Repository: lawlite19/MachineLearning_Python
Branch: master
Commit: f8d02fd9fb2b
Files: 45
Total size: 11.3 MB

Directory structure:
gitextract_y2eizqsm/

├── .github/
│   └── FUNDING.yml
├── .gitignore
├── AnomalyDetection/
│   ├── AnomalyDetection.py
│   ├── data1.mat
│   └── data2.mat
├── K-Means/
│   ├── K-Means_scikit-learn.py
│   ├── K-Menas.py
│   ├── bird.mat
│   └── data.mat
├── LICENSE
├── LinearRegression/
│   ├── LinearRegression.py
│   ├── LinearRegression_scikit-learn.py
│   ├── data.csv
│   ├── data.npy
│   └── data.txt
├── LogisticRegression/
│   ├── LogisticRegression.py
│   ├── LogisticRegression_OneVsAll.py
│   ├── LogisticRegression_OneVsAll_scikit-learn.py
│   ├── LogisticRegression_scikit-learn.py
│   ├── class_y.csv
│   ├── data1.npy
│   ├── data1.txt
│   ├── data2.txt
│   ├── data_digits.mat
│   └── predict.csv
├── NeuralNetwok/
│   ├── NeuralNetwork.py
│   ├── data_digits.mat
│   └── predict.csv
├── PCA/
│   ├── PCA.py
│   ├── PCA_scikit-learn.py
│   ├── data.mat
│   └── data_faces.mat
├── SVM/
│   ├── SVM_scikit-learn.py
│   ├── data.txt
│   ├── data1.mat
│   ├── data2.mat
│   └── data3.mat
├── formula/
│   ├── AnomalyDetection.wmf
│   ├── K-Means.wmf
│   ├── LinearRegression_01.wmf
│   ├── LogisticRegression_01.wmf
│   ├── NeuralNetwork.wmf
│   ├── PCA.wmf
│   └── SVM.wmf
└── readme.md

================================================
FILE CONTENTS
================================================

================================================
FILE: .github/FUNDING.yml
================================================
# These are supported funding model platforms

github: # Replace with up to 4 GitHub Sponsors-enabled usernames e.g., [user1, user2]
patreon: # Replace with a single Patreon username
open_collective: # Replace with a single Open Collective username
ko_fi: # Replace with a single Ko-fi username
tidelift: # Replace with a single Tidelift platform-name/package-name e.g., npm/babel
community_bridge: # Replace with a single Community Bridge project-name e.g., cloud-foundry
liberapay: # Replace with a single Liberapay username
issuehunt: # Replace with a single IssueHunt username
otechie: # Replace with a single Otechie username
custom: # Replace with up to 4 custom sponsorship URLs e.g., ['link1', 'link2']


================================================
FILE: .gitignore
================================================
.idea/
venv/


================================================
FILE: AnomalyDetection/AnomalyDetection.py
================================================
#-*- coding: utf-8 -*-
# Author: Bob
# Date:   2016.12.22
from __future__ import print_function
import numpy as np
from matplotlib import pyplot as plt
from scipy import io as spio

'''异常检测主运行程序'''
def anomalyDetection_example():
    '''加载并显示数据'''
    data = spio.loadmat('data1.mat')
    X = data['X']
    plt = display_2d_data(X, 'bx')
    plt.title("origin data")
    plt.show()
    '''多元高斯分布函数，并可视化拟合的边界'''
    mu,sigma2 = estimateGaussian(X)         # 参数估计（求均值和方差）
    #print (mu,sigma2)
    p = multivariateGaussian(X,mu,sigma2)   # 多元高斯分布函数
    #print (p)
    visualizeFit(X,mu,sigma2)  # 显示图像
    
    '''选择异常点（在交叉验证CV上训练得到最好的epsilon）'''
    Xval = data['Xval']
    yval = data['yval']   # y=1代表异常
    pval = multivariateGaussian(Xval, mu, sigma2) # 计算CV上的概率密度值
    epsilon,F1 = selectThreshold(yval,pval)       # 选择最优的epsilon临界值
    print(u'在CV上得到的最好的epsilon是：%e'%epsilon)
    print(u'对应的F1Score值为：%f'%F1)
    outliers = np.where(p<epsilon)  # 找到小于临界值的异常点，并作图
    plt.plot(X[outliers,0],X[outliers,1],'o',markeredgecolor='r',markerfacecolor='w',markersize=10.)
    plt = display_2d_data(X, 'bx')
    plt.show()
    
