{09F1} \pdfglyphtounicode{ramiddlediagonalbengali}{09F0} \pdfglyphtounicode{ramshorn}{0264} \pdfglyphtounicode{ratio}{2236} \pdfglyphtounicode{rbopomofo}{3116} \pdfglyphtounicode{rcaron}{0159} \pdfglyphtounicode{rcedilla}{0157} \pdfglyphtounicode{rcircle}{24E1} \pdfglyphtounicode{rcommaaccent}{0157} \pdfglyphtounicode{rdblgrave}{0211} \pdfglyphtounicode{rdotaccent}{1E59} \pdfglyphtounicode{rdotbelow}{1E5B} \pdfglyphtounicode{rdotbelowmacron}{1E5D} \pdfglyphtounicode{referencemark}{203B} \pdfglyphtounicode{reflexsubset}{2286} \pdfglyphtounicode{reflexsuperset}{2287} \pdfglyphtounicode{registered}{00AE} \pdfglyphtounicode{registersans}{F8E8} \pdfglyphtounicode{registerserif}{F6DA} \pdfglyphtounicode{reharabic}{0631} \pdfglyphtounicode{reharmenian}{0580} \pdfglyphtounicode{rehfinalarabic}{FEAE} \pdfglyphtounicode{rehiragana}{308C} % rehyehaleflamarabic;0631 FEF3 FE8E 0644 \pdfglyphtounicode{rekatakana}{30EC} \pdfglyphtounicode{rekatakanahalfwidth}{FF9A} \pdfglyphtounicode{resh}{05E8} \pdfglyphtounicode{reshdageshhebrew}{FB48} % reshhatafpatah;05E8 05B2 % reshhatafpatahhebrew;05E8 05B2 % reshhatafsegol;05E8 05B1 % reshhatafsegolhebrew;05E8 05B1 \pdfglyphtounicode{reshhebrew}{05E8} % reshhiriq;05E8 05B4 % reshhiriqhebrew;05E8 05B4 % reshholam;05E8 05B9 % reshholamhebrew;05E8 05B9 % reshpatah;05E8 05B7 % reshpatahhebrew;05E8 05B7 % reshqamats;05E8 05B8 % reshqamatshebrew;05E8 05B8 % reshqubuts;05E8 05BB % reshqubutshebrew;05E8 05BB % reshsegol;05E8 05B6 % reshsegolhebrew;05E8 05B6 % reshsheva;05E8 05B0 % reshshevahebrew;05E8 05B0 % reshtsere;05E8 05B5 % reshtserehebrew;05E8 05B5 \pdfglyphtounicode{reversedtilde}{223D} \pdfglyphtounicode{reviahebrew}{0597} \pdfglyphtounicode{reviamugrashhebrew}{0597} \pdfglyphtounicode{revlogicalnot}{2310} \pdfglyphtounicode{rfishhook}{027E} \pdfglyphtounicode{rfishhookreversed}{027F} \pdfglyphtounicode{rhabengali}{09DD} \pdfglyphtounicode{rhadeva}{095D} \pdfglyphtounicode{rho}{03C1} \pdfglyphtounicode{rhook}{027D} \pdfglyphtounicode{rhookturned}{027B} \pdfglyphtounicode{rhookturnedsuperior}{02B5} \pdfglyphtounicode{rhosymbolgreek}{03F1} \pdfglyphtounicode{rhotichookmod}{02DE} \pdfglyphtounicode{rieulacirclekorean}{3271} \pdfglyphtounicode{rieulaparenkorean}{3211} \pdfglyphtounicode{rieulcirclekorean}{3263} \pdfglyphtounicode{rieulhieuhkorean}{3140} \pdfglyphtounicode{rieulkiyeokkorean}{313A} \pdfglyphtounicode{rieulkiyeoksioskorean}{3169} \pdfglyphtounicode{rieulkorean}{3139} \pdfglyphtounicode{rieulmieumkorean}{313B} \pdfglyphtounicode{rieulpansioskorean}{316C} \pdfglyphtounicode{rieulparenkorean}{3203} \pdfglyphtounicode{rieulphieuphkorean}{313F} \pdfglyphtounicode{rieulpieupkorean}{313C} \pdfglyphtounicode{rieulpieupsioskorean}{316B} \pdfglyphtounicode{rieulsioskorean}{313D} \pdfglyphtounicode{rieulthieuthkorean}{313E} \pdfglyphtounicode{rieultikeutkorean}{316A} \pdfglyphtounicode{rieulyeorinhieuhkorean}{316D} \pdfglyphtounicode{rightangle}{221F} \pdfglyphtounicode{righttackbelowcmb}{0319} \pdfglyphtounicode{righttriangle}{22BF} \pdfglyphtounicode{rihiragana}{308A} \pdfglyphtounicode{rikatakana}{30EA} \pdfglyphtounicode{rikatakanahalfwidth}{FF98} \pdfglyphtounicode{ring}{02DA} \pdfglyphtounicode{ringbelowcmb}{0325} \pdfglyphtounicode{ringcmb}{030A} \pdfglyphtounicode{ringhalfleft}{02BF} \pdfglyphtounicode{ringhalfleftarmenian}{0559} \pdfglyphtounicode{ringhalfleftbelowcmb}{031C} \pdfglyphtounicode{ringhalfleftcentered}{02D3} \pdfglyphtounicode{ringhalfright}{02BE} \pdfglyphtounicode{ringhalfrightbelowcmb}{0339} \pdfglyphtounicode{ringhalfrightcentered}{02D2} \pdfglyphtounicode{rinvertedbreve}{0213} \pdfglyphtounicode{rittorusquare}{3351} \pdfglyphtounicode{rlinebelow}{1E5F} \pdfglyphtounicode{rlongleg}{027C} \pdfglyphtounicode{rlonglegturned}{027A} \pdfglyphtounicode{rmonospace}{FF52} \pdfglyphtounicode{rohiragana}{308D} \pdfglyphtounicode{rokatakana}{30ED} \pdfglyphtounicode{rokatakanahalfwidth}{FF9B} \pdfglyphtounicode{roruathai}{0E23} \pdfglyphtounicode{rparen}{24AD} \pdfglyphtounicode{rrabengali}{09DC} \pdfglyphtounicode{rradeva}{0931} \pdfglyphtounicode{rragurmukhi}{0A5C} \pdfglyphtounicode{rreharabic}{0691} \pdfglyphtounicode{rrehfinalarabic}{FB8D} \pdfglyphtounicode{rrvocalicbengali}{09E0} \pdfglyphtounicode{rrvocalicdeva}{0960} \pdfglyphtounicode{rrvocalicgujarati}{0AE0} \pdfglyphtounicode{rrvocalicvowelsignbengali}{09C4} \pdfglyphtounicode{rrvocalicvowelsigndeva}{0944} \pdfglyphtounicode{rrvocalicvowelsigngujarati}{0AC4} \pdfglyphtounicode{rsuperior}{F6F1} \pdfglyphtounicode{rtblock}{2590} \pdfglyphtounicode{rturned}{0279} \pdfglyphtounicode{rturnedsuperior}{02B4} \pdfglyphtounicode{ruhiragana}{308B} \pdfglyphtounicode{rukatakana}{30EB} \pdfglyphtounicode{rukatakanahalfwidth}{FF99} \pdfglyphtounicode{rupeemarkbengali}{09F2} \pdfglyphtounicode{rupeesignbengali}{09F3} \pdfglyphtounicode{rupiah}{F6DD} \pdfglyphtounicode{ruthai}{0E24} \pdfglyphtounicode{rvocalicbengali}{098B} \pdfglyphtounicode{rvocalicdeva}{090B} \pdfglyphtounicode{rvocalicgujarati}{0A8B} \pdfglyphtounicode{rvocalicvowelsignbengali}{09C3} \pdfglyphtounicode{rvocalicvowelsigndeva}{0943} \pdfglyphtounicode{rvocalicvowelsigngujarati}{0AC3} \pdfglyphtounicode{s}{0073} \pdfglyphtounicode{sabengali}{09B8} \pdfglyphtounicode{sacute}{015B} \pdfglyphtounicode{sacutedotaccent}{1E65} \pdfglyphtounicode{sadarabic}{0635} \pdfglyphtounicode{sadeva}{0938} \pdfglyphtounicode{sadfinalarabic}{FEBA} \pdfglyphtounicode{sadinitialarabic}{FEBB} \pdfglyphtounicode{sadmedialarabic}{FEBC} \pdfglyphtounicode{sagujarati}{0AB8} \pdfglyphtounicode{sagurmukhi}{0A38} \pdfglyphtounicode{sahiragana}{3055} \pdfglyphtounicode{sakatakana}{30B5} \pdfglyphtounicode{sakatakanahalfwidth}{FF7B} \pdfglyphtounicode{sallallahoualayhewasallamarabic}{FDFA} \pdfglyphtounicode{samekh}{05E1} \pdfglyphtounicode{samekhdagesh}{FB41} \pdfglyphtounicode{samekhdageshhebrew}{FB41} \pdfglyphtounicode{samekhhebrew}{05E1} \pdfglyphtounicode{saraaathai}{0E32} \pdfglyphtounicode{saraaethai}{0E41} \pdfglyphtounicode{saraaimaimalaithai}{0E44} \pdfglyphtounicode{saraaimaimuanthai}{0E43} \pdfglyphtounicode{saraamthai}{0E33} \pdfglyphtounicode{saraathai}{0E30} \pdfglyphtounicode{saraethai}{0E40} \pdfglyphtounicode{saraiileftthai}{F886} \pdfglyphtounicode{saraiithai}{0E35} \pdfglyphtounicode{saraileftthai}{F885} \pdfglyphtounicode{saraithai}{0E34} \pdfglyphtounicode{saraothai}{0E42} \pdfglyphtounicode{saraueeleftthai}{F888} \pdfglyphtounicode{saraueethai}{0E37} \pdfglyphtounicode{saraueleftthai}{F887} \pdfglyphtounicode{sarauethai}{0E36} \pdfglyphtounicode{sarauthai}{0E38} \pdfglyphtounicode{sarauuthai}{0E39} \pdfglyphtounicode{sbopomofo}{3119} \pdfglyphtounicode{scaron}{0161} \pdfglyphtounicode{scarondotaccent}{1E67} \pdfglyphtounicode{scedilla}{015F} \pdfglyphtounicode{schwa}{0259} \pdfglyphtounicode{schwacyrillic}{04D9} \pdfglyphtounicode{schwadieresiscyrillic}{04DB} \pdfglyphtounicode{schwahook}{025A} \pdfglyphtounicode{scircle}{24E2} \pdfglyphtounicode{scircumflex}{015D} \pdfglyphtounicode{scommaaccent}{0219} \pdfglyphtounicode{sdotaccent}{1E61} \pdfglyphtounicode{sdotbelow}{1E63} \pdfglyphtounicode{sdotbelowdotaccent}{1E69} \pdfglyphtounicode{seagullbelowcmb}{033C} \pdfglyphtounicode{second}{2033} \pdfglyphtounicode{secondtonechinese}{02CA} \pdfglyphtounicode{section}{00A7} \pdfglyphtounicode{seenarabic}{0633} \pdfglyphtounicode{seenfinalarabic}{FEB2} \pdfglyphtounicode{seeninitialarabic}{FEB3} \pdfglyphtounicode{seenmedialarabic}{FEB4} \pdfglyphtounicode{segol}{05B6} \pdfglyphtounicode{segol13}{05B6} \pdfglyphtounicode{segol1f}{05B6} \pdfglyphtounicode{segol2c}{05B6} \pdfglyphtounicode{segolhebrew}{05B6} \pdfglyphtounicode{segolnarrowhebrew}{05B6} \pdfglyphtounicode{segolquarterhebrew}{05B6} \pdfglyphtounicode{segoltahebrew}{0592} \pdfglyphtounicode{segolwidehebrew}{05B6} \pdfglyphtounicode{seharmenian}{057D} \pdfglyphtounicode{sehiragana}{305B} \pdfglyphtounicode{sekatakana}{30BB} \pdfglyphtounicode{sekatakanahalfwidth}{FF7E} \pdfglyphtounicode{semicolon}{003B} \pdfglyphtounicode{semicolonarabic}{061B} \pdfglyphtounicode{semicolonmonospace}{FF1B} \pdfglyphtounicode{semicolonsmall}{FE54} \pdfglyphtounicode{semivoicedmarkkana}{309C} \pdfglyphtounicode{semivoicedmarkkanahalfwidth}{FF9F} \pdfglyphtounicode{sentisquare}{3322} \pdfglyphtounicode{sentosquare}{3323} \pdfglyphtounicode{seven}{0037} \pdfglyphtounicode{sevenarabic}{0667} \pdfglyphtounicode{sevenbengali}{09ED} \pdfglyphtounicode{sevencircle}{2466} \pdfglyphtounicode{sevencircleinversesansserif}{2790} \pdfglyphtounicode{sevendeva}{096D} \pdfglyphtounicode{seveneighths}{215E} \pdfglyphtounicode{sevengujarati}{0AED} \pdfglyphtounicode{sevengurmukhi}{0A6D} \pdfglyphtounicode{sevenhackarabic}{0667} \pdfglyphtounicode{sevenhangzhou}{3027} \pdfglyphtounicode{sevenideographicparen}{3226} \pdfglyphtounicode{seveninferior}{2087} \pdfglyphtounicode{sevenmonospace}{FF17} \pdfglyphtounicode{sevenoldstyle}{F737} \pdfglyphtounicode{sevenparen}{247A} \pdfglyphtounicode{sevenperiod}{248E} \pdfglyphtounicode{sevenpersian}{06F7} \pdfglyphtounicode{sevenroman}{2176} \pdfglyphtounicode{sevensuperior}{2077} \pdfglyphtounicode{seventeencircle}{2470} \pdfglyphtounicode{seventeenparen}{2484} \pdfglyphtounicode{seventeenperiod}{2498} \pdfglyphtounicode{seventhai}{0E57} \pdfglyphtounicode{sfthyphen}{00AD} \pdfglyphtounicode{shaarmenian}{0577} \pdfglyphtounicode{shabengali}{09B6} \pdfglyphtounicode{shacyrillic}{0448} \pdfglyphtounicode{shaddaarabic}{0651} \pdfglyphtounicode{shaddadammaarabic}{FC61} \pdfglyphtounicode{shaddadammatanarabic}{FC5E} \pdfglyphtounicode{shaddafathaarabic}{FC60} % shaddafathatanarabic;0651 064B \pdfglyphtounicode{shaddakasraarabic}{FC62} \pdfglyphtounicode{shaddakasratanarabic}{FC5F} \pdfglyphtounicode{shade}{2592} \pdfglyphtounicode{shadedark}{2593} \pdfglyphtounicode{shadelight}{2591} \pdfglyphtounicode{shademedium}{2592} \pdfglyphtounicode{shadeva}{0936} \pdfglyphtounicode{shagujarati}{0AB6} \pdfglyphtounicode{shagurmukhi}{0A36} \pdfglyphtounicode{shalshelethebrew}{0593} \pdfglyphtounicode{shbopomofo}{3115} \pdfglyphtounicode{shchacyrillic}{0449} \pdfglyphtounicode{sheenarabic}{0634} \pdfglyphtounicode{sheenfinalarabic}{FEB6} \pdfglyphtounicode{sheeninitialarabic}{FEB7} \pdfglyphtounicode{sheenmedialarabic}{FEB8} \pdfglyphtounicode{sheicoptic}{03E3} \pdfglyphtounicode{sheqel}{20AA} \pdfglyphtounicode{sheqelhebrew}{20AA} \pdfglyphtounicode{sheva}{05B0} \pdfglyphtounicode{sheva115}{05B0} \pdfglyphtounicode{sheva15}{05B0} \pdfglyphtounicode{sheva22}{05B0} \pdfglyphtounicode{sheva2e}{05B0} \pdfglyphtounicode{shevahebrew}{05B0} \pdfglyphtounicode{shevanarrowhebrew}{05B0} \pdfglyphtounicode{shevaquarterhebrew}{05B0} \pdfglyphtounicode{shevawidehebrew}{05B0} \pdfglyphtounicode{shhacyrillic}{04BB} \pdfglyphtounicode{shimacoptic}{03ED} \pdfglyphtounicode{shin}{05E9} \pdfglyphtounicode{shindagesh}{FB49} \pdfglyphtounicode{shindageshhebrew}{FB49} \pdfglyphtounicode{shindageshshindot}{FB2C} \pdfglyphtounicode{shindageshshindothebrew}{FB2C} \pdfglyphtounicode{shindageshsindot}{FB2D} \pdfglyphtounicode{shindageshsindothebrew}{FB2D} \pdfglyphtounicode{shindothebrew}{05C1} \pdfglyphtounicode{shinhebrew}{05E9} \pdfglyphtounicode{shinshindot}{FB2A} \pdfglyphtounicode{shinshindothebrew}{FB2A} \pdfglyphtounicode{shinsindot}{FB2B} \pdfglyphtounicode{shinsindothebrew}{FB2B} \pdfglyphtounicode{shook}{0282} \pdfglyphtounicode{sigma}{03C3} \pdfglyphtounicode{sigma1}{03C2} \pdfglyphtounicode{sigmafinal}{03C2} \pdfglyphtounicode{sigmalunatesymbolgreek}{03F2} \pdfglyphtounicode{sihiragana}{3057} \pdfglyphtounicode{sikatakana}{30B7} \pdfglyphtounicode{sikatakanahalfwidth}{FF7C} \pdfglyphtounicode{siluqhebrew}{05BD} \pdfglyphtounicode{siluqlefthebrew}{05BD} \pdfglyphtounicode{similar}{223C} \pdfglyphtounicode{sindothebrew}{05C2} \pdfglyphtounicode{siosacirclekorean}{3274} \pdfglyphtounicode{siosaparenkorean}{3214} \pdfglyphtounicode{sioscieuckorean}{317E} \pdfglyphtounicode{sioscirclekorean}{3266} \pdfglyphtounicode{sioskiyeokkorean}{317A} \pdfglyphtounicode{sioskorean}{3145} \pdfglyphtounicode{siosnieunkorean}{317B} \pdfglyphtounicode{siosparenkorean}{3206} \pdfglyphtounicode{siospieupkorean}{317D} \pdfglyphtounicode{siostikeutkorean}{317C} \pdfglyphtounicode{six}{0036} \pdfglyphtounicode{sixarabic}{0666} \pdfglyphtounicode{sixbengali}{09EC} \pdfglyphtounicode{sixcircle}{2465} \pdfglyphtounicode{sixcircleinversesansserif}{278F} \pdfglyphtounicode{sixdeva}{096C} \pdfglyphtounicode{sixgujarati}{0AEC} \pdfglyphtounicode{sixgurmukhi}{0A6C} \pdfglyphtounicode{sixhackarabic}{0666} \pdfglyphtounicode{sixhangzhou}{3026} \pdfglyphtounicode{sixideographicparen}{3225} \pdfglyphtounicode{sixinferior}{2086} \pdfglyphtounicode{sixmonospace}{FF16} \pdfglyphtounicode{sixoldstyle}{F736} \pdfglyphtounicode{sixparen}{2479} \pdfglyphtounicode{sixperiod}{248D} \pdfglyphtounicode{sixpersian}{06F6} \pdfglyphtounicode{sixroman}{2175} \pdfglyphtounicode{sixsuperior}{2076} \pdfglyphtounicode{sixteencircle}{246F} \pdfglyphtounicode{sixteencurrencydenominatorbengali}{09F9} \pdfglyphtounicode{sixteenparen}{2483} \pdfglyphtounicode{sixteenperiod}{2497} \pdfglyphtounicode{sixthai}{0E56} \pdfglyphtounicode{slash}{002F} \pdfglyphtounicode{slashmonospace}{FF0F} \pdfglyphtounicode{slong}{017F} \pdfglyphtounicode{slongdotaccent}{1E9B} \pdfglyphtounicode{smileface}{263A} \pdfglyphtounicode{smonospace}{FF53} \pdfglyphtounicode{sofpasuqhebrew}{05C3} \pdfglyphtounicode{softhyphen}{00AD} \pdfglyphtounicode{softsigncyrillic}{044C} \pdfglyphtounicode{sohiragana}{305D} \pdfglyphtounicode{sokatakana}{30BD} \pdfglyphtounicode{sokatakanahalfwidth}{FF7F} \pdfglyphtounicode{soliduslongoverlaycmb}{0338} \pdfglyphtounicode{solidusshortoverlaycmb}{0337} \pdfglyphtounicode{sorusithai}{0E29} \pdfglyphtounicode{sosalathai}{0E28} \pdfglyphtounicode{sosothai}{0E0B} \pdfglyphtounicode{sosuathai}{0E2A} \pdfglyphtounicode{space}{0020} \pdfglyphtounicode{spacehackarabic}{0020} \pdfglyphtounicode{spade}{2660} \pdfglyphtounicode{spadesuitblack}{2660} \pdfglyphtounicode{spadesuitwhite}{2664} \pdfglyphtounicode{sparen}{24AE} \pdfglyphtounicode{squarebelowcmb}{033B} \pdfglyphtounicode{squarecc}{33C4} \pdfglyphtounicode{squarecm}{339D} \pdfglyphtounicode{squarediagonalcrosshatchfill}{25A9} \pdfglyphtounicode{squarehorizontalfill}{25A4} \pdfglyphtounicode{squarekg}{338F} \pdfglyphtounicode{squarekm}{339E} \pdfglyphtounicode{squarekmcapital}{33CE} \pdfglyphtounicode{squareln}{33D1} \pdfglyphtounicode{squarelog}{33D2} \pdfglyphtounicode{squaremg}{338E} \pdfglyphtounicode{squaremil}{33D5} \pdfglyphtounicode{squaremm}{339C} \pdfglyphtounicode{squaremsquared}{33A1} \pdfglyphtounicode{squareorthogonalcrosshatchfill}{25A6} \pdfglyphtounicode{squareupperlefttolowerrightfill}{25A7} \pdfglyphtounicode{squareupperrighttolowerleftfill}{25A8} \pdfglyphtounicode{squareverticalfill}{25A5} \pdfglyphtounicode{squarewhitewithsmallblack}{25A3} \pdfglyphtounicode{srsquare}{33DB} \pdfglyphtounicode{ssabengali}{09B7} \pdfglyphtounicode{ssadeva}{0937} \pdfglyphtounicode{ssagujarati}{0AB7} \pdfglyphtounicode{ssangcieuckorean}{3149} \pdfglyphtounicode{ssanghieuhkorean}{3185} \pdfglyphtounicode{ssangieungkorean}{3180} \pdfglyphtounicode{ssangkiyeokkorean}{3132} \pdfglyphtounicode{ssangnieunkorean}{3165} \pdfglyphtounicode{ssangpieupkorean}{3143} \pdfglyphtounicode{ssangsioskorean}{3146} \pdfglyphtounicode{ssangtikeutkorean}{3138} \pdfglyphtounicode{ssuperior}{F6F2} \pdfglyphtounicode{sterling}{00A3} \pdfglyphtounicode{sterlingmonospace}{FFE1} \pdfglyphtounicode{strokelongoverlaycmb}{0336} \pdfglyphtounicode{strokeshortoverlaycmb}{0335} \pdfglyphtounicode{subset}{2282} \pdfglyphtounicode{subsetnotequal}{228A} \pdfglyphtounicode{subsetorequal}{2286} \pdfglyphtounicode{succeeds}{227B} \pdfglyphtounicode{suchthat}{220B} \pdfglyphtounicode{suhiragana}{3059} \pdfglyphtounicode{sukatakana}{30B9} \pdfglyphtounicode{sukatakanahalfwidth}{FF7D} \pdfglyphtounicode{sukunarabic}{0652} \pdfglyphtounicode{summation}{2211} \pdfglyphtounicode{sun}{263C} \pdfglyphtounicode{superset}{2283} \pdfglyphtounicode{supersetnotequal}{228B} \pdfglyphtounicode{supersetorequal}{2287} \pdfglyphtounicode{svsquare}{33DC} \pdfglyphtounicode{syouwaerasquare}{337C} \pdfglyphtounicode{t}{0074} \pdfglyphtounicode{tabengali}{09A4} \pdfglyphtounicode{tackdown}{22A4} \pdfglyphtounicode{tackleft}{22A3} \pdfglyphtounicode{tadeva}{0924} \pdfglyphtounicode{tagujarati}{0AA4} \pdfglyphtounicode{tagurmukhi}{0A24} \pdfglyphtounicode{taharabic}{0637} \pdfglyphtounicode{tahfinalarabic}{FEC2} \pdfglyphtounicode{tahinitialarabic}{FEC3} \pdfglyphtounicode{tahiragana}{305F} \pdfglyphtounicode{tahmedialarabic}{FEC4} \pdfglyphtounicode{taisyouerasquare}{337D} \pdfglyphtounicode{takatakana}{30BF} \pdfglyphtounicode{takatakanahalfwidth}{FF80} \pdfglyphtounicode{tatweelarabic}{0640} \pdfglyphtounicode{tau}{03C4} \pdfglyphtounicode{tav}{05EA} \pdfglyphtounicode{tavdages}{FB4A} \pdfglyphtounicode{tavdagesh}{FB4A} \pdfglyphtounicode{tavdageshhebrew}{FB4A} \pdfglyphtounicode{tavhebrew}{05EA} \pdfglyphtounicode{tbar}{0167} \pdfglyphtounicode{tbopomofo}{310A} \pdfglyphtounicode{tcaron}{0165} \pdfglyphtounicode{tccurl}{02A8} \pdfglyphtounicode{tcedilla}{0163} \pdfglyphtounicode{tcheharabic}{0686} \pdfglyphtounicode{tchehfinalarabic}{FB7B} \pdfglyphtounicode{tchehinitialarabic}{FB7C} \pdfglyphtounicode{tchehmedialarabic}{FB7D} % tchehmeeminitialarabic;FB7C FEE4 \pdfglyphtounicode{tcircle}{24E3} \pdfglyphtounicode{tcircumflexbelow}{1E71} \pdfglyphtounicode{tcommaaccent}{0163} \pdfglyphtounicode{tdieresis}{1E97} \pdfglyphtounicode{tdotaccent}{1E6B} \pdfglyphtounicode{tdotbelow}{1E6D} \pdfglyphtounicode{tecyrillic}{0442} \pdfglyphtounicode{tedescendercyrillic}{04AD} \pdfglyphtounicode{teharabic}{062A} \pdfglyphtounicode{tehfinalarabic}{FE96} \pdfglyphtounicode{tehhahinitialarabic}{FCA2} \pdfglyphtounicode{tehhahisolatedarabic}{FC0C} \pdfglyphtounicode{tehinitialarabic}{FE97} \pdfglyphtounicode{tehiragana}{3066} \pdfglyphtounicode{tehjeeminitialarabic}{FCA1} \pdfglyphtounicode{tehjeemisolatedarabic}{FC0B} \pdfglyphtounicode{tehmarbutaarabic}{0629} \pdfglyphtounicode{tehmarbutafinalarabic}{FE94} \pdfglyphtounicode{tehmedialarabic}{FE98} \pdfglyphtounicode{tehmeeminitialarabic}{FCA4} \pdfglyphtounicode{tehmeemisolatedarabic}{FC0E} \pdfglyphtounicode{tehnoonfinalarabic}{FC73} \pdfglyphtounicode{tekatakana}{30C6} \pdfglyphtounicode{tekatakanahalfwidth}{FF83} \pdfglyphtounicode{telephone}{2121} \pdfglyphtounicode{telephoneblack}{260E} \pdfglyphtounicode{telishagedolahebrew}{05A0} \pdfglyphtounicode{telishaqetanahebrew}{05A9} \pdfglyphtounicode{tencircle}{2469} \pdfglyphtounicode{tenideographicparen}{3229} \pdfglyphtounicode{tenparen}{247D} \pdfglyphtounicode{tenperiod}{2491} \pdfglyphtounicode{tenroman}{2179} \pdfglyphtounicode{tesh}{02A7} \pdfglyphtounicode{tet}{05D8} \pdfglyphtounicode{tetdagesh}{FB38} \pdfglyphtounicode{tetdageshhebrew}{FB38} \pdfglyphtounicode{tethebrew}{05D8} \pdfglyphtounicode{tetsecyrillic}{04B5} \pdfglyphtounicode{tevirhebrew}{059B} \pdfglyphtounicode{tevirlefthebrew}{059B} \pdfglyphtounicode{thabengali}{09A5} \pdfglyphtounicode{thadeva}{0925} \pdfglyphtounicode{thagujarati}{0AA5} \pdfglyphtounicode{thagurmukhi}{0A25} \pdfglyphtounicode{thalarabic}{0630} \pdfglyphtounicode{thalfinalarabic}{FEAC} \pdfglyphtounicode{thanthakhatlowleftthai}{F898} \pdfglyphtounicode{thanthakhatlowrightthai}{F897} \pdfglyphtounicode{thanthakhatthai}{0E4C} \pdfglyphtounicode{thanthakhatupperleftthai}{F896} \pdfglyphtounicode{theharabic}{062B} \pdfglyphtounicode{thehfinalarabic}{FE9A} \pdfglyphtounicode{thehinitialarabic}{FE9B} \pdfglyphtounicode{thehmedialarabic}{FE9C} \pdfglyphtounicode{thereexists}{2203} \pdfglyphtounicode{therefore}{2234} \pdfglyphtounicode{theta}{03B8} \pdfglyphtounicode{theta1}{03D1} \pdfglyphtounicode{thetasymbolgreek}{03D1} \pdfglyphtounicode{thieuthacirclekorean}{3279} \pdfglyphtounicode{thieuthaparenkorean}{3219} \pdfglyphtounicode{thieuthcirclekorean}{326B} \pdfglyphtounicode{thieuthkorean}{314C} \pdfglyphtounicode{thieuthparenkorean}{320B} \pdfglyphtounicode{thirteencircle}{246C} \pdfglyphtounicode{thirteenparen}{2480} \pdfglyphtounicode{thirteenperiod}{2494} \pdfglyphtounicode{thonangmonthothai}{0E11} \pdfglyphtounicode{thook}{01AD} \pdfglyphtounicode{thophuthaothai}{0E12} \pdfglyphtounicode{thorn}{00FE} \pdfglyphtounicode{thothahanthai}{0E17} \pdfglyphtounicode{thothanthai}{0E10} \pdfglyphtounicode{thothongthai}{0E18} \pdfglyphtounicode{thothungthai}{0E16} \pdfglyphtounicode{thousandcyrillic}{0482} \pdfglyphtounicode{thousandsseparatorarabic}{066C} \pdfglyphtounicode{thousandsseparatorpersian}{066C} \pdfglyphtounicode{three}{0033} \pdfglyphtounicode{threearabic}{0663} \pdfglyphtounicode{threebengali}{09E9} \pdfglyphtounicode{threecircle}{2462} \pdfglyphtounicode{threecircleinversesansserif}{278C} \pdfglyphtounicode{threedeva}{0969} \pdfglyphtounicode{threeeighths}{215C} \pdfglyphtounicode{threegujarati}{0AE9} \pdfglyphtounicode{threegurmukhi}{0A69} \pdfglyphtounicode{threehackarabic}{0663} \pdfglyphtounicode{threehangzhou}{3023} \pdfglyphtounicode{threeideographicparen}{3222} \pdfglyphtounicode{threeinferior}{2083} \pdfglyphtounicode{threemonospace}{FF13} \pdfglyphtounicode{threenumeratorbengali}{09F6} \pdfglyphtounicode{threeoldstyle}{F733} \pdfglyphtounicode{threeparen}{2476} \pdfglyphtounicode{threeperiod}{248A} \pdfglyphtounicode{threepersian}{06F3} \pdfglyphtounicode{threequarters}{00BE} \pdfglyphtounicode{threequartersemdash}{F6DE} \pdfglyphtounicode{threeroman}{2172} \pdfglyphtounicode{threesuperior}{00B3} \pdfglyphtounicode{threethai}{0E53} \pdfglyphtounicode{thzsquare}{3394} \pdfglyphtounicode{tihiragana}{3061} \pdfglyphtounicode{tikatakana}{30C1} \pdfglyphtounicode{tikatakanahalfwidth}{FF81} \pdfglyphtounicode{tikeutacirclekorean}{3270} \pdfglyphtounicode{tikeutaparenkorean}{3210} \pdfglyphtounicode{tikeutcirclekorean}{3262} \pdfglyphtounicode{tikeutkorean}{3137} \pdfglyphtounicode{tikeutparenkorean}{3202} \pdfglyphtounicode{tilde}{02DC} \pdfglyphtounicode{tildebelowcmb}{0330} \pdfglyphtounicode{tildecmb}{0303} \pdfglyphtounicode{tildecomb}{0303} \pdfglyphtounicode{tildedoublecmb}{0360} \pdfglyphtounicode{tildeoperator}{223C} \pdfglyphtounicode{tildeoverlaycmb}{0334} \pdfglyphtounicode{tildeverticalcmb}{033E} \pdfglyphtounicode{timescircle}{2297} \pdfglyphtounicode{tipehahebrew}{0596} \pdfglyphtounicode{tipehalefthebrew}{0596} \pdfglyphtounicode{tippigurmukhi}{0A70} \pdfglyphtounicode{titlocyrilliccmb}{0483} \pdfglyphtounicode{tiwnarmenian}{057F} \pdfglyphtounicode{tlinebelow}{1E6F} \pdfglyphtounicode{tmonospace}{FF54} \pdfglyphtounicode{toarmenian}{0569} \pdfglyphtounicode{tohiragana}{3068} \pdfglyphtounicode{tokatakana}{30C8} \pdfglyphtounicode{tokatakanahalfwidth}{FF84} \pdfglyphtounicode{tonebarextrahighmod}{02E5} \pdfglyphtounicode{tonebarextralowmod}{02E9} \pdfglyphtounicode{tonebarhighmod}{02E6} \pdfglyphtounicode{tonebarlowmod}{02E8} \pdfglyphtounicode{tonebarmidmod}{02E7} \pdfglyphtounicode{tonefive}{01BD} \pdfglyphtounicode{tonesix}{0185} \pdfglyphtounicode{tonetwo}{01A8} \pdfglyphtounicode{tonos}{0384} \pdfglyphtounicode{tonsquare}{3327} \pdfglyphtounicode{topatakthai}{0E0F} \pdfglyphtounicode{tortoiseshellbracketleft}{3014} \pdfglyphtounicode{tortoiseshellbracketleftsmall}{FE5D} \pdfglyphtounicode{tortoiseshellbracketleftvertical}{FE39} \pdfglyphtounicode{tortoiseshellbracketright}{3015} \pdfglyphtounicode{tortoiseshellbracketrightsmall}{FE5E} \pdfglyphtounicode{tortoiseshellbracketrightvertical}{FE3A} \pdfglyphtounicode{totaothai}{0E15} \pdfglyphtounicode{tpalatalhook}{01AB} \pdfglyphtounicode{tparen}{24AF} \pdfglyphtounicode{trademark}{2122} \pdfglyphtounicode{trademarksans}{F8EA} \pdfglyphtounicode{trademarkserif}{F6DB} \pdfglyphtounicode{tretroflexhook}{0288} \pdfglyphtounicode{triagdn}{25BC} \pdfglyphtounicode{triaglf}{25C4} \pdfglyphtounicode{triagrt}{25BA} \pdfglyphtounicode{triagup}{25B2} \pdfglyphtounicode{ts}{02A6} \pdfglyphtounicode{tsadi}{05E6} \pdfglyphtounicode{tsadidagesh}{FB46} \pdfglyphtounicode{tsadidageshhebrew}{FB46} \pdfglyphtounicode{tsadihebrew}{05E6} \pdfglyphtounicode{tsecyrillic}{0446} \pdfglyphtounicode{tsere}{05B5} \pdfglyphtounicode{tsere12}{05B5} \pdfglyphtounicode{tsere1e}{05B5} \pdfglyphtounicode{tsere2b}{05B5} \pdfglyphtounicode{tserehebrew}{05B5} \pdfglyphtounicode{tserenarrowhebrew}{05B5} \pdfglyphtounicode{tserequarterhebrew}{05B5} \pdfglyphtounicode{tserewidehebrew}{05B5} \pdfglyphtounicode{tshecyrillic}{045B} \pdfglyphtounicode{tsuperior}{F6F3} \pdfglyphtounicode{ttabengali}{099F} \pdfglyphtounicode{ttadeva}{091F} \pdfglyphtounicode{ttagujarati}{0A9F} \pdfglyphtounicode{ttagurmukhi}{0A1F} \pdfglyphtounicode{tteharabic}{0679} \pdfglyphtounicode{ttehfinalarabic}{FB67} \pdfglyphtounicode{ttehinitialarabic}{FB68} \pdfglyphtounicode{ttehmedialarabic}{FB69} \pdfglyphtounicode{tthabengali}{09A0} \pdfglyphtounicode{tthadeva}{0920} \pdfglyphtounicode{tthagujarati}{0AA0} \pdfglyphtounicode{tthagurmukhi}{0A20} \pdfglyphtounicode{tturned}{0287} \pdfglyphtounicode{tuhiragana}{3064} \pdfglyphtounicode{tukatakana}{30C4} \pdfglyphtounicode{tukatakanahalfwidth}{FF82} \pdfglyphtounicode{tusmallhiragana}{3063} \pdfglyphtounicode{tusmallkatakana}{30C3} \pdfglyphtounicode{tusmallkatakanahalfwidth}{FF6F} \pdfglyphtounicode{twelvecircle}{246B} \pdfglyphtounicode{twelveparen}{247F} \pdfglyphtounicode{twelveperiod}{2493} \pdfglyphtounicode{twelveroman}{217B} \pdfglyphtounicode{twentycircle}{2473} \pdfglyphtounicode{twentyhangzhou}{5344} \pdfglyphtounicode{twentyparen}{2487} \pdfglyphtounicode{twentyperiod}{249B} \pdfglyphtounicode{two}{0032} \pdfglyphtounicode{twoarabic}{0662} \pdfglyphtounicode{twobengali}{09E8} \pdfglyphtounicode{twocircle}{2461} \pdfglyphtounicode{twocircleinversesansserif}{278B} \pdfglyphtounicode{twodeva}{0968} \pdfglyphtounicode{twodotenleader}{2025} \pdfglyphtounicode{twodotleader}{2025} \pdfglyphtounicode{twodotleadervertical}{FE30} \pdfglyphtounicode{twogujarati}{0AE8} \pdfglyphtounicode{twogurmukhi}{0A68} \pdfglyphtounicode{twohackarabic}{0662} \pdfglyphtounicode{twohangzhou}{3022} \pdfglyphtounicode{twoideographicparen}{3221} \pdfglyphtounicode{twoinferior}{2082} \pdfglyphtounicode{twomonospace}{FF12} \pdfglyphtounicode{twonumeratorbengali}{09F5} \pdfglyphtounicode{twooldstyle}{F732} \pdfglyphtounicode{twoparen}{2475} \pdfglyphtounicode{twoperiod}{2489} \pdfglyphtounicode{twopersian}{06F2} \pdfglyphtounicode{tworoman}{2171} \pdfglyphtounicode{twostroke}{01BB} \pdfglyphtounicode{twosuperior}{00B2} \pdfglyphtounicode{twothai}{0E52} \pdfglyphtounicode{twothirds}{2154} \pdfglyphtounicode{u}{0075} \pdfglyphtounicode{uacute}{00FA} \pdfglyphtounicode{ubar}{0289} \pdfglyphtounicode{ubengali}{0989} \pdfglyphtounicode{ubopomofo}{3128} \pdfglyphtounicode{ubreve}{016D} \pdfglyphtounicode{ucaron}{01D4} \pdfglyphtounicode{ucircle}{24E4} \pdfglyphtounicode{ucircumflex}{00FB} \pdfglyphtounicode{ucircumflexbelow}{1E77} \pdfglyphtounicode{ucyrillic}{0443} \pdfglyphtounicode{udattadeva}{0951} \pdfglyphtounicode{udblacute}{0171} \pdfglyphtounicode{udblgrave}{0215} \pdfglyphtounicode{udeva}{0909} \pdfglyphtounicode{udieresis}{00FC} \pdfglyphtounicode{udieresisacute}{01D8} \pdfglyphtounicode{udieresisbelow}{1E73} \pdfglyphtounicode{udieresiscaron}{01DA} \pdfglyphtounicode{udieresiscyrillic}{04F1} \pdfglyphtounicode{udieresisgrave}{01DC} \pdfglyphtounicode{udieresismacron}{01D6} \pdfglyphtounicode{udotbelow}{1EE5} \pdfglyphtounicode{ugrave}{00F9} \pdfglyphtounicode{ugujarati}{0A89} \pdfglyphtounicode{ugurmukhi}{0A09} \pdfglyphtounicode{uhiragana}{3046} \pdfglyphtounicode{uhookabove}{1EE7} \pdfglyphtounicode{uhorn}{01B0} \pdfglyphtounicode{uhornacute}{1EE9} \pdfglyphtounicode{uhorndotbelow}{1EF1} \pdfglyphtounicode{uhorngrave}{1EEB} \pdfglyphtounicode{uhornhookabove}{1EED} \pdfglyphtounicode{uhorntilde}{1EEF} \pdfglyphtounicode{uhungarumlaut}{0171} \pdfglyphtounicode{uhungarumlautcyrillic}{04F3} \pdfglyphtounicode{uinvertedbreve}{0217} \pdfglyphtounicode{ukatakana}{30A6} \pdfglyphtounicode{ukatakanahalfwidth}{FF73} \pdfglyphtounicode{ukcyrillic}{0479} \pdfglyphtounicode{ukorean}{315C} \pdfglyphtounicode{umacron}{016B} \pdfglyphtounicode{umacroncyrillic}{04EF} \pdfglyphtounicode{umacrondieresis}{1E7B} \pdfglyphtounicode{umatragurmukhi}{0A41} \pdfglyphtounicode{umonospace}{FF55} \pdfglyphtounicode{underscore}{005F} \pdfglyphtounicode{underscoredbl}{2017} \pdfglyphtounicode{underscoremonospace}{FF3F} \pdfglyphtounicode{underscorevertical}{FE33} \pdfglyphtounicode{underscorewavy}{FE4F} \pdfglyphtounicode{union}{222A} \pdfglyphtounicode{universal}{2200} \pdfglyphtounicode{uogonek}{0173} \pdfglyphtounicode{uparen}{24B0} \pdfglyphtounicode{upblock}{2580} \pdfglyphtounicode{upperdothebrew}{05C4} \pdfglyphtounicode{upsilon}{03C5} \pdfglyphtounicode{upsilondieresis}{03CB} \pdfglyphtounicode{upsilondieresistonos}{03B0} \pdfglyphtounicode{upsilonlatin}{028A} \pdfglyphtounicode{upsilontonos}{03CD} \pdfglyphtounicode{uptackbelowcmb}{031D} \pdfglyphtounicode{uptackmod}{02D4} \pdfglyphtounicode{uragurmukhi}{0A73} \pdfglyphtounicode{uring}{016F} \pdfglyphtounicode{ushortcyrillic}{045E} \pdfglyphtounicode{usmallhiragana}{3045} \pdfglyphtounicode{usmallkatakana}{30A5} \pdfglyphtounicode{usmallkatakanahalfwidth}{FF69} \pdfglyphtounicode{ustraightcyrillic}{04AF} \pdfglyphtounicode{ustraightstrokecyrillic}{04B1} \pdfglyphtounicode{utilde}{0169} \pdfglyphtounicode{utildeacute}{1E79} \pdfglyphtounicode{utildebelow}{1E75} \pdfglyphtounicode{uubengali}{098A} \pdfglyphtounicode{uudeva}{090A} \pdfglyphtounicode{uugujarati}{0A8A} \pdfglyphtounicode{uugurmukhi}{0A0A} \pdfglyphtounicode{uumatragurmukhi}{0A42} \pdfglyphtounicode{uuvowelsignbengali}{09C2} \pdfglyphtounicode{uuvowelsigndeva}{0942} \pdfglyphtounicode{uuvowelsigngujarati}{0AC2} \pdfglyphtounicode{uvowelsignbengali}{09C1} \pdfglyphtounicode{uvowelsigndeva}{0941} \pdfglyphtounicode{uvowelsigngujarati}{0AC1} \pdfglyphtounicode{v}{0076} \pdfglyphtounicode{vadeva}{0935} \pdfglyphtounicode{vagujarati}{0AB5} \pdfglyphtounicode{vagurmukhi}{0A35} \pdfglyphtounicode{vakatakana}{30F7} \pdfglyphtounicode{vav}{05D5} \pdfglyphtounicode{vavdagesh}{FB35} \pdfglyphtounicode{vavdagesh65}{FB35} \pdfglyphtounicode{vavdageshhebrew}{FB35} \pdfglyphtounicode{vavhebrew}{05D5} \pdfglyphtounicode{vavholam}{FB4B} \pdfglyphtounicode{vavholamhebrew}{FB4B} \pdfglyphtounicode{vavvavhebrew}{05F0} \pdfglyphtounicode{vavyodhebrew}{05F1} \pdfglyphtounicode{vcircle}{24E5} \pdfglyphtounicode{vdotbelow}{1E7F} \pdfglyphtounicode{vecyrillic}{0432} \pdfglyphtounicode{veharabic}{06A4} \pdfglyphtounicode{vehfinalarabic}{FB6B} \pdfglyphtounicode{vehinitialarabic}{FB6C} \pdfglyphtounicode{vehmedialarabic}{FB6D} \pdfglyphtounicode{vekatakana}{30F9} \pdfglyphtounicode{venus}{2640} \pdfglyphtounicode{verticalbar}{007C} \pdfglyphtounicode{verticallineabovecmb}{030D} \pdfglyphtounicode{verticallinebelowcmb}{0329} \pdfglyphtounicode{verticallinelowmod}{02CC} \pdfglyphtounicode{verticallinemod}{02C8} \pdfglyphtounicode{vewarmenian}{057E} \pdfglyphtounicode{vhook}{028B} \pdfglyphtounicode{vikatakana}{30F8} \pdfglyphtounicode{viramabengali}{09CD} \pdfglyphtounicode{viramadeva}{094D} \pdfglyphtounicode{viramagujarati}{0ACD} \pdfglyphtounicode{visargabengali}{0983} \pdfglyphtounicode{visargadeva}{0903} \pdfglyphtounicode{visargagujarati}{0A83} \pdfglyphtounicode{vmonospace}{FF56} \pdfglyphtounicode{voarmenian}{0578} \pdfglyphtounicode{voicediterationhiragana}{309E} \pdfglyphtounicode{voicediterationkatakana}{30FE} \pdfglyphtounicode{voicedmarkkana}{309B} \pdfglyphtounicode{voicedmarkkanahalfwidth}{FF9E} \pdfglyphtounicode{vokatakana}{30FA} \pdfglyphtounicode{vparen}{24B1} \pdfglyphtounicode{vtilde}{1E7D} \pdfglyphtounicode{vturned}{028C} \pdfglyphtounicode{vuhiragana}{3094} \pdfglyphtounicode{vukatakana}{30F4} \pdfglyphtounicode{w}{0077} \pdfglyphtounicode{wacute}{1E83} \pdfglyphtounicode{waekorean}{3159} \pdfglyphtounicode{wahiragana}{308F} \pdfglyphtounicode{wakatakana}{30EF} \pdfglyphtounicode{wakatakanahalfwidth}{FF9C} \pdfglyphtounicode{wakorean}{3158} \pdfglyphtounicode{wasmallhiragana}{308E} \pdfglyphtounicode{wasmallkatakana}{30EE} \pdfglyphtounicode{wattosquare}{3357} \pdfglyphtounicode{wavedash}{301C} \pdfglyphtounicode{wavyunderscorevertical}{FE34} \pdfglyphtounicode{wawarabic}{0648} \pdfglyphtounicode{wawfinalarabic}{FEEE} \pdfglyphtounicode{wawhamzaabovearabic}{0624} \pdfglyphtounicode{wawhamzaabovefinalarabic}{FE86} \pdfglyphtounicode{wbsquare}{33DD} \pdfglyphtounicode{wcircle}{24E6} \pdfglyphtounicode{wcircumflex}{0175} \pdfglyphtounicode{wdieresis}{1E85} \pdfglyphtounicode{wdotaccent}{1E87} \pdfglyphtounicode{wdotbelow}{1E89} \pdfglyphtounicode{wehiragana}{3091} \pdfglyphtounicode{weierstrass}{2118} \pdfglyphtounicode{wekatakana}{30F1} \pdfglyphtounicode{wekorean}{315E} \pdfglyphtounicode{weokorean}{315D} \pdfglyphtounicode{wgrave}{1E81} \pdfglyphtounicode{whitebullet}{25E6} \pdfglyphtounicode{whitecircle}{25CB} \pdfglyphtounicode{whitecircleinverse}{25D9} \pdfglyphtounicode{whitecornerbracketleft}{300E} \pdfglyphtounicode{whitecornerbracketleftvertical}{FE43} \pdfglyphtounicode{whitecornerbracketright}{300F} \pdfglyphtounicode{whitecornerbracketrightvertical}{FE44} \pdfglyphtounicode{whitediamond}{25C7} \pdfglyphtounicode{whitediamondcontainingblacksmalldiamond}{25C8} \pdfglyphtounicode{whitedownpointingsmalltriangle}{25BF} \pdfglyphtounicode{whitedownpointingtriangle}{25BD} \pdfglyphtounicode{whiteleftpointingsmalltriangle}{25C3} \pdfglyphtounicode{whiteleftpointingtriangle}{25C1} \pdfglyphtounicode{whitelenticularbracketleft}{3016} \pdfglyphtounicode{whitelenticularbracketright}{3017} \pdfglyphtounicode{whiterightpointingsmalltriangle}{25B9} \pdfglyphtounicode{whiterightpointingtriangle}{25B7} \pdfglyphtounicode{whitesmallsquare}{25AB} \pdfglyphtounicode{whitesmilingface}{263A} \pdfglyphtounicode{whitesquare}{25A1} \pdfglyphtounicode{whitestar}{2606} \pdfglyphtounicode{whitetelephone}{260F} \pdfglyphtounicode{whitetortoiseshellbracketleft}{3018} \pdfglyphtounicode{whitetortoiseshellbracketright}{3019} \pdfglyphtounicode{whiteuppointingsmalltriangle}{25B5} \pdfglyphtounicode{whiteuppointingtriangle}{25B3} \pdfglyphtounicode{wihiragana}{3090} \pdfglyphtounicode{wikatakana}{30F0} \pdfglyphtounicode{wikorean}{315F} \pdfglyphtounicode{wmonospace}{FF57} \pdfglyphtounicode{wohiragana}{3092} \pdfglyphtounicode{wokatakana}{30F2} \pdfglyphtounicode{wokatakanahalfwidth}{FF66} \pdfglyphtounicode{won}{20A9} \pdfglyphtounicode{wonmonospace}{FFE6} \pdfglyphtounicode{wowaenthai}{0E27} \pdfglyphtounicode{wparen}{24B2} \pdfglyphtounicode{wring}{1E98} \pdfglyphtounicode{wsuperior}{02B7} \pdfglyphtounicode{wturned}{028D} \pdfglyphtounicode{wynn}{01BF} \pdfglyphtounicode{x}{0078} \pdfglyphtounicode{xabovecmb}{033D} \pdfglyphtounicode{xbopomofo}{3112} \pdfglyphtounicode{xcircle}{24E7} \pdfglyphtounicode{xdieresis}{1E8D} \pdfglyphtounicode{xdotaccent}{1E8B} \pdfglyphtounicode{xeharmenian}{056D} \pdfglyphtounicode{xi}{03BE} \pdfglyphtounicode{xmonospace}{FF58} \pdfglyphtounicode{xparen}{24B3} \pdfglyphtounicode{xsuperior}{02E3} \pdfglyphtounicode{y}{0079} \pdfglyphtounicode{yaadosquare}{334E} \pdfglyphtounicode{yabengali}{09AF} \pdfglyphtounicode{yacute}{00FD} \pdfglyphtounicode{yadeva}{092F} \pdfglyphtounicode{yaekorean}{3152} \pdfglyphtounicode{yagujarati}{0AAF} \pdfglyphtounicode{yagurmukhi}{0A2F} \pdfglyphtounicode{yahiragana}{3084} \pdfglyphtounicode{yakatakana}{30E4} \pdfglyphtounicode{yakatakanahalfwidth}{FF94} \pdfglyphtounicode{yakorean}{3151} \pdfglyphtounicode{yamakkanthai}{0E4E} \pdfglyphtounicode{yasmallhiragana}{3083} \pdfglyphtounicode{yasmallkatakana}{30E3} \pdfglyphtounicode{yasmallkatakanahalfwidth}{FF6C} \pdfglyphtounicode{yatcyrillic}{0463} \pdfglyphtounicode{ycircle}{24E8} \pdfglyphtounicode{ycircumflex}{0177} \pdfglyphtounicode{ydieresis}{00FF} \pdfglyphtounicode{ydotaccent}{1E8F} \pdfglyphtounicode{ydotbelow}{1EF5} \pdfglyphtounicode{yeharabic}{064A} \pdfglyphtounicode{yehbarreearabic}{06D2} \pdfglyphtounicode{yehbarreefinalarabic}{FBAF} \pdfglyphtounicode{yehfinalarabic}{FEF2} \pdfglyphtounicode{yehhamzaabovearabic}{0626} \pdfglyphtounicode{yehhamzaabovefinalarabic}{FE8A} \pdfglyphtounicode{yehhamzaaboveinitialarabic}{FE8B} \pdfglyphtounicode{yehhamzaabovemedialarabic}{FE8C} \pdfglyphtounicode{yehinitialarabic}{FEF3} \pdfglyphtounicode{yehmedialarabic}{FEF4} \pdfglyphtounicode{yehmeeminitialarabic}{FCDD} \pdfglyphtounicode{yehmeemisolatedarabic}{FC58} \pdfglyphtounicode{yehnoonfinalarabic}{FC94} \pdfglyphtounicode{yehthreedotsbelowarabic}{06D1} \pdfglyphtounicode{yekorean}{3156} \pdfglyphtounicode{yen}{00A5} \pdfglyphtounicode{yenmonospace}{FFE5} \pdfglyphtounicode{yeokorean}{3155} \pdfglyphtounicode{yeorinhieuhkorean}{3186} \pdfglyphtounicode{yerahbenyomohebrew}{05AA} \pdfglyphtounicode{yerahbenyomolefthebrew}{05AA} \pdfglyphtounicode{yericyrillic}{044B} \pdfglyphtounicode{yerudieresiscyrillic}{04F9} \pdfglyphtounicode{yesieungkorean}{3181} \pdfglyphtounicode{yesieungpansioskorean}{3183} \pdfglyphtounicode{yesieungsioskorean}{3182} \pdfglyphtounicode{yetivhebrew}{059A} \pdfglyphtounicode{ygrave}{1EF3} \pdfglyphtounicode{yhook}{01B4} \pdfglyphtounicode{yhookabove}{1EF7} \pdfglyphtounicode{yiarmenian}{0575} \pdfglyphtounicode{yicyrillic}{0457} \pdfglyphtounicode{yikorean}{3162} \pdfglyphtounicode{yinyang}{262F} \pdfglyphtounicode{yiwnarmenian}{0582} \pdfglyphtounicode{ymonospace}{FF59} \pdfglyphtounicode{yod}{05D9} \pdfglyphtounicode{yoddagesh}{FB39} \pdfglyphtounicode{yoddageshhebrew}{FB39} \pdfglyphtounicode{yodhebrew}{05D9} \pdfglyphtounicode{yodyodhebrew}{05F2} \pdfglyphtounicode{yodyodpatahhebrew}{FB1F} \pdfglyphtounicode{yohiragana}{3088} \pdfglyphtounicode{yoikorean}{3189} \pdfglyphtounicode{yokatakana}{30E8} \pdfglyphtounicode{yokatakanahalfwidth}{FF96} \pdfglyphtounicode{yokorean}{315B} \pdfglyphtounicode{yosmallhiragana}{3087} \pdfglyphtounicode{yosmallkatakana}{30E7} \pdfglyphtounicode{yosmallkatakanahalfwidth}{FF6E} \pdfglyphtounicode{yotgreek}{03F3} \pdfglyphtounicode{yoyaekorean}{3188} \pdfglyphtounicode{yoyakorean}{3187} \pdfglyphtounicode{yoyakthai}{0E22} \pdfglyphtounicode{yoyingthai}{0E0D} \pdfglyphtounicode{yparen}{24B4} \pdfglyphtounicode{ypogegrammeni}{037A} \pdfglyphtounicode{ypogegrammenigreekcmb}{0345} \pdfglyphtounicode{yr}{01A6} \pdfglyphtounicode{yring}{1E99} \pdfglyphtounicode{ysuperior}{02B8} \pdfglyphtounicode{ytilde}{1EF9} \pdfglyphtounicode{yturned}{028E} \pdfglyphtounicode{yuhiragana}{3086} \pdfglyphtounicode{yuikorean}{318C} \pdfglyphtounicode{yukatakana}{30E6} \pdfglyphtounicode{yukatakanahalfwidth}{FF95} \pdfglyphtounicode{yukorean}{3160} \pdfglyphtounicode{yusbigcyrillic}{046B} \pdfglyphtounicode{yusbigiotifiedcyrillic}{046D} \pdfglyphtounicode{yuslittlecyrillic}{0467} \pdfglyphtounicode{yuslittleiotifiedcyrillic}{0469} \pdfglyphtounicode{yusmallhiragana}{3085} \pdfglyphtounicode{yusmallkatakana}{30E5} \pdfglyphtounicode{yusmallkatakanahalfwidth}{FF6D} \pdfglyphtounicode{yuyekorean}{318B} \pdfglyphtounicode{yuyeokorean}{318A} \pdfglyphtounicode{yyabengali}{09DF} \pdfglyphtounicode{yyadeva}{095F} \pdfglyphtounicode{z}{007A} \pdfglyphtounicode{zaarmenian}{0566} \pdfglyphtounicode{zacute}{017A} \pdfglyphtounicode{zadeva}{095B} \pdfglyphtounicode{zagurmukhi}{0A5B} \pdfglyphtounicode{zaharabic}{0638} \pdfglyphtounicode{zahfinalarabic}{FEC6} \pdfglyphtounicode{zahinitialarabic}{FEC7} \pdfglyphtounicode{zahiragana}{3056} \pdfglyphtounicode{zahmedialarabic}{FEC8} \pdfglyphtounicode{zainarabic}{0632} \pdfglyphtounicode{zainfinalarabic}{FEB0} \pdfglyphtounicode{zakatakana}{30B6} \pdfglyphtounicode{zaqefgadolhebrew}{0595} \pdfglyphtounicode{zaqefqatanhebrew}{0594} \pdfglyphtounicode{zarqahebrew}{0598} \pdfglyphtounicode{zayin}{05D6} \pdfglyphtounicode{zayindagesh}{FB36} \pdfglyphtounicode{zayindageshhebrew}{FB36} \pdfglyphtounicode{zayinhebrew}{05D6} \pdfglyphtounicode{zbopomofo}{3117} \pdfglyphtounicode{zcaron}{017E} \pdfglyphtounicode{zcircle}{24E9} \pdfglyphtounicode{zcircumflex}{1E91} \pdfglyphtounicode{zcurl}{0291} \pdfglyphtounicode{zdot}{017C} \pdfglyphtounicode{zdotaccent}{017C} \pdfglyphtounicode{zdotbelow}{1E93} \pdfglyphtounicode{zecyrillic}{0437} \pdfglyphtounicode{zedescendercyrillic}{0499} \pdfglyphtounicode{zedieresiscyrillic}{04DF} \pdfglyphtounicode{zehiragana}{305C} \pdfglyphtounicode{zekatakana}{30BC} \pdfglyphtounicode{zero}{0030} \pdfglyphtounicode{zeroarabic}{0660} \pdfglyphtounicode{zerobengali}{09E6} \pdfglyphtounicode{zerodeva}{0966} \pdfglyphtounicode{zerogujarati}{0AE6} \pdfglyphtounicode{zerogurmukhi}{0A66} \pdfglyphtounicode{zerohackarabic}{0660} \pdfglyphtounicode{zeroinferior}{2080} \pdfglyphtounicode{zeromonospace}{FF10} \pdfglyphtounicode{zerooldstyle}{F730} \pdfglyphtounicode{zeropersian}{06F0} \pdfglyphtounicode{zerosuperior}{2070} \pdfglyphtounicode{zerothai}{0E50} \pdfglyphtounicode{zerowidthjoiner}{FEFF} \pdfglyphtounicode{zerowidthnonjoiner}{200C} \pdfglyphtounicode{zerowidthspace}{200B} \pdfglyphtounicode{zeta}{03B6} \pdfglyphtounicode{zhbopomofo}{3113} \pdfglyphtounicode{zhearmenian}{056A} \pdfglyphtounicode{zhebrevecyrillic}{04C2} \pdfglyphtounicode{zhecyrillic}{0436} \pdfglyphtounicode{zhedescendercyrillic}{0497} \pdfglyphtounicode{zhedieresiscyrillic}{04DD} \pdfglyphtounicode{zihiragana}{3058} \pdfglyphtounicode{zikatakana}{30B8} \pdfglyphtounicode{zinorhebrew}{05AE} \pdfglyphtounicode{zlinebelow}{1E95} \pdfglyphtounicode{zmonospace}{FF5A} \pdfglyphtounicode{zohiragana}{305E} \pdfglyphtounicode{zokatakana}{30BE} \pdfglyphtounicode{zparen}{24B5} \pdfglyphtounicode{zretroflexhook}{0290} \pdfglyphtounicode{zstroke}{01B6} \pdfglyphtounicode{zuhiragana}{305A} \pdfglyphtounicode{zukatakana}{30BA} % entries from zapfdingbats.txt: \pdfglyphtounicode{a100}{275E} \pdfglyphtounicode{a101}{2761} \pdfglyphtounicode{a102}{2762} \pdfglyphtounicode{a103}{2763} \pdfglyphtounicode{a104}{2764} \pdfglyphtounicode{a105}{2710} \pdfglyphtounicode{a106}{2765} \pdfglyphtounicode{a107}{2766} \pdfglyphtounicode{a108}{2767} \pdfglyphtounicode{a109}{2660} \pdfglyphtounicode{a10}{2721} \pdfglyphtounicode{a110}{2665} \pdfglyphtounicode{a111}{2666} \pdfglyphtounicode{a112}{2663} \pdfglyphtounicode{a117}{2709} \pdfglyphtounicode{a118}{2708} \pdfglyphtounicode{a119}{2707} \pdfglyphtounicode{a11}{261B} \pdfglyphtounicode{a120}{2460} \pdfglyphtounicode{a121}{2461} \pdfglyphtounicode{a122}{2462} \pdfglyphtounicode{a123}{2463} \pdfglyphtounicode{a124}{2464} \pdfglyphtounicode{a125}{2465} \pdfglyphtounicode{a126}{2466} \pdfglyphtounicode{a127}{2467} \pdfglyphtounicode{a128}{2468} \pdfglyphtounicode{a129}{2469} \pdfglyphtounicode{a12}{261E} \pdfglyphtounicode{a130}{2776} \pdfglyphtounicode{a131}{2777} \pdfglyphtounicode{a132}{2778} \pdfglyphtounicode{a133}{2779} \pdfglyphtounicode{a134}{277A} \pdfglyphtounicode{a135}{277B} \pdfglyphtounicode{a136}{277C} \pdfglyphtounicode{a137}{277D} \pdfglyphtounicode{a138}{277E} \pdfglyphtounicode{a139}{277F} \pdfglyphtounicode{a13}{270C} \pdfglyphtounicode{a140}{2780} \pdfglyphtounicode{a141}{2781} \pdfglyphtounicode{a142}{2782} \pdfglyphtounicode{a143}{2783} \pdfglyphtounicode{a144}{2784} \pdfglyphtounicode{a145}{2785} \pdfglyphtounicode{a146}{2786} \pdfglyphtounicode{a147}{2787} \pdfglyphtounicode{a148}{2788} \pdfglyphtounicode{a149}{2789} \pdfglyphtounicode{a14}{270D} \pdfglyphtounicode{a150}{278A} \pdfglyphtounicode{a151}{278B} \pdfglyphtounicode{a152}{278C} \pdfglyphtounicode{a153}{278D} \pdfglyphtounicode{a154}{278E} \pdfglyphtounicode{a155}{278F} \pdfglyphtounicode{a156}{2790} \pdfglyphtounicode{a157}{2791} \pdfglyphtounicode{a158}{2792} \pdfglyphtounicode{a159}{2793} \pdfglyphtounicode{a15}{270E} \pdfglyphtounicode{a160}{2794} \pdfglyphtounicode{a161}{2192} \pdfglyphtounicode{a162}{27A3} \pdfglyphtounicode{a163}{2194} \pdfglyphtounicode{a164}{2195} \pdfglyphtounicode{a165}{2799} \pdfglyphtounicode{a166}{279B} \pdfglyphtounicode{a167}{279C} \pdfglyphtounicode{a168}{279D} \pdfglyphtounicode{a169}{279E} \pdfglyphtounicode{a16}{270F} \pdfglyphtounicode{a170}{279F} \pdfglyphtounicode{a171}{27A0} \pdfglyphtounicode{a172}{27A1} \pdfglyphtounicode{a173}{27A2} \pdfglyphtounicode{a174}{27A4} \pdfglyphtounicode{a175}{27A5} \pdfglyphtounicode{a176}{27A6} \pdfglyphtounicode{a177}{27A7} \pdfglyphtounicode{a178}{27A8} \pdfglyphtounicode{a179}{27A9} \pdfglyphtounicode{a17}{2711} \pdfglyphtounicode{a180}{27AB} \pdfglyphtounicode{a181}{27AD} \pdfglyphtounicode{a182}{27AF} \pdfglyphtounicode{a183}{27B2} \pdfglyphtounicode{a184}{27B3} \pdfglyphtounicode{a185}{27B5} \pdfglyphtounicode{a186}{27B8} \pdfglyphtounicode{a187}{27BA} \pdfglyphtounicode{a188}{27BB} \pdfglyphtounicode{a189}{27BC} \pdfglyphtounicode{a18}{2712} \pdfglyphtounicode{a190}{27BD} \pdfglyphtounicode{a191}{27BE} \pdfglyphtounicode{a192}{279A} \pdfglyphtounicode{a193}{27AA} \pdfglyphtounicode{a194}{27B6} \pdfglyphtounicode{a195}{27B9} \pdfglyphtounicode{a196}{2798} \pdfglyphtounicode{a197}{27B4} \pdfglyphtounicode{a198}{27B7} \pdfglyphtounicode{a199}{27AC} \pdfglyphtounicode{a19}{2713} \pdfglyphtounicode{a1}{2701} \pdfglyphtounicode{a200}{27AE} \pdfglyphtounicode{a201}{27B1} \pdfglyphtounicode{a202}{2703} \pdfglyphtounicode{a203}{2750} \pdfglyphtounicode{a204}{2752} \pdfglyphtounicode{a205}{276E} \pdfglyphtounicode{a206}{2770} \pdfglyphtounicode{a20}{2714} \pdfglyphtounicode{a21}{2715} \pdfglyphtounicode{a22}{2716} \pdfglyphtounicode{a23}{2717} \pdfglyphtounicode{a24}{2718} \pdfglyphtounicode{a25}{2719} \pdfglyphtounicode{a26}{271A} \pdfglyphtounicode{a27}{271B} \pdfglyphtounicode{a28}{271C} \pdfglyphtounicode{a29}{2722} \pdfglyphtounicode{a2}{2702} \pdfglyphtounicode{a30}{2723} \pdfglyphtounicode{a31}{2724} \pdfglyphtounicode{a32}{2725} \pdfglyphtounicode{a33}{2726} \pdfglyphtounicode{a34}{2727} \pdfglyphtounicode{a35}{2605} \pdfglyphtounicode{a36}{2729} \pdfglyphtounicode{a37}{272A} \pdfglyphtounicode{a38}{272B} \pdfglyphtounicode{a39}{272C} \pdfglyphtounicode{a3}{2704} \pdfglyphtounicode{a40}{272D} \pdfglyphtounicode{a41}{272E} \pdfglyphtounicode{a42}{272F} \pdfglyphtounicode{a43}{2730} \pdfglyphtounicode{a44}{2731} \pdfglyphtounicode{a45}{2732} \pdfglyphtounicode{a46}{2733} \pdfglyphtounicode{a47}{2734} \pdfglyphtounicode{a48}{2735} \pdfglyphtounicode{a49}{2736} \pdfglyphtounicode{a4}{260E} \pdfglyphtounicode{a50}{2737} \pdfglyphtounicode{a51}{2738} \pdfglyphtounicode{a52}{2739} \pdfglyphtounicode{a53}{273A} \pdfglyphtounicode{a54}{273B} \pdfglyphtounicode{a55}{273C} \pdfglyphtounicode{a56}{273D} \pdfglyphtounicode{a57}{273E} \pdfglyphtounicode{a58}{273F} \pdfglyphtounicode{a59}{2740} \pdfglyphtounicode{a5}{2706} \pdfglyphtounicode{a60}{2741} \pdfglyphtounicode{a61}{2742} \pdfglyphtounicode{a62}{2743} \pdfglyphtounicode{a63}{2744} \pdfglyphtounicode{a64}{2745} \pdfglyphtounicode{a65}{2746} \pdfglyphtounicode{a66}{2747} \pdfglyphtounicode{a67}{2748} \pdfglyphtounicode{a68}{2749} \pdfglyphtounicode{a69}{274A} \pdfglyphtounicode{a6}{271D} \pdfglyphtounicode{a70}{274B} \pdfglyphtounicode{a71}{25CF} \pdfglyphtounicode{a72}{274D} \pdfglyphtounicode{a73}{25A0} \pdfglyphtounicode{a74}{274F} \pdfglyphtounicode{a75}{2751} \pdfglyphtounicode{a76}{25B2} \pdfglyphtounicode{a77}{25BC} \pdfglyphtounicode{a78}{25C6} \pdfglyphtounicode{a79}{2756} \pdfglyphtounicode{a7}{271E} \pdfglyphtounicode{a81}{25D7} \pdfglyphtounicode{a82}{2758} \pdfglyphtounicode{a83}{2759} \pdfglyphtounicode{a84}{275A} \pdfglyphtounicode{a85}{276F} \pdfglyphtounicode{a86}{2771} \pdfglyphtounicode{a87}{2772} \pdfglyphtounicode{a88}{2773} \pdfglyphtounicode{a89}{2768} \pdfglyphtounicode{a8}{271F} \pdfglyphtounicode{a90}{2769} \pdfglyphtounicode{a91}{276C} \pdfglyphtounicode{a92}{276D} \pdfglyphtounicode{a93}{276A} \pdfglyphtounicode{a94}{276B} \pdfglyphtounicode{a95}{2774} \pdfglyphtounicode{a96}{2775} \pdfglyphtounicode{a97}{275B} \pdfglyphtounicode{a98}{275C} \pdfglyphtounicode{a99}{275D} \pdfglyphtounicode{a9}{2720} % entries from texglyphlist.txt: % Delta;2206 \pdfglyphtounicode{Ifractur}{2111} \pdfglyphtounicode{FFsmall}{D804} \pdfglyphtounicode{FFIsmall}{D807} \pdfglyphtounicode{FFLsmall}{D808} \pdfglyphtounicode{FIsmall}{D805} \pdfglyphtounicode{FLsmall}{D806} \pdfglyphtounicode{Germandbls}{D800} \pdfglyphtounicode{Germandblssmall}{D803} \pdfglyphtounicode{Ng}{014A} % Omega;2126 \pdfglyphtounicode{Rfractur}{211C} \pdfglyphtounicode{SS}{D800} \pdfglyphtounicode{SSsmall}{D803} \pdfglyphtounicode{altselector}{D802} \pdfglyphtounicode{angbracketleft}{27E8} \pdfglyphtounicode{angbracketright}{27E9} \pdfglyphtounicode{arrowbothv}{2195} \pdfglyphtounicode{arrowdblbothv}{21D5} \pdfglyphtounicode{arrowleftbothalf}{21BD} \pdfglyphtounicode{arrowlefttophalf}{21BC} \pdfglyphtounicode{arrownortheast}{2197} \pdfglyphtounicode{arrownorthwest}{2196} \pdfglyphtounicode{arrowrightbothalf}{21C1} \pdfglyphtounicode{arrowrighttophalf}{21C0} \pdfglyphtounicode{arrowsoutheast}{2198} \pdfglyphtounicode{arrowsouthwest}{2199} \pdfglyphtounicode{ascendercompwordmark}{D80A} \pdfglyphtounicode{asteriskcentered}{2217} \pdfglyphtounicode{bardbl}{2225} \pdfglyphtounicode{capitalcompwordmark}{D809} \pdfglyphtounicode{ceilingleft}{2308} \pdfglyphtounicode{ceilingright}{2309} \pdfglyphtounicode{circlecopyrt}{20DD} \pdfglyphtounicode{circledivide}{2298} \pdfglyphtounicode{circledot}{2299} \pdfglyphtounicode{circleminus}{2296} \pdfglyphtounicode{coproduct}{2A3F} \pdfglyphtounicode{cwm}{200C} \pdfglyphtounicode{dblbracketleft}{27E6} \pdfglyphtounicode{dblbracketright}{27E7} % diamond;2662 % diamondmath;22C4 % dotlessj;0237 % emptyset;2205 \pdfglyphtounicode{emptyslot}{D801} \pdfglyphtounicode{epsilon1}{03F5} \pdfglyphtounicode{equivasymptotic}{224D} \pdfglyphtounicode{flat}{266D} \pdfglyphtounicode{floorleft}{230A} \pdfglyphtounicode{floorright}{230B} \pdfglyphtounicode{follows}{227B} \pdfglyphtounicode{followsequal}{227D} \pdfglyphtounicode{greatermuch}{226B} % heart;2661 \pdfglyphtounicode{interrobang}{203D} \pdfglyphtounicode{interrobangdown}{D80B} \pdfglyphtounicode{intersectionsq}{2293} \pdfglyphtounicode{latticetop}{22A4} \pdfglyphtounicode{lessmuch}{226A} \pdfglyphtounicode{lscript}{2113} \pdfglyphtounicode{natural}{266E} \pdfglyphtounicode{negationslash}{0338} \pdfglyphtounicode{ng}{014B} \pdfglyphtounicode{owner}{220B} \pdfglyphtounicode{pertenthousand}{2031} % phi;03D5 % phi1;03C6 \pdfglyphtounicode{pi1}{03D6} \pdfglyphtounicode{precedesequal}{227C} \pdfglyphtounicode{prime}{2032} \pdfglyphtounicode{rho1}{03F1} \pdfglyphtounicode{ringfitted}{D80D} \pdfglyphtounicode{sharp}{266F} \pdfglyphtounicode{similarequal}{2243} \pdfglyphtounicode{slurabove}{2322} \pdfglyphtounicode{slurbelow}{2323} \pdfglyphtounicode{star}{22C6} \pdfglyphtounicode{subsetsqequal}{2291} \pdfglyphtounicode{supersetsqequal}{2292} \pdfglyphtounicode{triangle}{25B3} \pdfglyphtounicode{triangleinv}{25BD} \pdfglyphtounicode{triangleleft}{25B9} \pdfglyphtounicode{triangleright}{25C3} \pdfglyphtounicode{turnstileleft}{22A2} \pdfglyphtounicode{turnstileright}{22A3} \pdfglyphtounicode{twelveudash}{D80C} \pdfglyphtounicode{unionmulti}{228E} \pdfglyphtounicode{unionsq}{2294} \pdfglyphtounicode{vector}{20D7} \pdfglyphtounicode{visualspace}{2423} \pdfglyphtounicode{wreathproduct}{2240} \pdfglyphtounicode{Dbar}{0110} \pdfglyphtounicode{compwordmark}{200C} \pdfglyphtounicode{dbar}{0111} \pdfglyphtounicode{rangedash}{2013} \pdfglyphtounicode{hyphenchar}{002D} \pdfglyphtounicode{punctdash}{2014} \pdfglyphtounicode{visiblespace}{2423} % entries from additional.tex: \pdfglyphtounicode{ff}{0066 0066} \pdfglyphtounicode{fi}{0066 0069} \pdfglyphtounicode{fl}{0066 006C} \pdfglyphtounicode{ffi}{0066 0066 0069} \pdfglyphtounicode{ffl}{0066 0066 006C} \pdfglyphtounicode{IJ}{0049 004A} \pdfglyphtounicode{ij}{0069 006A} \pdfglyphtounicode{longs}{0073} ================================================ FILE: misc/original-template/Declaration/declaration.tex ================================================ % ******************************* Thesis Declaration ******************************** \begin{declaration} I hereby declare that except where specific reference is made to the work of others, the contents of this dissertation are original and have not been submitted in whole or in part for consideration for any other degree or qualification in this, or any other University. This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration, except where specifically indicated in the text. This dissertation contains less than 65,000 words including appendices, bibliography, footnotes, tables and equations and has less than 150 figures. % Author and date will be inserted automatically from thesis.tex \author \degreedate \end{declaration} ================================================ FILE: misc/original-template/Dedication/dedication.tex ================================================ % ******************************* Thesis Dedidcation ******************************** \begin{dedication} I would like to dedicate this thesis to my loving parents ... \end{dedication} ================================================ FILE: misc/original-template/Figs/University_Crest.eps ================================================ %!PS-Adobe-3.0 EPSF-3.0 %%Creator: cairo 1.12.8 (http://cairographics.org) %%CreationDate: Sat Nov 16 18:22:13 2013 %%Pages: 1 %%DocumentData: Clean7Bit %%LanguageLevel: 2 %%BoundingBox: 0 -1 59 68 %%EndComments %%BeginProlog save 50 dict begin /q { gsave } 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37.598 38.1 37.551 38.123 37.551 38.166 c 37.551 38.221 37.598 38.268 37.707 38.327 c 38.422 38.698 38.934 40.034 38.934 41.526 c 38.934 41.686 l 38.875 41.76 l 38.711 41.909 l 38.543 42.077 l 38.449 42.151 l 38.527 42.217 38.582 42.264 38.617 42.299 c 38.641 42.323 38.695 42.385 38.785 42.487 c 38.914 42.635 l 39.043 42.803 l f 46.121 42.803 m 46.25 42.635 l 46.398 42.487 l 46.566 42.299 l 46.715 42.151 l 46.621 42.077 l 46.457 41.909 l 46.289 41.76 l 46.23 41.686 l 46.23 41.526 l 46.23 40.045 46.742 38.717 47.457 38.327 c 47.57 38.264 47.617 38.213 47.617 38.162 c 47.605 38.139 l 47.457 38.1 l 47.293 38.1 l 47.074 38.1 46.688 38.17 46.438 38.252 c 46.25 36.912 l 45.992 36.912 l 45.805 38.252 l 45.504 38.155 45.176 38.1 44.879 38.1 c 44.676 38.1 44.633 38.112 44.633 38.174 c 44.633 38.233 44.68 38.272 44.801 38.327 c 44.953 38.389 45.219 38.666 45.379 38.92 c 45.672 39.389 45.844 39.795 45.844 40.034 c 45.844 40.436 45.902 40.76 45.973 40.76 c 46.012 41.448 l 46.027 41.686 l 45.996 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37.565 l 49.41 37.787 l 49.223 37.971 l 49.148 38.045 l 49.215 38.112 49.266 38.162 49.297 38.194 c 49.555 38.452 49.582 38.479 49.723 38.678 c f 53.254 42.803 m 53.387 42.635 l 53.531 42.487 l 53.699 42.299 l 53.848 42.151 l 53.758 42.077 l 53.59 41.909 l 53.422 41.76 l 53.367 41.686 l 53.363 41.526 l 53.363 40.045 53.875 38.717 54.594 38.327 c 54.703 38.264 54.754 38.213 54.754 38.162 c 54.742 38.139 l 54.594 38.1 l 54.426 38.1 l 54.207 38.1 53.82 38.17 53.57 38.252 c 53.387 36.912 l 53.125 36.912 l 52.938 38.252 l 52.637 38.155 52.309 38.1 52.012 38.1 c 51.809 38.1 51.766 38.112 51.766 38.174 c 51.766 38.233 51.812 38.272 51.938 38.327 c 52.086 38.389 52.355 38.666 52.512 38.92 c 52.805 39.389 52.977 39.795 52.977 40.034 c 52.977 40.436 53.035 40.76 53.105 40.76 c 53.145 41.448 l 53.16 41.686 l 53.129 41.721 53.102 41.744 53.086 41.76 c 53.004 41.846 52.957 41.893 52.938 41.909 c 52.754 42.077 l 52.68 42.151 l 52.828 42.299 l 52.992 42.487 l 53.145 42.635 l 53.254 42.803 l f 25.762 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9.42 l 32.574 8.811 32.828 8.03 33.137 7.58 c 33.34 7.284 33.566 7.049 33.711 6.987 c 33.836 6.932 33.883 6.893 33.883 6.834 c 33.883 6.776 33.836 6.764 33.637 6.764 c 33.395 6.764 33.102 6.803 32.711 6.893 c 32.523 5.577 l 32.266 5.577 l 32.078 6.893 l 31.641 6.803 31.371 6.764 31.223 6.764 c 31.055 6.764 l 30.945 6.764 30.898 6.784 30.898 6.827 c 30.898 6.881 30.949 6.932 31.055 6.987 c 31.234 7.08 31.488 7.342 31.633 7.58 c 31.957 8.127 32.113 8.487 32.113 8.698 c 32.113 9.096 32.164 9.42 32.227 9.42 c 32.266 10.108 l 32.281 10.35 l 32.227 10.424 l 32.059 10.573 l 31.891 10.741 l 31.797 10.815 l 31.949 10.963 l 32.113 11.147 l 32.266 11.295 l 32.395 11.463 l f 29.012 29.666 m 29.125 29.502 l 29.273 29.354 l 29.441 29.166 l 29.59 29.018 l 29.516 28.944 l 29.328 28.776 l 29.312 28.76 29.262 28.713 29.18 28.627 c 29.164 28.612 29.137 28.588 29.105 28.553 c 29.125 28.311 l 29.16 27.623 l 29.215 26.619 29.793 25.42 30.332 25.19 c 30.453 25.139 30.496 25.104 30.496 25.049 c 30.496 24.987 30.445 24.967 30.262 24.967 c 30.016 24.967 29.723 25.01 29.328 25.1 c 29.125 23.76 l 28.883 23.76 l 28.695 25.1 l 28.258 25.01 27.992 24.967 27.844 24.967 c 27.676 24.967 l 27.566 24.967 27.52 24.987 27.52 25.034 c 27.52 25.088 27.566 25.135 27.676 25.19 c 27.852 25.284 28.105 25.545 28.25 25.787 c 28.578 26.334 28.734 26.69 28.734 26.901 c 28.734 27.299 28.785 27.623 28.844 27.623 c 28.883 28.311 l 28.902 28.553 l 28.844 28.627 l 28.68 28.776 l 28.512 28.944 l 28.418 29.018 l 28.566 29.166 l 28.734 29.354 l 28.883 29.502 l 29.012 29.666 l f 25.668 63.569 m 25.727 63.506 25.77 63.455 25.797 63.42 c 25.867 63.342 25.918 63.284 25.949 63.252 c 25.969 63.229 26.027 63.174 26.113 63.088 c 26.148 63.053 26.195 63.006 26.262 62.94 c 26.188 62.866 l 26.004 62.698 l 25.988 62.682 25.93 62.627 25.836 62.53 c 25.824 62.518 25.805 62.498 25.781 62.475 c 25.797 62.213 l 25.836 61.545 l 25.871 60.901 26.16 59.998 26.43 59.69 c 26.801 59.26 l 26.82 59.241 26.891 59.182 27.008 59.092 c 27.164 58.971 l 27.008 58.869 l 26.84 58.889 l 26.484 58.928 l 26.168 58.983 l 25.984 59.018 l 25.781 57.737 l 25.559 57.682 l 25.371 59.018 l 25.188 58.983 l 24.871 58.928 l 24.516 58.889 l 24.32 58.869 l 24.254 58.869 24.199 58.897 24.199 58.928 c 24.199 58.963 24.227 59.006 24.277 59.045 c 24.293 59.061 24.387 59.131 24.555 59.26 c 24.793 59.444 24.863 59.541 25.203 60.209 c 25.34 60.467 25.406 60.678 25.41 60.823 c 25.41 61.221 25.461 61.545 25.52 61.545 c 25.559 62.213 l 25.574 62.475 l 25.5 62.53 l 25.352 62.698 l 25.168 62.866 l 25.094 62.94 l 25.16 63.006 25.207 63.053 25.242 63.088 c 25.5 63.342 25.523 63.369 25.668 63.569 c f 32.395 63.569 m 32.539 63.369 32.562 63.342 32.82 63.088 c 32.855 63.053 32.902 63.006 32.969 62.94 c 32.895 62.866 l 32.711 62.698 l 32.559 62.53 l 32.484 62.475 l 32.504 62.213 l 32.543 61.545 l 32.598 60.561 33.152 59.397 33.711 59.092 c 33.809 59.041 33.871 58.994 33.871 58.971 c 33.871 58.92 33.836 58.881 33.797 58.881 c 33.562 58.889 l 33.191 58.928 l 33.113 58.936 33.008 58.952 32.875 58.983 c 32.711 59.018 l 32.523 57.682 l 32.266 57.682 l 32.078 59.018 l 31.871 59.037 l 31.578 58.928 l 31.223 58.889 l 31.055 58.869 l 31.031 58.869 l 30.961 58.869 30.906 58.897 30.906 58.928 c 30.906 58.959 30.938 59.006 30.992 59.045 c 31.262 59.26 l 31.508 59.455 31.59 59.573 31.91 60.209 c 32.043 60.471 32.113 60.682 32.113 60.823 c 32.117 61.221 32.168 61.545 32.227 61.545 c 32.266 62.213 l 32.281 62.475 l 32.059 62.698 l 31.977 62.784 31.918 62.838 31.891 62.866 c 31.875 62.877 31.844 62.905 31.797 62.94 c 31.863 63.006 31.914 63.053 31.949 63.088 c 32.035 63.174 32.094 63.229 32.113 63.252 c 32.145 63.284 32.191 63.342 32.266 63.42 c 32.293 63.455 32.336 63.506 32.395 63.569 c f 28.977 57.756 m 29.105 57.608 l 29.254 57.44 l 29.344 57.338 29.398 57.276 29.422 57.256 c 29.453 57.221 29.504 57.17 29.57 57.108 c 29.477 57.03 l 29.309 56.866 l 29.145 56.717 l 29.086 56.643 l 29.105 56.401 l 29.125 55.768 l 29.18 54.752 29.727 53.6 30.312 53.28 c 30.426 53.217 30.473 53.17 30.473 53.116 c 30.461 53.092 l 30.312 53.057 l 30.145 53.057 l 29.926 53.057 29.539 53.123 29.289 53.205 c 29.105 51.866 l 28.844 51.866 l 28.66 53.205 l 28.41 53.116 28.043 53.057 27.734 53.057 c 27.531 53.057 27.488 53.069 27.488 53.127 c 27.488 53.186 27.531 53.225 27.656 53.28 c 27.805 53.346 28.074 53.623 28.23 53.873 c 28.527 54.346 28.695 54.748 28.695 54.987 c 28.699 55.389 28.758 55.713 28.828 55.713 c 28.863 56.401 l 28.883 56.643 l 28.848 56.674 28.824 56.698 28.809 56.717 c 28.727 56.799 28.676 56.85 28.66 56.866 c 28.473 57.03 l 28.398 57.108 l 28.547 57.256 l 28.715 57.44 l 28.801 57.537 28.852 57.592 28.863 57.608 c 28.891 57.639 28.926 57.69 28.977 57.756 c f 25.648 51.959 m 25.781 51.811 l 25.91 51.643 l 25.941 51.604 25.996 51.549 26.078 51.479 c 26.246 51.33 l 26.191 51.287 26.16 51.264 26.152 51.256 c 25.762 50.866 l 25.762 50.69 l 25.762 49.713 26.051 48.487 26.375 48.131 c 26.785 47.651 l 26.785 47.647 26.855 47.592 26.988 47.483 c 27.137 47.362 l 26.988 47.26 l 26.82 47.28 l 26.469 47.315 l 26.152 47.373 l 25.965 47.409 l 25.781 46.073 l 25.52 46.073 l 25.336 47.409 l 25.148 47.373 l 24.832 47.315 l 24.48 47.28 l 24.312 47.26 l 24.289 47.26 l 24.219 47.26 24.16 47.284 24.16 47.319 c 24.16 47.35 24.195 47.393 24.246 47.436 c 24.516 47.651 l 25.137 48.143 25.539 49.342 25.539 50.69 c 25.539 50.866 l 25.316 51.088 l 25.234 51.174 25.176 51.229 25.148 51.256 c 25.133 51.268 25.102 51.291 25.055 51.33 c 25.129 51.393 25.184 51.444 25.223 51.479 c 25.301 51.549 25.359 51.604 25.391 51.643 c 25.52 51.811 l 25.648 51.959 l f 32.355 51.959 m 32.414 51.893 32.457 51.846 32.484 51.811 c 32.555 51.733 32.605 51.674 32.637 51.643 c 32.656 51.619 32.715 51.565 32.801 51.479 c 32.836 51.444 32.887 51.393 32.949 51.33 c 32.898 51.287 32.867 51.264 32.859 51.256 c 32.469 50.866 l 32.484 50.604 l 32.523 49.916 l 32.578 48.893 32.977 48.069 33.695 47.483 c 33.844 47.362 l 33.695 47.26 l 33.527 47.28 l 33.172 47.315 l 32.672 47.409 l 32.484 46.073 l 32.227 46.073 l 32.023 47.463 l 31.684 47.389 31.672 47.327 31.559 47.315 c 31.188 47.28 l 31.02 47.26 l 31 47.26 l 30.934 47.26 30.887 47.284 30.887 47.319 c 30.887 47.358 30.914 47.401 30.961 47.436 c 31.242 47.651 l 31.484 47.834 31.551 47.936 31.891 48.596 c 32.027 48.858 32.094 49.069 32.098 49.209 c 32.098 49.6 32.148 49.916 32.207 49.916 c 32.246 50.604 l 32.266 50.866 l 32.188 50.92 l 32.039 51.088 l 31.855 51.256 l 31.781 51.33 l 31.848 51.393 31.895 51.444 31.93 51.479 c 32.188 51.733 32.211 51.76 32.355 51.959 c f 18.441 54.952 m 18.516 54.67 l 18.582 54.42 18.633 54.221 18.664 54.077 c 18.898 53.018 19.098 52.737 19.613 52.737 c 19.906 52.737 20.152 52.901 20.227 53.151 c 20.336 53.522 l 20.363 53.608 20.426 53.655 20.516 53.655 c 20.715 53.655 20.809 53.448 20.82 52.983 c 20.824 52.846 20.867 52.545 20.949 52.073 c 21 51.791 21.023 51.522 21.023 51.26 c 21.023 51.022 21.004 50.924 20.914 50.678 c 20.977 50.315 21.027 49.823 21.027 49.541 c 21.027 49.08 20.941 48.655 20.801 48.432 c 20.879 48.123 20.895 47.998 20.895 47.733 c 20.895 47.534 20.875 47.319 20.84 47.112 c 20.758 46.666 20.742 46.553 20.742 46.467 c 20.742 46.385 20.754 46.299 20.781 46.221 c 20.82 46.116 20.84 46.037 20.84 45.998 c 20.84 45.944 20.812 45.905 20.781 45.905 c 20.41 46.053 l 20.004 46.276 l 19.871 46.346 19.781 46.42 19.574 46.612 c 19.215 46.284 19.109 46.237 18.164 45.979 c 17.977 45.928 17.836 45.869 17.754 45.811 c 17.637 45.725 17.551 45.682 17.508 45.682 c 17.348 45.682 17.273 45.94 17.273 46.467 c 17.273 46.862 17.312 47.229 17.383 47.502 c 17.105 47.608 16.871 47.67 16.719 47.67 c 16.691 47.67 16.598 47.659 16.438 47.631 c 16.367 47.627 l 16.312 47.627 16.266 47.705 16.266 47.803 c 16.266 47.877 16.316 48.057 16.383 48.209 c 16.566 48.655 l 16.633 48.815 16.785 48.998 17.012 49.194 c 17.195 49.35 17.363 49.444 17.496 49.471 c 18.164 49.6 l 18.562 49.678 18.797 49.889 18.797 50.162 c 18.797 50.432 18.617 50.612 18.219 50.752 c 17.422 51.03 l 17.215 51.104 16.973 51.221 16.695 51.385 c 16.254 51.647 16.18 51.682 16.074 51.682 c 15.914 51.682 15.82 51.561 15.82 51.354 c 15.82 51.022 16.035 50.791 16.34 50.791 c 16.406 50.791 16.469 50.803 16.527 50.827 c 16.617 50.866 16.684 50.881 16.719 50.881 c 16.805 50.881 16.852 50.834 16.852 50.744 c 16.844 50.659 l 16.809 50.342 l 16.777 50.073 16.602 49.748 16.215 49.229 c 16.047 49.006 l 15.895 48.409 15.523 47.866 15.043 47.537 c 14.922 47.459 14.855 47.409 14.84 47.389 c 14.824 47.373 14.773 47.284 14.691 47.131 c 14.652 47.061 14.527 46.909 14.32 46.686 c 14.16 46.514 14.008 46.373 13.855 46.256 c 13.648 46.1 13.52 45.987 13.484 45.924 c 13.297 45.608 l 13.262 45.549 13.207 45.514 13.148 45.514 c 12.984 45.514 12.832 45.686 12.738 45.979 c 12.672 46.198 12.609 46.256 12.469 46.256 c 12.398 46.256 12.289 46.237 12.145 46.202 c 11.887 46.166 l 11.215 46.166 