    
    
    
# 显示二维数据    
def display_2d_data(X,marker):
    plt.plot(X[:,0],X[:,1],marker)
    plt.axis('square')
    return plt

# 参数估计函数（就是求均值和方差）
def estimateGaussian(X):
    m,n = X.shape
    mu = np.zeros((n,1))
    sigma2 = np.zeros((n,1))
    
    mu = np.mean(X, axis=0) # axis=0表示列，每列的均值
    sigma2 = np.var(X,axis=0) # 求每列的方差
    return mu,sigma2
   
# 多元高斯分布函数    
def multivariateGaussian(X,mu,Sigma2):
    k = len(mu)
    if (Sigma2.shape[0]>1):
        Sigma2 = np.diag(Sigma2)
    '''多元高斯分布函数'''    
    X = X-mu
    argu = (2*np.pi)**(-k/2)*np.linalg.det(Sigma2)**(-0.5)
    p = argu*np.exp(-0.5*np.sum(np.dot(X,np.linalg.inv(Sigma2))*X,axis=1))  # axis表示每行
    return p
    
# 可视化边界
def visualizeFit(X,mu,sigma2):
    x = np.arange(0, 36, 0.5) # 0-36，步长0.5
    y = np.arange(0, 36, 0.5)
    X1,X2 = np.meshgrid(x,y)  # 要画等高线，所以meshgird
    Z = multivariateGaussian(np.hstack((X1.reshape(-1,1),X2.reshape(-1,1))), mu, sigma2)  # 计算对应的高斯分布函数
    Z = Z.reshape(X1.shape)  # 调整形状
    plt.plot(X[:,0],X[:,1],'bx')
    
    if np.sum(np.isinf(Z).astype(float)) == 0:   # 如果计算的为无穷，就不用画了
        #plt.contourf(X1,X2,Z,10.**np.arange(-20, 0, 3),linewidth=.5)
        CS = plt.contour(X1,X2,Z,10.**np.arange(-20, 0, 3),color='black',linewidth=.5)   # 画等高线，Z的值在10.**np.arange(-20, 0, 3)
        #plt.clabel(CS)
            
    plt.show()

# 选择最优的epsilon，即：使F1Score最大    
def selectThreshold(yval,pval):
    '''初始化所需变量'''
    bestEpsilon = 0.
    bestF1 = 0.
    F1 = 0.
    step = (np.max(pval)-np.min(pval))/1000
    '''计算'''
    for epsilon in np.arange(np.min(pval),np.max(pval),step):
        cvPrecision = pval<epsilon
        tp = np.sum((cvPrecision == 1) & (yval == 1).ravel()).astype(float)  # sum求和是int型的，需要转为float
        fp = np.sum((cvPrecision == 1) & (yval == 0).ravel()).astype(float)
        fn = np.sum((cvPrecision == 0) & (yval == 1).ravel()).astype(float)
        precision = tp/(tp+fp)  # 精准度
        recision = tp/(tp+fn)   # 召回率
        F1 = (2*precision*recision)/(precision+recision)  # F1Score计算公式
        if F1 > bestF1:  # 修改最优的F1 Score
            bestF1 = F1
            bestEpsilon = epsilon
    return bestEpsilon,bestF1
        


if __name__ == '__main__':
    anomalyDetection_example()
    

================================================
FILE: K-Means/K-Means_scikit-learn.py
================================================
#-*- coding: utf-8 -*-
import numpy as np
from scipy import io as spio
from matplotlib import pyplot as plt
from sklearn.cluster import KMeans


def kMenas():
    data = spio.loadmat("data.mat")
    X = data['X']   
    model = KMeans(n_clusters=3).fit(X) # n_clusters指定3类，拟合数据
    centroids = model.cluster_centers_  # 聚类中心
    
    plt.scatter(X[:,0], X[:,1])     # 原数据的散点图
    plt.plot(centroids[:,0],centroids[:,1],'r^',markersize=10)  # 聚类中心
    plt.show()

if __name__ == "__main__":
    kMenas()