l 10.883 46.166 10.41 46.08 10.195 45.979 c 10.109 45.94 10.043 45.916 10.004 45.916 c 9.957 45.916 9.93 45.944 9.879 46.034 c 9.844 46.096 9.836 46.123 9.836 46.182 c 9.836 46.522 10.078 47.162 10.398 47.67 c 10.602 47.991 10.664 48.065 10.902 48.264 c 10.824 48.58 10.621 48.823 10.379 48.897 c 10.066 48.987 l 10.02 49.002 9.988 49.037 9.988 49.077 c 9.988 49.174 10.141 49.307 10.418 49.452 c 10.781 49.643 11.145 49.748 11.438 49.748 c 11.746 49.748 12.047 49.627 12.387 49.358 c 12.891 48.963 13.027 48.893 13.273 48.893 c 13.668 48.893 14.047 49.256 14.047 49.639 c 14.047 49.893 13.91 50.166 13.559 50.623 c 13.039 51.291 12.883 51.639 12.883 52.131 c 12.883 52.44 12.977 52.827 13.094 52.998 c 13.277 53.28 l 13.32 53.338 13.34 53.401 13.34 53.459 c 13.34 53.58 13.199 53.67 13.004 53.67 c 12.711 53.67 12.148 53.463 11.699 53.186 c 11.477 53.049 11.324 52.889 11.324 52.791 c 11.324 52.733 11.352 52.678 11.402 52.647 c 11.605 52.518 l 11.633 52.502 11.645 52.471 11.645 52.44 c 11.645 52.412 11.613 52.354 11.57 52.295 c 11.199 51.811 l 11.004 51.561 10.711 51.252 10.418 50.994 c 10.32 50.909 10.258 50.846 10.23 50.807 c 10.23 50.803 10.18 50.713 10.082 50.53 c 9.949 50.28 9.598 49.92 9.266 49.694 c 9.129 49.6 9.043 49.53 9.008 49.491 c 8.98 49.459 8.93 49.369 8.855 49.229 c 8.809 49.139 8.73 49.006 8.617 48.838 c 8.434 48.573 8.285 48.377 8.168 48.264 c 7.648 47.744 l 7.488 47.584 7.348 47.272 7.129 46.592 c 6.996 46.182 6.879 45.94 6.809 45.94 c 6.781 45.94 6.727 45.983 6.664 46.053 c 6.297 46.475 5.98 46.721 5.812 46.721 c 5.773 46.721 5.703 46.69 5.605 46.627 c 5.469 46.541 5.301 46.479 5.105 46.444 c 4.418 46.315 l 3.898 46.213 3.695 46.104 3.395 45.756 c 3.324 45.674 3.266 45.623 3.234 45.623 c 3.176 45.623 3.133 45.717 3.133 45.842 c 3.156 46.014 l 3.246 46.518 l 3.348 47.053 3.469 47.452 3.617 47.725 c 3.934 48.299 l 3.996 48.412 4.027 48.514 4.027 48.604 c 4.027 49.03 3.586 49.42 3.043 49.471 c 2.906 49.483 2.855 49.518 2.855 49.592 c 2.855 49.686 3.004 49.873 3.211 50.03 c 3.582 50.311 4.07 50.51 4.383 50.51 c 4.57 50.51 4.859 50.436 5.215 50.287 c 5.625 50.119 l 5.777 50.061 5.918 50.026 6.039 50.026 c 6.473 50.026 6.891 50.42 7.406 51.311 c 7.641 51.709 7.742 52.002 7.742 52.264 c 7.742 52.733 7.52 53.018 7.152 53.018 c 6.875 53.018 6.57 52.885 6.352 52.666 c 6.16 52.479 6.062 52.284 6.062 52.092 c 6.07 51.885 l 6.074 51.838 l 6.074 51.666 6.016 51.569 5.91 51.569 c 5.738 51.569 5.48 51.819 4.75 52.702 c 4.207 53.362 4.086 53.643 4.062 54.299 c 3.434 55.104 3.277 55.444 3.277 56.018 c 3.301 56.401 l 2.867 56.924 2.742 57.209 2.742 57.694 c 2.742 57.866 2.773 57.971 2.875 58.182 c 2.656 58.506 2.582 58.655 2.523 58.909 c 2.43 59.299 l 2.367 59.565 2.266 59.795 2.133 59.987 c 1.965 60.225 1.945 60.256 1.945 60.342 c 1.945 60.557 2.094 60.643 2.746 60.803 c 2.719 60.983 2.707 61.119 2.707 61.213 c 2.707 61.323 2.719 61.428 2.766 61.659 c 2.895 62.346 l 2.945 62.623 2.973 62.823 2.973 62.94 c 2.973 63.069 2.953 63.159 2.84 63.534 c 2.773 63.744 2.719 63.889 2.672 63.959 c 2.609 64.057 2.559 64.174 2.559 64.217 c 2.559 64.28 2.605 64.319 2.688 64.319 c 2.871 64.319 3.359 64.022 3.859 63.608 c 4.926 62.725 5.145 62.373 5.289 61.303 c 5.301 61.221 5.32 61.096 5.348 60.932 c 5.859 60.557 6.125 60.104 6.125 59.616 c 6.125 59.233 5.988 58.784 5.754 58.424 c 5.383 57.85 l 5.273 57.678 5.215 57.467 5.215 57.233 c 5.215 56.881 5.316 56.565 5.57 56.119 c 5.855 55.619 6.094 55.416 6.398 55.416 c 6.887 55.416 7.316 56.268 7.316 57.237 c 7.316 57.366 l 7.312 57.702 l 7.312 57.889 7.324 58.077 7.352 58.315 c 7.387 58.604 7.406 58.78 7.406 58.85 c 7.426 59.299 l 7.434 59.592 l 7.434 59.698 7.375 59.799 7.242 59.948 c 6.832 60.393 l 6.242 61.037 5.918 61.643 5.918 62.096 c 5.918 62.662 6.324 63.084 6.867 63.084 c 6.973 63.084 7.07 63.069 7.258 63.014 c 7.258 63.139 l 7.258 63.959 7.426 64.565 7.652 64.565 c 7.742 64.537 l 8.078 64.241 l 8.484 63.979 l 8.598 63.869 l 8.598 64.096 8.609 64.119 8.84 64.389 c 8.895 64.452 8.98 64.569 9.098 64.741 c 9.191 64.877 9.254 64.971 9.285 65.018 c 9.312 65.065 9.344 65.123 9.379 65.186 c 9.426 65.284 9.477 65.327 9.531 65.327 c 9.582 65.327 9.625 65.256 9.691 65.057 c 9.738 64.928 9.766 64.897 10.082 64.612 c 10.195 64.51 10.27 64.428 10.305 64.369 c 10.336 64.319 10.387 64.194 10.453 63.998 c 10.465 63.963 10.484 63.916 10.512 63.85 c 10.648 64.022 10.715 64.084 10.883 64.182 c 11.16 64.35 l 11.266 64.412 11.336 64.448 11.367 64.448 c 11.391 64.448 11.426 64.401 11.457 64.33 c 11.488 64.268 11.535 64.174 11.605 64.053 c 11.805 63.709 11.871 63.541 11.871 63.381 c 11.867 63.198 l 11.867 62.975 l 12.477 62.784 12.703 62.467 12.703 61.811 c 12.703 61.338 12.457 60.795 11.977 60.225 c 11.66 59.846 11.621 59.772 11.621 59.506 c 11.621 58.131 12.578 57.334 14.262 57.311 c 15.508 57.291 l 16.309 57.28 16.785 57.202 17.531 56.955 c 18.414 56.67 18.672 56.604 18.926 56.604 c 19.016 56.604 19.098 56.631 19.168 56.678 c 19.238 56.729 19.277 56.787 19.277 56.842 c 19.277 57.104 18.82 57.315 17.328 57.737 c 14.266 58.608 12.961 59.838 12.961 61.869 c 12.961 63.436 14.266 64.651 15.953 64.651 c 16.496 64.651 17.332 64.506 17.809 64.33 c 18.574 64.053 l 18.711 64.002 18.848 63.975 18.973 63.975 c 19.23 63.975 19.363 64.194 19.371 64.627 c 19.375 64.866 19.43 64.944 19.582 64.944 c 19.797 64.944 20.129 64.678 20.43 64.256 c 20.734 63.834 20.934 63.358 20.934 63.037 c 20.934 62.616 20.734 62.323 20.281 62.065 c 19.863 61.432 19.352 61.135 18.66 61.135 c 18.125 61.135 17.363 61.44 16.102 62.159 c 15.66 62.409 15.438 62.494 15.207 62.494 c 14.879 62.494 14.664 62.303 14.664 62.01 c 14.664 61.741 14.871 61.569 15.414 61.377 c 16.215 61.096 16.414 60.991 16.734 60.655 c 17.68 60.491 18.094 60.327 18.684 59.873 c 20.316 59.358 21.324 58.244 21.324 56.967 c 21.324 55.729 20.488 54.838 19.32 54.838 c 19.043 54.838 18.871 54.862 18.441 54.952 c f 53.105 54.952 m 53.18 54.67 l 53.246 54.42 53.297 54.221 53.328 54.077 c 53.562 53.018 53.762 52.737 54.277 52.737 c 54.57 52.737 54.816 52.901 54.891 53.151 c 55 53.522 l 55.027 53.608 55.09 53.655 55.176 53.655 c 55.379 53.655 55.488 53.432 55.504 52.983 c 55.508 52.811 55.547 52.506 55.613 52.073 c 55.664 51.76 55.688 51.471 55.688 51.213 c 55.688 51.022 55.664 50.912 55.578 50.678 c 55.613 50.455 l 55.676 50.088 55.691 49.948 55.691 49.776 c 55.691 48.979 55.652 48.741 55.465 48.432 c 55.543 48.127 55.562 47.983 55.562 47.764 c 55.562 47.623 55.543 47.424 55.504 47.112 c 55.43 46.553 l 55.445 46.221 l 55.496 46.096 55.52 46.022 55.52 45.998 c 55.52 45.944 55.488 45.905 55.445 45.905 c 55.074 46.053 l 54.664 46.276 l 54.535 46.346 54.441 46.42 54.238 46.612 c 53.883 46.291 53.828 46.26 53.402 46.147 c 52.785 45.975 52.559 45.897 52.438 45.811 c 52.316 45.725 52.227 45.682 52.18 45.682 c 52.035 45.682 51.934 45.959 51.934 46.369 c 51.934 46.963 51.969 47.272 52.066 47.502 c 51.703 47.623 51.523 47.67 51.391 47.67 c 51.34 47.67 51.312 47.666 51.117 47.631 c 51.035 47.623 l 50.98 47.623 50.949 47.69 50.949 47.803 c 50.949 47.983 51.078 48.346 51.25 48.655 c 51.48 49.069 51.863 49.412 52.16 49.471 c 52.828 49.6 l 53.227 49.678 53.461 49.889 53.461 50.166 c 53.461 50.432 53.285 50.608 52.883 50.752 c 52.102 51.03 l 51.84 51.123 51.594 51.244 51.359 51.385 c 51.164 51.502 51.027 51.584 50.949 51.627 c 50.883 51.662 50.812 51.682 50.75 51.682 c 50.586 51.682 50.484 51.561 50.484 51.366 c 50.484 51.049 50.727 50.787 51.023 50.787 c 51.09 50.787 51.152 50.803 51.211 50.827 c 51.297 50.866 51.352 50.881 51.383 50.881 c 51.469 50.881 51.527 50.799 51.527 50.686 c 51.527 50.354 51.32 49.784 51.098 49.506 c 50.879 49.229 l 50.711 49.006 l 50.648 48.53 50.18 47.842 49.707 47.537 c 49.594 47.467 49.523 47.416 49.504 47.389 c 49.488 47.373 49.438 47.284 49.352 47.131 c 49.316 47.061 49.191 46.909 48.98 46.686 c 48.824 46.514 48.668 46.373 48.52 46.256 c 48.309 46.1 48.184 45.987 48.145 45.924 c 47.961 45.608 l 47.922 45.541 47.875 45.51 47.816 45.51 c 47.648 45.51 47.496 45.686 47.402 45.979 c 47.336 46.198 47.273 46.256 47.133 46.256 c 47.062 46.256 46.953 46.237 46.809 46.202 c 46.547 46.166 l 45.898 46.166 l 45.547 46.166 45.074 46.08 44.859 45.979 c 44.773 45.94 44.707 45.92 44.664 45.92 c 44.57 45.92 44.523 46.018 44.523 46.202 c 44.523 46.791 44.973 47.686 45.562 48.264 c 45.477 48.608 45.297 48.834 45.062 48.897 c 44.852 48.952 44.738 48.983 44.727 48.987 c 44.695 49.006 44.672 49.037 44.672 49.077 c 44.672 49.198 44.805 49.311 45.098 49.452 c 45.625 49.702 45.781 49.748 46.078 49.748 c 46.41 49.748 46.699 49.635 47.051 49.358 c 47.551 48.967 47.691 48.893 47.934 48.893 c 48.332 48.893 48.711 49.256 48.711 49.635 c 48.711 49.893 48.57 50.178 48.219 50.623 c 47.727 51.256 47.551 51.666 47.551 52.202 c 47.551 52.561 47.621 52.756 47.941 53.28 c 47.992 53.362 48.02 53.432 48.02 53.475 c 48.02 53.58 47.852 53.67 47.664 53.67 c 47.465 53.67 47.219 53.6 46.883 53.448 c 46.324 53.19 45.992 52.952 45.992 52.803 c 45.992 52.748 46.027 52.69 46.082 52.647 c 46.27 52.518 l 46.293 52.498 46.309 52.471 46.309 52.44 c 46.309 52.409 46.273 52.342 46.23 52.295 c 46.195 52.252 46.07 52.092 45.859 51.811 c 45.602 51.467 45.352 51.205 45.082 50.994 c 44.984 50.916 44.918 50.854 44.895 50.807 c 44.746 50.53 l 44.621 50.295 44.258 49.912 43.949 49.694 c 43.809 49.592 43.719 49.526 43.688 49.491 c 43.648 49.448 43.594 49.362 43.52 49.229 c 43.332 48.889 43.031 48.467 42.832 48.264 c 42.332 47.744 l 42.188 47.596 41.988 47.162 41.793 46.592 c 41.641 46.143 41.543 45.94 41.473 45.94 c 41.445 45.94 41.391 45.983 41.328 46.053 c 40.969 46.467 40.645 46.721 40.484 46.721 c 40.445 46.721 40.371 46.69 40.27 46.627 c 40.129 46.541 39.957 46.479 39.77 46.444 c 39.098 46.315 l 38.539 46.205 38.316 46.084 38.059 45.756 c 37.992 45.674 37.941 45.627 37.91 45.627 c 37.844 45.627 37.797 45.721 37.797 45.854 c 37.816 46.014 l 37.82 46.034 37.852 46.202 37.91 46.518 c 38.008 47.045 38.141 47.483 38.281 47.725 c 38.617 48.299 l 38.676 48.405 38.711 48.51 38.711 48.6 c 38.711 49.03 38.309 49.377 37.707 49.471 c 37.582 49.491 37.52 49.537 37.52 49.608 c 37.52 49.694 37.664 49.858 37.891 50.03 c 38.285 50.327 38.723 50.51 39.031 50.51 c 39.234 50.51 39.512 50.44 39.879 50.287 c 40.289 50.119 l 40.441 50.061 40.582 50.026 40.703 50.026 c 41.137 50.026 41.555 50.42 42.07 51.311 c 42.305 51.713 42.406 51.998 42.406 52.268 c 42.406 52.737 42.188 53.018 41.824 53.018 c 41.535 53.018 41.258 52.897 41.031 52.666 c 40.828 52.459 40.723 52.256 40.723 52.077 c 40.734 51.885 l 40.738 51.787 l 40.738 51.635 40.691 51.569 40.578 51.569 c 40.438 51.569 40.277 51.705 39.953 52.088 c 39.82 52.248 39.648 52.452 39.434 52.702 c 38.867 53.362 38.75 53.627 38.727 54.299 c 38.098 55.104 37.941 55.549 37.941 56.018 c 37.965 56.401 l 37.516 56.944 37.426 57.151 37.426 57.686 c 37.426 57.928 37.441 57.991 37.539 58.182 c 37.316 58.506 37.246 58.655 37.188 58.909 c 37.094 59.299 l 37.031 59.565 36.93 59.791 36.797 59.987 c 36.629 60.225 36.609 60.26 36.609 60.33 c 36.609 60.526 36.793 60.651 37.262 60.764 c 37.41 60.803 l 37.383 60.983 37.371 61.119 37.371 61.213 c 37.371 61.323 37.383 61.428 37.426 61.659 c 37.559 62.346 l 37.609 62.612 37.633 62.819 37.633 62.955 c 37.633 63.116 37.609 63.241 37.52 63.534 c 37.465 63.721 37.398 63.866 37.336 63.959 c 37.262 64.065 37.223 64.151 37.223 64.205 c 37.223 64.268 37.285 64.319 37.371 64.319 c 37.414 64.319 37.492 64.291 37.559 64.256 c 38.184 63.916 38.867 63.373 39.359 62.827 c 39.781 62.362 39.914 61.971 40.008 60.932 c 40.578 60.506 40.793 60.123 40.793 59.522 c 40.793 59.162 40.676 58.819 40.418 58.424 c 40.191 58.077 40.066 57.885 40.047 57.85 c 39.938 57.662 39.879 57.444 39.879 57.221 c 39.879 56.881 39.98 56.569 40.23 56.119 c 40.512 55.627 40.758 55.416 41.055 55.416 c 41.539 55.416 41.984 56.256 41.984 57.182 c 41.98 57.366 l 41.98 57.866 l 42.035 58.315 l 42.07 58.85 l 42.109 59.299 l 42.121 59.545 l 42.121 59.729 42.086 59.784 41.516 60.393 c 40.883 61.069 40.586 61.639 40.586 62.166 c 40.586 62.666 41.016 63.084 41.531 63.084 c 41.637 63.084 41.734 63.069 41.922 63.014 c 41.922 63.237 l 41.941 63.627 l 42.051 64.108 l 42.141 64.494 42.184 64.565 42.316 64.565 c 42.406 64.537 l 42.738 64.241 l 43.148 63.979 l 43.262 63.869 l 43.285 64.1 43.309 64.147 43.5 64.389 c 43.598 64.506 43.684 64.627 43.762 64.741 c 43.949 65.018 l 44.059 65.186 l 44.133 65.299 44.164 65.327 44.215 65.327 c 44.266 65.327 44.297 65.26 44.355 65.057 c 44.379 64.971 44.438 64.885 44.523 64.815 c 44.766 64.612 l 44.961 64.444 45.094 64.202 45.172 63.85 c 45.32 64.034 45.391 64.096 45.543 64.182 c 45.844 64.35 l 45.953 64.412 46.02 64.444 46.039 64.444 c 46.059 64.444 46.09 64.401 46.121 64.33 c 46.164 64.237 46.215 64.143 46.27 64.053 c 46.512 63.67 46.531 63.616 46.531 63.354 c 46.531 62.975 l 47.133 62.811 47.367 62.487 47.367 61.819 c 47.367 61.319 47.137 60.803 46.66 60.225 c 46.344 59.846 46.289 59.725 46.289 59.44 c 46.289 58.127 47.277 57.334 48.945 57.311 c 50.172 57.291 l 50.973 57.28 51.461 57.198 52.195 56.955 c 53.055 56.674 53.336 56.604 53.598 56.604 c 53.785 56.604 53.941 56.709 53.941 56.834 c 53.941 57.104 53.492 57.315 51.992 57.737 c 48.934 58.604 47.625 59.838 47.625 61.869 c 47.625 63.436 48.934 64.651 50.633 64.651 c 51.164 64.651 52 64.506 52.473 64.33 c 53.234 64.053 l 53.375 64.002 53.512 63.975 53.637 63.975 c 53.898 63.975 53.996 64.139 54.035 64.627 c 54.051 64.85 54.121 64.944 54.262 64.944 c 54.457 64.944 54.836 64.639 55.113 64.256 c 55.395 63.866 55.594 63.35 55.594 63.01 c 55.594 62.616 55.387 62.315 54.945 62.065 c 54.527 61.432 54.016 61.135 53.324 61.135 c 52.797 61.135 51.898 61.494 50.766 62.159 c 50.32 62.42 50.125 62.494 49.879 62.494 c 49.543 62.494 49.328 62.303 49.328 62.01 c 49.328 61.741 49.527 61.577 50.078 61.377 c 50.957 61.065 51.145 60.963 51.398 60.655 c 52.344 60.491 52.754 60.327 53.348 59.873 c 55.043 59.33 55.988 58.284 55.988 56.944 c 55.988 55.721 55.156 54.838 54.008 54.838 c 53.703 54.838 53.543 54.858 53.105 54.952 c f 18.797 19.877 m 18.957 19.467 19.012 19.268 19.035 19.006 c 19.074 18.225 l 19.102 17.444 19.363 17.057 19.852 17.057 c 20.031 17.057 20.18 17.1 20.262 17.186 c 20.469 17.389 l 20.547 17.412 l 20.68 17.412 20.75 17.327 20.75 17.17 c 20.727 16.869 l 20.711 16.733 20.738 16.448 20.801 16.014 c 20.84 15.764 20.859 15.51 20.859 15.264 c 20.859 14.869 20.828 14.717 20.691 14.362 c 20.82 13.994 20.895 13.584 20.895 13.241 c 20.895 12.834 20.828 12.561 20.672 12.338 c 20.746 12.018 20.766 11.885 20.766 11.705 c 20.766 11.522 20.738 11.284 20.691 10.998 c 20.625 10.612 20.598 10.311 20.617 10.108 c 20.617 10.061 l 20.617 9.944 20.582 9.885 20.512 9.885 c 20.441 9.885 20.355 9.94 20.281 10.034 c 20.168 10.178 20.094 10.233 20.008 10.233 c 19.879 10.233 19.84 10.162 19.648 9.608 c 19.59 9.436 19.441 9.202 19.203 8.92 c 19.02 8.698 18.906 8.514 18.871 8.381 c 18.809 8.159 18.773 8.026 18.758 7.991 c 18.738 7.94 18.695 7.905 18.648 7.897 c 18.621 7.897 l 18.316 7.897 17.965 8.549 17.605 9.791 c 17.504 10.139 17.375 10.299 17.188 10.299 c 17.051 10.299 16.914 10.209 16.863 10.088 c 16.75 9.83 l 16.73 9.784 16.695 9.756 16.656 9.756 c 16.508 9.756 16.281 10.022 16.141 10.369 c 16.023 10.651 15.957 10.959 15.957 11.221 c 15.957 11.803 16.387 12.237 17.68 12.967 c 18.242 13.284 18.461 13.69 18.461 14.416 c 18.461 14.994 18.289 15.323 17.793 15.698 c 17.402 15.994 17.172 16.108 16.965 16.108 c 16.789 16.108 16.652 15.991 16.652 15.842 c 16.652 15.67 16.75 15.561 16.992 15.475 c 17.281 15.373 17.406 15.276 17.406 15.147 c 17.406 15.03 17.285 14.776 17.086 14.471 c 16.875 14.151 16.785 14.057 16.492 13.862 c 16.406 13.35 16.164 13.022 15.602 12.651 c 15.488 12.334 15.426 12.217 15.23 11.967 c 14.84 11.463 l 14.715 11.307 14.578 11.022 14.559 10.889 c 14.52 10.604 14.492 10.553 14.383 10.553 c 14.238 10.553 13.934 10.815 13.633 11.205 c 13.453 11.436 13.41 11.541 13.391 11.815 c 13.195 11.666 13.137 11.643 12.758 11.592 c 12.531 11.565 12.363 11.518 12.258 11.463 c 11.973 11.315 11.879 11.272 11.824 11.272 c 11.75 11.272 11.723 11.315 11.719 11.444 c 11.699 11.854 l 11.699 12.416 11.918 13.069 12.258 13.506 c 11.93 13.787 11.605 13.959 11.348 13.991 c 11.234 14.002 11.207 14.014 11.207 14.057 c 11.207 14.127 11.281 14.268 11.383 14.397 c 11.523 14.573 11.633 14.698 11.719 14.772 c 11.98 14.994 12.438 15.159 12.793 15.159 c 13.035 15.159 13.391 15.073 13.855 14.901 c 14.02 14.838 14.168 14.807 14.297 14.807 c 14.504 14.807 14.617 14.94 14.617 15.174 c 14.617 15.51 14.449 16.041 14.078 16.889 c 13.824 17.463 13.707 17.846 13.707 18.116 c 13.707 18.76 14.012 19.428 14.395 19.619 c 14.215 19.709 14.133 19.729 13.977 19.729 c 13.793 19.729 13.645 19.682 13.406 19.545 c 13.133 19.385 12.98 19.244 12.98 19.147 c 12.98 19.112 13.035 19.053 13.129 18.987 c 13.273 18.885 13.297 18.846 13.297 18.713 c 13.297 18.252 12.918 17.639 12.387 17.241 c 12.27 16.592 11.969 16.198 11.383 15.92 c 11.305 15.569 11.258 15.467 10.977 15.069 c 10.488 14.385 10.375 14.17 10.289 13.784 c 10.242 13.588 10.195 13.502 10.137 13.502 c 10.055 13.502 9.891 13.627 9.711 13.823 c 9.555 13.991 9.457 14.112 9.414 14.174 c 9.336 14.299 9.293 14.416 9.211 14.772 c 9.078 14.733 9.039 14.725 8.945 14.725 c 8.672 14.733 l 8.488 14.737 l 8.078 14.737 7.648 14.647 7.426 14.51 c 7.348 14.459 7.293 14.436 7.266 14.436 c 7.145 14.436 7.047 14.69 7.047 14.998 c 7.047 15.67 7.234 16.225 7.668 16.834 c 7.508 16.967 7.457 17.014 7.316 17.166 c 7.254 17.233 7.203 17.276 7.168 17.295 c 6.961 17.409 l 6.895 17.448 6.852 17.491 6.852 17.53 c 6.852 17.627 7.121 17.885 7.445 18.096 c 7.801 18.327 8.086 18.432 8.344 18.432 c 8.648 18.432 8.812 18.354 9.191 18.022 c 9.453 17.791 9.637 17.686 9.762 17.686 c 9.852 17.686 9.949 17.748 10.027 17.854 c 10.09 17.94 10.121 18.037 10.121 18.131 c 10.121 18.487 9.699 19.084 9.23 19.397 c 8.879 19.627 8.496 19.748 8.121 19.748 c 7.605 19.748 7.16 19.491 6.961 19.08 c 6.758 18.655 l 6.711 18.553 6.648 18.502 6.582 18.502 c 6.461 18.502 6.398 18.557 5.977 19.041 c 5.496 19.6 l 5.348 19.772 5.246 19.936 5.199 20.084 c 5.152 20.221 5.121 20.299 5.105 20.323 c 5.086 20.346 5.023 20.397 4.918 20.471 c 4.547 20.737 4.277 21.26 4.121 22.014 c 4.109 22.069 4.09 22.151 4.062 22.256 c 3.609 22.815 3.336 23.502 3.336 24.104 c 3.336 24.182 3.344 24.217 3.375 24.354 c 3.113 24.623 3.082 24.686 2.875 25.397 c 2.727 25.905 2.633 26.119 2.449 26.362 c 2.398 26.424 2.375 26.479 2.375 26.506 c 2.375 26.608 2.531 26.69 2.801 26.733 c 3.055 26.776 3.07 26.776 3.176 26.776 c 3.543 26.752 l 3.523 27.26 l 3.523 27.561 3.543 27.756 3.617 28.108 c 3.656 28.284 3.676 28.416 3.676 28.494 c 3.676 28.53 3.664 28.635 3.637 28.815 c 3.617 28.952 3.598 29.045 3.582 29.092 c 3.555 29.162 3.543 29.217 3.543 29.252 c 3.543 29.311 3.582 29.342 3.656 29.342 c 3.707 29.342 3.781 29.315 3.898 29.26 c 4.223 29.1 4.488 28.94 4.695 28.776 c 5.336 28.276 5.84 27.506 5.84 27.026 c 5.828 26.881 l 5.805 26.717 5.793 26.623 5.793 26.608 c 5.793 26.526 5.836 26.42 5.922 26.307 c 6.125 26.026 l 6.309 25.78 6.426 25.491 6.426 25.299 c 6.426 25.123 6.387 25.049 6.09 24.616 c 5.863 24.291 5.773 24.049 5.773 23.784 c 5.773 23.452 5.906 23.03 6.145 22.612 c 6.363 22.225 6.617 22.01 6.859 22.01 c 7.273 22.01 7.453 22.381 7.52 23.369 c 7.555 23.889 7.602 24.354 7.668 24.784 c 7.688 25.037 l 7.688 25.151 7.652 25.221 7.5 25.397 c 6.945 26.045 l 6.41 26.666 6.25 26.987 6.25 27.432 c 6.25 27.944 6.59 28.295 7.074 28.295 c 7.297 28.276 l 7.293 28.42 l 7.293 28.537 7.324 28.709 7.391 28.924 c 7.504 29.327 7.652 29.631 7.73 29.631 c 7.754 29.631 7.816 29.6 7.91 29.537 c 7.938 29.518 8.051 29.475 8.242 29.409 c 8.348 29.373 8.422 29.334 8.617 29.221 c 8.703 29.545 8.77 29.69 8.914 29.873 c 9.152 30.182 9.426 30.432 9.52 30.432 c 9.578 30.432 9.688 30.35 9.824 30.205 c 10.176 29.834 l 10.234 29.772 10.305 29.655 10.379 29.483 c 10.453 29.315 l 10.555 29.373 10.629 29.416 10.676 29.444 c 10.824 29.534 10.922 29.588 10.977 29.612 c 11.203 29.717 11.32 29.776 11.328 29.78 c 11.422 29.842 11.492 29.873 11.52 29.873 c 11.582 29.873 11.617 29.764 11.754 29.147 c 11.848 28.729 11.863 28.639 11.887 28.33 c 12.477 28.248 12.758 27.909 12.758 27.272 c 12.758 26.67 12.465 26.077 11.828 25.397 c 11.656 25.209 11.613 25.123 11.613 24.952 c 11.613 23.307 12.188 22.791 14.246 22.573 c 15.508 22.44 15.883 22.358 17.402 21.905 c 18.203 21.662 18.66 21.565 18.957 21.565 c 19.305 21.565 19.504 21.69 19.504 21.92 c 19.504 22.26 19.254 22.373 17.773 22.682 c 14.332 23.409 12.98 24.584 12.98 26.854 c 12.98 28.471 14.191 29.686 15.809 29.686 c 16.52 29.686 17.156 29.471 18.164 28.889 c 19.098 28.346 19.289 28.256 19.508 28.256 c 19.758 28.256 19.855 28.416 19.855 28.819 c 19.855 29.08 19.934 29.205 20.098 29.205 c 20.242 29.205 20.469 29.053 20.672 28.815 c 20.98 28.452 21.098 28.264 21.098 27.76 c 21.098 27.088 20.742 26.647 20.039 26.455 c 19.965 26.373 19.91 26.311 19.871 26.268 c 19.492 25.838 19.156 25.694 18.547 25.694 c 18.02 25.694 17.355 25.909 16.902 26.233 c 16.008 26.862 l 15.625 27.135 15.344 27.252 15.074 27.252 c 14.695 27.252 14.465 27.026 14.465 26.643 c 14.465 26.272 14.711 25.967 15.117 25.842 c 15.953 25.58 l 16.25 25.491 16.391 25.432 16.75 25.248 c 16.957 25.28 17.055 25.287 17.168 25.287 c 17.688 25.287 18.246 25.147 18.738 24.893 c 18.895 24.815 19.008 24.764 19.074 24.744 c 19.539 24.616 l 20.594 24.319 21.453 23.139 21.453 21.991 c 21.453 20.674 20.602 19.858 19.227 19.858 c 18.797 19.877 l f 51.117 20.159 m 51.441 19.729 51.898 19.448 52.27 19.448 c 52.387 19.448 52.586 19.479 52.773 19.526 c 52.871 19.549 52.953 19.565 53.02 19.565 c 53.109 19.565 53.145 19.53 53.145 19.444 c 53.145 19.381 53.102 19.221 53.051 19.096 c 52.992 18.959 52.953 18.846 52.938 18.764 c 52.863 18.373 l 52.68 18.041 l 52.418 17.799 l 52.344 17.725 l 52.344 17.471 l 52.344 17.018 52.211 16.76 51.844 16.518 c 51.812 16.498 51.77 16.467 51.711 16.424 c 51.676 16.276 l 51.594 15.932 51.531 15.713 51.488 15.623 c 51.441 15.526 51.324 15.35 51.137 15.104 c 50.914 14.811 50.766 14.569 50.672 14.342 c 50.598 14.159 50.539 14.104 50.418 14.104 c 50.234 14.104 50.055 14.33 50.023 14.604 c 49.98 14.545 49.949 14.51 49.93 14.491 c 49.707 14.342 l 49.352 14.1 l 49.199 13.994 49.098 13.92 49.055 13.877 c 48.852 13.674 l 48.816 13.639 48.777 13.619 48.746 13.619 c 48.652 13.619 48.555 13.776 48.461 14.084 c 48.277 14.698 l 48.227 14.862 48.203 15.061 48.203 15.295 c 48.203 15.604 48.223 15.713 48.332 15.959 c 47.961 15.92 47.855 15.901 47.664 15.83 c 47.605 15.807 47.535 15.791 47.504 15.791 c 47.438 15.791 47.402 15.834 47.402 15.905 c 47.402 15.948 47.438 16.077 47.477 16.182 c 47.719 16.815 l 47.906 17.303 48.512 17.748 48.992 17.748 c 49.105 17.748 49.266 17.717 49.465 17.651 c 49.586 17.612 49.684 17.588 49.746 17.588 c 49.961 17.588 50.133 17.748 50.133 17.948 c 50.133 18.041 50.066 18.151 49.984 18.19 c 49.953 18.202 49.73 18.272 49.316 18.393 c 49.004 18.483 48.852 18.6 48.5 19.006 c 48.242 19.299 48.145 19.381 48.051 19.381 c 47.98 19.381 47.941 19.338 47.941 19.252 c 47.961 19.006 l 47.969 18.893 l 47.969 18.377 47.812 17.881 47.441 17.205 c 46.918 16.256 l 46.672 15.807 46.566 15.467 46.566 15.123 c 46.566 14.748 46.727 14.416 46.938 14.362 c 47.219 14.287 l 47.254 14.28 47.285 14.26 47.309 14.233 c 47.348 14.194 47.367 14.155 47.367 14.123 c 47.367 14.084 47.332 14.022 47.273 13.952 c 47.238 13.916 47.129 13.772 46.938 13.526 c 46.617 13.108 46.418 12.955 46.027 12.838 c 45.805 12.291 45.438 11.975 44.785 11.76 c 44.711 11.631 l 44.488 11.241 43.996 10.885 43.684 10.885 c 43.613 10.889 l 43.57 10.545 43.328 10.194 42.965 9.94 c 42.711 9.764 42.566 9.659 42.535 9.623 c 42.516 9.6 42.48 9.545 42.445 9.455 c 42.359 9.272 42.281 9.198 42.18 9.198 c 41.98 9.198 41.816 9.381 41.699 9.737 c 41.625 9.686 41.57 9.651 41.531 9.623 c 41.324 9.475 41.203 9.397 41.18 9.381 c 41.094 9.342 40.879 9.28 40.531 9.198 c 40.074 9.088 39.68 8.897 39.582 8.733 c 39.414 8.455 l 39.34 8.424 l 39.246 8.424 39.145 8.549 39.098 8.713 c 38.996 9.1 38.93 9.502 38.93 9.756 c 38.93 10.416 39.148 10.846 39.824 11.518 c 39.824 11.682 l 39.824 12.705 40.605 13.401 41.758 13.401 c 41.941 13.397 l 42.574 13.358 l 42.633 13.358 l 42.973 13.358 43.289 13.487 43.633 13.768 c 44.086 14.139 44.219 14.369 44.219 14.784 c 44.219 14.912 44.18 15.065 44.039 15.534 c 43.949 15.838 43.902 16.127 43.902 16.405 c 43.902 17.17 44.465 18.123 45.23 18.655 c 45.426 18.787 45.574 18.967 45.574 19.073 c 45.574 19.116 45.551 19.151 45.523 19.151 c 45.473 19.151 45.391 19.1 45.285 19.006 c 45.195 18.924 45.047 18.83 44.84 18.729 c 44.543 18.577 l 44.207 18.448 l 44.105 18.409 44.055 18.354 44.055 18.28 c 44.055 18.241 44.07 18.182 44.098 18.116 c 44.121 18.053 44.133 18.006 44.133 17.979 c 44.133 17.936 44.102 17.862 44.039 17.76 c 43.781 17.334 l 43.527 16.916 43.344 16.737 43.02 16.592 c 42.809 15.979 42.512 15.67 42.07 15.608 c 41.723 14.901 41.414 14.655 40.789 14.584 c 40.703 14.315 40.555 14.084 40.23 13.729 c 40.102 13.584 40.02 13.487 39.992 13.432 c 39.848 13.17 39.836 13.155 39.766 13.155 c 39.68 13.155 39.488 13.28 39.34 13.432 c 39.273 13.502 39.219 13.577 39.172 13.655 c 39.008 13.952 l 38.953 14.049 38.898 14.092 38.828 14.092 c 38.711 14.084 l 38.301 14.026 l 38.043 13.991 37.68 13.784 37.559 13.6 c 37.336 13.264 l 37.312 13.229 37.277 13.209 37.246 13.209 c 37.078 13.209 36.977 13.577 36.977 14.186 c 36.977 14.717 37.059 15.073 37.242 15.327 c 37.762 16.053 l 37.836 16.155 37.875 16.252 37.875 16.334 c 37.875 16.416 37.84 16.498 37.781 16.573 c 37.672 16.705 37.566 16.784 37.484 16.795 c 37.094 16.85 l 37.023 16.862 36.996 16.877 36.996 16.909 c 36.996 16.94 37.016 16.987 37.055 17.037 c 37.184 17.198 37.344 17.334 37.52 17.444 c 37.98 17.725 38.281 17.838 38.594 17.838 c 38.906 17.838 39.25 17.725 39.562 17.518 c 40.102 17.166 l 40.18 17.116 40.27 17.092 40.359 17.092 c 40.609 17.092 40.797 17.241 40.977 17.577 c 41.145 17.901 41.219 18.162 41.219 18.436 c 41.219 19.135 40.723 19.635 40.027 19.635 c 39.707 19.635 39.379 19.51 39.062 19.264 c 38.969 19.19 38.902 19.119 38.875 19.061 c 38.727 18.729 l 38.691 18.647 38.641 18.596 38.594 18.596 c 38.5 18.596 38.289 18.866 38.113 19.209 c 38.023 19.393 37.934 19.616 37.855 19.877 c 37.742 20.244 37.695 20.518 37.695 20.807 c 37.727 21.307 l 37.473 21.702 37.387 21.991 37.387 22.459 c 37.387 22.627 37.414 22.76 37.52 23.112 c 37.363 23.369 37.316 23.561 37.316 23.94 c 37.336 24.17 l 37.391 24.432 l 37.426 24.655 l 37.246 24.889 37.195 25.041 37.02 25.862 c 36.918 26.323 36.855 26.522 36.777 26.639 c 36.742 26.694 36.723 26.733 36.723 26.752 c 36.723 26.862 36.98 26.955 37.277 26.955 c 37.539 26.936 l 37.473 27.112 37.461 27.178 37.461 27.315 c 37.461 27.541 37.508 27.795 37.594 28.053 c 37.691 28.334 37.711 28.416 37.711 28.506 c 37.711 28.725 37.68 28.952 37.633 29.073 c 37.484 29.444 l 37.449 29.53 37.438 29.584 37.438 29.635 c 37.438 29.674 37.461 29.686 37.527 29.686 c 37.895 29.686 38.59 29.295 39.137 28.776 c 39.68 28.264 39.855 27.85 39.855 27.084 c 39.824 26.678 l 40.332 26.237 40.551 25.869 40.551 25.467 c 40.551 25.178 40.457 24.987 40.176 24.709 c 39.676 24.209 39.562 23.952 39.562 23.311 c 39.562 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57.272 11.02 57.248 11.066 57.198 c 11.254 57.014 l 11.375 56.889 11.5 56.827 11.621 56.827 c 11.777 56.827 11.871 56.889 11.871 56.991 c 11.871 57.08 11.836 57.17 11.699 57.42 c 11.57 57.662 l 11.418 57.94 11.301 58.053 11.16 58.053 c 10.977 58.034 l 10.754 57.94 l 10.684 57.912 10.637 57.901 10.605 57.901 c 10.574 57.901 10.547 57.932 10.547 57.975 c 10.547 58.139 10.656 58.401 10.863 58.733 c 11.012 58.967 11.113 59.467 11.16 60.19 c 11.223 61.159 11.352 61.787 11.488 61.787 c 11.582 61.787 11.617 61.709 11.617 61.502 c 11.605 61.248 l 11.598 61.069 l 11.598 60.897 11.637 60.756 11.684 60.756 c 11.844 60.756 12.031 60.994 12.184 61.377 c 12.246 61.537 12.277 61.666 12.277 61.756 c 12.258 61.842 l 11.887 62.53 l 11.512 62.643 l 11.445 62.662 11.398 62.694 11.383 62.725 c 11.383 63.182 l 11.348 63.405 l 11.273 63.588 l 11.18 63.795 l 10.773 63.573 10.578 63.428 10.438 63.237 c 10.289 63.041 10.207 62.967 10.137 62.967 c 10.09 62.967 10.047 63.014 10.047 63.065 c 10.047 63.112 10.066 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58.35 3.898 58.256 c 3.988 58.209 4.051 58.139 4.051 58.084 c 4.051 58.002 3.98 57.94 3.891 57.94 c 3.879 57.94 l 3.637 57.979 l 3.5 57.994 l 3.43 57.994 3.395 57.928 3.395 57.787 c 3.395 57.284 3.793 56.705 4.492 56.198 c 4.574 56.135 4.641 56.045 4.641 55.998 c 4.641 55.936 4.578 55.877 4.504 55.877 c 4.461 55.877 4.352 55.912 4.27 55.955 c 4.215 55.979 4.172 55.994 4.141 55.994 c 4.047 55.994 3.988 55.916 3.988 55.803 c 3.988 55.67 4.094 55.409 4.324 54.971 c 4.477 54.682 4.699 54.463 4.957 54.358 c 5.793 54.002 l 5.836 53.983 5.867 53.94 5.867 53.893 c 5.867 53.827 5.816 53.768 5.754 53.76 c 5.586 53.744 l 5.438 53.725 l 5.199 53.744 l 5.16 53.744 l 5.047 53.744 5 53.721 5 53.662 c 5 53.569 5.102 53.338 5.27 53.057 c 5.469 52.725 5.645 52.534 5.75 52.534 c 5.793 52.534 5.84 52.573 5.867 52.627 c 6.07 53.057 l 6.105 53.131 6.273 53.26 6.555 53.428 c 6.73 53.534 6.906 53.604 7.074 53.631 c 7.406 53.686 l 7.496 53.702 7.543 53.764 7.543 53.866 c 7.539 53.909 l 7.48 54.264 l 7.445 54.655 l 7.406 55.006 l 7.371 55.416 l 7.367 55.522 7.336 55.569 7.277 55.569 c 7.234 55.569 7.16 55.506 7.109 55.436 c 6.82 54.987 6.617 54.838 6.305 54.838 c 5.316 54.838 4.566 56.58 4.566 58.885 c 4.566 59.088 4.578 59.342 4.602 59.651 c 4.652 60.241 4.66 60.389 4.66 60.608 c 4.66 61.608 4.449 62.311 4.027 62.733 c 3.766 62.994 l 3.602 63.182 l f 5.996 49.6 m 5.738 49.639 5.613 49.67 5.457 49.729 c 5.414 49.748 5.297 49.799 5.105 49.881 c 4.805 50.006 4.582 50.065 4.395 50.065 c 4.113 50.065 3.883 49.971 3.617 49.748 c 3.785 49.619 l 4.156 49.358 l 4.512 49.08 l 4.602 49.01 4.711 48.971 4.816 48.971 c 5.18 48.971 5.496 49.139 5.996 49.6 c f 13.148 46.221 m 13.25 46.452 13.301 46.514 13.52 46.702 c 14.039 47.147 14.281 47.479 14.281 47.744 c 14.281 48.151 l 14.281 48.389 14.32 48.467 14.445 48.467 c 14.5 48.467 14.547 48.448 14.559 48.412 c 14.652 48.19 l 14.676 48.139 14.734 48.104 14.809 48.104 c 14.879 48.104 14.988 48.19 15.082 48.319 c 15.266 48.58 l 15.359 48.709 15.402 48.819 15.402 48.94 c 15.402 49.014 15.387 49.088 15.359 49.155 c 15.344 49.194 15.301 49.291 15.23 49.452 c 15.188 49.549 15.152 49.674 15.152 49.744 c 15.152 49.866 15.215 49.936 15.32 49.936 c 15.375 49.936 15.422 49.916 15.453 49.881 c 15.602 49.694 l 15.633 49.655 15.684 49.631 15.742 49.631 c 15.809 49.631 15.887 49.694 15.934 49.787 c 16.082 50.084 l 16.121 50.159 16.141 50.217 16.141 50.252 c 16.141 50.307 16.082 50.369 15.973 50.416 c 15.73 50.53 l 15.41 50.678 15.172 51.19 15.172 51.721 c 15.172 51.791 15.184 51.862 15.211 51.924 c 15.34 52.241 l 15.379 52.33 15.398 52.428 15.398 52.537 c 15.398 52.729 15.332 52.924 15.211 53.112 c 15.062 53.342 14.98 53.471 14.969 53.502 c 14.859 53.795 14.801 53.877 14.711 53.877 c 14.617 53.854 l 14.484 53.78 l 14.281 53.67 l 14.039 53.373 l 13.781 53.241 l 13.625 53.162 13.477 52.834 13.477 52.557 c 13.477 52.077 13.531 51.944 14.168 50.846 c 14.418 50.42 14.547 50.053 14.547 49.784 c 14.547 49.022 13.867 48.479 12.496 48.151 c 11.645 47.948 l 11.016 47.795 10.645 47.366 10.512 46.627 c 10.77 46.764 10.871 46.784 11.332 46.784 c 11.605 46.776 l 11.648 46.776 l 11.898 46.776 11.93 46.795 12.09 47.018 c 12.273 47.276 12.453 47.428 12.703 47.537 c 13 47.67 l 13.242 47.799 l 13.316 47.819 l 13.383 47.819 13.418 47.791 13.418 47.744 c 13.418 47.67 13.348 47.549 13.242 47.428 c 13.027 47.19 12.945 46.987 12.945 46.709 c 12.945 46.53 12.988 46.424 13.148 46.221 c f 10.809 49.1 m 11.023 48.967 11.238 48.772 11.363 48.596 c 11.543 48.354 11.566 48.338 11.742 48.338 c 11.82 48.338 11.91 48.35 12.016 48.373 c 12.184 48.412 l 12.488 48.471 12.594 48.518 12.594 48.592 c 12.594 48.635 12.559 48.694 12.496 48.748 c 12.293 48.932 l 12.09 49.119 l 11.957 49.237 11.699 49.327 11.477 49.327 c 11.254 49.327 11.125 49.28 10.809 49.1 c f 16.828 48.096 m 16.938 48.116 l 17.105 48.135 l 17.273 48.116 l 17.355 48.112 l 17.77 48.112 18.164 48.459 18.164 48.827 c 18.164 48.955 18.047 49.026 17.836 49.026 c 17.336 49.026 17.023 48.737 16.828 48.096 c f 54.945 46.815 m 54.902 46.983 54.887 47.065 54.887 47.155 c 54.906 47.409 l 54.953 47.756 54.969 47.924 54.969 48.131 c 54.969 48.268 54.93 48.377 54.836 48.506 c 54.609 48.803 l 54.465 48.994 54.406 49.143 54.406 49.307 c 54.406 49.416 54.434 49.452 54.508 49.452 c 54.594 49.432 l 54.602 49.428 54.656 49.397 54.758 49.342 c 54.836 49.299 54.898 49.276 54.945 49.276 c 55.055 49.276 55.113 49.452 55.113 49.799 c 55.113 50.284 55.02 50.514 54.742 50.733 c 54.48 50.94 l 54.434 50.975 54.406 51.034 54.406 51.092 c 54.406 51.178 54.457 51.237 54.543 51.237 c 54.594 51.237 54.668 51.221 54.758 51.198 c 54.965 51.143 l 55.043 51.123 l 55.105 51.123 55.137 51.174 55.137 51.28 c 55.129 51.346 l 55.074 51.737 l 55.016 52.143 54.949 52.299 54.828 52.299 c 54.797 52.295 l 54.312 52.202 l 54.23 52.194 l 53.652 52.194 52.969 52.744 52.789 53.354 c 52.438 54.561 l 52.332 54.92 52.02 55.319 51.547 55.694 c 51.391 55.815 51.254 55.885 51.156 55.897 c 50.82 55.936 l 50.762 55.944 50.719 55.979 50.719 56.026 c 50.719 56.104 50.785 56.139 50.922 56.139 c 51.395 56.139 51.895 56.045 52.566 55.823 c 53.18 55.623 53.512 55.549 53.809 55.549 c 54.738 55.549 55.297 56.034 55.297 56.842 c 55.297 57.241 55.152 57.748 54.945 58.073 c 54.766 58.35 54.461 58.639 54.07 58.889 c 54.016 58.928 53.965 58.948 53.926 58.948 c 53.871 58.948 53.828 58.905 53.828 58.854 c 53.848 58.612 l 53.852 58.588 l 53.852 58.498 53.797 58.424 53.73 58.424 c 53.629 58.424 53.52 58.545 53.402 58.795 c 53.062 59.522 52.512 59.959 51.941 59.959 c 51.816 59.959 51.746 59.916 51.746 59.842 c 51.746 59.784 51.781 59.698 51.844 59.596 c 51.902 59.502 51.957 59.389 52.012 59.26 c 52.039 59.19 52.055 59.135 52.055 59.1 c 52.055 59.065 52.023 59.049 51.969 59.049 c 51.887 59.049 51.75 59.123 51.656 59.221 c 51.473 59.42 51.379 59.522 51.379 59.522 c 51.137 59.893 l 50.949 60.182 50.742 60.362 50.395 60.541 c 49.93 60.784 l 49.836 60.834 49.754 60.858 49.691 60.858 c 49.633 60.858 49.602 60.83 49.602 60.776 c 49.602 60.717 49.633 60.643 49.688 60.561 c 49.785 60.416 49.859 60.26 49.859 60.19 c 49.859 60.127 49.82 60.077 49.766 60.077 c 49.637 60.077 49.402 60.311 49.074 60.764 c 48.84 61.088 48.719 61.444 48.719 61.807 c 48.719 62.592 49.324 63.198 50.102 63.198 c 50.496 63.198 50.996 63.034 51.488 62.733 c 52.383 62.198 l 52.699 62.002 53.133 61.862 53.402 61.862 c 53.547 61.862 53.594 61.869 53.922 61.955 c 54.02 61.979 54.102 62.01 54.164 62.045 c 54.207 62.073 54.234 62.116 54.234 62.155 c 54.234 62.229 54.188 62.26 54.07 62.268 c 53.656 62.299 52.797 62.639 52.234 62.994 c 51.902 63.202 51.699 63.319 51.621 63.346 c 51.469 63.401 51.363 63.483 51.363 63.545 c 51.363 63.573 51.395 63.588 51.445 63.588 c 51.555 63.569 l 52.363 63.272 l 53.145 62.994 l 53.527 62.858 54.117 62.733 54.391 62.733 c 54.742 62.733 54.988 62.979 54.988 63.33 c 54.988 63.51 54.941 63.709 54.871 63.85 c 54.758 64.069 54.609 64.26 54.547 64.26 c 54.477 64.26 54.434 64.186 54.434 64.061 c 54.445 63.959 l 54.469 63.721 l 54.469 63.471 54.328 63.307 54.117 63.307 c 54.031 63.307 53.93 63.327 53.812 63.366 c 52.605 63.756 l 51.555 64.096 51.148 64.182 50.633 64.182 c 49.172 64.182 48.07 63.198 48.07 61.885 c 48.07 60.405 49.086 59.479 51.695 58.573 c 54.227 57.698 54.555 57.494 54.555 56.827 c 54.555 56.448 54.195 56.139 53.746 56.139 c 53.582 56.139 53.434 56.151 53.309 56.178 c 53.199 56.198 52.781 56.299 52.047 56.475 c 51.273 56.659 50.613 56.737 49.738 56.737 c 47.895 56.737 47.008 56.455 46.047 55.565 c 45.824 55.358 45.734 55.319 45.484 55.319 c 45.359 55.323 l 45.066 55.342 l 44.844 55.342 44.676 55.233 44.543 55.006 c 44.246 54.506 l 44.203 54.436 44.121 54.393 44.031 54.393 c 43.84 54.393 43.734 54.522 43.559 54.971 c 43.465 55.209 43.336 55.397 43.188 55.526 c 42.797 55.862 l 42.738 55.912 42.703 55.963 42.703 55.998 c 42.703 56.034 42.734 56.096 42.797 56.178 c 42.852 56.252 42.91 56.338 42.965 56.436 c 43.02 56.541 43.051 56.616 43.055 56.659 c 43.074 56.827 l 43.09 56.881 43.145 56.92 43.207 56.92 c 43.332 56.92 43.352 56.838 43.352 56.381 c 43.352 56.264 43.402 56.092 43.5 55.881 c 43.707 55.436 l 43.73 55.377 43.797 55.338 43.875 55.338 c 44.066 55.338 44.309 55.791 44.375 56.268 c 44.43 56.666 44.473 56.791 44.555 56.791 c 44.629 56.791 44.664 56.744 44.82 56.42 c 44.871 56.319 44.945 56.221 45.043 56.139 c 45.137 56.065 45.211 56.018 45.266 56.01 c 45.285 56.01 l 45.441 56.01 45.492 56.104 45.527 56.455 c 45.543 56.643 l 45.508 56.901 l 45.48 57.08 l 45.48 57.202 45.523 57.272 45.594 57.272 c 45.633 57.272 45.684 57.248 45.73 57.198 c 45.918 57.014 l 46.039 56.889 46.168 56.827 46.289 56.827 c 46.449 56.827 46.547 56.897 46.547 57.01 c 46.547 57.104 46.527 57.147 46.363 57.42 c 46.348 57.448 46.305 57.526 46.23 57.662 c 46.066 57.979 45.992 58.053 45.848 58.053 c 45.656 58.034 l 45.414 57.94 l 45.344 57.912 45.293 57.901 45.27 57.901 c 45.234 57.901 45.211 57.932 45.211 57.975 c 45.211 58.143 45.32 58.405 45.527 58.733 c 45.652 58.932 45.762 59.432 45.844 60.19 c 45.934 61.061 l 45.992 61.596 46.051 61.787 46.16 61.787 c 46.199 61.787 46.227 61.772 46.23 61.748 c 46.289 61.471 l 46.289 61.248 l 46.289 60.979 46.316 60.756 46.352 60.756 c 46.43 60.756 46.547 60.85 46.66 61.006 c 46.738 61.119 46.805 61.244 46.848 61.377 c 46.906 61.577 46.938 61.741 46.938 61.866 c 46.938 62.295 46.758 62.479 46.176 62.643 c 46.117 62.659 46.078 62.678 46.066 62.698 c 46.047 62.866 l 46.047 63.182 l 46.012 63.405 l 45.934 63.588 l 45.844 63.795 l 45.695 63.702 l 45.316 63.463 45.223 63.393 45.098 63.237 c 44.93 63.022 44.871 62.971 44.801 62.971 c 44.75 62.971 44.711 63.006 44.711 63.049 c 44.785 63.366 l 44.789 63.42 l 44.789 63.827 44.559 64.221 44.152 64.518 c 44.023 64.369 l 43.891 64.182 l 43.668 63.924 l 43.539 63.772 43.461 63.565 43.461 63.373 c 43.469 63.248 l 43.469 63.104 43.445 63.053 43.375 63.053 c 43.336 63.069 l 43.188 63.252 l 43 63.479 l 42.668 63.682 l 42.5 63.811 l 42.32 63.42 42.312 63.381 42.312 62.881 c 42.312 62.725 l 42.312 62.557 42.258 62.455 42.168 62.455 c 42.125 62.455 42.047 62.487 41.941 62.549 c 41.859 62.596 41.758 62.623 41.664 62.623 c 41.395 62.623 41.195 62.393 41.195 62.08 c 41.195 61.741 41.27 61.506 41.496 61.135 c 41.641 60.897 41.836 60.698 41.922 60.698 c 41.934 60.698 41.957 60.873 41.98 61.1 c 41.922 61.416 l 41.91 61.557 l 41.91 61.678 41.953 61.748 42.023 61.748 c 42.129 61.748 42.191 61.635 42.258 61.323 c 42.406 60.635 l 42.555 59.76 l 42.594 59.526 42.668 59.276 42.777 59.018 c 42.922 58.67 43.031 58.452 43.094 58.369 c 43.297 58.104 43.324 58.069 43.324 58.01 c 43.324 57.963 43.285 57.924 43.242 57.924 c 43.203 57.924 43.145 57.944 43.074 57.979 c 43.012 58.014 42.934 58.037 42.852 58.053 c 42.719 58.073 l 42.41 58.073 42.348 57.827 42.348 56.592 c 42.348 55.823 42.508 54.799 42.703 54.299 c 43.094 53.315 l 43.156 53.159 43.188 53.002 43.188 52.854 c 43.188 51.494 41.352 49.323 39.414 48.393 c 38.66 48.03 38.441 47.655 38.375 46.627 c 38.695 46.78 38.828 46.834 39.023 46.889 c 39.574 47.041 39.691 47.119 39.77 47.373 c 39.844 47.612 l 39.875 47.717 39.945 47.768 40.324 47.967 c 40.469 48.041 40.57 48.123 40.621 48.209 c 40.719 48.366 40.793 48.444 40.844 48.444 c 40.895 48.444 40.926 48.409 40.926 48.35 c 40.918 48.319 l 40.715 47.791 l 40.691 47.659 l 40.691 47.272 40.91 46.944 41.309 46.741 c 41.336 46.838 41.355 46.912 41.363 46.963 c 41.43 47.248 41.492 47.436 41.551 47.522 c 41.906 48.041 l 42.031 48.229 42.055 48.354 42.055 48.83 c 42.035 49.416 l 42.023 49.565 l 42.023 49.662 42.07 49.713 42.152 49.713 c 42.219 49.713 42.27 49.69 42.277 49.655 c 42.332 49.303 l 42.344 49.229 42.422 49.174 42.516 49.174 c 42.691 49.174 42.922 49.401 43.168 49.823 c 43.262 49.987 43.301 50.116 43.301 50.268 c 43.301 50.385 43.273 50.647 43.242 50.846 c 43.234 50.955 l 43.234 51.045 43.289 51.104 43.375 51.104 c 43.422 51.104 43.457 51.092 43.465 51.069 c 43.594 50.643 l 43.641 50.498 43.738 50.401 43.84 50.401 c 44.125 50.401 44.418 50.983 44.449 51.608 c 44.469 51.998 44.516 52.135 44.637 52.135 c 44.684 52.135 44.719 52.116 44.727 52.088 c 44.801 51.85 l 44.828 51.76 44.883 51.702 44.934 51.702 c 44.98 51.702 45.078 51.784 45.191 51.924 c 45.258 52.002 45.312 52.084 45.359 52.166 c 45.398 52.229 45.414 52.295 45.414 52.362 c 45.414 52.452 45.398 52.479 45.246 52.592 c 45.113 52.694 45.043 52.827 45.043 52.983 c 45.043 53.252 45.16 53.541 45.488 54.116 c 45.793 54.639 45.832 54.698 45.906 54.698 c 45.977 54.698 46.031 54.627 46.031 54.537 c 46.031 54.483 46.016 54.428 45.992 54.373 c 45.941 54.268 45.914 54.194 45.914 54.155 c 45.914 54.08 45.98 54.022 46.062 54.022 c 46.152 54.022 46.27 54.061 46.398 54.135 c 46.48 54.178 46.648 54.244 46.902 54.338 c 47.246 54.463 47.457 54.537 47.531 54.561 c 47.648 54.596 47.781 54.619 47.914 54.619 c 48.023 54.619 48.133 54.604 48.238 54.58 c 48.453 54.53 48.594 54.502 48.668 54.502 c 48.836 54.502 48.969 54.635 48.969 54.799 c 48.969 54.869 48.914 55.014 48.852 55.119 c 48.758 55.276 48.699 55.362 48.684 55.377 c 48.602 55.463 48.57 55.51 48.57 55.545 c 48.594 55.619 l 48.613 55.666 48.652 55.694 48.695 55.694 c 48.844 55.694 49.129 55.397 49.391 54.971 c 50.848 52.592 51.551 51.838 52.621 51.514 c 53.738 51.178 54 50.94 54 50.276 c 54 49.92 53.812 49.494 53.422 48.952 c 52.68 47.92 52.418 47.315 52.418 46.631 c 52.418 46.479 l 52.512 46.518 52.578 46.549 52.621 46.573 c 52.773 46.651 52.875 46.694 52.922 46.702 c 53.387 46.795 l 53.531 46.827 53.648 46.998 53.648 47.186 c 53.645 47.221 l 53.605 47.631 l 53.605 47.698 l 53.605 47.928 53.719 48.116 53.855 48.116 c 53.906 48.116 53.969 48.069 53.996 48.002 c 54.293 47.373 l 54.418 47.112 54.559 46.991 54.945 46.815 c f 39.711 59.557 m 39.695 59.26 l 39.688 59.08 l 39.688 58.776 39.734 58.647 39.848 58.647 c 39.973 58.647 40.164 58.877 40.25 59.131 c 40.305 59.287 40.348 59.506 40.348 59.635 c 40.348 59.725 40.324 59.807 40.25 59.987 c 40.168 60.19 40.082 60.284 39.977 60.284 c 39.82 60.284 39.746 60.08 39.711 59.557 c f 38.266 63.182 m 38.266 63.018 l 38.266 62.85 38.215 62.534 38.133 62.233 c 38.062 61.967 38.027 61.827 38.023 61.807 c 37.996 61.698 37.984 61.584 37.984 61.467 c 37.984 61.205 38.109 60.959 38.301 60.858 c 38.504 60.748 l 38.543 60.487 l 38.531 60.385 38.551 60.248 38.598 60.077 c 38.648 59.893 38.676 59.705 38.676 59.518 c 38.676 59.366 38.66 59.307 38.625 59.307 c 38.555 59.307 38.434 59.498 38.32 59.78 c 38.141 60.229 37.98 60.397 37.727 60.397 c 37.523 60.397 37.391 60.362 37.148 60.244 c 37.344 59.994 37.461 59.784 37.688 59.28 c 37.984 58.619 38.098 58.487 38.559 58.256 c 38.664 58.205 38.73 58.139 38.73 58.084 c 38.73 57.998 38.664 57.94 38.562 57.94 c 38.52 57.94 38.402 57.959 38.301 57.979 c 38.164 57.994 l 38.09 57.994 38.059 57.928 38.059 57.78 c 38.059 57.287 38.48 56.69 39.172 56.198 c 39.266 56.131 39.324 56.057 39.324 56.006 c 39.324 55.975 39.301 55.944 39.266 55.916 c 39.234 55.893 39.199 55.877 39.168 55.877 c 39.129 55.877 39.043 55.909 38.949 55.955 c 38.895 55.983 38.848 55.994 38.812 55.994 c 38.727 55.994 38.668 55.909 38.668 55.776 c 38.668 55.608 38.793 55.311 39.008 54.971 c 39.203 54.655 39.418 54.444 39.621 54.358 c 40.457 54.002 l 40.5 53.983 40.531 53.944 40.531 53.897 c 40.531 53.83 40.484 53.776 40.418 53.76 c 40.27 53.744 l 40.102 53.725 l 39.879 53.744 l 39.844 53.744 l 39.727 53.744 39.68 53.721 39.68 53.662 c 39.68 53.573 39.766 53.385 39.953 53.057 c 40.16 52.698 40.305 52.534 40.414 52.534 c 40.457 52.534 40.504 52.573 40.531 52.627 c 40.734 53.057 l 40.762 53.112 40.93 53.244 41.219 53.428 c 41.391 53.541 41.57 53.612 41.738 53.631 c 42.184 53.69 42.219 53.713 42.219 53.897 c 42.219 53.963 42.203 54.084 42.164 54.264 c 42.145 54.358 42.125 54.487 42.109 54.655 c 42.07 55.006 l 42.035 55.416 l 42.031 55.518 42 55.569 41.941 55.569 c 41.902 55.569 41.84 55.51 41.793 55.436 c 41.574 55.077 41.254 54.838 40.992 54.838 c 39.957 54.838 39.246 56.494 39.246 58.92 c 39.266 59.651 l 39.32 60.526 l 39.332 60.834 l 39.332 61.58 39.07 62.354 38.691 62.733 c 38.43 62.994 l 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24.057 18.145 23.28 c 19.5 22.983 20.098 22.588 20.098 21.983 c 20.098 21.432 19.652 21.084 18.945 21.084 c 18.73 21.084 18.562 21.1 18.441 21.123 c 18.324 21.147 17.684 21.323 16.527 21.643 c 15.73 21.866 14.84 21.991 14.047 21.991 c 12.652 21.991 11.707 21.577 11.066 20.678 c 10.992 20.573 10.918 20.51 10.867 20.51 c 10.832 20.51 10.777 20.534 10.715 20.584 c 10.453 20.787 l 10.273 20.932 9.98 21.041 9.789 21.041 c 9.648 21.041 9.512 20.987 9.395 20.881 c 9.301 20.799 9.234 20.752 9.207 20.752 c 9.102 20.752 8.91 20.987 8.766 21.291 c 8.555 21.725 8.449 22.159 8.449 22.553 c 8.449 22.741 8.504 22.869 8.59 22.869 c 8.652 22.869 8.699 22.819 8.727 22.721 c 8.875 22.182 l 8.957 21.881 9.125 21.643 9.254 21.643 c 9.312 21.643 9.41 21.725 9.527 21.866 c 9.594 21.952 9.645 22.041 9.676 22.127 c 9.789 22.467 9.82 22.518 9.91 22.518 c 10.02 22.518 10.074 22.452 10.148 22.108 c 10.16 22.049 10.184 21.952 10.27 21.791 c 10.344 21.651 10.418 21.537 10.492 21.459 c 10.547 21.397 10.617 21.362 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26.53 c 7.297 26.299 7.473 26.119 7.543 26.119 c 7.59 26.119 7.629 26.241 7.629 26.381 c 7.594 26.604 l 7.559 26.709 7.543 26.807 7.543 26.889 c 7.543 26.994 7.57 27.049 7.629 27.049 c 7.793 27.049 7.969 26.565 8.039 25.897 c 8.113 25.209 l 8.141 24.967 8.215 24.717 8.336 24.467 c 8.508 24.119 8.672 23.885 8.82 23.78 c 8.996 23.655 9.105 23.534 9.105 23.463 c 9.105 23.405 9.059 23.354 9.012 23.354 c 9.004 23.354 8.875 23.397 8.633 23.483 c 8.477 23.506 l 8.105 23.506 7.984 23.295 7.984 22.655 c 7.984 21.881 8.504 20.772 9.078 20.307 c 9.934 19.619 l 10.367 19.272 10.695 18.694 10.695 18.28 c 10.695 17.737 10.102 17.202 9.078 16.834 c 8.469 16.608 l 7.992 16.436 7.801 16.131 7.613 15.237 c 7.855 15.311 7.961 15.33 8.125 15.33 c 8.207 15.327 l 8.484 15.319 l 8.734 15.319 8.914 15.471 8.914 15.682 c 8.914 15.846 l 8.914 15.897 8.98 15.967 9.078 16.014 c 9.246 16.096 9.332 16.139 9.34 16.147 c 9.543 16.311 l 9.578 16.338 9.613 16.354 9.648 16.354 c 9.703 16.354 9.75 16.303 9.75 16.241 c 9.75 16.194 9.738 16.135 9.711 16.073 c 9.648 15.905 9.613 15.682 9.613 15.432 c 9.613 14.975 9.684 14.733 9.914 14.397 c 9.938 14.522 9.957 14.604 9.973 14.639 c 10.012 14.733 10.117 14.901 10.289 15.143 c 10.57 15.534 10.777 16.014 10.777 16.272 c 10.77 16.35 l 10.715 16.776 l 10.715 16.854 10.75 16.928 10.809 16.983 c 10.848 17.018 10.895 17.037 10.934 17.037 c 11.016 17.037 11.062 16.991 11.125 16.85 c 11.168 16.748 11.219 16.682 11.258 16.682 c 11.324 16.682 11.449 16.854 11.531 17.057 c 11.594 17.202 11.625 17.334 11.625 17.448 c 11.625 17.514 11.605 17.596 11.57 17.686 c 11.531 17.78 11.512 17.854 11.512 17.897 c 11.512 17.948 11.527 17.971 11.586 18.022 c 11.617 18.045 11.648 18.057 11.684 18.057 c 11.719 18.057 11.758 18.045 11.793 18.022 c 11.961 17.909 l 12.016 17.873 12.066 17.854 12.105 17.854 c 12.184 17.854 12.289 17.936 12.367 18.057 c 12.531 18.307 12.633 18.51 12.633 18.588 c 12.633 18.635 12.602 18.674 12.496 18.744 c 12.129 19.006 11.883 19.401 11.883 19.741 c 11.883 19.815 11.902 19.909 11.941 20.01 c 12.125 20.51 l 12.219 20.764 12.27 20.893 12.273 20.901 c 12.328 20.924 l 12.395 20.924 12.43 20.869 12.43 20.78 c 12.43 20.717 12.414 20.647 12.387 20.565 c 12.34 20.444 12.309 20.299 12.309 20.213 c 12.309 20.08 12.375 20.01 12.504 20.01 c 12.594 20.01 12.699 20.034 12.812 20.084 c 13.375 20.327 13.82 20.455 14.09 20.455 c 14.34 20.455 14.5 20.401 14.766 20.213 c 14.836 20.162 14.895 20.139 14.938 20.139 c 14.969 20.139 15.008 20.166 15.043 20.213 c 15.066 20.244 15.082 20.276 15.082 20.307 c 15.082 20.373 15.039 20.459 14.969 20.549 c 14.871 20.666 14.836 20.737 14.836 20.807 c 14.836 20.869 14.875 20.905 14.938 20.905 c 15.227 20.905 15.684 19.983 16.121 18.522 c 16.465 17.369 16.559 17.252 17.66 16.627 c 18.594 16.1 18.949 15.58 18.949 14.729 c 18.949 13.893 18.719 12.827 18.238 11.428 c 18.086 10.983 18.02 10.682 18.02 10.436 c 18.02 10.237 18.059 10.049 18.203 9.549 c 18.27 9.311 18.312 9.166 18.332 9.123 c 18.352 9.069 18.402 8.975 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24.225 l 3.969 24.221 3.926 24.186 3.926 24.147 c 3.926 24.084 3.977 23.897 4.082 23.592 c 4.27 23.057 l 4.586 22.756 l 4.918 22.518 l 5.016 22.444 5.078 22.366 5.078 22.303 c 5.078 22.256 5.047 22.205 5.012 22.202 c 4.824 22.182 l 4.766 22.174 4.73 22.123 4.73 22.034 c 4.73 21.893 4.863 21.643 5.105 21.307 c 5.391 20.916 5.594 20.787 5.922 20.787 c 6.199 20.787 l 6.332 20.787 6.414 20.725 6.414 20.619 c 6.414 20.534 6.359 20.479 6.238 20.455 c 5.98 20.401 5.902 20.346 5.902 20.229 c 5.902 20.182 5.934 20.112 5.996 20.026 c 6.145 19.823 l 6.27 19.655 6.406 19.541 6.496 19.541 c 6.547 19.541 6.664 19.592 6.77 19.713 c 6.934 19.905 7.125 20.057 7.371 20.119 c 8.016 20.284 8.152 20.35 8.152 20.498 c 8.152 20.553 8.125 20.627 8.078 20.713 c 7.871 21.08 7.797 21.323 7.797 21.623 c 7.797 21.854 7.738 22.014 7.652 22.014 c 7.617 22.014 7.562 21.959 7.5 21.866 c 7.348 21.631 7.09 21.475 6.852 21.475 c 6.207 21.475 5.434 22.483 5.051 23.819 c 4.961 24.123 4.914 24.479 4.914 24.819 c 4.914 25.17 4.961 25.561 5.051 25.952 c 5.172 26.498 5.242 26.803 5.254 26.862 c 5.277 27.116 l 5.277 27.577 4.852 28.17 4.195 28.627 c f 5.531 26.045 m 5.395 25.787 5.27 25.205 5.27 24.834 c 5.27 24.619 5.34 24.448 5.434 24.448 c 5.461 24.448 5.5 24.483 5.531 24.541 c 5.738 24.912 l 5.77 24.975 5.805 25.041 5.867 25.139 c 5.941 25.252 5.996 25.358 5.996 25.416 c 5.996 25.635 5.793 25.909 5.531 26.045 c f 7.574 17.666 m 7.617 17.627 7.648 17.596 7.668 17.577 c 7.75 17.494 7.797 17.444 7.816 17.428 c 8.18 17.053 8.23 17.014 8.406 17.014 c 8.57 17.014 8.906 17.108 9.098 17.205 c 9.18 17.244 9.23 17.295 9.23 17.33 c 9.23 17.354 9.16 17.432 9.043 17.537 c 8.84 17.725 l 8.684 17.866 8.453 17.994 8.344 17.994 c 8.133 17.994 7.914 17.901 7.574 17.666 c f 11.887 14.307 m 12.082 14.221 12.156 14.186 12.258 14.119 c 12.422 14.01 12.523 13.948 12.555 13.936 c 12.609 13.909 12.711 13.897 12.848 13.897 c 12.934 13.897 13.008 13.909 13.074 13.936 c 13.465 14.084 l 13.836 14.213 l 13.938 14.248 14.004 14.307 14.004 14.358 c 14.004 14.385 13.918 14.424 13.797 14.455 c 13.715 14.475 13.621 14.506 13.52 14.549 c 13.062 14.729 12.996 14.756 12.832 14.756 c 12.469 14.756 12.102 14.58 11.887 14.307 c f 16.586 10.573 m 16.711 10.869 16.793 10.92 17.215 10.944 c 17.695 10.967 18.148 11.823 18.148 12.401 c 18.148 12.459 18.129 12.506 18.102 12.506 c 18.082 12.506 18.035 12.487 17.977 12.448 c 17.844 12.362 17.75 12.303 17.699 12.28 c 17.16 12.037 l 16.801 11.877 16.484 11.401 16.484 11.022 c 16.484 10.893 16.508 10.787 16.586 10.573 c f 50.449 15.01 m 50.523 15.217 l 50.598 15.475 l 50.613 15.526 50.68 15.631 50.801 15.791 c 50.992 16.041 51.086 16.35 51.086 16.717 c 51.086 16.83 51.059 16.936 50.988 17.073 c 50.949 17.147 50.934 17.205 50.934 17.248 c 50.934 17.272 50.945 17.303 50.969 17.334 c 50.992 17.366 51.02 17.385 51.043 17.389 c 51.055 17.389 l 51.105 17.389 51.184 17.334 51.25 17.26 c 51.328 17.166 51.398 17.112 51.438 17.112 c 51.52 17.112 51.609 17.252 51.656 17.463 c 51.684 17.577 51.695 17.635 51.695 17.647 c 51.695 17.741 51.621 17.971 51.527 18.17 c 51.492 18.244 51.473 18.307 51.473 18.358 c 51.473 18.424 51.52 18.475 51.578 18.475 c 51.629 18.475 51.719 18.42 51.824 18.319 c 51.875 18.272 51.922 18.244 51.965 18.244 c 52.082 18.244 52.164 18.362 52.25 18.655 c 52.305 18.834 52.328 18.916 52.328 18.944 c 52.328 18.998 52.301 19.03 52.262 19.03 c 52.195 19.026 l 51.977 19.002 l 51.637 19.002 51.156 19.233 50.934 19.506 c 50.41 20.139 l 50.211 20.385 49.934 20.752 49.949 20.752 c 49.52 21.03 l 49.449 21.077 49.41 21.135 49.41 21.182 c 49.41 21.252 49.48 21.299 49.586 21.299 c 49.684 21.299 49.988 21.19 50.543 20.955 c 51.078 20.729 51.918 20.545 52.426 20.545 c 53.312 20.545 54.047 21.264 54.047 22.135 c 54.047 22.612 53.832 23.053 53.309 23.666 c 53.211 23.784 53.043 23.893 52.969 23.893 c 52.906 23.893 52.863 23.85 52.863 23.787 c 52.863 23.737 52.883 23.662 52.922 23.577 c 52.945 23.518 52.957 23.459 52.957 23.405 c 52.957 23.319 52.926 23.272 52.867 23.272 c 52.75 23.272 52.629 23.463 52.531 23.799 c 52.406 24.213 52.164 24.459 51.602 24.744 c 51.387 24.854 51.102 24.955 51.012 24.955 c 50.977 24.955 50.949 24.92 50.949 24.873 c 50.949 24.803 50.977 24.729 51.027 24.655 c 51.105 24.526 51.16 24.405 51.172 24.299 c 51.176 24.284 l 51.176 24.209 51.137 24.17 51.074 24.17 c 50.953 24.17 50.926 24.198 50.656 24.616 c 50.516 24.83 50.141 25.112 49.816 25.248 c 49.715 25.291 49.609 25.315 49.504 25.323 c 49.492 25.323 l 49.422 25.323 49.371 25.291 49.371 25.248 c 49.371 25.237 49.402 25.178 49.457 25.08 c 49.484 24.975 l 49.484 24.92 49.445 24.873 49.395 24.873 c 49.285 24.873 48.938 25.159 48.668 25.471 c 48.371 25.811 48.184 26.244 48.184 26.584 c 48.184 27.291 48.859 27.924 49.621 27.924 c 49.812 27.924 49.973 27.893 50.098 27.83 c 50.746 27.494 l 51.508 26.994 l 51.949 26.705 52.547 26.487 52.91 26.487 c 53.047 26.487 53.148 26.51 53.496 26.604 c 53.676 26.655 53.789 26.725 53.789 26.791 c 53.785 26.807 l 53.719 26.846 l 53.309 26.881 l 53.082 26.905 52.566 27.135 52.16 27.401 c 51.305 27.959 l 50.543 28.276 l 50.484 28.299 50.449 28.338 50.449 28.373 c 50.449 28.412 50.488 28.444 50.535 28.444 c 50.535 28.444 50.605 28.432 50.746 28.405 c 51.453 28.202 l 51.629 28.151 51.914 28.018 52.309 27.811 c 53.082 27.397 53.586 27.217 53.961 27.217 c 54.387 27.217 54.703 27.514 54.703 27.912 c 54.703 28.045 54.672 28.268 54.629 28.444 c 54.605 28.53 54.57 28.612 54.52 28.686 c 54.484 28.729 54.449 28.756 54.426 28.756 c 54.371 28.756 54.344 28.702 54.344 28.604 c 54.352 28.479 l 54.359 28.369 l 54.359 28.03 54.227 27.909 53.844 27.909 c 53.539 27.909 53.125 28.03 52.176 28.405 c 51.293 28.756 50.738 28.889 50.188 28.889 c 48.789 28.889 47.566 27.748 47.566 26.444 c 47.566 25.323 48.312 24.463 49.836 23.834 c 51.414 23.186 l 52.719 22.647 53.199 22.264 53.199 21.756 c 53.199 21.33 52.809 21.061 52.191 21.061 c 51.926 21.061 51.598 21.135 50.988 21.327 c 49.508 21.799 48.551 21.994 47.703 21.994 c 46.324 21.994 45.344 21.619 44.559 20.787 c 44.457 20.678 44.398 20.647 44.305 20.647 c 44.242 20.647 44.18 20.67 44.039 20.752 c 43.898 20.834 43.742 20.885 43.633 20.885 c 43.496 20.885 43.285 20.768 43.188 20.639 c 43 20.397 l 42.945 20.327 42.887 20.287 42.836 20.287 c 42.703 20.287 42.461 20.518 42.312 20.787 c 42.223 20.955 42.16 21.119 42.129 21.272 c 42.102 21.389 42.09 21.58 42.09 21.78 c 42.109 22.127 l 42.164 22.498 l 42.191 22.674 42.211 22.78 42.219 22.815 c 42.234 22.858 42.27 22.889 42.309 22.889 c 42.406 22.889 42.449 22.799 42.449 22.596 c 42.445 22.479 l 42.441 22.397 l 42.441 21.866 42.648 21.381 42.879 21.381 c 43.062 21.381 43.219 21.705 43.352 22.35 c 43.379 22.479 43.434 22.553 43.496 22.553 c 43.57 22.553 43.613 22.459 43.668 22.182 c 43.703 22.002 43.824 21.752 43.949 21.588 c 44.02 21.494 44.098 21.44 44.16 21.44 c 44.312 21.44 44.41 21.666 44.488 22.202 c 44.516 22.409 44.578 22.537 44.648 22.537 c 44.68 22.537 44.707 22.522 44.727 22.498 c 44.742 22.483 44.785 22.412 44.859 22.295 c 44.906 22.217 44.977 22.155 45.062 22.108 c 45.156 22.061 45.23 22.034 45.277 22.034 c 45.336 22.034 45.398 22.061 45.453 22.108 c 45.535 22.182 45.582 22.237 45.582 22.26 c 45.582 22.287 45.543 22.346 45.473 22.424 c 45.348 22.557 45.254 22.729 45.137 23.037 c 45.059 23.241 45.004 23.381 44.969 23.463 c 44.941 23.53 44.91 23.577 44.875 23.592 c 44.816 23.608 l 44.711 23.592 l 44.688 23.588 44.559 23.553 44.32 23.483 c 44.199 23.467 l 44.145 23.467 44.109 23.498 44.109 23.553 c 44.109 23.604 44.176 23.694 44.266 23.76 c 44.734 24.135 45.012 24.76 45.117 25.674 c 45.184 26.229 45.254 26.44 45.488 26.772 c 45.539 26.842 45.582 26.881 45.605 26.881 c 45.676 26.881 45.715 26.823 45.715 26.717 c 45.715 26.647 45.695 26.557 45.656 26.455 c 45.629 26.385 45.617 26.307 45.617 26.229 c 45.621 26.159 l 45.656 26.045 l 45.859 26.119 l 46.082 26.381 l 46.215 26.534 46.328 26.885 46.328 27.139 c 46.328 27.319 46.285 27.44 46.176 27.569 c 46.062 27.709 45.98 27.764 45.895 27.764 c 45.77 27.756 l 45.637 27.744 l 45.516 27.744 45.457 27.78 45.457 27.854 c 45.488 27.998 l 45.508 28.202 l 45.398 28.666 l 45.305 28.889 l 45.246 29.073 l 45.117 28.983 l 44.84 28.815 l 44.77 28.772 44.691 28.702 44.617 28.608 c 44.352 28.287 44.336 28.276 44.23 28.276 c 44.164 28.276 44.125 28.299 44.125 28.346 c 44.133 28.424 l 44.172 28.596 44.188 28.721 44.188 28.811 c 44.188 29.135 43.953 29.623 43.668 29.893 c 43.281 29.522 43.074 29.112 43.074 28.721 c 43.074 28.498 l 43.074 28.412 43.047 28.369 42.988 28.369 c 42.98 28.369 l 42.938 28.377 42.891 28.409 42.852 28.459 c 42.629 28.756 l 42.613 28.78 42.516 28.862 42.332 28.998 c 42.297 29.026 42.246 29.069 42.184 29.131 c 42.109 28.983 l 41.926 28.612 41.859 28.393 41.848 28.088 c 41.84 27.842 41.82 27.768 41.762 27.768 c 41.734 27.768 41.691 27.784 41.645 27.811 c 41.363 27.971 41.336 27.979 41.215 27.979 c 40.879 27.979 40.652 27.741 40.652 27.389 c 40.652 27.252 40.73 27.045 40.883 26.772 c 41.051 26.467 41.309 26.19 41.418 26.19 c 41.453 26.19 41.461 26.213 41.461 26.307 c 41.457 26.342 l 41.441 26.659 l 41.402 26.901 l 41.398 26.932 l 41.398 27.002 41.426 27.037 41.484 27.037 c 41.652 27.037 41.797 26.545 41.922 25.565 c 42.031 24.698 42.27 24.104 42.648 23.76 c 42.742 23.674 42.797 23.6 42.797 23.561 c 42.797 23.526 42.766 23.502 42.723 23.502 c 42.668 23.502 42.586 23.534 42.48 23.592 c 42.414 23.631 42.344 23.651 42.277 23.651 c 42.094 23.651 41.887 23.436 41.848 23.205 c 41.68 22.147 l 41.551 20.659 l 41.527 20.377 41.566 20.002 41.664 19.545 c 41.75 19.143 41.793 18.85 41.793 18.662 c 41.793 17.553 41.098 16.67 39.953 16.33 c 38.766 15.979 l 38.023 15.756 37.66 15.272 37.559 14.362 c 37.746 14.514 37.805 14.537 38.098 14.584 c 38.449 14.639 l 38.598 14.662 38.703 14.694 38.766 14.733 c 38.797 14.752 38.867 14.823 38.969 14.936 c 39.285 15.291 l 39.406 15.424 39.512 15.498 39.578 15.498 c 39.621 15.498 39.656 15.452 39.656 15.393 c 39.656 15.323 39.613 15.217 39.527 15.084 c 39.438 14.948 39.391 14.784 39.391 14.608 c 39.391 14.373 39.453 14.221 39.695 13.877 c 39.723 13.991 39.75 14.065 39.77 14.1 c 39.801 14.162 39.891 14.291 40.047 14.491 c 40.305 14.823 40.488 15.174 40.492 15.327 c 40.496 15.616 40.523 15.709 40.594 15.709 c 40.656 15.709 40.668 15.694 40.695 15.534 c 40.703 15.491 40.723 15.452 40.754 15.42 c 40.797 15.373 40.836 15.346 40.867 15.346 c 40.891 15.346 40.941 15.373 41.012 15.42 c 41.141 15.506 41.242 15.596 41.309 15.682 c 41.387 15.776 41.449 15.889 41.496 16.014 c 41.559 16.178 41.59 16.295 41.59 16.366 c 41.586 16.608 l 41.586 16.67 41.613 16.721 41.648 16.721 c 41.695 16.721 41.789 16.623 41.848 16.518 c 41.891 16.444 41.914 16.424 41.969 16.424 c 41.98 16.424 l 42.043 16.432 42.105 16.455 42.145 16.498 c 42.371 16.717 42.52 16.991 42.52 17.194 c 42.5 17.537 l 42.426 17.815 l 42.422 17.842 l 42.422 17.897 42.465 17.948 42.512 17.948 c 42.598 17.948 42.688 17.83 42.812 17.557 c 42.875 17.428 42.93 17.369 42.984 17.369 c 43.074 17.369 43.141 17.436 43.316 17.686 c 43.484 17.928 l 43.52 17.983 43.539 18.045 43.539 18.116 c 43.539 18.182 43.52 18.244 43.484 18.299 c 43.336 18.522 l 43.309 18.561 43.297 18.623 43.297 18.709 c 43.297 18.959 43.387 19.248 43.578 19.58 c 43.711 19.823 43.793 19.979 43.816 20.045 c 43.848 20.123 43.879 20.159 43.922 20.159 c 43.953 20.159 43.984 20.112 43.984 20.065 c 43.965 19.862 l 43.965 19.565 l 43.992 19.494 44.039 19.467 44.125 19.467 c 44.184 19.467 44.25 19.483 44.32 19.506 c 44.785 19.674 l 44.895 19.713 45.039 19.784 45.211 19.877 c 45.938 20.28 46.164 20.366 46.523 20.366 c 46.734 20.366 47.012 20.284 47.402 20.1 c 47.484 20.065 47.547 20.045 47.594 20.045 c 47.648 20.045 47.707 20.08 47.758 20.139 c 47.793 20.182 47.812 20.233 47.812 20.28 c 47.812 20.346 47.785 20.428 47.738 20.51 c 47.676 20.612 47.625 20.741 47.625 20.787 c 47.625 20.838 47.656 20.869 47.711 20.869 c 47.75 20.869 47.816 20.823 47.848 20.772 c 47.906 20.682 48.047 20.436 48.277 20.026 c 48.621 19.412 49.074 18.983 49.465 18.893 c 50.281 18.709 l 50.535 18.651 50.598 18.553 50.598 18.237 c 50.598 17.659 50.328 17.151 49.855 16.85 c 49.188 16.424 l 48.941 16.268 48.742 15.842 48.742 15.483 c 48.742 15.256 48.797 14.85 48.871 14.53 c 49 14.659 l 49.316 14.846 l 49.594 15.123 l 49.742 15.534 l 49.891 15.905 l 49.965 16.084 50.051 16.202 50.105 16.202 c 50.172 16.202 50.211 16.1 50.211 15.916 c 50.207 15.866 l 50.207 15.799 l 50.207 15.53 50.242 15.412 50.449 15.01 c f 39.469 9.381 m 39.695 9.592 39.734 9.608 40.363 9.737 c 40.871 9.838 41.109 10.073 41.219 10.573 c 41.297 10.94 41.348 11.01 41.68 11.205 c 41.789 11.264 41.871 11.327 41.922 11.389 c 41.965 11.436 41.996 11.463 42.016 11.463 c 42.051 11.463 42.082 11.428 42.082 11.381 c 42.082 11.319 42.039 11.209 41.961 11.073 c 41.91 10.983 41.887 10.873 41.887 10.717 c 41.887 10.42 41.945 10.217 42.109 9.979 c 42.223 10.225 42.285 10.295 42.574 10.498 c 42.941 10.756 43.082 11.057 43.148 11.725 c 43.176 12.006 43.23 12.155 43.309 12.155 c 43.344 12.155 43.375 12.131 43.391 12.096 c 43.52 11.815 l 43.539 11.78 43.582 11.756 43.637 11.756 c 43.691 11.756 43.789 11.811 43.871 11.893 c 44.133 12.131 l 44.242 12.233 44.316 12.491 44.355 12.912 c 44.371 13.065 44.422 13.174 44.484 13.174 c 44.523 13.174 44.562 13.135 44.578 13.08 c 44.691 12.709 l 44.719 12.619 44.77 12.561 44.824 12.561 c 44.848 12.561 44.898 12.592 44.969 12.651 c 45.07 12.737 45.156 12.83 45.23 12.932 c 45.316 13.049 45.359 13.178 45.359 13.303 c 45.359 13.619 l 45.359 13.686 45.387 13.76 45.434 13.823 c 45.469 13.869 45.5 13.897 45.516 13.897 c 45.559 13.897 45.598 13.862 45.621 13.803 c 45.695 13.6 l 45.727 13.51 45.797 13.452 45.867 13.452 c 45.965 13.452 46.066 13.534 46.215 13.748 c 46.258 13.807 46.293 13.858 46.324 13.897 c 46.41 13.998 46.457 14.084 46.457 14.131 c 46.457 14.159 46.43 14.205 46.383 14.268 c 46.109 14.619 45.973 15.045 45.973 15.53 c 45.973 16.147 46.207 16.666 46.918 17.631 c 47.348 18.205 47.59 18.748 47.59 19.119 c 47.59 19.229 47.539 19.28 47.441 19.28 c 47.375 19.28 47.309 19.26 47.219 19.209 c 47.148 19.174 47.102 19.155 47.078 19.155 c 47.043 19.155 47 19.182 46.957 19.229 c 46.828 19.377 l 46.734 19.526 l 46.699 19.584 46.641 19.619 46.582 19.619 c 46.57 19.619 46.523 19.608 46.438 19.58 c 46.27 19.565 l 46.078 19.557 46.004 19.487 46.004 19.303 c 46.012 19.19 l 46.012 19.108 l 46.012 18.948 45.871 18.698 45.676 18.522 c 45.324 18.205 l 44.887 17.819 44.551 17.198 44.551 16.784 c 44.551 16.608 44.605 16.323 44.711 15.979 c 44.875 15.42 44.934 15.159 44.934 14.979 c 44.934 14.084 43.574 12.912 41.051 11.631 c 39.77 10.979 39.449 10.596 39.449 9.713 c 39.469 9.381 l f 38.246 28.869 m 38.246 28.409 l 38.246 28.311 38.238 28.241 38.207 28.108 c 38.148 27.85 38.133 27.76 38.133 27.69 c 38.133 27.33 38.184 27.159 38.395 26.791 c 38.508 26.588 38.582 26.342 38.582 26.166 c 38.582 26.092 38.555 26.045 38.504 26.045 c 38.473 26.045 38.438 26.065 38.41 26.104 c 38.266 26.307 l 38.145 26.467 38.031 26.514 37.73 26.514 c 37.41 26.491 l 37.539 26.174 37.578 26.053 37.633 25.842 c 37.781 25.237 37.805 25.194 38.133 24.893 c 38.465 24.596 38.496 24.557 38.496 24.452 c 38.496 24.366 38.441 24.299 38.367 24.299 c 38.246 24.319 l 38.172 24.327 l 38.07 24.327 37.98 24.139 37.98 23.932 c 37.98 23.815 38.027 23.623 38.098 23.463 c 38.145 23.354 38.188 23.28 38.227 23.241 c 38.41 23.057 l 38.461 23.006 38.488 22.94 38.488 22.866 c 38.488 22.776 38.434 22.702 38.375 22.702 c 38.336 22.702 38.281 22.721 38.207 22.756 c 38.16 22.784 38.117 22.795 38.078 22.795 c 37.992 22.795 37.953 22.744 37.953 22.643 c 37.984 22.369 l 38.098 21.885 l 38.121 21.78 38.191 21.659 38.301 21.534 c 38.395 21.424 38.488 21.346 38.578 21.307 c 38.812 21.205 38.938 21.151 38.949 21.143 c 38.984 21.116 39.008 21.061 39.008 20.994 c 39.008 20.92 38.949 20.862 38.883 20.862 c 38.863 20.862 38.797 20.877 38.691 20.901 c 38.5 20.92 l 38.312 20.92 38.301 20.909 38.301 20.76 c 38.301 20.67 38.312 20.573 38.34 20.471 c 38.41 20.174 l 38.488 19.862 38.594 19.694 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42.787 28.961 42.787 c 28.629 42.787 28.414 42.514 28.414 42.088 c 28.414 41.135 l f 31.953 42.088 m 31.953 40.928 l 31.953 40.651 32.172 40.44 32.453 40.44 c 32.781 40.44 32.965 40.616 32.965 40.928 c 32.965 42.073 l 32.965 42.51 32.785 42.784 32.5 42.784 c 32.441 42.784 32.348 42.76 32.273 42.725 c 32.211 42.698 32.168 42.674 32.141 42.651 c 32.098 42.616 32.059 42.565 32.027 42.502 c 31.98 42.401 31.953 42.33 31.953 42.295 c 31.953 42.088 l f 24.5 32.291 1.102 -1.422 re f 24.5 30.549 1.102 -0.449 re f 32.027 32.291 1.105 -1.422 re f 32.027 30.549 1.105 -0.449 re f Q Q showpage %%Trailer end restore %%EOF ================================================ FILE: misc/original-template/LICENSE ================================================ The MIT License (MIT) Copyright (c) 2013 Krishna Kumar Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. ================================================ FILE: misc/original-template/Makefile ================================================ # Copyright 2004 Chris Monson (shiblon@gmail.com) # Latest version available at http://www.bouncingchairs.net/oss # # This file is part of ``Chris Monson's Free Software''. # # ``Chris Monson's Free Software'' is free software; you can redistribute it # and/or modify it under the terms of the GNU General Public License as # published by the Free Software Foundation, Version 2. # # ``Chris Monson's Free Software'' is distributed in the hope that it will # be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General # Public License for more details. # # You should have received a copy of the GNU General Public License along # with ``Chris Monson's Free Software''; if not, write to the Free Software # Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA # # It is also available on the web at http://www.gnu.org/copyleft/gpl.html # # Note that using this makefile to build your documents does NOT place them # under the GPL unless you, the author, specifically do so. In other words, # I, Chris Monson, the copyright holder and author of this makefile, # consider it impossible to ``link'' to this makefile in any way covered by # the GPL. # # # TO OBTAIN INSTRUCTIONS FOR USING THIS FILE, RUN: # make help # fileinfo := LaTeX Makefile author := Chris Monson version := 2.2.1-alpha9 # .DEFAULT_GOAL := all # Note that the user-global version is imported *after* the source directory, # so that you can use stuff like ?