================================================
FILE: K-Means/K-Menas.py
================================================
#-*- coding: utf-8 -*-
from __future__ import print_function
import numpy as np
from matplotlib import pyplot as plt
from matplotlib import colors
from scipy import io as spio
from scipy import misc      # 图片操作
from matplotlib.font_manager import FontProperties
font = FontProperties(fname=r"c:\windows\fonts\simsun.ttc", size=14)    # 解决windows环境下画图汉字乱码问题



def KMeans():
    '''二维数据聚类过程演示'''
    print(u'聚类过程展示...\n')
    data = spio.loadmat("data.mat")
    X = data['X']
    K = 3   # 总类数
    initial_centroids = np.array([[3,3],[6,2],[8,5]])   # 初始化类中心
    max_iters = 10
    runKMeans(X,initial_centroids,max_iters,True)       # 执行K-Means聚类算法
    '''
    图片压缩
    '''
    print(u'K-Means压缩图片\n')
    img_data = misc.imread("bird.png")  # 读取图片像素数据
    img_data = img_data/255.0             # 像素值映射到0-1
    img_size = img_data.shape
    X = img_data.reshape(img_size[0]*img_size[1],3)    # 调整为N*3的矩阵，N是所有像素点个数
    
    K = 16
    max_iters = 5
    initial_centroids = kMeansInitCentroids(X,K)
    centroids,idx = runKMeans(X, initial_centroids, max_iters, False)
    print(u'\nK-Means运行结束\n')
    print(u'\n压缩图片...\n')
    idx = findClosestCentroids(X, centroids)
    X_recovered = centroids[idx,:]
    X_recovered = X_recovered.reshape(img_size[0],img_size[1],3)
    
    print(u'绘制图片...\n')
    plt.subplot(1,2,1)
    plt.imshow(img_data)
    plt.title(u"原先图片",fontproperties=font)
    plt.subplot(1,2,2)
    plt.imshow(X_recovered)
    plt.title(u"压缩图像",fontproperties=font)
    plt.show()
    print(u'运行结束！')
    
    
# 找到每条数据距离哪个类中心最近    
def findClosestCentroids(X,initial_centroids):
    m = X.shape[0]                  # 数据条数
    K = initial_centroids.shape[0]  # 类的总数
    dis = np.zeros((m,K))           # 存储计算每个点分别到K个类的距离
    idx = np.zeros((m,1))           # 要返回的每条数据属于哪个类
    
    '''计算每个点到每个类中心的距离'''
    for i in range(m):
        for j in range(K):
            dis[i,j] = np.dot((X[i,:]-initial_centroids[j,:]).reshape(1,-1),(X[i,:]-initial_centroids[j,:]).reshape(-1,1))
    
    '''返回dis每一行的最小值对应的列号，即为对应的类别
    - np.min(dis, axis=1)返回每一行的最小值
    - np.where(dis == np.min(dis, axis=1).reshape(-1,1)) 返回对应最小值的坐标
     - 注意：可能最小值对应的坐标有多个，where都会找出来，所以返回时返回前m个需要的即可（因为对于多个最小值，属于哪个类别都可以）
    '''  
    dummy,idx = np.where(dis == np.min(dis, axis=1).reshape(-1,1))
    return idx[0:dis.shape[0]]  # 注意截取一下
             

# 计算类中心
def computerCentroids(X,idx,K):
    n = X.shape[1]
    centroids = np.zeros((K,n))
    for i in range(K):
        centroids[i,:] = np.mean(X[np.ravel(idx==i),:], axis=0).reshape(1,-1)   # 索引要是一维的,axis=0为每一列，idx==i一次找出属于哪一类的，然后计算均值
    return centroids

# 聚类算法
def runKMeans(X,initial_centroids,max_iters,plot_process):
    m,n = X.shape                   # 数据条数和维度
    K = initial_centroids.shape[0]  # 类数
    centroids = initial_centroids   # 记录当前类中心
    previous_centroids = centroids  # 记录上一次类中心
    idx = np.zeros((m,1))           # 每条数据属于哪个类
    
    for i in range(max_iters):      # 迭代次数
        print(u'迭代计算次数：%d'%(i+1))
        idx = findClosestCentroids(X, centroids)
        if plot_process:    # 如果绘制图像
            plt = plotProcessKMeans(X,centroids,previous_centroids) # 画聚类中心的移动过程
            previous_centroids = centroids  # 重置
        centroids = computerCentroids(X, idx, K)    # 重新计算类中心
    if plot_process:    # 显示最终的绘制结果
        plt.show()
    return centroids,idx    # 返回聚类中心和数据属于哪个类