= to get proper override behavior. .PHONY: Makefile GNUmakefile Makefile.ini $(HOME)/.latex-makefile/Makefile.ini -include Makefile.ini -include $(HOME)/.latex-makefile/Makefile.ini # Better names for these things .PHONY: Variables.ini $(HOME)/.latex-makefile/Variables.ini -include Variables.ini -include $(HOME)/.latex-makefile/Variables.ini # # This can be pdflatex or latex - you can change this by adding the following line to your Makefile.ini: # BUILD_STRATEGY := latex BUILD_STRATEGY ?= pdflatex # This can be used to pass extra options to latex. LATEX_OPTS ?= # # Sets LC_ALL=C, by default, so that the locale-aware tools, like sort, be # # immune to changes to the locale in the user environment. export LC_ALL ?= C # # # If you specify sources here, all other files with the same suffix # will be treated as if they were _include_ files. #onlysources.tex ?= main.tex #onlysources.lhs ?= #onlysources.tex.sh ?= #onlysources.tex.pl ?= #onlysources.tex.py ?= #onlysources.rst ?= #onlysources.mp ?= #onlysources.fig ?= #onlysources.gpi ?= #onlysources.dot ?= #onlysources.xvg ?= #onlysources.svg ?= #onlysources.eps.gz ?= #onlysources.eps ?= # # If you list files here, they will be treated as _include_ files #includes.tex ?= file1.tex file2.tex #includes.lhs ?= #includes.tex.sh ?= #includes.tex.pl ?= #includes.tex.py ?= #includes.rst ?= #includes.mp ?= #includes.fig ?= #includes.gpi ?= #includes.dot ?= #includes.xvg ?= #includes.svg ?= #includes.eps.gz ?= #includes.eps ?= # # If you list files or wildcards here, they will *not* be cleaned - default is # to allow everything to be cleaned. #neverclean ?= *.pdf # # Alternatively (recommended), you can add those lines to a Makefile.ini file # and it will get picked up automatically without your having to edit this # Makefile. # # KNOWN ISSUES: # * The following occurs: # file with: \usepackage{named}\bibliographystyle{named} # Compile # change to: \usepackage{apalike}\bibliographystyle{apalike} # Compile again -- BARF! # # The workaround: make clean-nographics; make # # Note that we may not be able to fix this. LaTeX itself barfs # on this, not the makefile. The very first invocation of LaTeX # (when something like this has happened) reads the existing .aux # file and discovers invalid commands like \citeauthoryear that # are only valid in the package that was just removed. It then # tries to parse them and explodes. It's not at all clear to me # how to fix this. I tried removing the .aux files on the first # run of LaTeX, but that necessarily requires more subsequent # rebuilds on common edits. There does not appear to be a # graceful solution to this issue. # # # CHANGES: # Krishna Kumar (2013-11-27): # * Added PS to rm_ext to remove PS files generated during LaTeX output # Krishna Kumar (2013-11-26): # * Changed PS_EMBED_OPTION ?= -dPDFSETTINGS=/prepress from /printer # # Chris Monson (2012-06-25): # * Bumped version to 2.2.1-alpha9 # * Built with Holger Dell's changes to fix multiple unnecessary compilations. # Chris Monson (2011-11-10): # * Issue 144: Help patch from girard.nicolas applied # Andrew McNabb (2011-09-30): # * Bumped version to 2.2.1-alpha8 # * Issue 141: No font embedding for gnuplot when not doing pdf # * Syntax error fixed for gpi handling code # Chris Monson (2011-09-06): # * Issue 140: clean mlt*, mlf*, and mtc* files # * Issue 136: initial support for metapost files # Chris Monson (2011-08-09): # * Bumped version to 2.2.1-alpha7 # * Issue 138: existing .eps files now included correctly # * Issue 139: added missing backslash to ps build rule # Chris Monson (2011-07-20): # * Added LATEX_OPTS # Chris Monson (2011-06-23): # * Bumped version to 2.2.1-alpha6 # * Issue 133: Set jobname to fix .fls generation to always have the source name # * Removed unnecessary (?) double-invocation of cygpath # Chris Monson (2011-06-16): # * Added support for keeping .rst and .lhs tex intermediates around. # * Separated scripts from source generation files (rst and lhs) # * Fixed run-script problem for lhs2tex (was invoked incorrectly) # * Issue 133: Fixed typo from literate Haskell support # Chris Monson (2011-06-13): # * Bumped version to 2.2.1-alpha5 # * Fixed problems with detecting graphics for very long source names. # Chris Monson (2011-06-13): # * Issue 134: name of self corrected for dependency graph # * Issue 133: Added literate Haskell support (lhs2tex) # Chris Monson (2011-05-31): # * Rewrote specials (%%COMMENTS) to be easier to extend and parse. # Chris Monson (2011-05-11): # * Bumped version to 2.2.1-alpha4 # * Issue 129: nomenclature dependency fix # Chris Monson (2011-05-09): # * Bumped version to 2.2.1-alpha3 # * Issue 112: Cygpath fixes # Chris Monson (2011-04-27): # * Bumped version to 2.2.1-alpha2 # * Issue 126: Broken log parsing for latex pipeline # * Fixed month in recent changes (had May, should be April) # * Noticed problems with some existing parsing (colorizing errors, notably) and # fixed them. # * New test case for specified graphic extensions. # * Added .bb generation for .eps files (when extensionless in latex pipeline) # Chris Monson (2011-04-22): # * Bumped version to 2.2.1-alpha1 # * Issue 105: add support for format file detection and compilation # Chris Monson (2011-04-20): # * Bumped version to 2.2.0 (release!) # Chris Monson (2011-04-19): # * Bumped version to 2.2.0-rc15 # * Issue 125: infinite recursion with nomenclature files # * Issue 125: removed .d as a target for .nls in get-log-index # * Cleaned up invocation of run-makeindex to take an optional .ist instead of flags. # Chris Monson (2011-04-06): # * Bumped version to 2.2.0-rc14 # * Issue 121: Added Targets.ini and corresponding help text for it. # * Issue 121: Added Variables.ini (Makefile.ini still works, though). # * Issue 121: Added .DEFAULT_GOAL optional setting. # * Issue 120: xindy compatibility # Chris Monson (2011-03-16): # * Bumped version to 2.2.0-rc13 # * Fixed a bug in kspewhich invocation - random characters and a missing pipe. # * Added font embedding to gnuplot output. # Chris Monson (2011-03-15): # * Bumped version to 2.2.0-rc12 # * Issue 119: Annoying warning from which if Gnuplot not installed. # * Fixed catchall error output to show more info from the log. Cutting off the # first line is too jarring. # * Issue 118: Better glossary support # Chris Monson (2011-03-03): # * Bumped version to 2.2.0-rc11 # * Issue 112: Fixed regression introduced by use of cygpath (ugly warnings) # Chris Monson (2011-02-03): # * Bumped version to 2.2.0-rc10 # * Issue 112: Added path normalization for cygwin systems # * Fixed a bug in get-missing-inputs where we weren't specifying target files # Chris Monson (2011-01-24): # * Issue 111: Added .jpeg as a possible image extension # Chris Monson (2011-01-21): # * Issue 110: Long filenames not produced correctly in .d file # * Fixed problem with unknown control sequence error parsing # * Fixed problem with \r in fatal output (was interpreted as LF by echo) # * Removed a spurious "hi" # Chris Monson (2011-01-14): # * Bumped version to 2.2.0-rc8 # * Issue 107: Removed comment with embedded newline, fixing MinGW on Windows 7. # Chris Monson (2011-01-07): # * Emit an error if .gpi.d files have dependencies with : in the name # Chris Monson (2011-01-05): # * Bumped version to 2.2.0-rc7 # * Issue 106: existing graphic dependencies not generated correctly # Chris Monson (2011-01-04): # * Issue 106: not cleaning eps log files properly # * Issue 106: not rebuilding after creating .pdf graphics from .eps # * Issue 94: svg going unnecessarily through eps (can't reproduce) # Chris Monson (2010-12-31): # * Issue 100: make hanging because of faulty graphics detection (sed bug) # * Issue 108: do not ignore fatal errors from pdftex # Chris Monson (2010-12-23): # * Added gpi_global to gnuplot dependencies so that changes are detected # Chris Monson (2010-12-20): # * Updated build file to be smarter about Python version detection # * Created a bunch of test files and supporting scripts # * Issue 72: added apacite capaability (thanks to matkarat) # Chris Monson (2010-11-23): # * Changed to multi-part makefile build (split out sed scripts) # * Added build script and supporting infrastructure # * Updated test directory format # * Added notes about needed test cases # * Changed to use -file-line-error and fixed multiple inclusion/error bugs # * Added run_sed.py to allow easy testing of sed scripts outside of make # Chris Monson (2010-11-11): # * Bumped version to 2.2.0-rc7 # * issue 92: broken hyperref driver detection fixed # * issue 101: Broken inkscape conversion # * issue 102: Broken specification of font size for gnuplot pdfcairo # * Added KEEP_TEMP so that we can avoid deleting useful temporaries for debugging # * Restructured gnuplot code to be easier to follow # * Fixed a bug in convert-gpi where we were using $< instead of $1 # Chris Monson (2010-11-03): # * Bumped version to 2.2.0-rc6 # * issue 96: Fix sed errors when using make variables in substitutions # Chris Monson (2010-07-28): # * Bumped version to 2.2.0-rc5 (rc4 is broken) # * Bail out when we find the use of the import.sty package # * Issue 90: Add -z to dvips invocation # * Issue 67: Add xelatex support (thanks to Nikolai Prokoschenko for the patch!) # * Issue 85: Add warning about make 3.80 # Chris Monson (2010-06-20): # * Bumped version to 2.2.0-rc3 # * Attempt to fix bug with ! error detection (issue 88) # * Added svg->pdf direct support (issue 89) # Chris Monson (2010-04-28): # * Bumped version to 2.2.0-rc2 # * Fixed %._show target # Chris Monson (2010-04-08): # * Bumped version to 2.2.0-rc1 # * Added back in the rst_style_file stuff that got broken when switching # rst -> tex to use the script mechanism # Chris Monson (2010-03-23): # * Bumped version to 2.2.0-beta8 # * Work on issue 76: bad backtick escape for some sed versions, failure # to clear out the hold buffer when outputting MISSING comment. # - Backed out 2>&1 to &> (doesn't work in sh) # - Backed out using . to source variables # Chris Monson (2010-03-22): # * Bumped version to 2.2.0-beta7 # * Issue 72: Fix latex/bibtex invocation order for annotated bib styles # * Fixed informational output to reflect which LaTeX run we're on # * Fixed graphic detection to include graphics that are already there in # .d files # * Tightened up the .d file output to only make .d depend on graphic # *source* files. This means that building foo.d no longer # builds all of the graphics files on which foo.tex depends. # Had to use .SECONDEXPANSION trickery to make it work. # * Changed get-graphics to only accept a stem. # * Fixed build-once logic for scripted .tex to work better # * Made get-inputs sed script more maintainable. # * Moved Makefile.ini import up higher. # * Changed bare stems to not recursively invoke make # * Updated diff output to be more silent everywhere # * Added a MISSING comment to the .d file if stuff isn't found - forces # removal of .1st.make file, which often forces it to try again. # * Fixed broken graphics-target function # * Added sleep to .d file generation when stuff is missing - if it # builds too fast, make doesn't realize it needs to be reloaded, # and thus never discovers some deeper dependencies (especially # evident when graphics are included from scripted include # files). # Chris Monson (2010-03-17): # * Bumped version to 2.2.0-beta6 # * Fixed bareword builds to actually work (requires static patterns) # * Fixed colorization to work with new paragraph stuff # Chris Monson (2010-03-17): # * Bumped version to 2.2.0-beta5 # * Fixed graphic detection to be much more focused - splits log file # into paragraphs before doing pattern matching. # * Fixed make foo to work properly (recursively calls make foo.pdf) # * Fixed gpi -> pdf generation to not waste time building .eps *after* # the pdf already exists. # * Changed log copies to include MAKE_RESTARTS as part of the name. # * Fixed missing include file detection (also makes use of the paragraph # stuff) to detect missing scripted include files. # Chris Monson (2010-03-16): # * Bumped version to 2.2.0-beta4 # * issue 70: .pdf not moved out of the way properly on first # compilation, resulting in early error detection failure. # * issue 74: fixed broken error on missing .aux files: the # implementation was masking real errors. # Chris Monson (2010-03-15): # * Bumped version to 2.2.0-beta3 # * issue 71: Made the tput dependency optional # * issue 73: Made .tex targets not pull in .d files (building them from # scripts should not require a .d) # * issue 74: Output a much saner error when a .aux file is not produced # (e.g., when you are typing "make" without arguments in a # directory with included .tex files that are not named with # ._include_.) # Chris Monson (2010-03-11): # * Bumped version to 2.2.0-beta2 # * Fixed clean-graphics to get rid of intermediate .eps files that may # be hanging around # * Added an automatic setting to use eps terminals in pdflatex mode for # gnuplot if it doesn't understand pdf. # * issue 66: Removed grayscale generation via magic suffix. Grayscale # generation is now only available via GRAY=1 # * issue 68: Added explicit handling of LC_ALL for locale-aware tools # like "sort" # Chris Monson (2010-03-10): # * Bumped version to 2.2.0-beta1 # * Fixed success message to handle output message in different places # * Added name of produced file to success message # Chris Monson (2010-03-10): # * Bumped version to 2.2.0-alpha3 # * Added meaningful error message for wrong hyperref options # * Added meaningful error message for incorrect graphics extensions # Chris Monson (2010-03-09): # * Bumped version to 2.2.0-alpha2 # * Updated graphics handling (gnuplot and fig generate pdf natively) # * Changed xmgrace to output monochrome natively # Chris Monson (2010-03-09): # * Bumped version to 2.2.0-alpha1 - major change! # * Support pdflatex natively and by default (issue 6 - a long time coming) # * Add ability to have a single $HOME/.latex-makefile/Makefile.ini for # all invocations # * Reworked graphic inclusion detection so that extensions need not be # specified for either build strategy (e.g., # \includegraphics{test1.eps} -> \includegrahpics{test1}) # * Changed log format to include filenames and line numbers # Chris Monson (2010-02-04): # * Bumped version to 2.1.43 # * All of the following are for issue 63 (thanks to mojoh81): # * Added documentation about fixing Makefile.ini default target # * Added perl and python script targets # * Fixed run logic to allow included .tex files to be scripted (the # run-again logic now detects missing .tex files, and the MV # command has been switched out for a command that only invokes # MV if the files exist) # * Changed scripted generation to only run once per make invocation # * Added dependency on expr # Chris Monson (2010-01-19): # * Bumped version to 2.1.42 # * issue 62: Added .brf extension to cleanable files (backrefs) # Chris Monson (2010-01-07): # * Bumped version to 2.1.41 # * issue 60: bad makeindex runs now error out on subsequent tries # Chris Monson (2009-12-01): # * Bumped version to 2.1.40 # * issue 36: build all indices (for e.g., splitidx usage) # * issue 59: clean up all generated files (including indices) # Chris Monson (2009-11-23): # * Bumped version to 2.1.39 # * issue 57: change ps2pdf invocations to just use gs directly # Chris Monson (2009-11-19): # * Bumped version to 2.1.38 # * issue 57: Added some limited support for Cygwin (spaces in filenames) # Chris Monson (2009-11-15): # * Bumped version to 2.1.37 # * Removed svninfo, since this is now managed by mercurial # * Fixed typo in changelist # * Issue 52: added jpg->eps conversion (thanks to brubakee) # * Issue 54: fix missing Overfull colorization due to lack of a blank # line preceding the first error. # * Issue 51: remove head.tmp and body.tmp in make clean invocation # * Issue 56: maintain multiple versions of log files (for debugging) # Chris Monson (2009-11-14): # * Bumped version to 2.1.36 # * Issues 53 and 49: added .brf, .mtc, and .maf to the cleanables # Chris Monson (2009-11-05): # * Bumped version to 2.1.35 # * Added nomenclature support (see issue 48) # Chris Monson (2009-10-29): # * Bumped version to 2.1.34 # * Fixed _out_ creation bug introduced in 2.1.33 (it was always created) # * Fixed erroneous help output for $HOME in BINARY_TARGET_DIR # * Changed contact email address - bring on the spam! # Chris Monson (2009-10-21): # * Bumped version to 2.1.33 # * Fixed issue 46, adding support for dot2tex (thanks to fdemesmay) # * Made all_files.* settable in Makefile.ini (using ?= instead of :=) # * Fixed issue 47, thanks to fdemesmay: add binary copy directory, copy # dvi, pdf, and ps if it exists # Chris Monson (2009-09-25): # * Bumped version to 2.1.32 # * Fixed so that a changed lol file will cause a rebuild # * Added .lol files to the cleanable list # Chris Monson (2009-09-08): # * Bumped version to 2.1.31 # * Closed issue 43: evince doesn't notice pdf change w/out touch # Chris Monson (2009-08-28): # * Bumped version to 2.1.30 # * Closed issue 39: Capture multi-line log warnings/errors to output # Chris Monson (2009-08-26): # * Bumped version to 2.1.29 # * Closed issue 42: add svg support using inkscape # Chris Monson (2009-08-17): # * Bumped version to 2.1.28 # * Patch from paul.biggar for issue 38: package warnings are overlooked # Chris Monson (2009-08-07): # * Bumped version to 2.1.27 # * Included patch for issue 37 - removes pdf/ps files before copying, # allowing some broken viewers to see changes properly. # Chris Monson (2009-05-15): # * Bumped version to 2.1.26 # * Included patch for issue 9 from favonia - detects .fig changes for # pstex files during regular compilation, so long as the pstex # has been built at least once with make all-pstex. # Chris Monson (2009-03-27): # * Bumped version to 2.1.25 # * Cleaned up a bunch of variable setting stuff - more stuff is now # settable from Makefile.ini # * Cleaned up documentation for various features, especially settable # variables. # * issue 28: support for png -> eps conversion (it even looks good!) # * issue 29: support for "neverclean" files in Makefile.ini # * issue 30: make ps2pdf14 the default - fall back when not there # Chris Monson (2009-03-09): # * Bumped version to 2.1.24 # * issue 27: xmgrace support (thanks to rolandschulzhd) # Chris Monson (2008-10-23): # * Bumped version to 2.1.23 # * issue 23: fixed _check_programs to not use bash string subs # Chris Monson (2008-09-02): # * Bumped version to 2.1.22 # * Appled patch from Holger to add include # sources and some documentation updates. # * Updated backup_patterns to be a bit more aggressive (also thanks to # Holger) # Chris Monson (2008-08-30): # * Bumped version to 2.1.21 # * Added ability to specify onlysources.* variables to indicate the only # files that should *not* be considered includes. Thanks to Holger # for this patch. # * Added an automatic include of Makefile.ini if it exists. Allows # settings to be made outside of this makefile. # Chris Monson (2008-05-21): # * Bumped version to 2.1.20 # * Added manual pstex compilation support (run make all-pstex first) # * Removed all automatic pstex support. It was totally breaking # everything and is very hard to incorporate into the makefile # concept because it requires LaTeX to *fail* before it can # determine that it needs the files. # Chris Monson (2008-04-17): # * Bumped version to 2.1.19 # * Changed the pstex build hack to be on by default # Chris Monson (2008-04-09): # * Bumped version to 2.1.18 # * issue 16: fixed pstex build problems, seems nondeterministic. Added # gratuitious hack for testing: set PSTEX_BUILD_ALL_HACK=1. # Chris Monson (2008-04-09): # * Bumped version to 2.1.17 # * issue 20: fixed accumulation of *.make files - wildcard was # refusing to work on files that are very recently created. # Chris Monson (2008-04-02): # * Bumped version to 2.1.16 # * issue 19: Removed the use of "type" to fix broken "echo" settings # Chris Monson (2008-03-27): # * Bumped version to 2.1.15 # * issue 18: Favors binary echo over builtin, as binary understands -n # * issue 16: Fixed handling of missing pstex_t files in the log # * issue 9: Added .SECONDARY target for .pstex files # Chris Monson (2008-03-21): # * Bumped version to 2.1.14 # * Fixed broken aux file flattening, which caused included bibs to be # missed. # Chris Monson (2008-03-20): # * Bumped version to 2.1.13 # * Changed error output colorization to show errors for missing files # that are not graphics files. # Chris Monson (2008-03-20): # * Bumped version to 2.1.12 # * Fixed a regression introduced in r28 that makes bibtex fail when # there is no index file present # Chris Monson (2008-03-03): # * Bumped version to 2.1.11 # * Fixed issue 11 (handle index files, reported by abachn) # * Cleaned up some comments and help text # Chris Monson (2008-01-24): # * Bumped version to 2.1.10 # * Fixed to work when 'sh' is a POSIX shell like 'dash' # Chris Monson (2007-12-12): # * Bumped version to 2.1.9 # * Fixed documentation and dependency graph for pstex files # Chris Monson (2007-12-12): # * Bumped version to 2.1.8 # * Added basic pstex_t support for fig files (Issue 9 by favonia) # I still suggest that psfrag be used instead. # Chris Monson (2007-10-16): # * Bumped version to 2.1.7 # * Removed todo item: allow other comment directives for rst conversion # * Added ability to use global rst style file _rststyle_._include_.tex # * Added help text to that effect # Chris Monson (2007-05-20): # * Bumped version to 2.1.6 # * Changed default paper size for rst files # * Added todo item: fix paper size for rst files # * Added todo item: allow other comment directives for rst conversion # Chris Monson (2007-04-02): # * Bumped version to 2.1.5 # * Addressed Issue 7, incorrect .gpi.d generation in subdirectories # Chris Monson (2007-03-28): # * Bumped version to 2.1.4 # * Fixed syntax error in dot output # Chris Monson (2007-03-01): # * Bumped version to 2.1.3 # * Added reST to the included documentation # * Fixed graphics and script generation to be settable in the # environment. # Chris Monson (2007-02-23): # * Bumped version to 2.1.2 # * Added the ability to generate .tex files from .rst files # Chris Monson (2006-10-17): # * Bumped version to 2.1.1 # * Fixed includes from subdirectories (sed-to-sed slash escape problem) # Chris Monson (2006-10-05): # * Bumped version to 2.1.0 (pretty serious new feature added) # * New feature: bib files can now be anywhere on the BIBINPUTS path # * New programs: kpsewhich (with tetex) and xargs (BSD) # Chris Monson (2006-09-28): # * Bumped version to 2.0.9 # * Added ability to parse more than one bibliography # Chris Monson (2006-06-01): # * Bumped version to 2.0.8 # * Added .vrb to the list of cleaned files # Chris Monson (2006-04-26): # * Bumped version to 2.0.7 # * Fixed so that clean-nographics does not remove .gpi.d files # * Removed jpg -> eps hack (not working properly -- just pre-convert) # * Fixed so that postscript grayscale can be done with BSD sed # Chris Monson (2006-04-25): # * Bumped version to 2.0.6 # * Fixed so that changed toc, lot, lof, or out causes a rebuild # Chris Monson (2006-04-17): # * Bumped version to 2.0.5 # * Added jpg -> eps conversion target # Chris Monson (2006-04-12): # * Bumped version to 2.0.4 # * Fixed BSD sed invocation to not use \| as a branch delimiter # * Added a comment section on what is and is not allowed in BSD sed # * Made paper size handling more robust while I was at it # * Fixed postscript RGB grayscale to use a weighted average # * Fixed postscript HSB grayscale to convert to RGB first # * Fixed a problem with rebuilding .bbl files # Chris Monson (2006-04-11): # * Bumped version to 2.0.3 # * Fixed some BSD sed problems: can't use \n in substitutions # Chris Monson (2006-04-10): # * Bumped version to 2.0.2 # * Once again removed ability to create .tex files from scripts # * \includeonly works again # Chris Monson (2006-04-09): # * Bumped version to 2.0.1 # * Fixed grayscale postscript handling to be more robust # * Added ability to generate ._gray_. files from eps and eps.gz # * Added ability to clean ._gray_.eps files created from .eps files # Chris Monson (2006-04-07): # * Bumped version to 2.0.0 # * Removed clunky ability to create included .tex files from scripts # * Added note in the help about included tex scripting not working # * Fixed the .eps generation to delete %.gpihead.make when finished # * Abandoned designs to use shell variables to create sed scripts # * Abandoned __default__.tex.sh idea: it causes recursion with %: . # * Removed web page to-do. All items are now complete. # * Added better grayscale conversion for dot figures (direct ps fixup). # * Include files can now be scripted (at the expense of \includeonly). # * Updated dependency graph to contain better node names. # Chris Monson (2006-04-06): # * Bumped version to 2.0b3 # * Top level includes now fail if there is no rule to build them # * A helpful message is printed when they do fail # * Grayscale has been changed to be ._gray_, other phonies use _ now, too # * Grayscale handling has been completed # * Changed _include_stems target to _includes target. # * Fixed _includes target to be useful by itself. # * Removed the ability to specify clean and build targets at once # * Verified that epsfig works fine with current code # * Fixed included scripts so that they are added to the dep files # * Fixed so that graphics includes don't happen if they aren't for gpi # * Fixed dot output to allow grayscale. # Chris Monson (2006-04-05): # * Bumped version to 2.0b2 # * Removed automatic -gray output. It needs fixing in a bad way. # * Revamped dependency creation completely. # * Fixed conditional inclusion to actually work (test.nobuild.d, test.d). # * Fixed clean target to remove log targets # * Added the 'monochrome' word for gray gpi output # * Added a _check_gpi_files target that checks for common problems # * Changed the _version target into the version target (no _) # * Added better handling of grayscale files. Use the .gray.pdf target. # * Fixed testing for rebuilds # Chris Monson (2006-04-04): # * Bumped version to 2.0b1 # * Changed colorization of output # * Made .auxbbl and .auxtex .make files secondary targets # * Shortened and simplified the final latex invocation loop # * Added version-specific output ($$i vs. $$$$i) in latex loop # * Added a build message for the first .dvi run (Building .dvi (0)) # * Removed some build messages that most people don't care about. # * Simplified procedure for user-set colors -- simple text specification # * Fixed diff output to...not output. # * Fixed rerun bug -- detect not only when preceded with LaTeX Warning # * Sped up gpi plotting # * Added error handling and colorized output for gpi failure # * Documented color changing stuff. # * Now sort the flattened aux file to avoid false recompilation needs # * Added clean-nographics target # * Don't remove self.dvi file if self.aux is missing in the log # * Clarified some code. Did some very minor adjusting. # Chris Monson (2006-04-03): # * Bumped version to 2.0a7 # * Added .dvi and .ps files as secondary files. # * Fixed handling of multiple run detection when includeonly is in use. # * Added code to flatten .aux files. # * Added more files as .SECONDARY prerequisites to avoid recompilation. # * Fixed the inputs generation to be much simpler and to use pipes. # * Added the dependency graph directly into the makefile. # * Changed flatten-aux to remove \@writefile \relax \newlabel, etc. # * Undid pipe changes with sed usage (BSD sed doesn't know -f-). # * Added a _check_programs target that tells you what your system has. # * Fixed an error in colorization that made unnecessary errors appear # * Added view targets. # * Updated help text. # * Augmented cookies so that .aux can trigger .bbl and .dvi rebuilds # * Added more informative error handling for dvips and ps2pdf # Chris Monson (2006-04-02): # * Bumped version to 2.0a6 # * Added indirection to .bbl dependencies to avoid rebuilding .bbl files # * Streamlined the diff invocation to eliminate an existence test # * Removed special shell quote escape variables # * Moved includes to a more prominent location # * Fixed .inputs.make to not contain .aux files # * Fixed embedding to use a file instead of always grepping. # * Added *.make.temp to the list of cleanable files # * Fixed Ruby. It should now be supported properly. # * Now differentiate between all, default, and buildable files. # * Fixed to bail out on serious errors. # * Revised the handling of includable files. Still working on it. # Chris Monson (2006-03-31): # * Bumped version to 2.0a5 # * Fixed a bug with LaTeX error detection (there can be spaces) # * Added .bbl support, simplifying everything and making it more correct # * Refactored some tests that muddy the code # * Did a little cleanup of some shell loops that can safely be make loops # * Added support for graphviz .dot files # * Made _all_programs output easier to read # * Added the ruby support that has long been advertised # * Font embedding was screwed up for PostScript -- now implicit # * Changed the generation of -gray.gpi files to a single command # * Changed any make-generated file that is not included from .d to .make # Chris Monson (2006-03-30): # * Bumped version to 2.0a4 # * Fixed a bug with very long graphics file names # * Added a todo entry for epsfig support # * Fixed a bug paper size bug: sometimes more than one entry appears # * Fixed DVI build echoing to display the number instead of process ID # * DVI files are now removed on first invocation if ANY file is missing # * Added a simple grayscale approach: if a file ends with -gray.gpi, it # is created from the corresponding .gpi file with a special # comment ##GRAY in its header, which causes coloring to be # turned off. # * Fixed a bug in the handling of .tex.sh files. For some reason I had # neglected to define file stems for scripted output. # * Removed a trailing ; from the %.graphics dependencies # * Added dvips embedding (I think it works, anyway) # Chris Monson (2006-03-29): # * Bumped version to 2.0a3 # * Fixed error in make 3.79 with MAKEFILE_LIST usage # * Added the presumed filename to the _version output # * Added a vim macro for converting sed scripts to make commands # * Added gpi dependency support (plotting external files and loading gpi) # * Allow .gpi files to be ignored if called .include.gpi or .nobuild.gpi # * Fixed sed invocations where \+ was used. BSD sed uses \{1,\}. # Chris Monson (2006-03-28): # * Bumped version to 2.0a2 # * Added SHELL_DEBUG and VERBOSE options # * Changed the default shell back to /bin/sh (unset, in other words) # * Moved .PHONY declarations closer to their targets # * Moved help text into its own define block to obtain better formatting # * Removed need for double-entry when adding a new program invocation # * Moved .SECONDARY declaration closer to its relevant occurrence # * Commented things more heavily # * Added help text about setting terminal and output in gnuplot # * Created more fine-grained clean targets # * Added a %.graphics target that generates all of %'s graphics # * Killed backward-compatible graphics generation (e.g., eps.gpi=gpi.eps) # * For now, we're just GPL 2, not 3. Maybe it will change later # * Made the version and svninfo into variables # Chris Monson (2006-03-27): # * Bumped version to 2.0a1 # * Huge, sweeping changes -- automatic dependencies # # IMPORTANT! # # When adding to the following list, do not introduce any blank lines. The # list is extracted for documentation using sed and is terminated by a blank # line. # # EXTERNAL PROGRAMS: # = ESSENTIAL PROGRAMS = # == Basic Shell Utilities == CAT ?