# 画图，聚类中心的移动过程        
def plotProcessKMeans(X,centroids,previous_centroids):
    plt.scatter(X[:,0], X[:,1])     # 原数据的散点图
    plt.plot(previous_centroids[:,0],previous_centroids[:,1],'rx',markersize=10,linewidth=5.0)  # 上一次聚类中心
    plt.plot(centroids[:,0],centroids[:,1],'rx',markersize=10,linewidth=5.0)                    # 当前聚类中心
    for j in range(centroids.shape[0]): # 遍历每个类，画类中心的移动直线
        p1 = centroids[j,:]
        p2 = previous_centroids[j,:]
        plt.plot([p1[0],p2[0]],[p1[1],p2[1]],"->",linewidth=2.0)
    return plt


# 初始化类中心--随机取K个点作为聚类中心
def kMeansInitCentroids(X,K):
    m = X.shape[0]
    m_arr = np.arange(0,m)      # 生成0-m-1
    centroids = np.zeros((K,X.shape[1]))
    np.random.shuffle(m_arr)    # 打乱m_arr顺序    
    rand_indices = m_arr[:K]    # 取前K个
    centroids = X[rand_indices,:]
    return centroids

if __name__ == "__main__":
    KMeans()
    

================================================
FILE: LICENSE
================================================
MIT License

Copyright (c) 2017 lawlite

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: LinearRegression/LinearRegression.py
================================================
#-*- coding: utf-8 -*-
from __future__ import print_function
import numpy as np
from matplotlib import pyplot as plt
from matplotlib.font_manager import FontProperties
font = FontProperties(fname=r"c:\windows\fonts\simsun.ttc", size=14)    # 解决windows环境下画图汉字乱码问题


def linearRegression(alpha=0.01,num_iters=400):
    print(u"加载数据...\n")
    
    data = loadtxtAndcsv_data("data.txt",",",np.float64)  #读取数据
    X = data[:,0:-1]      # X对应0到倒数第2列                  
    y = data[:,-1]        # y对应最后一列  
    m = len(y)            # 总的数据条数
    col = data.shape[1]      # data的列数
    
    X,mu,sigma = featureNormaliza(X)    # 归一化
    plot_X1_X2(X)         # 画图看一下归一化效果
    
    X = np.hstack((np.ones((m,1)),X))    # 在X前加一列1
    
    print(u"\n执行梯度下降算法....\n")
    
    theta = np.zeros((col,1))
    y = y.reshape(-1,1)   #将行向量转化为列
    theta,J_history = gradientDescent(X, y, theta, alpha, num_iters)
    
    plotJ(J_history, num_iters)
    
    return mu,sigma,theta   #返回均值mu,标准差sigma,和学习的结果theta
    
   
# 加载txt和csv文件
def loadtxtAndcsv_data(fileName,split,dataType):
    return np.loadtxt(fileName,delimiter=split,dtype=dataType)

# 加载npy文件
def loadnpy_data(fileName):
    return np.load(fileName)

# 归一化feature
def featureNormaliza(X):
    X_norm = np.array(X)            #将X转化为numpy数组对象，才可以进行矩阵的运算
    #定义所需变量
    mu = np.zeros((1,X.shape[1]))   
    sigma = np.zeros((1,X.shape[1]))
    
    mu = np.mean(X_norm,0)          # 求每一列的平均值（0指定为列，1代表行）
    sigma = np.std(X_norm,0)        # 求每一列的标准差
    for i in range(X.shape[1]):     # 遍历列
        X_norm[:,i] = (X_norm[:,i]-mu[i])/sigma[i]  # 归一化
    
    return X_norm,mu,sigma

# 画二维图
def plot_X1_X2(X):
    plt.scatter(X[:,0],X[:,1])
    plt.show()


# 梯度下降算法
def gradientDescent(X,y,theta,alpha,num_iters):
    m = len(y)      
    n = len(theta)
    
    temp = np.matrix(np.zeros((n,num_iters)))   # 暂存每次迭代计算的theta，转化为矩阵形式
    
    
    J_history = np.zeros((num_iters,1)) #记录每次迭代计算的代价值
    
    for i in range(num_iters):  # 遍历迭代次数    
        h = np.dot(X,theta)     # 计算内积，matrix可以直接乘
        temp[:,i] = theta - ((alpha/m)*(np.dot(np.transpose(X),h-y)))   #梯度的计算
        theta = temp[:,i]
        J_history[i] = computerCost(X,y,theta)      #调用计算代价函数
        print('.', end=' ')      
    return theta,J_history  

# 计算代价函数
def computerCost(X,y,theta):
    m = len(y)
    J = 0
    
    J = (np.transpose(X*theta-y))*(X*theta-y)/(2*m) #计算代价J
    return J

# 画每次迭代代价的变化图
def plotJ(J_history,num_iters):
    x = np.arange(1,num_iters+1)
    plt.plot(x,J_history)
    plt.xlabel(u"迭代次数",fontproperties=font) # 注意指定字体，要不然出现乱码问题
    plt.ylabel(u"代价值",fontproperties=font)
    plt.title(u"代价随迭代次数的变化",fontproperties=font)
    plt.show()

# 测试linearRegression函数
def testLinearRegression():
    mu,sigma,theta = linearRegression(0.01,400)
    #print u"\n计算的theta值为：\n",theta
    #print u"\n预测结果为：%f"%predict(mu, sigma, theta)
    
# 测试学习效果（预测）
def predict(mu,sigma,theta):
    result = 0
    # 注意归一化
    predict = np.array([1650,3])
    norm_predict = (predict-mu)/sigma
    final_predict = np.hstack((np.ones((1)),norm_predict))
    
    result = np.dot(final_predict,theta)    # 预测结果
    return result
    
    
if __name__ == "__main__":
    testLinearRegression()

================================================
FILE: LinearRegression/LinearRegression_scikit-learn.py
================================================
#-*- coding: utf-8 -*-
from __future__ import print_function
import numpy as np
from sklearn import linear_model
from sklearn.preprocessing import StandardScaler    #引入归一化的包

def linearRegression():
    print(u"加载数据...\n")
    data = loadtxtAndcsv_data("data.txt",",",np.float64)  #读取数据
    X = np.array(data[:,0:-1],dtype=np.float64)      # X对应0到倒数第2列                  
    y = np.array(data[:,-1],dtype=np.float64)        # y对应最后一列  
        
    # 归一化操作
    scaler = StandardScaler()   
    scaler.fit(X)
    x_train = scaler.transform(X)
    x_test = scaler.transform(np.array([1650,3]))
    
    # 线性模型拟合
    model = linear_model.LinearRegression()
    model.fit(x_train, y)
    
    #预测结果
    result = model.predict(x_test)
    print(model.coef_)       # Coefficient of the features 决策函数中的特征系数
    print(model.intercept_)  # 又名bias偏置,若设置为False，则为0
    print(result)            # 预测结果


# 加载txt和csv文件
def loadtxtAndcsv_data(fileName,split,dataType):
    return np.loadtxt(fileName,delimiter=split,dtype=dataType)

# 加载npy文件
def loadnpy_data(fileName):
    return np.load(fileName)




if __name__ == "__main__":
    linearRegression()

================================================
FILE: LinearRegression/data.csv
================================================
2104,3,399900
1600,3,329900
2400,3,369000
1416,2,232000
3000,4,539900
1985,4,299900
1534,3,314900
1427,3,198999
1380,3,212000
1494,3,242500
1940,4,239999
2000,3,347000
1890,3,329999
4478,5,699900
1268,3,259900
2300,4,449900
1320,2,299900
1236,3,199900
2609,4,499998
3031,4,599000
1767,3,252900
1888,2,255000
1604,3,242900
1962,4,259900
3890,3,573900
1100,3,249900
1458,3,464500
2526,3,469000
2200,3,475000
2637,3,299900
1839,2,349900
1000,1,169900
2040,4,314900
3137,3,579900
1811,4,285900
1437,3,249900
1239,3,229900
2132,4,345000
4215,4,549000
2162,4,287000
1664,2,368500
2238,3,329900
2567,4,314000
1200,3,299000
852,2,179900
1852,4,299900
1203,3,239500