= cat CP ?= cp -f DIFF ?= diff ECHO ?= echo EGREP ?= egrep ENV ?= env EXPR ?= expr MV ?= mv -f SED ?= sed SORT ?= sort TOUCH ?= touch UNIQ ?= uniq WHICH ?= which XARGS ?= xargs SLEEP ?= sleep # == LaTeX (tetex-provided) == BIBTEX ?= bibtex DVIPS ?= dvips LATEX ?= latex PDFLATEX ?= pdflatex XELATEX ?= xelatex EPSTOPDF ?= epstopdf MAKEINDEX ?= makeindex XINDY ?= xindy KPSEWHICH ?= kpsewhich GS ?= gs # = OPTIONAL PROGRAMS = # == For MikTex under Cygwin, to get path names right CYGPATH ?= cygpath # == Makefile Color Output == TPUT ?= tput # == TeX Generation == PERL ?= perl PYTHON ?= python RST2LATEX ?= rst2latex.py LHS2TEX ?= lhs2tex # == EPS Generation == CONVERT ?= convert # ImageMagick DOT ?= dot # GraphViz DOT2TEX ?= dot2tex # dot2tex - add options (not -o) as needed MPOST ?= mpost # MetaPost FIG2DEV ?= fig2dev # XFig GNUPLOT ?= gnuplot # GNUplot INKSCAPE ?= inkscape # Inkscape (svg support) XMGRACE ?= xmgrace # XMgrace PNGTOPNM ?= pngtopnm # From NetPBM - step 1 for png -> eps PPMTOPGM ?= ppmtopgm # From NetPBM - (gray) step 2 for png -> eps PNMTOPS ?= pnmtops # From NetPBM - step 3 for png -> eps GUNZIP ?= gunzip # GZipped EPS # == Beamer Enlarged Output == PSNUP ?= psnup # == Viewing Stuff == VIEW_POSTSCRIPT ?= gv VIEW_PDF ?= xpdf VIEW_GRAPHICS ?= display # Xindy glossaries XINDYLANG ?= english XINDYENC ?= utf8 # If cygpath is present, then we create a path-norm function that uses it, # otherwise the function is just a no-op. Issue 112 has details. USE_CYGPATH := $(if $(shell $(WHICH) $(CYGPATH) 2>/dev/null),yes,) define path-norm $(if $(USE_CYGPATH),$(shell $(CYGPATH) -u "$1"),$1) endef # Command options for embedding fonts and postscript->pdf conversion PS_EMBED_OPTIONS ?= -dPDFSETTINGS=/prepress -dEmbedAllFonts=true -dSubsetFonts=true -dMaxSubsetPct=100 PS_COMPATIBILITY ?= 1.4 # If set to something, will cause temporary files to not be deleted immediately KEEP_TEMP ?= # Defaults for GPI DEFAULT_GPI_EPS_FONTSIZE ?= 22 DEFAULT_GPI_PDF_FONTSIZE ?= 12 # Style file for ReST RST_STYLE_FILE ?= $(wildcard _rststyle_._include_.tex) # This ensures that even when echo is a shell builtin, we still use the binary # (the builtin doesn't always understand -n) FIXED_ECHO := $(if $(findstring -n,$(shell $(ECHO) -n)),$(shell which echo),$(ECHO)) ECHO := $(if $(FIXED_ECHO),$(FIXED_ECHO),$(ECHO)) define determine-gnuplot-output-extension $(if $(shell $(WHICH) $(GNUPLOT) 2>/dev/null), $(if $(findstring unknown or ambiguous, $(shell $(GNUPLOT) -e "set terminal pdf" 2>&1)), eps, pdf), none) endef GNUPLOT_OUTPUT_EXTENSION ?= $(strip $(call determine-gnuplot-output-extension)) # Internal code should use this because of :=. This means that the potentially # expensive script invocation used to determine whether pdf is available will # only be run once. GPI_OUTPUT_EXTENSION := $(strip $(GNUPLOT_OUTPUT_EXTENSION)) # Note, if the terminal *does* understand fsize, then we expect this call to # create a specific error here: "fsize: expecting font size". Otherwise, we # assume that fsize is not understood. GPI_FSIZE_SYNTAX := $(strip \ $(if \ $(filter pdf,$(GPI_OUTPUT_EXTENSION)),\ $(if \ $(findstring fsize: expecting font size,$(shell $(GNUPLOT) -e "set terminal pdf fsize" 2>&1)),\ fsize FONTSIZE,\ font ",FONTSIZE"),\ FONTSIZE)) # Directory into which we place "binaries" if it exists. # Note that this can be changed on the commandline or in Makefile.ini: # # Command line: # make BINARY_TARGET_DIR=$HOME/pdfs myfile.pdf # # Also, you can specify a relative directory (relative to the Makefile): # make BINARY_TARGET_DIR=pdfs myfile.pdf # # Or, you can use Makefile.ini: # # BINARY_TARGET_DIR := $(HOME)/bin_out # BINARY_TARGET_DIR ?= _out_ RESTARTS := $(if $(MAKE_RESTARTS),$(MAKE_RESTARTS),0) # SH NOTES # # On some systems, /bin/sh, which is the default shell, is not linked to # /bin/bash. While bash is supposed to be sh-compatible when invoked as sh, it # just isn't. This section details some of the things you have to stay away # from to remain sh-compatible. # # * File pattern expansion does not work for {} # * [ "$x" = "$y" ] has to be [ x"$x" x"$y" ] # * &> for stderr redirection doesn't work, use 2>&1 instead # # BSD SED NOTES # # BSD SED is not very nice compared to GNU sed, but it is the most # commonly-invoked sed on Macs (being based on BSD), so we have to cater to # it or require people to install GNU sed. It seems like the GNU # requirement isn't too bad since this makefile is really a GNU makefile, # but apparently GNU sed is much less common than GNU make in general, so # I'm supporting it here. # # Sad experience has taught me the following about BSD sed: # # * \+ is not understood to mean \{1,\} # * \| is meaningless (does not branch) # * \n cannot be used as a substitution character # * ? does not mean \{0,1\}, but is literal # * a\ works, but only reliably for a single line if subsequent lines # have forward slashes in them (as is the case in postscript) # # For more info (on the Mac) you can consult # # man -M /usr/share/man re_format # # And look for the word "Obsolete" near the bottom. # # EXTERNAL PROGRAM DOCUMENTATION SCRIPT # # $(call output-all-programs,[]) define output-all-programs [ -f '$(this_file)' ] && \ $(SED) \ -e '/^[[:space:]]*#[[:space:]]*EXTERNAL PROGRAMS:/,/^$$/!d' \ -e '/EXTERNAL PROGRAMS/d' \ -e '/^$$/d' \ -e '/^[[:space:]]*#/i\ '\ -e 's/^[[:space:]]*#[[:space:]][^=]*//' \ $(this_file) $(if $1,> '$1',) || \ $(ECHO) "Cannot determine the name of this makefile." endef # If they misspell gray, it should still work. GRAY ?= $(call get-default,$(GREY),) # # Utility Functions and Definitions # # # Transcript # For debug/testing purposes: writes a message to # filename.transcript.make for each command that was run, including # some human-readable justification for why it had to be run. # For example: "Running latex (log-file indicated that this is necessary)" # Set WRITE_TRANSCRIPT to something to activate WRITE_TRANSCRIPT ?= # Set reason for the next run call # $(call set-run-reason,message) set-run-reason = export run_reason="$1" # Log command to the transcript file # $(call set-run-reason,command,job_name) define transcript $(if $(WRITE_TRANSCRIPT), \ $(ECHO) "Running $1 ($$run_reason)" >> $2.transcript.make; \ export run_reason="", \ $(sh_true)) endef # Don't call this directly - it is here to avoid calling wildcard more than # once in remove-files. remove-files-helper = $(if $1,$(RM) $1,$(sh_true)) # $(call remove-files,file1 file2) remove-files = $(call remove-files-helper,$(wildcard $1)) # Removes all cleanable files in the given list # $(call clean-files,file1 file2 file3 ...) # Works exactly like remove-files, but filters out files in $(neverclean) clean-files = \ $(call remove-files-helper,$(call cleanable-files,$(wildcard $1))) # Outputs all generated files to STDOUT, along with some others that are # created by these (e.g., .idx files end up producing .ilg and .ind files). # Discovered by reading *.fls OUTPUT lines and producing corresponding .ind # filenames as needed. # # $(call get-generated-names,) define get-generated-names [ -f '$1' ] && \ $(SED) \ -e '/^OUTPUT /{' \ -e ' s///' \ -e ' p' \ -e ' s/\.idx/\.ind/p' \ -e ' s/\.ind/\.ilg/p' \ -e '}' \ -e 'd' \ '$1' \ | $(SORT) | $(UNIQ) endef # This removes files without checking whether they are there or not. This # sometimes has to be used when the file is created by a series of shell # commands, but there ends up being a race condition: make doesn't know about # the file generation as quickly as the system does, so $(wildcard ...) doesn't # work right. Blech. # $(call remove-temporary-files,filenames) remove-temporary-files = $(if $(KEEP_TEMP),:,$(if $1,$(RM) $1,:)) # Create an identifier from a file name # $(call cleanse-filename,filename) cleanse-filename = $(subst .,_,$(subst /,__,$1)) # Escape dots # $(call escape-fname-regex,str) escape-fname-regex = $(subst /,\\/,$(subst .,\\.,$1)) # Test that a file exists # $(call test-exists,file) test-exists = [ -e '$1' ] # $(call test-not-exists,file) test-not-exists = [ ! -e '$1' ] # $(call move-files,source,destination) move-if-exists = $(call test-exists,$1) && $(MV) '$1' '$2' # Copy file1 to file2 only if file2 doesn't exist or they are different # $(call copy-if-different,sfile,dfile) copy-if-different = $(call test-different,$1,$2) && $(CP) '$1' '$2' copy-if-exists = $(call test-exists,$1) && $(CP) '$1' '$2' move-if-different = $(call test-different,$1,$2) && $(MV) '$1' '$2' replace-if-different-and-remove = \ $(call test-different,$1,$2) \ && $(MV) '$1' '$2' \ || $(call remove-files,'$1') # Note that $(DIFF) returns success when the files are the SAME.... # $(call test-different,sfile,dfile) test-different = ! $(DIFF) -q '$1' '$2' >/dev/null 2>&1 test-exists-and-different = \ $(call test-exists,$2) && $(call test-different,$1,$2) # Return value 1, or value 2 if value 1 is empty # $(call get-default,,) get-default = $(if $1,$1,$2) # Copy a file and log what's going on # $(call copy-with-logging,,) define copy-with-logging if [ -d '$2/' ]; then \ if $(CP) '$1' '$2/'; then \ $(ECHO) "$(C_INFO)Copied '$1' to '$2/'$(C_RESET)"; \ else \ $(ECHO) "$(C_ERROR)Failed to copy '$1' to '$2/'$(C_RESET)"; \ fi; \ fi endef # Gives a reassuring message about the failure to find include files # $(call include-message,) define include-message $(strip \ $(if $(filter-out $(wildcard $1),$1),\ $(shell $(ECHO) \ "$(C_INFO)NOTE: You may ignore warnings about the"\ "following files:" >&2;\ $(ECHO) >&2; \ $(foreach s,$(filter-out $(wildcard $1),$1),$(ECHO) ' $s' >&2;)\ $(ECHO) "$(C_RESET)" >&2) )) endef # Characters that are hard to specify in certain places space := $(empty) $(empty) colon := \: comma := , # Useful shell definitions sh_true := : sh_false := ! : # Clear out the standard interfering make suffixes .SUFFIXES: # Turn off forceful rm (RM is usually mapped to rm -f) ifdef SAFE_RM RM := rm endif # Turn command echoing back on with VERBOSE=1 ifndef VERBOSE QUIET := @ endif # Turn on shell debugging with SHELL_DEBUG=1 # (EVERYTHING is echoed, even $(shell ...) invocations) ifdef SHELL_DEBUG SHELL += -x endif # Get the name of this makefile (always right in 3.80, often right in 3.79) # This is only really used for documentation, so it isn't too serious. ifdef MAKEFILE_LIST this_file := $(firstword $(MAKEFILE_LIST)) else this_file := $(wildcard GNUmakefile makefile Makefile) endif # Terminal color definitions REAL_TPUT := $(if $(NO_COLOR),,$(shell $(WHICH) $(TPUT))) # $(call get-term-code,codeinfo) # e.g., # $(call get-term-code,setaf 0) get-term-code = $(if $(REAL_TPUT),$(shell $(REAL_TPUT) $1),) black := $(call get-term-code,setaf 0) red := $(call get-term-code,setaf 1) green := $(call get-term-code,setaf 2) yellow := $(call get-term-code,setaf 3) blue := $(call get-term-code,setaf 4) magenta := $(call get-term-code,setaf 5) cyan := $(call get-term-code,setaf 6) white := $(call get-term-code,setaf 7) bold := $(call get-term-code,bold) uline := $(call get-term-code,smul) reset := $(call get-term-code,sgr0) # # User-settable definitions # LATEX_COLOR_WARNING ?= magenta LATEX_COLOR_ERROR ?= red LATEX_COLOR_INFO ?= green LATEX_COLOR_UNDERFULL ?= magenta LATEX_COLOR_OVERFULL ?= red bold LATEX_COLOR_PAGES ?= bold LATEX_COLOR_BUILD ?= cyan LATEX_COLOR_GRAPHIC ?= yellow LATEX_COLOR_DEP ?= green LATEX_COLOR_SUCCESS ?= green bold LATEX_COLOR_FAILURE ?= red bold # Gets the real color from a simple textual definition like those above # $(call get-color,ALL_CAPS_COLOR_NAME) # e.g., $(call get-color,WARNING) get-color = $(subst $(space),,$(foreach c,$(LATEX_COLOR_$1),$($c))) # # STANDARD COLORS # C_WARNING := $(call get-color,WARNING) C_ERROR := $(call get-color,ERROR) C_INFO := $(call get-color,INFO) C_UNDERFULL := $(call get-color,UNDERFULL) C_OVERFULL := $(call get-color,OVERFULL) C_PAGES := $(call get-color,PAGES) C_BUILD := $(call get-color,BUILD) C_GRAPHIC := $(call get-color,GRAPHIC) C_DEP := $(call get-color,DEP) C_SUCCESS := $(call get-color,SUCCESS) C_FAILURE := $(call get-color,FAILURE) C_RESET := $(reset) # # PRE-BUILD TESTS # # Check that clean targets are not combined with other targets (weird things # happen, and it's not easy to fix them) hascleangoals := $(if $(sort $(filter clean clean-%,$(MAKECMDGOALS))),1) hasbuildgoals := $(if $(sort $(filter-out clean clean-%,$(MAKECMDGOALS))),1) ifneq "$(hasbuildgoals)" "" ifneq "$(hascleangoals)" "" $(error $(C_ERROR)Clean and build targets specified together$(C_RESET))) endif endif # # VARIABLE DECLARATIONS # # Names of sed scripts that morph gnuplot files -- only the first found is used GNUPLOT_SED := global-gpi.sed gnuplot.sed GNUPLOT_GLOBAL := global._include_.gpi gnuplot.global ifeq "$(strip $(BUILD_STRATEGY))" "latex" default_graphic_extension ?= eps latex_build_program ?= $(LATEX) build_target_extension ?= dvi hyperref_driver_pattern ?= hdvips hyperref_driver_error ?= Using dvips: specify ps2pdf in the hyperref options. endif ifeq "$(strip $(BUILD_STRATEGY))" "pdflatex" default_graphic_extension ?= pdf latex_build_program ?= $(PDFLATEX) build_target_extension ?= pdf hyperref_driver_pattern ?= hpdf.* hyperref_driver_error ?= Using pdflatex: specify pdftex in the hyperref options (or leave it blank). endif ifeq "$(strip $(BUILD_STRATEGY))" "xelatex" default_graphic_extension ?= pdf latex_build_program ?= $(XELATEX) build_target_extension ?= pdf hyperref_driver_pattern ?= hdvipdf.* hyperref_driver_error ?= Using pdflatex: specify pdftex in the hyperref options (or leave it blank). endif # Files of interest all_files.tex ?= $(wildcard *.tex) all_files.tex.sh ?= $(wildcard *.tex.sh) all_files.tex.pl ?= $(wildcard *.tex.pl) all_files.tex.py ?= $(wildcard *.tex.py) all_files.rst ?= $(wildcard *.rst) all_files.lhs ?= $(wildcard *.lhs) all_files.mp ?= $(wildcard *.mp) all_files.fig ?= $(wildcard *.fig) all_files.gpi ?= $(wildcard *.gpi) all_files.dot ?= $(wildcard *.dot) all_files.xvg ?= $(wildcard *.xvg) all_files.svg ?= $(wildcard *.svg) all_files.png ?= $(wildcard *.png) all_files.jpg ?= $(wildcard *.jpg) all_files.jpeg ?= $(wildcard *.jpeg) all_files.eps.gz ?= $(wildcard *.eps.gz) all_files.eps ?= $(wildcard *.eps) # Utility function for obtaining all files not specified in $(neverclean) # $(call cleanable-files,file1 file2 file3 ...) # Returns the list of files that is not in $(wildcard $(neverclean)) cleanable-files = $(filter-out $(wildcard $(neverclean)), $1) # Utility function for getting all .$1 files that are to be ignored # * files listed in $(includes.$1) # * files not listed in $(onlysources.$1) if it is defined ignore_files = \ $(includes.$1) \ $(if $(onlysources.$1),$(filter-out $(onlysources.$1), $(all_files.$1))) # Patterns to never be allowed as source targets ignore_patterns := %._include_ # Patterns allowed as source targets but not included in 'all' builds nodefault_patterns := %._nobuild_ $(ignore_patterns) # Utility function for getting targets suitable building # $(call filter-buildable,suffix) filter-buildable = \ $(filter-out $(call ignore_files,$1) \ $(addsuffix .$1,$(ignore_patterns)),$(all_files.$1)) # Utility function for getting targets suitable for 'all' builds # $(call filter-default,suffix) filter-default = \ $(filter-out $(call ignore_files,$1) \ $(addsuffix .$1,$(nodefault_patterns)),$(all_files.$1)) # Top level sources that can be built even when they are not by default files.tex := $(call filter-buildable,tex) files.tex.sh := $(call filter-buildable,tex.sh) files.tex.pl := $(call filter-buildable,tex.pl) files.tex.py := $(call filter-buildable,tex.py) files.rst := $(call filter-buildable,rst) files.lhs := $(call filter-buildable,lhs) files.gpi := $(call filter-buildable,gpi) files.dot := $(call filter-buildable,dot) files.mp := $(call filter-buildable,mp) files.fig := $(call filter-buildable,fig) files.xvg := $(call filter-buildable,xvg) files.svg := $(call filter-buildable,svg) files.png := $(call filter-buildable,png) files.jpg := $(call filter-buildable,jpg) files.jpeg := $(call filter-buildable,jpeg) files.eps.gz := $(call filter-buildable,eps.gz) files.eps := $(call filter-buildable,eps) # Make all pstex targets secondary. The pstex_t target requires the pstex # target, and nothing else really depends on it, so it often gets deleted. # This avoids that by allowing *all* fig files to be pstex targets, which is # perfectly valid and causes no problems even if they're going to become eps # files in the end. .SECONDARY: $(patsubst %.fig,%.pstex,$(files.fig)) # Make all .tex targets secondary that result .rst and .lhs: .SECONDARY: $(patsubst %.rst,%.tex,$(files.rst)) .SECONDARY: $(patsubst %.lhs,%.tex,$(files.lhs)) # Top level sources that are built by default targets default_files.tex := $(call filter-default,tex) default_files.tex.sh := $(call filter-default,tex.sh) default_files.tex.pl := $(call filter-default,tex.pl) default_files.tex.py := $(call filter-default,tex.py) default_files.rst := $(call filter-default,rst) default_files.lhs := $(call filter-default,lhs) default_files.gpi := $(call filter-default,gpi) default_files.dot := $(call filter-default,dot) default_files.mp := $(call filter-default,mp) default_files.fig := $(call filter-default,fig) default_files.xvg := $(call filter-default,xvg) default_files.svg := $(call filter-default,svg) default_files.png := $(call filter-default,png) default_files.jpg := $(call filter-default,jpg) default_files.jpeg := $(call filter-default,jpeg) default_files.eps.gz := $(call filter-default,eps.gz) default_files.eps := $(call filter-default,eps) # Utility function for creating larger lists of files # $(call concat-files,suffixes,[prefix]) concat-files = $(foreach s,$1,$($(if $2,$2_,)files.$s)) # Useful file groupings all_files_source := $(call concat-files,tex,all) all_files_source_gen := $(call concat-files,rst rhs,all) all_files_scripts := $(call concat-files,tex.sh tex.pl tex.py,all) .PHONY: $(all_files_scripts) default_files_source := $(call concat-files,tex,default) default_files_source_gen := $(call concat-files,rhs lhs,default) default_files_scripts := $(call concat-files,tex.sh tex.pl tex.py,default) files_source := $(call concat-files,tex) files_source_gen := $(call concat-files,rst lhs) files_scripts := $(call concat-files,tex.sh tex.pl tex.py) # Utility function for obtaining stems # $(call get-stems,suffix,[prefix]) get-stems = $(sort $($(if $2,$2_,)files.$1:%.$1=%)) # List of all stems (including ._include_ and ._nobuild_ file stems) all_stems.tex := $(call get-stems,tex,all) all_stems.tex.sh := $(call get-stems,tex.sh,all) all_stems.tex.pl := $(call get-stems,tex.pl,all) all_stems.tex.py := $(call get-stems,tex.py,all) all_stems.rst := $(call get-stems,rst,all) all_stems.lhs := $(call get-stems,lhs,all) all_stems.mp := $(call get-stems,mp,all) all_stems.fig := $(call get-stems,fig,all) all_stems.gpi := $(call get-stems,gpi,all) all_stems.dot := $(call get-stems,dot,all) all_stems.xvg := $(call get-stems,xvg,all) all_stems.svg := $(call get-stems,svg,all) all_stems.png := $(call get-stems,png,all) all_stems.jpg := $(call get-stems,jpg,all) all_stems.jpeg := $(call get-stems,jpeg,all) all_stems.eps.gz := $(call get-stems,eps.gz,all) all_stems.eps := $(call get-stems,eps,all) # List of all default stems (all default PDF targets): default_stems.tex := $(call get-stems,tex,default) default_stems.tex.sh := $(call get-stems,tex.sh,default) default_stems.tex.pl := $(call get-stems,tex.pl,default) default_stems.tex.py := $(call get-stems,tex.py,default) default_stems.rst := $(call get-stems,rst,default) default_stems.lhs := $(call get-stems,lhs,default) default_stems.mp := $(call get-stems,mp,default) default_stems.fig := $(call get-stems,fig,default) default_stems.gpi := $(call get-stems,gpi,default) default_stems.dot := $(call get-stems,dot,default) default_stems.xvg := $(call get-stems,xvg,default) default_stems.svg := $(call get-stems,svg,default) default_stems.png := $(call get-stems,png,default) default_stems.jpg := $(call get-stems,jpg,default) default_stems.jpeg := $(call get-stems,jpeg,default) default_stems.eps.gz := $(call get-stems,eps.gz,default) default_stems.eps := $(call get-stems,eps,default) # List of all stems (all possible bare PDF targets created here): stems.tex := $(call get-stems,tex) stems.tex.sh := $(call get-stems,tex.sh) stems.tex.pl := $(call get-stems,tex.pl) stems.tex.py := $(call get-stems,tex.py) stems.rst := $(call get-stems,rst) stems.lhs := $(call get-stems,lhs) stems.mp := $(call get-stems,mp) stems.fig := $(call get-stems,fig) stems.gpi := $(call get-stems,gpi) stems.dot := $(call get-stems,dot) stems.xvg := $(call get-stems,xvg) stems.svg := $(call get-stems,svg) stems.png := $(call get-stems,png) stems.jpg := $(call get-stems,jpg) stems.jpeg := $(call get-stems,jpeg) stems.eps.gz := $(call get-stems,eps.gz) stems.eps := $(call get-stems,eps) # Utility function for creating larger lists of stems # $(call concat-stems,suffixes,[prefix]) concat-stems = $(sort $(foreach s,$1,$($(if $2,$2_,)stems.$s))) # The most likely to be source but not finished product go first graphic_source_extensions := mp \ fig \ gpi \ xvg \ svg \ dot \ eps.gz ifeq "$(strip $(BUILD_STRATEGY))" "latex" graphic_source_extensions += png jpg jpeg eps graphic_target_extensions := eps ps endif ifeq "$(strip $(BUILD_STRATEGY))" "pdflatex" graphic_source_extensions += eps graphic_target_extensions := pdf png jpg jpeg mps tif endif ifeq "$(strip $(BUILD_STRATEGY))" "xelatex" graphic_source_extensions += eps graphic_target_extensions := pdf png jpg jpeg mps tif endif all_stems_source := $(call concat-stems,tex,all) all_stems_source_gen := $(call concat-stems,rst lhs,all) all_stems_script := $(call concat-stems,tex.sh tex.pl tex.py,all) all_stems_graphic := $(call concat-stems,$(graphic_source_extensions),all) all_stems_ss := $(sort $(all_stems_source) $(all_stems_source_gen) $(all_stems_script)) all_stems_sg := $(sort $(all_stems_script) $(all_stems_source_gen)) all_stems_ssg := $(sort $(all_stems_ss)) default_stems_source := $(call concat-stems,tex,default) default_stems_source_gen := $(call concat-stems,rst lhs,default) default_stems_script := $(call concat-stems,tex.sh tex.pl tex.py,default) default_stems_ss := $(sort $(default_stems_source) $(default_stems_source_gen) $(default_stems_script)) default_stems_sg := $(sort $(default_stems_script) $(default_stems_source_gen)) default_stems_ssg := $(sort $(default_stems_ss)) stems_source := $(call concat-stems,tex) stems_source_gen := $(call concat-stems,rst lhs) stems_script := $(call concat-stems,tex.sh tex.pl tex.py) stems_graphic := $(call concat-stems,$(graphic_source_extensions)) stems_gg := $(sort $(stems_graphic)) stems_ss := $(sort $(stems_source) $(stems_source_gen) $(stems_script)) stems_sg := $(sort $(stems_script) $(stems_source_gen)) stems_ssg := $(sort $(stems_ss)) # Calculate names that can generate the need for an include file. We can't # really do this with patterns because it's too easy to screw up, so we create # an exhaustive list. allowed_source_suffixes := \ pdf \ ps \ dvi \ ind \ nls \ bbl \ aux \ aux.make \ d \ auxbbl.make \ _graphics \ _show allowed_source_patterns := $(addprefix %.,$(allowed_source_suffixes)) allowed_graphic_suffixes := \ pdf \ eps \ gpihead.make \ gpi.d allowed_graphic_patterns := $(addprefix %.,$(allowed_graphic_suffixes)) # All targets allowed to build documents allowed_source_targets := \ $(foreach suff,$(allowed_source_suffixes),\ $(addsuffix .$(suff),$(stems_ssg))) # All targets allowed to build graphics allowed_graphic_targets := \ $(foreach suff,$(allowed_graphic_suffixes),\ $(addsuffix .$(suff),$(stems_gg))) # All targets that build multiple documents (like 'all') allowed_batch_source_targets := \ all \ all-pdf \ all-ps \ all-dvi \ all-bbl \ all-ind \ all-gls \ all-nls \ show # All targets that build multiple graphics (independent of document) allowed_batch_graphic_targets := \ all-graphics \ all-pstex \ all-dot2tex \ show-graphics # Now we figure out which stuff is available as a make target for THIS RUN. real_goals := $(call get-default,$(filter-out _includes,$(MAKECMDGOALS)),\ all) specified_source_targets := $(strip \ $(filter $(allowed_source_targets) $(stems_ssg),$(real_goals)) \ ) specified_batch_source_targets := $(strip \ $(filter $(allowed_batch_source_targets),$(real_goals)) \ ) specified_graphic_targets := $(strip \ $(filter $(allowed_graphic_targets),$(real_goals)) \ ) specified_batch_graphic_targets := $(strip \ $(filter $(allowed_batch_graphic_targets),$(real_goals)) \ ) specified_gpi_targets := $(patsubst %.gpi,%.$(default_graphic_extension),\ $(filter $(patsubst %.$(default_graphic_extension),%.gpi,$(specified_graphic_targets)),\ $(all_files.gpi)) \ ) # Determine which .d files need including from the information gained above. # This is done by first checking whether a batch target exists. If it does, # then all *default* stems are used to create possible includes (nobuild need # not apply for batch status). If no batch targets exist, then the individual # targets are considered and appropriate includes are taken from them. source_stems_to_include := \ $(sort\ $(if $(specified_batch_source_targets),\ $(default_stems_ss),\ $(foreach t,$(specified_source_targets),\ $(foreach p,$(allowed_source_patterns),\ $(patsubst $p,%,$(filter $p $(stems_ssg),$t)) \ )) \ )) # Determine which .gpi.d files are needed using the above information. We # first check whether a batch target is specified, then check individual # graphics that may have been specified. graphic_stems_to_include := \ $(sort\ $(if $(specified_batch_graphic_targets),\ $(default_stems.gpi),\ $(foreach t,$(specified_gpi_targets),\ $(foreach p,$(allowed_graphic_patterns),\ $(patsubst $p,%,$(filter $p,$t)) \ )) \ )) # All dependencies for the 'all' targets all_pdf_targets := $(addsuffix .pdf,$(stems_ssg)) all_ps_targets := $(addsuffix .ps,$(stems_ssg)) all_dvi_targets := $(addsuffix .dvi,$(stems_ssg)) all_tex_targets := $(addsuffix .tex,$(stems_sg)) all_d_targets := $(addsuffix .d,$(stems_ssg)) all_graphics_targets := $(addsuffix .$(default_graphic_extension),$(stems_gg)) all_pstex_targets := $(addsuffix .pstex_t,$(stems.fig)) all_dot2tex_targets := $(addsuffix .dot_t,$(stems.dot)) all_known_graphics := $(sort $(all_graphics_targets) $(wildcard *.$(default_graphic_extension))) default_pdf_targets := $(addsuffix .pdf,$(default_stems_ss)) ifeq "$(strip $(BUILD_STRATEGY))" "latex" default_ps_targets := $(addsuffix .ps,$(default_stems_ss)) default_dvi_targets := $(addsuffix .dvi,$(default_stems_ss)) pre_pdf_extensions := dvi ps endif # Extensions generated by LaTeX invocation that can be removed when complete rm_ext := \ log *.log aux $(pre_pdf_extensions) ps pdf blg bbl out nav snm toc lof lot lol pfg \ fls vrb idx ind ilg glg glo gls lox nls nlo nlg brf mtc* mlf* mlt* maf brf ist fmt backup_patterns := *~ *.bak *.backup body.tmp head.tmp graph_stem := _graph # All LaTeX-generated files that can be safely removed rm_tex := \ $(foreach e,$(rm_ext),$(addsuffix .$e,$(all_stems_source))) \ $(foreach e,$(rm_ext) tex,$(addsuffix .$e,$(all_stems_sg))) \ $(addsuffix .log,$(all_ps_targets) $(all_pdf_targets)) \ $(addsuffix .*.log,$(stems_graphic)) # These are the files that will affect .gpi transformation for all .gpi files. # # Use only the first one found. Backward compatible values are at the end. # Note that we use foreach, even though wildcard also returns a list, to ensure # that the order in the uppercase variables is preserved. Directory listings # provide no such guarantee, so we avoid relying on them. gpi_sed := $(strip \ $(firstword $(foreach f,$(GNUPLOT_SED),$(wildcard $f)))) gpi_global := $(strip \ $(firstword $(foreach f,$(GNUPLOT_GLOBAL),$(wildcard $f)))) # # Functions used in generating output # # Outputs all source dependencies to stdout. The first argument is the file to # be parsed, the second is a list of files that will show up as dependencies in # the new .d file created here. # # $(call get-inputs,,) define get-inputs $(SED) \ -e '/^INPUT/!d' \ -e 's!^INPUT \(\./\)\{0,1\}!!' \ -e 's/[[:space:]]/\\ /g' \ -e 's/\(.*\)\.aux$$/\1.tex/' \ -e '/\.tex$$/b addtargets' \ -e '/\.cls$$/b addtargets' \ -e '/\.sty$$/b addtargets' \ -e '/\.pstex_t$$/b addtargets' \ -e '/\.dot_t$$/b addtargets' \ -e 'd' \ -e ':addtargets' \ -e 's!.*!$2: $$(call path-norm,&)!' \ '$1' | $(SORT) | $(UNIQ) endef # $(call get-format,,) define get-format $(SED) \ -e '1!d' \ -e '/^%&\([[:alnum:]]\{1,\}\)\( .*\)*$$/{' \ -e ' s!!\1!' \ -e ' h' \ -e ' s/.*/# MISSING format "&.fmt" (comment forces rebuild of target file)/' \ -e ' p' \ -e ' g' \ -e ' s!.*!$2: $$(call path-norm,&.fmt)!' \ -e ' p' \ -e '}' \ -e 'd' \ '$1' endef # $(call get-missing-inputs,,) define get-missing-inputs $(SED) \ -e '$${' \ -e ' /^$$/!{' \ -e ' H' \ -e ' s/.*//' \ -e ' }' \ -e '}' \ -e '/^$$/!{' \ -e ' H' \ -e ' d' \ -e '}' \ -e '/^$$/{' \ -e ' x' \ -e ' s/^\(\n\)\(.*\)/\2\1/' \ -e '}' \ -e '/^::P\(P\{1,\}\)::/{' \ -e ' s//::\1::/' \ -e ' G' \ -e ' h' \ -e ' d' \ -e '}' \ -e '/^::P::/{' \ -e ' s//::0::/' \ -e ' G' \ -e '}' \ -e 'b start' \ -e ':needonemore' \ -e 's/^/::P::/' \ -e 'G' \ -e 'h' \ -e 'd' \ -e ':needtwomore' \ -e 's/^/::PP::/' \ -e 'G' \ -e 'h' \ -e 'd' \ -e ':needthreemore' \ -e 's/^/::PPP::/' \ -e 'G' \ -e 'h' \ -e 'd' \ -e ':start' \ -e '/^! LaTeX Error: File `/{' \ -e ' b needtwomore' \ -e '}' \ -e '/^::0::\(.*\)/{' \ -e ' s//\1/' \ -e ' /Default extension: /!d' \ -e ' s/.*File `\([^'"'"']*\)'"'"' not found.*/\1/' \ -e ' s/[[:cntrl:]]//' \ -e ' /\.tex/!s/$$/.tex/' \ -e ' s/[[:space:]]/\\ /g' \ -e ' h' \ -e ' s/.*/# MISSING input "&" - (presence of comment affects build)/' \ -e ' p' \ -e ' s/.*//' \ -e ' x' \ -e ' s!^.*!$2: $$(call path-norm,&)!' \ -e ' p' \ -e '}' \ -e 'd' \ '$1' | $(SORT) | $(UNIQ) endef # Get source file for specified graphics stem. # # $(call graphics-source,) define graphics-source $(strip $(firstword \ $(wildcard \ $(addprefix $1.,\ $(graphic_source_extensions))) \ $1 \ )) endef # Get the target file for the specified graphics file/stem # # $(call graphics-target,) define graphics-target $(strip $(if $(filter $(addprefix %.,$(graphic_target_extensions)),$1), $1, $(firstword $(patsubst $(addprefix %.,$(graphic_source_extensions) $(graphic_target_extensions)), %, $1).$(default_graphic_extension) $1.$(default_graphic_extension)))) endef # Outputs all of the graphical dependencies to stdout. The first argument is # the stem of the source file being built, the second is a list of suffixes # that will show up as dependencies in the generated .d file. # # Note that we try to escape spaces in filenames where possible. We have to do # it with three backslashes so that as the name percolates through the makefile # it eventually ends up with the proper escaping when the build rule is found. # Ugly, but it appears to work. Note that graphicx doesn't allow filenames # with spaces, so this could in many ways be moot unless you're using something # like grffile. # # For pdflatex, we really need the missing file to be specified without an # extension, otherwise compilation barfs on the first missing file. Truly # annoying, but there you have it. # # It turns out that the graphics errors, although they have lines with empty # space, are only made of two paragraphs. So, we just use some sed magic to # get everything into paragraphs, detect when it's a paragraph that interests # us, and double it up. Then we get the filename only if we're missing # extensions (a sign that it's graphicx complaining). # # $(call get-graphics,) define get-graphics $(SED) \ -e '/^File: \(.*\) Graphic file (type [^)]*)/{' \ -e ' s//\1/' \ -e ' s/\.e\{0,1\}ps$$//' \ -e ' b addtargets' \ -e '}' \ -e '$${' \ -e ' /^$$/!{' \ -e ' H' \ -e ' s/.*//' \ -e ' }' \ -e '}' \ -e '/^$$/!{' \ -e ' H' \ -e ' d' \ -e '}' \ -e '/^$$/{' \ -e ' x' \ -e ' s/^\(\n\)\(.*\)/\2\1/' \ -e '}' \ -e '/^::P\(P\{1,\}\)::/{' \ -e ' s//::\1::/' \ -e ' G' \ -e ' h' \ -e ' d' \ -e '}' \ -e '/^::P::/{' \ -e ' s//::0::/' \ -e ' G' \ -e '}' \ -e 'b start' \ -e ':needonemore' \ -e 's/^/::P::/' \ -e 'G' \ -e 'h' \ -e 'd' \ -e ':needtwomore' \ -e 's/^/::PP::/' \ -e 'G' \ -e 'h' \ -e 'd' \ -e ':needthreemore' \ -e 's/^/::PPP::/' \ -e 'G' \ -e 'h' \ -e 'd' \ -e ':start' \ -e '/^[^[:cntrl:]:]*:[[:digit:]]\{1,\}:[[:space:][:cntrl:]]*LaTeX[[:space:][:cntrl:]]*Error:[[:space:][:cntrl:]]*File `/{' \ -e ' s/\n//g' \ -e ' b needonemore' \ -e '}' \ -e '/^::0::.*: LaTeX Error: File `/{' \ -e ' /\n\n$$/{' \ -e ' s/^::0:://' \ -e ' b needonemore' \ -e ' }' \ -e ' s/\n\{1,\}/ /g' \ -e ' s/[[:space:]]\{1,\}/ /g' \ -e ' s/^.*File `//' \ -e ' /extensions: /{' \ -e ' s/'"'"' not found\..*extensions: \([^[:space:]]*\).*/::::\1/' \ -e ' b fileparsed' \ -e ' }' \ -e ' s/'"'"' not found\..*/::::/' \ -e ' :fileparsed' \ -e ' s/\.e\{0,1\}ps::::$$/::::/' \ -e ' h' \ -e ' s/\(.*\)::::\(.*\)/# MISSING stem "\1" - allowed extensions are "\2" - leave comment here - it affects the build/' \ -e ' p' \ -e ' g' \ -e ' s/::::.*//' \ -e ' b addtargets' \ -e '}' \ -e 'd' \ -e ':addtargets' \ -e 's/[[:space:]]/\\\\\\&/g' \ -e 'h' \ -e 's/.*/-include &.gpi.d/' \ -e 'p' \ -e 'g' \ -e 's!.*!$1.d: $$$$(call graphics-source,&)!' \ -e 'p' \ -e 's/.*//' \ -e 'x' \ -e 's!.*!$1.$(build_target_extension) $1._graphics: $$$$(call graphics-target,&)!' \ -e 'p' \ -e 'd' \ $1.log endef # Get some special comments out of the source file (e.g., paper size, beamer # usage, etc.) # # $(call get-source-specials,,) define get-source-specials $(SED) \ -e '/^%%[[:space:]]*\([^%].*\)[[:space:]]*$$/{' \ -e ' s//\1/' \ -e ' s/[[:space:]]//g' \ -e ' /BEAMER/{' \ -e ' /BEAMERLARGE/!d' \ -e ' s/.*/BEAMER/' \ -e ' }' \ -e ' p' \ -e ' d' \ -e '}' \ -e '/\\documentclass[^{]*{.*}/b procclass' \ -e '/\\documentclass/,/}/{' \ -e ' s/%.*//' \ -e ' H' \ -e ' /}/{' \ -e ' s/.*//' \ -e ' x' \ -e ' b procclass' \ -e ' }' \ -e ' d' \ -e '}' \ -e 'd' \ -e ':procclass' \ -e 's/\\documentclass//' \ -e 's/[][]//g' \ -e 's/{.*}//' \ -e 's/[[:cntrl:][:space:]]*//g' \ -e 's/,/ /g' \ -e 's/^/ /' \ -e 's/$$/ /' \ -e '/.* \([[:alnum:]]\{1,\}\)paper .*/{' \ -e ' h' \ -e ' s//PAPERSIZE=\1/' \ -e ' p' \ -e ' g' \ -e '}' \ -e '/.* landscape .*/{' \ -e ' h' \ -e ' s//ORIENTATION=landscape/' \ -e ' p' \ -e ' g' \ -e '}' \ -e 'd' \ $1 > '$2' endef # Checks for build failure due to pstex inclusion, and gives instructions. # # $(call die-on-pstexs,) define die-on-pstexs if $(EGREP) -q '^! LaTeX Error: File .*\.pstex.* not found' $1; then \ $(ECHO) "$(C_ERROR)Missing pstex_t file(s)$(C_RESET)"; \ $(ECHO) "$(C_ERROR)Please run$(C_RESET)"; \ $(ECHO) "$(C_ERROR) make all-pstex$(C_RESET)"; \ $(ECHO) "$(C_ERROR)before proceeding.$(C_RESET)"; \ exit 1; \ fi endef # Checks for the use of import.sty and bails - we don't support subdirectories # # $(call die-on-import-sty,) define die-on-import-sty if $(EGREP) -s '/import.sty\)' '$1'; then \ $(ECHO) "$(C_ERROR)import.sty is not supported - included files must"; \ $(ECHO) "$(C_ERROR)be in the same directory as the primary document$(C_RESET)"; \ exit 1; \ fi endef # Checks for build failure due to dot2tex, and gives instructions. # # $(call die-on-dot2tex,) define die-on-dot2tex if $(EGREP) -q ' LaTeX Error: File .*\.dot_t.* not found' $1; then \ $(ECHO) "$(C_ERROR)Missing dot_t file(s)$(C_RESET)"; \ $(ECHO) "$(C_ERROR)Please run$(C_RESET)"; \ $(ECHO) "$(C_ERROR) make all-dot2tex$(C_RESET)"; \ $(ECHO) "$(C_ERROR)before proceeding.$(C_RESET)"; \ exit 1; \ fi endef # Checks for the existence of a .aux file, and dies with an error message if it # isn't there. Note that we pass the file stem in, not the full filename, # e.g., to check for foo.aux, we call it thus: $(call die-on-no-aux,foo) # # $(call die-on-no-aux,) define die-on-no-aux if $(call test-not-exists,$1.aux); then \ $(call colorize-latex-errors,$1.log); \ $(ECHO) "$(C_ERROR)Error: failed to create $1.aux$(C_RESET)"; \ exit 1; \ fi endef # Outputs all index files to stdout. Arg 1 is the source file stem, arg 2 is # the list of targets for the discovered dependency. # # $(call get-log-index,,) define get-log-index $(SED) \ -e 's/^No file \(.*\.ind\)\.$$/TARGETS=\1/' \ -e 's/^No file \(.*\.[gn]ls\)\.$$/TARGETS=\1/' \ -e 's/[[:space:]]/\\&/g' \ -e '/^TARGETS=/{' \ -e ' h' \ -e ' s!^TARGETS=!$2: !p' \ -e ' g' \ -e ' s!^TARGETS=\(.*\)!\1: $1.tex!p' \ -e '}' \ -e 'd' \ '$1.log' | $(SORT) | $(UNIQ) endef # Outputs all bibliography files to stdout. Arg 1 is the source stem, arg 2 is # a list of targets for each dependency found. # # The script kills all lines that do not contain bibdata. Remaining lines have # the \bibdata macro and delimiters removed to create a dependency list. A # trailing comma is added, then all adjacent commas are collapsed into a single # comma. Then commas are replaced with the string .bib[space], and the # trailing space is killed off. Finally, all filename spaces are escaped. # This produces a list of space-delimited .bib filenames, which is what the # make dep file expects to see. # # Note that we give kpsewhich a bogus argument so that a failure of sed to # produce output will not cause an error. # # $(call get-bibs,,) define get-bibs $(SED) \ -e '/^\\bibdata/!d' \ -e 's/\\bibdata{\([^}]*\)}/\1,/' \ -e 's/,\{2,\}/,/g' \ -e 's/[[:space:]]/\\&/g' \ -e 's/,/.bib /g' \ -e 's/ \{1,\}$$//' \ '$1' | $(XARGS) $(KPSEWHICH) '#######' | \ $(SED) \ -e 's!.*!$2: $$(call path-norm,&)!' | \ $(SORT) | $(UNIQ) endef # Makes a an aux file that only has stuff relevant to the target in it # $(call make-auxtarget-file,,) define make-auxtarget-file $(SED) \ -e '/^\\newlabel/!d' \ $1 > $2 endef # Makes an aux file that only has stuff relevant to the bbl in it # $(call make-auxbbl-file,,) define make-auxbbl-file $(SED) \ -e '/^\\newlabel/d' \ $1 > $2 endef # Makes a .gpi.d file from a .gpi file # $(call make-gpi-d,<.gpi>,<.gpi.d>) define make-gpi-d $(ECHO) '# vim: ft=make' > $2; \ $(ECHO) 'ifndef INCLUDED_$(call cleanse-filename,$2)' >> $2; \ $(ECHO) 'INCLUDED_$(call cleanse-filename,$2) := 1' >> $2; \ $(call get-gpi-deps,$1,$(addprefix $(2:%.gpi.d=%).,$(GPI_OUTPUT_EXTENSION) gpi.d)) >> $2; \ $(ECHO) 'endif' >> $2; endef # Parse .gpi files for data and loaded dependencies, output to stdout # # The sed script here tries to be clever about obtaining valid # filenames from the gpi file. It assumes that the plot command starts its own # line, which is not too difficult a constraint to satisfy. # # This command script also generates 'include' directives for every 'load' # command in the .gpi file. The load command must appear on a line by itself # and the file it loads must have the suffix .gpi. If you don't want it to be # compiled when running make graphics, then give it a suffix of ._include_.gpi. # # $(call get-gpi-deps,,) define get-gpi-deps $(SED) \ -e '/^[[:space:]]*s\{0,1\}plot/,/[^\\]$$/{' \ -e ' H' \ -e ' /[^\\]$$/{' \ -e ' s/.*//' \ -e ' x' \ -e ' s/\\\{0,1\}\n//g' \ -e ' s/^[[:space:]]*s\{0,1\}plot[[:space:]]*\(\[[^]]*\][[:space:]]*\)*/,/' \ -e ' s/[[:space:]]*\(['"'"'\'"'"''"'"'"][^'"'"'\'"'"''"'"'"]*['"'"'\'"'"''"'"'"]\)\{0,1\}[^,]*/\1/g' \ -e ' s/,['"'"'\'"'"''"'"'"]-\{0,1\}['"'"'\'"'"''"'"'"]//g' \ -e ' s/[,'"'"'\'"'"''"'"'"]\{1,\}/ /g' \ -e ' s/.*:.*/$$(error Error: Filenames with colons are not allowed: &)/' \ -e ' s!.*!$2: &!' \ -e ' p' \ -e ' }' \ -e ' d' \ -e '}' \ -e 's/^[[:space:]]*load[[:space:]]*['"'"'\'"'"''"'"'"]\([^'"'"'\'"'"''"'"'"]*\.gpi\)['"'"'\'"'"''"'"'"].*$$/-include \1.d/p' \ -e 'd' \ '$1' endef # Colorizes real, honest-to-goodness LaTeX errors that can't be overcome with # recompilation. # # Note that we only ignore file not found errors for things that we know how to # build, like graphics files. # # Also note that the output of this is piped through sed again to escape any # backslashes that might have made it through. This is to avoid sending things # like "\right" to echo, which interprets \r as LF. In bash, we could just do # ${var//\\/\\\\}, but in other popular sh variants (like dash), this doesn't # work. # # $(call colorize-latex-errors,) define colorize-latex-errors $(SED) \ -e '$${' \ -e ' /^$$/!{' \ -e ' H' \ -e ' s/.*//' \ -e ' }' \ -e '}' \ -e '/^$$/!{' \ -e ' H' \ -e ' d' \ -e '}' \ -e '/^$$/{' \ -e ' x' \ -e ' s/^\(\n\)\(.*\)/\2\1/' \ -e '}' \ -e '/^::P\(P\{1,\}\)::/{' \ -e ' s//::\1::/' \ -e ' G' \ -e ' h' \ -e ' d' \ -e '}' \ -e '/^::P::/{' \ -e ' s//::0::/' \ -e ' G' \ -e '}' \ -e 'b start' \ -e ':needonemore' \ -e 's/^/::P::/' \ -e 'G' \ -e 'h' \ -e 'd' \ -e ':needtwomore' \ -e 's/^/::PP::/' \ -e 'G' \ -e 'h' \ -e 'd' \ -e ':needthreemore' \ -e 's/^/::PPP::/' \ -e 'G' \ -e 'h' \ -e 'd' \ -e ':start' \ -e '/^! LaTeX Error: File /{' \ -e ' s/\n//g' \ -e ' b needtwomore' \ -e '}' \ -e 's/^[^[:cntrl:]:]*:[[:digit:]]\{1,\}:/!!! &/' \ -e 's/^\(.*\n\)\([^[:cntrl:]:]*:[[:digit:]]\{1,\}: .*\)/\1!!! \2/' \ -e '/^!!! .*[[:space:][:cntrl:]]LaTeX[[:space:][:cntrl:]]Error:[[:space:][:cntrl:]]*File /{' \ -e ' s/\n//g' \ -e ' b needonemore' \ -e '}' \ -e '/^::0::! LaTeX Error: File .*/{' \ -e ' /\n\n$$/{' \ -e ' s/^::0:://' \ -e ' b needonemore' \ -e ' }' \ -e ' s/^::0::! //' \ -e ' s/^\(.*not found.\).*Enter file name:.*\n\(.*[[:digit:]]\{1,\}\): Emergency stop.*/\2: \1/' \ -e ' b error' \ -e '}' \ -e '/^::0::!!! .*LaTeX Error: File .*/{' \ -e ' /\n\n$$/{' \ -e ' s/^::0:://' \ -e ' b needonemore' \ -e ' }' \ -e ' s/::0::!!! //' \ -e ' /File .*\.e\{0,1\}ps'"'"' not found/b skip' \ -e ' /could not locate.*any of these extensions:/{' \ -e ' b skip' \ -e ' }' \ -e ' s/\(not found\.\).*/\1/' \ -e ' s/^/!!! /' \ -e ' b error' \ -e '}' \ -e '/^\(.* LaTeX Error: Missing .begin.document.\.\).*/{' \ -e ' s//\1 --- Are you trying to build an include file?/' \ -e ' b error' \ -e '}' \ -e '/.*\(!!! .*Undefined control sequence\)[^[:cntrl:]]*\(.*\)/{' \ -e ' s//\1: \2/' \ -e ' s/\nl\.[[:digit:]][^[:cntrl:]]*\(\\[^\\[:cntrl:]]*\).*/\1/' \ -e ' b error' \ -e '}' \ -e '/^\(!pdfTeX error:.*\)s*/{' \ -e ' b error' \ -e '}' \ -e '/.*\(!!! .*\)/{' \ -e ' s//\1/' \ -e ' s/[[:cntrl:]]//' \ -e ' s/[[:cntrl:]]$$//' \ -e ' /Cannot determine size of graphic .*(no BoundingBox)/b skip' \ -e ' b error' \ -e '}' \ -e ':skip' \ -e 'd' \ -e ':error' \ -e 's/^!\(!! \)\{0,1\}\(.*\)/$(C_ERROR)\2$(C_RESET)/' \ -e 'p' \ -e 'd' \ '$1' | $(SED) -e 's/\\\\/\\\\\\\\/g' endef # Colorize Makeindex errors define colorize-makeindex-errors $(SED) \ -e '/^!! /{' \ -e ' N' \ -e ' s/^.*$$/$(C_ERROR)&$(C_RESET)/' \ -e ' p' \ -e '}' \ -e 'd' \ '$1' endef # Colorize xindy errors # $(call colorize-xindy-errors,) define colorize-xindy-errors $(SED) \ -e 's/^xindy:.*/$(C_ERROR)&$(C_RESET)/p' \ -e 'd' \ '$1' endef # Colorize epstopdf errors # # $(call colorize-epstopdf-errors,) define colorize-epstopdf-errors $(SED) \ -e '/^Error:/,/^Execution stack:/{' \ -e ' /^Execution stack:/d' \ -e ' s/.*/$(C_ERROR)&$(C_RESET)/' \ -e ' p' \ -e '}' \ -e 'd' \ '$1' endef # Colorize GNUplot errors # # $(call colorize-gnuplot-errors,) define colorize-gnuplot-errors $(SED) \ -e '/, line [0-9]*:/!{' \ -e ' H' \ -e ' x' \ -e ' s/.*\n\(.*\n.*\)$$/\1/' \ -e ' x' \ -e '}' \ -e '/, line [0-9]*:/{' \ -e ' H' \ -e ' /unknown.*terminal type/{' \ -e ' s/.*/--- Try changing the GNUPLOT_OUTPUT_EXTENSION variable to '"'"'eps'"'"'./' \ -e ' H' \ -e ' }' \ -e ' /gpihead/{' \ -e ' s/.*/--- This could be a Makefile bug - contact the maintainer./' \ -e ' H' \ -e ' }' \ -e ' g' \ -e ' s/.*/$(C_ERROR)&$(C_RESET)/' \ -e ' p' \ -e '}' \ -e '/^gnuplot>/,/^$$/{' \ -e ' s/^gnuplot.*/$(C_ERROR)&/' \ -e ' s/^$$/$(C_RESET)/' \ -e ' p' \ -e '}' \ -e 'd' \ $1 endef # Colorize GraphViz errors # # $(call colorize-dot-errors,) define colorize-dot-errors $(SED) \ -e 's/.*not found.*/$(C_ERROR)&$(C_RESET)/p' \ -e '/^Error:/,/context:/s/.*/$(C_ERROR)&$(C_RESET)/p' \ -e 's/^Warning:.*/$(C_WARNING)&$(C_RESET)/p' \ -e 'd' \ '$1' endef # Get all important .aux files from the top-level .aux file and merges them all # into a single file, which it outputs to stdout. # It does this by finding all \input commands and creating a new sed script # that reads those files inline when it encounters one. # It then runs that sed script, generating a flattened aux file, and # then it cleans up the flattened file by removing some crufty things. # # $(call flatten-aux,,) define flatten-aux $(SED) \ -e '/\\@input{\(.*\)}/{' \ -e ' s//\1/' \ -e ' s![.:]!\\&!g' \ -e ' h' \ -e ' s!.*!\\:\\\\@input{&}:{!' \ -e ' p' \ -e ' x' \ -e ' s/\\././g' \ -e ' s/.*/r &/p' \ -e ' s/.*/d/p' \ -e ' s/.*/}/p' \ -e ' d' \ -e '}' \ -e 'd' \ '$1' > "$1.$$$$.sed.make"; \ $(SED) -f "$1.$$$$.sed.make" '$1' > "$1.$$$$.make"; \ $(SED) \ -e '/^\\relax/d' \ -e '/^\\bibcite/d' \ -e 's/^\(\\newlabel{[^}]\{1,\}}\).*/\1/' \ "$1.$$$$.make" | $(SORT) > '$2'; \ $(call remove-temporary-files,$1.$$$$.make $1.$$$$.sed.make) endef # Generate pdf from postscript # # Note that we don't just call ps2pdf, since there are so many versions of that # script on various systems. Instead, we call the postscript interpreter # directly. # # $(call ps2pdf,infile,outfile,[embed fonts]) define ps2pdf $(GS) \ -dSAFER -dCompatibilityLevel=$(PS_COMPATIBILITY) \ $(if $3,$(PS_EMBED_OPTIONS)) \ -q -dNOPAUSE -dBATCH \ -sDEVICE=pdfwrite -sstdout=%stderr \ '-sOutputFile=$2' \ -dSAFER -dCompatibilityLevel=$(PS_COMPATIBILITY) \ $(if $3,$(PS_EMBED_OPTIONS)) \ -c .setpdfwrite \ -f '$1' endef # Colorize LaTeX output. color_tex := \ $(SED) \ -e '$${' \ -e ' /^$$/!{' \ -e ' H' \ -e ' s/.*//' \ -e ' }' \ -e '}' \ -e '/^$$/!{' \ -e ' H' \ -e ' d' \ -e '}' \ -e '/^$$/{' \ -e ' x' \ -e ' s/^\n//' \ -e ' /Output written on /{' \ -e ' s/.*Output written on \([^(]*\) (\([^)]\{1,\}\)).*/Success! Wrote \2 to \1/' \ -e ' s/[[:digit:]]\{1,\}/$(C_PAGES)&$(C_RESET)/g' \ -e ' s/Success!/$(C_SUCCESS)&$(C_RESET)/g' \ -e ' s/to \(.*\)$$/to $(C_SUCCESS)\1$(C_RESET)/' \ -e ' b end' \ -e ' }' \ -e ' / *LaTeX Error:.*/{' \ -e ' s/.*\( *LaTeX Error:.*\)/\1/' \ -e ' b error' \ -e ' }' \ -e ' /^[^[:cntrl:]:]*:[[:digit:]]\{1,\}:/b error' \ -e ' /.*Warning:.*/{' \ -e ' s//$(C_WARNING)&$(C_RESET)/' \ -e ' b end' \ -e ' }' \ -e ' /Underfull.*/{' \ -e ' s/.*\(Underfull.*\)/$(C_UNDERFULL)\1$(C_RESET)/' \ -e ' b end' \ -e ' }' \ -e ' /Overfull.*/{' \ -e ' s/.*\(Overfull.*\)/$(C_OVERFULL)\1$(C_RESET)/' \ -e ' b end' \ -e ' }' \ -e ' d' \ -e ' :error' \ -e ' s/.*/$(C_ERROR)&$(C_RESET)/' \ -e ' b end' \ -e ' :end' \ -e ' G' \ -e '}' # Colorize BibTeX output. color_bib := \ $(SED) \ -e 's/^Warning--.*/$(C_WARNING)&$(C_RESET)/' \ -e 't' \ -e '/---/,/^.[^:]/{' \ -e ' H' \ -e ' /^.[^:]/{' \ -e ' x' \ -e ' s/\n\(.*\)/$(C_ERROR)\1$(C_RESET)/' \ -e ' p' \ -e ' s/.*//' \ -e ' h' \ -e ' d' \ -e ' }' \ -e ' d' \ -e '}' \ -e '/(.*error.*)/s//$(C_ERROR)&$(C_RESET)/' \ -e 'd' # Make beamer output big enough to print on a full page. Landscape doesn't # seem to work correctly. enlarge_beamer = $(PSNUP) -l -1 -W128mm -H96mm -pletter # $(call test-run-again,) define test-run-again $(EGREP) '^(.*Rerun .*|No file $1\.[^.]+\.)$$' $1.log \ | $(EGREP) -q -v '^(Package: rerunfilecheck.*Rerun checks for auxiliary files.*)$$' endef # This tests whether the build target commands should be run at all, from # viewing the log file. # $(call test-log-for-need-to-run,) define test-log-for-need-to-run $(SED) \ -e '/^No file $(call escape-fname-regex,$1)\.aux\./d' \ $1.log \ | $(EGREP) '^(.*Rerun .*|No file $1\.[^.]+\.|No file .+\.tex\.|LaTeX Warning: File.*)$$' \ | $(EGREP) -q -v '^(Package: rerunfilecheck.*Rerun checks for auxiliary files.*)$$' endef # LaTeX invocations # # Note that we use # # -recorder: generates .fls files for things that are input and output # -jobname: to make sure that .fls files are named after the source that created them # -file-line-error: to make sure that we have good line information for error output # -interaction=batchmode: so that we don't stop on errors (we'll parse logs for that) # # $(call latex,,[extra LaTeX args]) define run-latex $(latex_build_program) -jobname='$1' -interaction=batchmode -file-line-error $(LATEX_OPTS) $(if $2,$2,) $1 > /dev/null; \ $(call transcript,latex,$1) endef # $(call latex-color-log,) latex-color-log = $(color_tex) $1.log # $(call run-makeindex,,,,[.ist style file]) define run-makeindex success=1; \ if ! $(MAKEINDEX) -q $1 -t $3 -o $2 $(if $4,-s $4,) > /dev/null || $(EGREP) -q '^!!' $3; \ then \ $(call colorize-makeindex-errors,$3); \ $(RM) -f '$2'; \ success=0; \ $(call transcript,makeindex,$1); \ fi; \ [ "$$success" = "1" ] && $(sh_true) || $(sh_false); endef # $(call run-xindy,,,,) define run-xindy success=1; \ if ! $(XINDY) -q -o $2 -L $(XINDYLANG) -C $(XINDYENC) -I xindy -M $3 -t $4 $1 > /dev/null || $(EGREP) -q '^xindy:' $4; then \ $(call colorize-xindy-errors,$4); \ $(RM) -f '$2'; \ success=0; \ $(call transcript,xindy,$1); \ fi; \ [ "$$success" = "1" ] && $(sh_true) || $(sh_false); endef # This runs the given script to generate output, and it uses MAKE_RESTARTS to # ensure that it never runs it more than once for a particular root make # invocation. # # $(call run-script,,,) define run-script $(call test-not-exists,$2.cookie) && $(ECHO) "restarts=$(RESTARTS)" \ > $2.cookie && $(ECHO) "level=$(MAKELEVEL)" >> $2.cookie; \ restarts=`$(SED) -n -e 's/^restarts=//p' $2.cookie`; \ level=`$(SED) -n -e 's/^level=//p' $2.cookie`; \ if $(EXPR) $(MAKELEVEL) '<=' $$level '&' $(RESTARTS) '<=' $$restarts >/dev/null; then \ $(call echo-build,$2,$3,$(RESTARTS)-$(MAKELEVEL)); \ $1 '$2' '$3'; \ $(ECHO) "restarts=$(RESTARTS)" > '$2.cookie'; \ $(ECHO) "level=$(MAKELEVEL)" >> '$2.cookie'; \ fi endef # BibTeX invocations # # $(call run-bibtex,) run-bibtex = $(BIBTEX) $1 | $(color_bib); $(call transcript,bibtex,$1) # $(call convert-eps-to-pdf,,,[gray]) # Note that we don't use the --filter flag because it has trouble with bounding boxes that way. define convert-eps-to-pdf $(if $3,$(CAT) '$1' | $(call kill-ps-color) > '$1.cookie',$(CP) '$1' '$1.cookie'); \ $(EPSTOPDF) '$1.cookie' --outfile='$2' > $1.log; \ $(call colorize-epstopdf-errors,$1.log); endef # $(call default-gpi-fontsize,) # # Find the default fontsize given the *output* file (it is based on the output extension) # default-gpi-fontsize = $(if $(filter %.pdf,$1),$(DEFAULT_GPI_PDF_FONTSIZE),$(DEFAULT_GPI_EPS_FONTSIZE)) # $(call gpi-fontsize,,) # # Find out what the gnuplot fontsize should be. Tries, in this order: # - ##FONTSIZE comment in gpi file # - ##FONTSIZE comment in global gpi file # - default fontsize based on output type define gpi-fontsize $(strip $(firstword \ $(shell $(SED) -e 's/^\#\#FONTSIZE=\([[:digit:]]\{1,\}\)/\1/p' -e 'd' $1 $(strip $(gpi_global))) \ $(call default-gpi-fontsize,$2))) endef # $(call gpi-monochrome,,[gray]) define gpi-monochrome $(strip $(if $2,monochrome,$(if $(shell $(EGREP) '^\#\#[[:space:]]*GRAY[[:space:]]*$$' $1 $(gpi_global)),monochrome,color))) endef # $(call gpi-font-entry,,) # # Get the font entry given the output file (type) and the font size. For PDF # it uses fsize or font, for eps it just uses the bare number. gpi-font-entry = $(if $(filter %.pdf,$1),$(subst FONTSIZE,$2,$(GPI_FSIZE_SYNTAX)),$2) # $(call gpi-terminal,,[gray]) # # Get the terminal settings for a given gpi and its intended output file define gpi-terminal $(if $(filter %.pdf,$2),pdf enhanced,postscript enhanced eps) \ $(call gpi-font-entry,$2,$(call gpi-fontsize,$1,$2)) \ $(call gpi-monochrome,$1,$3) endef # $(call gpi-embed-pdf-fonts,,) # define gpi-embed-pdf-fonts $(GS) \ -q \ -dSAFER \ -dNOPAUSE \ -dBATCH \ -sDEVICE=$(if $(filter pdf,$(GPI_OUTPUT_EXTENSION)),pdfwrite,pswrite) \ -sOutputFile='$2' \ -sstdout=%stderr \ -dColorConversionStrategy=/LeaveColorUnchanged \ -dCompatibilityLevel=1.5 \ -dPDFSETTINGS=/prepress \ -c .setpdfwrite \ -f '$1' endef # $(call convert-gpi,,,[gray]) # define convert-gpi $(ECHO) 'set terminal $(call gpi-terminal,$1,$2,$3)' > $1head.make; \ $(ECHO) 'set output "$2"' >> $1head.make; \ $(if $(gpi_global),$(CAT) $(gpi_global) >> $1head.make;,) \ fnames='$1head.make $1';\ $(if $(gpi_sed),\ $(SED) -f '$(gpi_sed)' $$fnames > $1.temp.make; \ fnames=$1.temp.make;,\ ) \ success=1; \ if ! $(GNUPLOT) $$fnames 2>$1.log; then \ $(call colorize-gnuplot-errors,$1.log); \ success=0; \ elif [ x"$(suffix $2)" = x".pdf" ]; then \ if ! $(call gpi-embed-pdf-fonts,$2,$2.embed.tmp.make); then \ success=0; \ else \ $(call move-if-exists,$2.embed.tmp.make,$2); \ fi; \ fi; \ $(if $(gpi_sed),$(call remove-temporary-files,$1.temp.make);,) \ $(call remove-temporary-files,$1head.make); \ [ "$$success" = "1" ] && $(sh_true) || $(sh_false); endef # Creation of .eps files from .png files # # The intermediate step of PNM (using NetPBM) produces much nicer output than # ImageMagick's "convert" binary. I couldn't get the right combination of # flags to make it look nice, anyway. # # To handle gray scale conversion, we pipe things through ppmtopgm in the # middle. # # $(call convert-png,,) define convert-png $(PNGTOPNM) "$1" \ $(if $3,| $(PPMTOPGM),) \ | $(PNMTOPS) -noturn \ > "$2" endef # Creation of .eps files from .jpg/.jpeg files # # Thanks to brubakee for this solution. # # Uses Postscript level 2 to avoid file size bloat # $(call convert-jpg,,) define convert-jpg $(CONVERT) $(if $3,-type Grayscale,) '$1' eps2:'$2' endef # Creation of .eps files from .fig files # $(call convert-fig,,,[gray]) convert-fig = $(FIG2DEV) -L $(if $(filter %.pdf,$2),pdf,eps) $(if $3,-N,) $1 $2 # Creation of .pstex files from .fig files # $(call convert-fig-pstex,,) convert-fig-pstex = $(FIG2DEV) -L pstex $1 $2 > /dev/null 2>&1 # Creation of .pstex_t files from .fig files # $(call convert-fig-pstex-t,,,) convert-fig-pstex-t = $(FIG2DEV) -L pstex_t -p $3 $1 $2 > /dev/null 2>&1 # Creation of .dot_t files from .dot files # #(call convert-dot-tex,,) convert-dot-tex = $(DOT2TEX) '$1' > '$2' # Converts svg files into .eps files # # $(call convert-svg,,,[gray]) convert-svg = $(INKSCAPE) --without-gui $(if $(filter %.pdf,$2),--export-pdf,--export-eps)='$2' '$1' # Converts xvg files into .eps files # # $(call convert-xvg,,,[gray]) convert-xvg = $(XMGRACE) '$1' -printfile - -hardcopy -hdevice $(if $3,-mono,) EPS > '$2' # Converts .eps.gz files into .eps files # # $(call convert-epsgz,,,[gray]) convert-epsgz = $(GUNZIP) -c '$1' $(if $3,| $(call kill-ps-color)) > '$2' # Generates a .bb file from a .eps file (sometimes latex really wants this) # # $(call eps-bb,,) define eps-bb $(SED) \ -e '/^%%Title:/p' \ -e '/^%%Creator:/p' \ -e '/^%%BoundingBox:/p' \ -e '/^%%CreationDate:/p' \ -e '/^%%EndComments/{' \ -e ' d' \ -e ' q' \ -e '}' \ -e 'd' \ $1 > "$2" endef # Converts .eps files into .eps files (usually a no-op, but can make grayscale) # # $(call convert-eps,,,[gray]) convert-eps = $(if $3,$(call kill-ps-color) $1 > $2) # The name of the file containing special postscript commands for grayscale gray_eps_file := gray.eps.make # Changes sethsbcolor and setrgbcolor calls in postscript to always produce # grayscale. In general, this is accomplished by writing new versions of those # functions into the user dictionary space, which is looked up before the # global or system dictionaries (userdict is one of the permanent dictionaries # in postscript and is not read-only like systemdict). # # For setrgbcolor, the weighted average of the triple is computed and the # triple is replaced with three copies of that average before the original # procedure is called: .299R + .587G + .114B # # For sethsbcolor, the color is first converted to RGB, then to grayscale by # the new setrgbcolor operator as described above. Why is this done? # Because simply using the value component will tend to make pure colors # white, a very undesirable thing. Pure blue should not translate to white, # but to some level of gray. Conversion to RGB does the right thing. It's # messy, but it works. # # From # http://en.wikipedia.org/wiki/HSV_color_space#Transformation_from_HSV_to_RGB, # HSB = HSV (Value = Brightness), and the formula used to convert to RGB is # as follows: # # Hi = int(floor(6 * H)) mod 6 # f = 6 * H - Hi # p = V(1-S) # q = V(1-fS) # t = V(1-(1-f)S) # if Hi = 0: R G B <-- V t p # if Hi = 1: R G B <-- q V p # if Hi = 2: R G B <-- p V t # if Hi = 3: R G B <-- p q V # if Hi = 4: R G B <-- t p V # if Hi = 5: R G B <-- V p q # # The messy stack-based implementation is below # $(call create-gray-eps-file,filename) define create-gray-eps-file $(ECHO) -n -e '\ /OLDRGB /setrgbcolor load def\n\ /setrgbcolor {\n\ .114 mul exch\n\ .587 mul add exch\n\ .299 mul add\n\ dup dup\n\ OLDRGB\n\ } bind def\n\ /OLDHSB /sethsbcolor load def\n\ /sethsbcolor {\n\ 2 index % H V S H\n\ 6 mul floor cvi 6 mod % Hi V S H\n\ 3 index % H Hi V S H\n\ 6 mul % 6H Hi V S H\n\ 1 index % Hi 6H Hi V S H\n\ sub % f Hi V S H\n\ 2 index 1 % 1 V f Hi V S H\n\ 4 index % S 1 V f Hi V S H\n\ sub mul % p f Hi V S H\n\ 3 index 1 % 1 V p f Hi V S H\n\ 6 index % S 1 V p f Hi V S H\n\ 4 index % f S 1 V p f Hi V S H\n\ mul sub mul % q p f Hi V S H\n\ 4 index 1 1 % 1 1 V q p f Hi V S H\n\ 5 index % f 1 1 V q p f Hi V S H\n\ sub % (1-f) 1 V q p f Hi V S H\n\ 8 index % S (1-f) 1 V q p f Hi V S H\n\ mul sub mul % t q p f Hi V S H\n\ 4 -1 roll pop % t q p Hi V S H\n\ 7 -2 roll pop pop % t q p Hi V\n\ 5 -2 roll % Hi V t q p\n\ dup 0 eq\n\ {1 index 3 index 6 index}\n\ {\n\ dup 1 eq\n\ {3 index 2 index 6 index}\n\ {\n\ dup 2 eq\n\ {4 index 2 index 4 index}\n\ {\n\ dup 3 eq\n\ {4 index 4 index 3 index}\n\ {\n\ dup 4 eq\n\ {2 index 5 index 3 index}\n\ {\n\ dup 5 eq\n\ {1 index 5 index 5 index}\n\ {0 0 0}\n\ ifelse\n\ }\n\ ifelse\n\ }\n\ ifelse\n\ }\n\ ifelse\n\ }\n\ ifelse\n\ }\n\ ifelse % B G R Hi V t q p\n\ setrgbcolor\n\ 5 {pop} repeat\n\ } bind def\n'\ > $1 endef # This actually inserts the color-killing code into a postscript file # $(call kill-ps-color) define kill-ps-color $(SED) -e '/%%EndComments/r $(gray_eps_file)' endef # Converts graphviz .dot files into .eps files # Grayscale is not directly supported by dot, so we pipe it through fig2dev in # that case. # $(call convert-dot,,,,[gray]) define convert-dot $(DOT) -Tps '$1' 2>'$3' $(if $4,| $(call kill-ps-color)) > $2; \ $(call colorize-dot-errors,$3) endef # Convert DVI to Postscript # $(call make-ps,,,,[],[]) make-ps = \ $(DVIPS) -z -o '$2' $(if $(strip $4),-t$(strip $4),) '$1' \ $(if $5,| $(enlarge_beamer)) > $3 2>&1 # Convert Postscript to PDF # $(call make-pdf,,,,) make-pdf = \ $(call ps2pdf,$1,$2,$(filter 1,$(shell $(CAT) '$4'))) > '$3' 2>&1 # Display information about what is being done # $(call echo-build,,,[]) echo-build = $(ECHO) "$(C_BUILD)= $1 --> $2$(if $3, ($3),) =$(C_RESET)" echo-graphic = $(ECHO) "$(C_GRAPHIC)= $1 --> $2 =$(C_RESET)" echo-dep = $(ECHO) "$(C_DEP)= $1 --> $2 =$(C_RESET)" # Display a list of something # $(call echo-list,) echo-list = for x in $1; do $(ECHO) "$$x"; done # # DEFAULT TARGET # .PHONY: all all: $(default_pdf_targets) ; .PHONY: all-pdf all-pdf: $(default_pdf_targets) ; ifeq "$(strip $(BUILD_STRATEGY))" "latex" .PHONY: all-ps all-ps: $(default_ps_targets) ; .PHONY: all-dvi all-dvi: $(default_dvi_targets) ; endif # # VIEWING TARGET # .PHONY: show show: all $(QUIET)for x in $(default_pdf_targets); do \ [ -e "$$x" ] && $(VIEW_PDF) $$x & \ done # # INCLUDES # source_includes := $(addsuffix .d,$(source_stems_to_include)) graphic_includes := $(addsuffix .gpi.d,$(graphic_stems_to_include)) # Check the version of the makefile ifneq "" "$(filter 3.79 3.80,$(MAKE_VERSION))" $(warning $(C_WARNING)Your version of make is too old. Please upgrade.$(C_RESET)) endif # Include only the dependencies used ifneq "" "$(source_includes)" include $(source_includes)$(call include-message,$(source_includes)) endif ifneq "" "$(graphic_includes)" include $(graphic_includes)$(call include-message,$(graphic_includes)) endif # # MAIN TARGETS # # Note that we don't just say %: %.pdf here - this can tend to mess up our # includes, which detect what kind of file we are asking for. For example, # asking to build foo.pdf is much different than asking to build foo when # foo.gpi exists, because we look through all of the goals for *.pdf that # matches *.gpi, then use that to determine which include files we need to # build. # # Thus, we invoke make recursively with better arugments instead, restarting # all of the appropriate machinery. .PHONY: $(default_stems_ss) $(default_stems_ss): %: %.pdf ; # This builds and displays the wanted file. .PHONY: $(addsuffix ._show,$(stems_ssg)) $(addsuffix ._show,$(stems_ssg)): %._show: %.pdf $(QUIET)$(VIEW_PDF) $< & ifeq "$(strip $(BUILD_STRATEGY))" "latex" .SECONDARY: $(all_pdf_targets) %.pdf: %.ps %.embed.make $(QUIET)$(call echo-build,$<,$@) $(QUIET)$(call make-pdf,$<,$@.temp,$@.log,$*.embed.make); \ if [ x"$$?" = x"0" ]; then \ $(if $(VERBOSE),$(CAT) $@.log,:); \ $(RM) -f '$@'; \ $(MV) '$@.temp' '$@'; \ $(TOUCH) '$@'; \ $(call copy-with-logging,$@,$(BINARY_TARGET_DIR)); \ else \ $(CAT) $@.log; \ $(call remove-temporary-files,'$@.temp'); \ $(sh_false); \ fi .SECONDARY: $(all_ps_targets) %.ps: %.dvi %.paper.make %.beamer.make $(QUIET)$(call echo-build,$<,$@) $(QUIET)$(call make-ps,$<,$@.temp,$@.log,\ $(strip $(shell $(CAT) $*.paper.make)),\ $(strip $(shell $(CAT) $*.beamer.make))); \ if [ x"$$?" = x"0" ]; then \ $(if $(VERBOSE),$(CAT) $@.log,:); \ $(RM) -f '$@'; \ $(MV) '$@.temp' '$@'; \ $(TOUCH) '$@'; \ $(call copy-with-logging,$@,$(BINARY_TARGET_DIR)); \ else \ $(CAT) $@.log; \ $(call remove-temporary-files,'$@.temp'); \ $(sh_false); \ fi endif # Build the final target (dvi or pdf) file. This is a very tricky rule because # of the way that latex runs multiple times, needs graphics after the first run # (or maybe already has them), and relies on bibliographies or indices that may # not exist. # # Check the log for fatal errors. If they exist, colorize and bail. # # Create the .auxtarget.cookie file. (Needed for next time if not present) # # If any of the following are true, we must rebuild at least one time: # # * the .bbl was recently rebuilt # # check a cookie, then delete it # # * any of several output files was created or changed: # # check $*.run.cookie, then delete it # # * the .aux file changed in a way that necessitates attention # # Note that if the .auxtarget.make file doesn't exist, this means # that we are doing a clean build, so it doesn't figure into the # test for running again. # # compare against .auxtarget.make # # move if different, remove if not # # * the .log file has errors or warnings requiring at least one more run # # We use a loop over a single item to simplify the process of breaking # out when we find one of the conditions to be true. # # If we do NOT need to run latex here, then we move the $@.1st.make file # over to $@ because the target file has already been built by the first # dependency run and is valid. # # If we do, we delete that cookie file and do the normal multiple-runs # routine. # ifeq "$(strip $(BUILD_STRATEGY))" "latex" .SECONDARY: $(all_dvi_targets) endif %.