================================================
FILE: LinearRegression/data.txt
================================================
2104,3,399900
1600,3,329900
2400,3,369000
1416,2,232000
3000,4,539900
1985,4,299900
1534,3,314900
1427,3,198999
1380,3,212000
1494,3,242500
1940,4,239999
2000,3,347000
1890,3,329999
4478,5,699900
1268,3,259900
2300,4,449900
1320,2,299900
1236,3,199900
2609,4,499998
3031,4,599000
1767,3,252900
1888,2,255000
1604,3,242900
1962,4,259900
3890,3,573900
1100,3,249900
1458,3,464500
2526,3,469000
2200,3,475000
2637,3,299900
1839,2,349900
1000,1,169900
2040,4,314900
3137,3,579900
1811,4,285900
1437,3,249900
1239,3,229900
2132,4,345000
4215,4,549000
2162,4,287000
1664,2,368500
2238,3,329900
2567,4,314000
1200,3,299000
852,2,179900
1852,4,299900
1203,3,239500


================================================
FILE: LogisticRegression/LogisticRegression.py
================================================
#-*- coding: utf-8 -*-
from __future__ import print_function
import numpy as np
import matplotlib.pyplot as plt
from scipy import optimize
from matplotlib.font_manager import FontProperties
font = FontProperties(fname=r"c:\windows\fonts\simsun.ttc", size=14)    # 解决windows环境下画图汉字乱码问题


def LogisticRegression():
    data = loadtxtAndcsv_data("data2.txt", ",", np.float64) 
    X = data[:,0:-1]
    y = data[:,-1]
    
    plot_data(X,y)  # 作图
    
    X = mapFeature(X[:,0],X[:,1])           #映射为多项式
    initial_theta = np.zeros((X.shape[1],1))#初始化theta
    initial_lambda = 0.1                    #初始化正则化系数，一般取0.01,0.1,1.....
    
    J = costFunction(initial_theta,X,y,initial_lambda)  #计算一下给定初始化的theta和lambda求出的代价J
    
    print(J)  #输出一下计算的值，应该为0.693147
    #result = optimize.fmin(costFunction, initial_theta, args=(X,y,initial_lambda))    #直接使用最小化的方法，效果不好
    '''调用scipy中的优化算法fmin_bfgs（拟牛顿法Broyden-Fletcher-Goldfarb-Shanno）
    - costFunction是自己实现的一个求代价的函数，
    - initial_theta表示初始化的值,
    - fprime指定costFunction的梯度
    - args是其余测参数，以元组的形式传入，最后会将最小化costFunction的theta返回 
    '''
    result = optimize.fmin_bfgs(costFunction, initial_theta, fprime=gradient, args=(X,y,initial_lambda))    
    p = predict(X, result)   #预测
    print(u'在训练集上的准确度为%f%%'%np.mean(np.float64(p==y)*100))   # 与真实值比较，p==y返回True，转化为float   
    
    X = data[:,0:-1]
    y = data[:,-1]    
    plotDecisionBoundary(result,X,y)    #画决策边界  
    
    

# 加载txt和csv文件
def loadtxtAndcsv_data(fileName,split,dataType):
    return np.loadtxt(fileName,delimiter=split,dtype=dataType)

# 加载npy文件
def loadnpy_data(fileName):
    return np.load(fileName)

# 显示二维图形
def plot_data(X,y):
    pos = np.where(y==1)    #找到y==1的坐标位置
    neg = np.where(y==0)    #找到y==0的坐标位置
    #作图
    plt.figure(figsize=(15,12))
    plt.plot(X[pos,0],X[pos,1],'ro')        # red o
    plt.plot(X[neg,0],X[neg,1],'bo')        # blue o
    plt.title(u"两个类别散点图",fontproperties=font)
    plt.show()