$(build_target_extension): %.bbl %.aux %.$(build_target_extension).1st.make $(QUIET)\ fatal=`$(call colorize-latex-errors,$*.log)`; \ if [ x"$$fatal" != x"" ]; then \ $(ECHO) "$$fatal"; \ exit 1; \ fi; \ $(call make-auxtarget-file,$*.aux.make,$*.auxtarget.cookie); \ run=0; \ for i in 1; do \ if $(call test-exists,$*.bbl.cookie); then \ $(call set-run-reason,$*.bbl.cookie is present); \ run=1; \ break; \ fi; \ if $(call test-exists,$*.run.cookie); then \ $(call set-run-reason,$*.run.cookie is present); \ run=1; \ break; \ fi; \ if $(call \ test-exists-and-different,$*.auxtarget.cookie,$*.auxtarget.make);\ then \ $(call set-run-reason,$*.auxtarget.cookie differs from $*.auxtarget.make); \ run=1; \ break; \ fi; \ if $(call test-log-for-need-to-run,$*); then \ $(call set-run-reason,$*.log indicated that this is necessary); \ run=1; \ break; \ fi; \ if $(call test-not-exists,$@.1st.make); then \ $(call set-run-reason,$@.1st.make does not exist); \ run=1; \ break; \ fi; \ done; \ $(call remove-temporary-files,$*.bbl.cookie $*.run.cookie); \ $(MV) $*.auxtarget.cookie $*.auxtarget.make; \ if [ x"$$run" = x"1" ]; then \ $(call remove-files,$@.1st.make); \ for i in 2 3 4 5; do \ $(if $(findstring 3.79,$(MAKE_VERSION)),\ $(call echo-build,$*.tex,$@,$(RESTARTS)-$$$$i),\ $(call echo-build,$*.tex,$@,$(RESTARTS)-$$i)\ ); \ $(call run-latex,$*); \ $(CP) '$*.log' '$*.'$(RESTARTS)-$$i'.log'; \ if $(call test-run-again,$*); then \ $(call set-run-reason,rerun requested by $*.log); \ else \ break; \ fi; \ done; \ else \ $(MV) '$@.1st.make' '$@'; \ fi; \ $(call copy-with-logging,$@,$(BINARY_TARGET_DIR)); \ $(call latex-color-log,$*) # Build the .bbl file. When dependencies are included, this will (or will # not!) depend on something.bib, which we detect, acting accordingly. The # dependency creation also produces the %.auxbbl.make file. BibTeX is a bit # finicky about what you call the actual files, but we can rest assured that if # a .auxbbl.make file exists, then the .aux file does, as well. The # .auxbbl.make file is a cookie indicating whether the .bbl needs to be # rewritten. It only changes if the .aux file changes in ways relevant to .bbl # creation. # # Note that we do NOT touch the .bbl file if there is no need to # create/recreate it. We would like to leave existing files alone if they # don't need to be changed, thus possibly avoiding a rebuild trigger. %.bbl: %.auxbbl.make $(QUIET)\ $(if $(filter %.bib,$^),\ $(call echo-build,$(filter %.bib,$?) $*.aux,$@); \ $(call set-run-reason,dependencies of $@ changed); \ $(call run-bibtex,$*); \ $(TOUCH) $@.cookie; \ ) \ if $(EGREP) -q 'bibstyle.(apacite|apacann|chcagoa|[^}]*annot)' '$*.aux'; then \ $(call echo-build,** annotated extra latex **,output ignored,$(RESTARTS)-1); \ $(call run-latex,$*); \ $(CP) '$*.log' '$*.$(RESTARTS)-annotated.log'; \ $(if $(filter %.bib,$^),\ $(call echo-build,** annotated extra bibtex ** $(filter %.bib,$?) $*.aux,$@); \ $(call run-bibtex,$*); \ $(TOUCH) $@.cookie; \ ) \ $(call echo-build,** annotated extra latex **,output ignored,$(RESTARTS)-2); \ $(call run-latex,$*); \ fi # Create the index file - note that we do *not* depend on %.tex here, since # that unnecessarily restricts the kinds of indices that we can build to those # with exactly the same stem as the source file. Things like splitidx create # idx files with other names. # # Therefore, we add the .tex dependency in the sourcestem.d file in the call to # get index file dependencies from the logs. %.ind: %.idx $(QUIET)$(call echo-build,$<,$@) $(QUIET)$(call run-makeindex,$<,$@,$*.ilg) # Create a glossary file from a .ist file %.gls: %.glo %.tex %.ist $(QUIET)$(call echo-build,$<,$@) $(QUIET)$(call run-makeindex,$<,$@,$*.glg,$*.ist) # Create a glossary file from a glossary input formatted for xindy %.gls: %.glo %.tex %.xdy $(QUIET)$(call echo-build,$<,$@) $(QUIET)$(call run-xindy,$<,$@,$*,$*.glg) # Create the glossary file from a nomenclature file %.gls: %.glo %.tex $(call path-norm,$(shell $(KPSEWHICH) nomencl.ist || $(ECHO) nomencl.ist)) $(QUIET)$(call echo-build,$<,$@) $(QUIET)$(call run-makeindex,$<,$@,$*.glg,nomencl.ist) # Create the nomenclature file %.nls: %.nlo %.tex $(call path-norm,$(shell $(KPSEWHICH) nomencl.ist || $(ECHO) nomencl.ist)) $(QUIET)$(call echo-build,$<,$@) $(QUIET)$(call run-makeindex,$<,$@,$*.nlg,nomencl.ist) # Precompile the format file %.fmt: %.tex $(QUIET)$(call echo-build,$<,$@) $(QUIET)\ $(call run-latex,$*,-ini "&$(latex_build_program) $*.tex\dump"); \ fatal=`$(call colorize-latex-errors,$*.log)`; \ if [ x"$$fatal" != x"" ]; then \ $(ECHO) "$$fatal"; \ exit 1; \ fi; # SCRIPTED LaTeX TARGETS # # Keep the generated .tex files around for debugging if needed. .SECONDARY: $(all_tex_targets) %.tex:: %.tex.sh $(QUIET)$(call run-script,$(SHELL),$<,$@) %.tex:: %.tex.py $(QUIET)$(call run-script,$(PYTHON),$<,$@) %.tex:: %.tex.pl $(QUIET)$(call run-script,$(PERL),$<,$@) %.tex:: %.rst $(RST_STYLE_FILE) $(QUIET)\ $(call run-script,$(RST2LATEX)\ --documentoptions=letterpaper\ $(if $(RST_STYLE_FILE),--stylesheet=$(RST_STYLE_FILE),),$<,$@) %.tex:: %.lhs $(QUIET)\ function run_lhs2tex() { $(LHS2TEX) -o "$$2" "$$1"; }; \ $(call run-script,run_lhs2tex,$<,$@) # # GRAPHICS TARGETS # .PHONY: all-graphics all-graphics: $(all_graphics_targets); ifeq "$(strip $(BUILD_STRATEGY))" "latex" .PHONY: all-pstex all-pstex: $(all_pstex_targets); endif .PHONY: all-dot2tex all-dot2tex: $(all_dot2tex_targets); .PHONY: show-graphics show-graphics: all-graphics $(VIEW_GRAPHICS) $(all_known_graphics) $(gray_eps_file): $(QUIET)$(call echo-build,$^,$@) $(QUIET)$(call create-gray-eps-file,$@) ifeq "$(strip $(BUILD_STRATEGY))" "pdflatex" %.pdf: %.eps $(if $(GRAY),$(gray_eps_file)) $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call convert-eps-to-pdf,$<,$@,$(GRAY)) ifeq "$(strip $(GPI_OUTPUT_EXTENSION))" "pdf" %.pdf: %.gpi %.gpi.d $(gpi_sed) $(gpi_global) $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call convert-gpi,$<,$@,$(GRAY)) endif %.pdf: %.fig $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call convert-fig,$<,$@,$(GRAY)) %.pdf: %.svg $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call convert-svg,$<,$@,$(GRAY)) endif ifeq "$(strip $(BUILD_STRATEGY))" "xelatex" %.pdf: %.eps $(if $(GRAY),$(gray_eps_file)) $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call convert-eps-to-pdf,$<,$@,$(GRAY)) ifeq "$(strip $(GPI_OUTPUT_EXTENSION))" "pdf" %.pdf: %.gpi %.gpi.d $(gpi_sed) $(gpi_global) $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call convert-gpi,$<,$@,$(GRAY)) endif %.pdf: %.fig $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call convert-fig,$<,$@,$(GRAY)) endif # TODO: capture mpost output and display errors # TODO: figure out why pdf generation is erroring out (but working anyway) %.eps %.mps %.mpx %.log: %.mp $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(MPOST) $* $(QUIET)$(call move-if-exists,$*.mps,$*.eps) $(QUIET)$(call move-if-exists,$*.log,$*.log.make) $(QUIET)$(call move-if-exists,$*.mpx,$*.mpx.make) %.eps: %.gpi %.gpi.d $(gpi_sed) $(gpi_global) $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call convert-gpi,$<,$@,$(GRAY)) %.eps: %.fig $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call convert-fig,$<,$@,$(GRAY)) %.eps: %.dot $(if $(GRAY),$(gray_eps_file)) $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call convert-dot,$<,$@,$<.log,$(GRAY)) %.eps: %.xvg $(if $(GRAY),$(gray_eps_file)) $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call convert-xvg,$<,$@,$(GRAY)) ifneq "$(default_graphic_extension)" "pdf" # We have a perfectly good build rule for svg to pdf, so we eliminate this to # avoid confusing make (it sometimes chooses to go svg -> eps -> pdf). %.eps: %.svg $(if $(GRAY),$(gray_eps_file)) $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call convert-svg,$<,$@,$(GRAY)) # Similarly for these, we don't need eps if we have supported extensions # already. %.eps: %.jpg $(if $(GRAY),$(gray_eps_file)) $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call convert-jpg,$<,$@,$(GRAY)) %.eps: %.jpeg $(if $(GRAY),$(gray_eps_file)) $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call convert-jpg,$<,$@,$(GRAY)) %.eps: %.png $(if $(GRAY),$(gray_eps_file)) $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call convert-png,$<,$@,$(GRAY)) endif %.eps.bb: %.eps $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call eps-bb,$<,$@) %.eps: %.eps.gz $(if $(GRAY),$(gray_eps_file)) $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call convert-epsgz,$<,$@,$(GRAY)) %.pstex: %.fig $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call convert-fig-pstex,$<,$@,$(GRAY)) %.pstex_t: %.fig %.pstex $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call convert-fig-pstex-t,$<,$@,$*.pstex,$(GRAY)) %.dot_t: %.dot $(QUIET)$(call echo-graphic,$^,$@) $(QUIET)$(call convert-dot-tex,$<,$@) # # DEPENDENCY-RELATED TARGETS. # # Generate all of the information needed to get dependencies # As a side effect, this creates a .dvi or .pdf file (depending on the build # strategy). We need to be sure to remove it if there are errors. Errors can # take several forms and all of them are found within the log file: # * There was a LaTeX error # * A needed file was not found # * Cross references need adjustment # # Behavior: # This rule is responsible for generating the following: # %.aux # %.d # %.aux.make # %.(pdf|dvi).1st.make (the .pdf or .dvi output file, moved) # # Steps: # # Run latex # Move .pdf or .dvi somewhere else (make no judgements about success) # Flatten the .aux file into another file # Add source dependencies # Add graphic dependencies # Add bib dependencies # # Create cookies for various suffixes that may represent files that # need to be read by LaTeX in order for it to function properly. # # Note that if some of the dependencies are discovered because they turn # up missing in the log file, we really need the .d file to be reloaded. # Adding a sleep command helps with this. Otherwise make is extremely # nondeterministic, sometimes working, sometimes not. # # Usually we can force this by simply removing the generated pdf file and # not creating a .1st.make file.. # %.$(build_target_extension).1st.make %.d %.aux %.aux.make %.fls: %.tex $(QUIET)$(call echo-build,$<,$*.d $*.$(build_target_extension).1st.make,$(RESTARTS)-1) $(QUIET)\ $(call set-run-reason,need to build .d and .$(build_target_extension).1st.make); \ $(call run-latex,$*,-recorder) || $(sh_true); \ $(CP) '$*.log' '$*.$(RESTARTS)-1.log'; \ $(call die-on-import-sty,$*.log); \ $(call die-on-dot2tex,$*.log); \ $(call die-on-no-aux,$*); \ $(call flatten-aux,$*.aux,$*.aux.make); \ $(ECHO) "# vim: ft=make" > $*.d; \ $(ECHO) ".PHONY: $*._graphics" >> $*.d; \ $(call get-inputs,$*.fls,$(addprefix $*.,aux aux.make d $(build_target_extension))) >> $*.d; \ $(call get-format,$<,$(addprefix $*.,fls aux aux.make d $(build_target_extension) $(build_target_extension).1st.make)) >> $*.d; \ $(call get-missing-inputs,$*.log,$(addprefix $*.,aux aux.make d $(build_target_extension))) >> $*.d; \ $(ECHO) ".SECONDEXPANSION:" >> $*.d; \ $(call get-graphics,$*) >> $*.d; \ $(call get-log-index,$*,$(addprefix $*.,aux aux.make)) >> $*.d; \ $(call get-bibs,$*.aux.make,$(addprefix $*.,bbl aux aux.make)) >> $*.d; \ $(EGREP) -q "# MISSING stem" $*.d && $(SLEEP) 1 && $(RM) $*.pdf; \ $(EGREP) -q "# MISSING format" $*.d && $(RM) $*.pdf; \ $(call move-if-exists,$*.$(build_target_extension),$*.$(build_target_extension).1st.make); \ for s in toc out lot lof lol nav; do \ if [ -e "$*.$$s" ]; then \ if ! $(DIFF) -q $*.$$s $*.$$s.make >/dev/null 2>&1; then \ $(TOUCH) $*.run.cookie; \ fi; \ $(CP) $*.$$s $*.$$s.make; \ fi; \ done # This is a cookie that is updated if the flattened aux file has changed in a # way that affects the bibliography generation. .SECONDARY: $(addsuffix .auxbbl.make,$(stems_ssg)) %.auxbbl.make: %.aux.make $(QUIET)\ $(call make-auxbbl-file,$<,$@.temp); \ $(call replace-if-different-and-remove,$@.temp,$@) # Build a dependency file for .gpi files. These often plot data files that # also reside in the directory, so if a data file changes, it's nice to know # about it. This also handles loaded .gpi files, whose filename should have # _include_. in it. %.gpi.d: %.gpi $(QUIET)$(call echo-build,$<,$@) $(QUIET)$(call make-gpi-d,$<,$@) # Get source specials, e.g., paper size and special %% comments. %.specials.make: %.tex $(QUIET)$(call get-source-specials,$<,$@) # Get info about whether to enlarge beamer postscript files (for use with # dvips, and requires a special comment in the source file). .SECONDARY: $(addsuffix .specials.make,$(all_stems_ss)) %.beamer.make: %.specials.make $(QUIET)$(SED) -e 's/^BEAMER.*/BEAMER/p' -e 'd' '$<' > '$@' # Store the paper size for this document -- note that if beamer is used we set # it to the special BEAMER paper size. We only do this, however, if the # special comment exists, in which case we enlarge the output with psnup. # # The paper size is extracted from a documentclass attribute. %.paper.make: %.specials.make $(QUIET)$(SED) -e 's/^PAPERSIZE=//p' -e 'd' '$<' > '$@' # Store embedding instructions for this document using a special comment %.embed.make: %.specials.make $(QUIET)$(EGREP) '^NOEMBED$$' $< \ && $(ECHO) '' > $@ \ || $(ECHO) '1' > $@; # # HELPFUL PHONY TARGETS # .PHONY: _all_programs _all_programs: $(QUIET)$(ECHO) "== All External Programs Used ==" $(QUIET)$(call output-all-programs) .PHONY: _check_programs _check_programs: $(QUIET)$(ECHO) "== Checking Makefile Dependencies =="; $(ECHO) $(QUIET) \ allprogs=`\ ($(call output-all-programs)) | \ $(SED) \ -e 's/^[[:space:]]*//' \ -e '/^#/d' \ -e 's/[[:space:]]*#.*//' \ -e '/^=/s/[[:space:]]/_/g' \ -e '/^[[:space:]]*$$/d' \ -e 's/^[^=].*=[[:space:]]*\([^[:space:]]\{1,\}\).*$$/\\1/' \ `; \ spaces=' '; \ for p in $${allprogs}; do \ case $$p in \ =*) $(ECHO); $(ECHO) "$$p";; \ *) \ $(ECHO) -n "$$p:$$spaces" | $(SED) -e 's/^\(.\{0,20\}\).*$$/\1/'; \ loc=`$(WHICH) $$p`; \ if [ x"$$?" = x"0" ]; then \ $(ECHO) "$(C_SUCCESS)Found:$(C_RESET) $$loc"; \ else \ $(ECHO) "$(C_FAILURE)Not Found$(C_RESET)"; \ fi; \ ;; \ esac; \ done .PHONY: _check_gpi_files _check_gpi_files: $(QUIET)$(ECHO) "== Checking all .gpi files for common errors =="; \ $(ECHO); \ for f in $(files.gpi); do \ result=`$(EGREP) '^([^#]*set terminal |set output )' $$f`; \ $(ECHO) -n "$$f: "; \ if [ x"$$result" = x"" ]; then \ $(ECHO) "$(C_SUCCESS)Okay$(C_RESET)"; \ else \ $(ECHO) "$(C_FAILURE)Warning: Problematic commands:$(C_RESET)";\ $(ECHO) "$(C_ERROR)$$result$(C_RESET)"; \ fi; \ done; \ $(ECHO) .PHONY: _all_stems _all_stems: $(QUIET)$(ECHO) "== All Stems ==" $(QUIET)$(call echo-list,$(sort $(default_stems_ss))) .PHONY: _includes _includes: $(QUIET)$(ECHO) "== Include Stems ==" $(QUIET)$(ECHO) "=== Sources ===" $(QUIET)$(call echo-list,$(sort $(source_includes))) $(QUIET)$(ECHO) "=== Graphics ===" $(QUIET)$(call echo-list,$(sort $(graphic_includes))) .PHONY: _all_sources _all_sources: $(QUIET)$(ECHO) "== All Sources ==" $(QUIET)$(call echo-list,$(sort $(all_files.tex))) .PHONY: _dependency_graph _dependency_graph: $(QUIET)$(ECHO) "/* LaTeX Dependency Graph */" $(QUIET)$(call output-dependency-graph) .PHONY: _show_dependency_graph _show_dependency_graph: $(QUIET)$(call output-dependency-graph,$(graph_stem).dot) $(QUIET)$(DOT) -Tps -o $(graph_stem).eps $(graph_stem).dot $(QUIET)$(VIEW_POSTSCRIPT) $(graph_stem).eps $(QUIET)$(call remove-temporary-files,$(graph_stem).*) .PHONY: _sources _sources: $(QUIET)$(ECHO) "== Sources ==" $(QUIET)$(call echo-list,$(sort $(files.tex))) .PHONY: _source_gens _source_gens: $(QUIET)$(ECHO) "== Generated Sources ==" $(QUIET)$(call echo-list,$(sort $(files_source_gen))) .PHONY: _scripts _scripts: $(QUIET)$(ECHO) "== Scripts ==" $(QUIET)$(call echo-list,$(sort $(files_scripts))) .PHONY: _graphic_outputs _graphic_outputs: $(QUIET)$(ECHO) "== Graphic Outputs ==" $(QUIET)$(call echo-list,$(sort $(all_graphics_targets))) .PHONY: _env _env: ifdef .VARIABLES $(QUIET)$(ECHO) "== MAKE VARIABLES ==" $(QUIET)$(call echo-list,$(foreach var,$(sort $(.VARIABLES)),'$(var)')) endif $(QUIET)$(ECHO) "== ENVIRONMENT ==" $(QUIET)$(ENV) # # CLEAN TARGETS # # clean-generated is somewhat unique - it relies on the .fls file being # properly built so that it can determine which of the files was generated, and # which was not. Expect it to silently fail if the .fls file is missing. # # This is used to, e.g., clean up index files that are generated by the LaTeX. .PHONY: clean-generated clean-generated: $(QUIET)$(call clean-files,$(foreach e,$(addsuffix .fls,$(all_stems_source)),\ $(shell $(call get-generated-names,$e)))) .PHONY: clean-deps clean-deps: $(QUIET)$(call clean-files,$(all_d_targets) *.make *.make.temp *.cookie) .PHONY: clean-tex clean-tex: clean-deps $(QUIET)$(call clean-files,$(rm_tex)) .PHONY: clean-graphics # TODO: This *always* deletes pstex files, even if they were not generated by # anything.... In other words, if you create a pstex and pstex_t pair by hand # an drop them in here without the generating fig file, they will be deleted # and you won't get them back. It's a hack put in here because I'm not sure we # even want to keep pstex functionality, so my motivation is not terribly high # for doing it right. clean-graphics: $(QUIET)$(call clean-files,$(all_graphics_targets) *.gpi.d *.pstex *.pstex_t *.dot_t) .PHONY: clean-backups clean-backups: $(QUIET)$(call clean-files,$(backup_patterns) *.temp) .PHONY: clean-auxiliary clean-auxiliary: $(QUIET)$(call clean-files,$(graph_stem).*) .PHONY: clean-nographics clean-nographics: clean-tex clean-deps clean-backups clean-auxiliary ; .PHONY: clean clean: clean-generated clean-tex clean-graphics clean-deps clean-backups clean-auxiliary ; # # HELP TARGETS # .PHONY: help help: $(help_text) .PHONY: version version: $(QUIET)\ $(ECHO) "$(fileinfo) Version $(version)"; \ $(ECHO) "by $(author)"; \ # # HELP TEXT # define help_text # $(fileinfo) Version $(version) # # by $(author) # # Generates a number of possible output files from a LaTeX document and its # various dependencies. Handles .bib files, \include and \input, and .eps # graphics. All dependencies are handled automatically by running LaTeX over # the source. # # USAGE: # # make [GRAY=1] [VERBOSE=1] [SHELL_DEBUG=1] # # STANDARD OPTIONS: # GRAY: # Setting this variable forces all recompiled graphics to be grayscale. # It is useful when creating a document for printing. The default is # to allow colors. Note that it only changes graphics that need to be # rebuilt! It is usually a good idea to do a 'make clean' first. # # VERBOSE: # This turns off all @ prefixes for commands invoked by make. Thus, # you get to see all of the gory details of what is going on. # # SHELL_DEBUG: # This enables the -x option for sh, meaning that everything it does is # echoed to stderr. This is particularly useful for debugging # what is going on in $$(shell ...) invocations. One of my favorite # debugging tricks is to do this: # # make -d SHELL_DEBUG=1 VERBOSE=1 2>&1 | less # # KEEP_TEMP: # When set, this allows .make and other temporary files to stick around # long enough to do some debugging. This can be useful when trying to # figure out why gnuplot is not doing the right things, for example # (e.g., look for *head.make). # # STANDARD AUXILIARY FILES: # # Variables.ini (formerly Makefile.ini, which still works) # # This file can contain variable declarations that override various # aspects of the makefile. For example, one might specify # # neverclean := *.pdf *.ps # onlysources.tex := main.tex # LATEX_COLOR_WARNING := 'bold red uline' # # And this would override the neverclean setting to ensure that pdf # and ps files always remain behind, set the makefile to treat all # .tex files that are not "main.tex" as includes (and therefore not # default targets). It also changes the LaTeX warning output to be # red, bold, and underlined. # # There are numerous variables in this file that can be overridden in # this way. Search for '?=' to find them all. # # The Variables.ini is imported before *anything else* is done, so go # wild with your ideas for changes to this makefile in there. It # makes it easy to test them before submitting patches. # # If you're adding rules or targets, however, see Targets.ini below. # # Targets.ini # # This is included much later in the makefile, after all variables and # targets are defined. This is where you would put new make rules, # e.g., # # generated.tex: generating_script.weird_lang depA depB # ./generating_script.weird_lang > $$@ # # In this file, you have access to all of the variables that the # makefile creates, like $$(onlysources.tex). While accessing those can # be somewhat brittle (they are implementation details and may change), # it is a great way to test your ideas when submitting feature requests. # # STANDARD ENVIRONMENT VARIABLES: # # LATEX_COLOR_WARNING '$(LATEX_COLOR_WARNING)' # LATEX_COLOR_ERROR '$(LATEX_COLOR_ERROR)' # LATEX_COLOR_UNDERFULL '$(LATEX_COLOR_UNDERFULL)' # LATEX_COLOR_OVERFULL '$(LATEX_COLOR_OVERFULL)' # LATEX_COLOR_PAGES '$(LATEX_COLOR_PAGES)' # LATEX_COLOR_BUILD '$(LATEX_COLOR_BUILD)' # LATEX_COLOR_GRAPHIC '$(LATEX_COLOR_GRAPHIC)' # LATEX_COLOR_DEP '$(LATEX_COLOR_DEP)' # LATEX_COLOR_SUCCESS '$(LATEX_COLOR_SUCCESS)' # LATEX_COLOR_FAILURE '$(LATEX_COLOR_FAILURE)' # # These may be redefined in your environment to be any of the following: # # black # red # green # yellow # blue # magenta # cyan # white # # Bold or underline may be used, as well, either alone or in combination # with colors: # # bold # uline # # Order is not important. You may want, for example, to specify: # # export LATEX_COLOR_SUCCESS='bold blue uline' # # in your .bashrc file. I don't know why, but you may want to. # # STANDARD TARGETS: # # all: # Make all possible documents in this directory. The documents are # determined by scanning for .tex and .tex.sh (described in more detail # later) and omitting any file that ends in ._include_.tex or # ._nobuild_.tex. The output is a set of .pdf files. # # If you wish to omit files without naming them with the special # underscore names, set the following near the top of the Makefile, # or (this is recommended) within a Makefile.ini in the same directory: # # includes.tex := file1.tex file2.tex # # This will cause the files listed to be considered as include files. # # If you have only few source files, you can set # # onlysources.tex := main.tex # # This will cause only the source files listed to be considered in # dependency detection. All other .tex files will be considered as # include files. Note that these options work for *any* source type, # so you could do something similar with includes.gpi, for example. # Note that this works for *any valid source* target. All of the # onlysources.* variables are commented out in the shipping version of # this file, so it does the right thing when they simply don't exist. # The comments are purely documentation. If you know, for example, that # file.mycoolformat is supported by this Makefile, but don't see the # "onlysources.mycoolformat" declared in the comments, that doesn't mean # you can't use it. Go ahead and set "onlysources.mycoolformat" and it # should do the right thing. # # show: # Builds and displays all documents in this directory. It uses the # environment-overridable value of VIEW_PDF (currently $(VIEW_PDF)) to # do its work. # # all-graphics: # Make all of the graphics in this directory. # # all-pstex (only for BUILD_STRATEGY=latex): # Build all fig files into pstex and pstex_t files. Gray DOES NOT WORK. # # all-gray-pstex (only for BUILD_STRATEGY=latex): # Build all fig files into grayscale pstex and pstex_t files. # # all-dot2tex: # Build all dot files into tex files. # # show-graphics: # Builds and displays all graphics in this directory. Uses the # environment-overridable value of VIEW_GRAPHICS (currently # $(VIEW_GRAPHICS)) to do its work. # # clean: # Remove ALL generated files, leaving only source intact. # This will *always* skip files mentioned in the "neverclean" variable, # either in this file or specified in Makefile.ini: # # neverclean := *.pdf *.ps # # The neverclean variable works on all "clean" targets below, as well. # # clean-graphics: # Remove all generated graphics files. # # clean-backups: # Remove all backup files: $(backup_patterns) # (XFig and other editors have a nasty habit of leaving them around) # Also removes Makefile-generated .temp files # # clean-tex: # Remove all files generated from LaTeX invocations except dependency # information. Leaves graphics alone. # # clean-deps: # Removes all auto-generated dependency information. # # clean-auxiliary: # Removes extra files created by various targets (like the dependency # graph output). # # clean-nographics: # Cleans everything *except* the graphics files. # # help: # This help text. # # version: # Version information about this LaTeX makefile. # # DEBUG TARGETS: # # _all_programs: # A list of the programs used by this makefile. # # _check_programs: # Checks your system for the needed software and reports what it finds. # # _check_gpi_files: # Checks the .gpi files in the current directory for common errors, such # as specification of the terminal or output file inside of the gpi file # itself. # # _dependency_graph: # Outputs a .dot file to stdout that represents a graph of LaTeX # dependencies. To see it, use the _show_dependency_graph target or # direct the output to a file, run dot on it, and view the output, e.g.: # # make _dependency_graph > graph.dot # dot -T ps -o graph.eps graph.dot # gv graph.eps # # _show_dependency_graph: # Makes viewing the graph simple: extracts, builds and displays the # dependency graph given in the _dependency_graph target using the value # of the environment-overridable VIEW_POSTSCRIPT variable (currently set # to $(VIEW_POSTSCRIPT)). The postscript viewer is used because it # makes it easier to zoom in on the graph, a critical ability for # something so dense and mysterious. # # _all_sources: # List all .tex files in this directory. # # _sources: # Print out a list of all compilable sources in this directory. This is # useful for determining what make thinks it will be using as the # primary source for 'make all'. # # _scripts: # Print out a list of scripts that make knows can be used to generate # .tex files (described later). # # _all_stems: # Print a list of stems. These represent bare targets that can be # executed. Listing as a bare target will produce .pdf. # # _includes: # A list of .d files that would be included in this run if _includes # weren't specified. This target may be used alone or in conjunction # with other targets. # # _graphic_outputs: # A list of all generated .eps files # # _env: # A list of environment variables and their values. If supported by # your version of make, also a list of variables known to make. # # FILE TARGETS: # # %, %.pdf: # Build a PDF file from the corresponding %.tex file. # # If BUILD_STRATEGY=pdflatex, then this builds the pdf directly. # Otherwise, it uses this old-school but effective approach: # # latex -> dvips -> ps2pdf # # The BUILD_STRATEGY can be overridden in Makefile.ini in the same # directory. The default is pdflatex. # # Reasons for using latex -> dvips include the "psfrag" package, and the # generation of postscript instead of PDF. Arguments for using pdflatex # include "new and shiny" and "better supported." I can't argue with # either of those, and supporting them both didn't turn out to be that # difficult, so there you have it. Choices. # # %._show: # A phony target that builds the pdf file and then displays it using the # environment-overridable value of VIEW_PDF ($(VIEW_PDF)). # # %._graphics: # A phony target that generates all graphics on which %.pdf (or %.dvi) # depends. # # %.ps (only for BUILD_STRATEGY=latex): # Build a Postscript file from the corresponding %.tex file. # This is done using dvips. Paper size is automatically # extracted from the declaration # # \documentclass[paper] # # or it is the system default. # # If using beamer (an excellent presentation class), the paper # size is ignored. More on this later. # # %.dvi (only for BUILD_STRATEGY=latex): # Build the DVI file from the corresponding %.tex file. # # %.ind: # Build the index for this %.tex file. # # %.gls: # Build the nomenclature glossary for this %.tex file. # # %.nls: # Build the (newer) nomenclature file for this %.tex file. # # %.eps: # Build an eps file from one of the following file types: # # .dot : graphviz # .gpi : gnuplot # .fig : xfig # .xvg : xmgrace # .svg : scalable vector graphics (goes through inkscape) # .png : png (goes through NetPBM) # .jpg : jpeg (goes through ImageMagick) # .eps.gz : gzipped eps # # The behavior of this makefile with each type is described in # its own section below. # # %.pstex{,_t} (only for BUILD_STRATEGY=latex): # Build a .pstex_t file from a .fig file. # # FEATURES: # # Optional Binary Directory: # If you create the _out_ directory in the same place as the makefile, # it will automatically be used as a dumping ground for .pdf (or .dvi, # .ps, and .pdf) output files. # # Alternatively, you can set the BINARY_TARGET_DIR variable, either as a # make argument or in Makefile.ini, to point to your directory of # choice. Note that no pathname wildcard expansion is done in the # makefile, so make sure that the path is complete before going in # there. E.g., if you want to specify something in your home directory, # use $$HOME/ instead of ~/ so that the shell expands it before it gets # to the makefile. # # External Program Dependencies: # Every external program used by the makefile is represented by an # ALLCAPS variable at the top of this file. This should allow you to # make judgments about whether your system supports the use of this # makefile. The list is available in the ALL_PROGRAMS variable and, # provided that you are using GNU make 3.80 or later (or you haven't # renamed this file to something weird like "mylatexmakefile" and like # invoking it with make -f) can be viewed using # # make _all_programs # # Additionally, the availability of these programs can be checked # automatically for you by running # # make _check_programs # # The programs are categorized according to how important they are and # what function they perform to help you decide which ones you really # need. # # Colorized Output: # The output of commands is colorized to highlight things that are often # important to developers. This includes {underfull,overfull} # {h,v}boxes, general LaTeX Errors, each stage of document building, and # the number of pages in the final document. The colors are obtained # using 'tput', so colorization should work pretty well on any terminal. # # The colors can be customized very simply by setting any of the # LATEX_COLOR_ variables in your environment (see above). # # Predecessors to TeX Files: # Given a target , if no .tex file exists but a # corresponding script or predecessor file exists, then appropriate # action will be taken to generate the tex file. # # Currently supported script or predecessor languages are: # # sh: %.tex.sh # perl: %.tex.pl # python: %.tex.py # # Calls the script using the appropriate interpreter, assuming that # its output is a .tex file. # # The script is called thus: # #