# 映射为多项式 
def mapFeature(X1,X2):
    degree = 2;                     # 映射的最高次方
    out = np.ones((X1.shape[0],1))  # 映射后的结果数组（取代X）
    '''
    这里以degree=2为例，映射为1,x1,x2,x1^2,x1,x2,x2^2
    '''
    for i in np.arange(1,degree+1): 
        for j in range(i+1):
            temp = X1**(i-j)*(X2**j)    #矩阵直接乘相当于matlab中的点乘.*
            out = np.hstack((out, temp.reshape(-1,1)))
    return out

# 代价函数
def costFunction(initial_theta,X,y,inital_lambda):
    m = len(y)
    J = 0
    
    h = sigmoid(np.dot(X,initial_theta))    # 计算h(z)
    theta1 = initial_theta.copy()           # 因为正则化j=1从1开始，不包含0，所以复制一份，前theta(0)值为0 
    theta1[0] = 0   
    
    temp = np.dot(np.transpose(theta1),theta1)
    J = (-np.dot(np.transpose(y),np.log(h))-np.dot(np.transpose(1-y),np.log(1-h))+temp*inital_lambda/2)/m   # 正则化的代价方程
    return J

# 计算梯度
def gradient(initial_theta,X,y,inital_lambda):
    m = len(y)
    grad = np.zeros((initial_theta.shape[0]))
    
    h = sigmoid(np.dot(X,initial_theta))# 计算h(z)
    theta1 = initial_theta.copy()
    theta1[0] = 0

    grad = np.dot(np.transpose(X),h-y)/m+inital_lambda/m*theta1 #正则化的梯度
    return grad

# S型函数    
def sigmoid(z):
    h = np.zeros((len(z),1))    # 初始化，与z的长度一置
    
    h = 1.0/(1.0+np.exp(-z))
    return h


#画决策边界
def plotDecisionBoundary(theta,X,y):
    pos = np.where(y==1)    #找到y==1的坐标位置
    neg = np.where(y==0)    #找到y==0的坐标位置
    #作图
    plt.figure(figsize=(15,12))
    plt.plot(X[pos,0],X[pos,1],'ro')        # red o
    plt.plot(X[neg,0],X[neg,1],'bo')        # blue o
    plt.title(u"决策边界",fontproperties=font)
    
    #u = np.linspace(30,100,100)
    #v = np.linspace(30,100,100)
    
    u = np.linspace(-1,1.5,50)  #根据具体的数据，这里需要调整
    v = np.linspace(-1,1.5,50)
    
    z = np.zeros((len(u),len(v)))
    for i in range(len(u)):
        for j in range(len(v)):
            z[i,j] = np.dot(mapFeature(u[i].reshape(1,-1),v[j].reshape(1,-1)),theta)    # 计算对应的值，需要map
    
    z = np.transpose(z)
    plt.contour(u,v,z,[0,0.01],linewidth=2.0)   # 画等高线，范围在[0,0.01]，即近似为决策边界
    #plt.legend()
    plt.show()

# 预测
def predict(X,theta):
    m = X.shape[0]
    p = np.zeros((m,1))
    p = sigmoid(np.dot(X,theta))    # 预测的结果，是个概率值
    
    for i in range(m):
        if p[i] > 0.5:  #概率大于0.5预测为1，否则预测为0
            p[i] = 1
        else:
            p[i] = 0
    return p


# 测试逻辑回归函数
def testLogisticRegression():
    LogisticRegression()


if __name__ == "__main__":
    testLogisticRegression()
    



================================================
FILE: LogisticRegression/LogisticRegression_OneVsAll.py
================================================
#-*- coding: utf-8 -*-
from __future__ import print_function
import numpy as np
import matplotlib.pyplot as plt
import scipy.io as spio
from scipy import optimize
from matplotlib.font_manager import FontProperties
font = FontProperties(fname=r"c:\windows\fonts\simsun.ttc", size=14)    # 解决windows环境下画图汉字乱码问题


def logisticRegression_OneVsAll():
    data = loadmat_data("data_digits.mat") 
    X = data['X']   # 获取X数据，每一行对应一个数字20x20px
    y = data['y']
    m,n = X.shape
    num_labels = 10 # 数字个数，0-9
    
    ##　随机显示几行数据
    rand_indices = [t for t in [np.random.randint(x-x, m) for x in range(100)]]  # 生成100个0-m的随机数
    display_data(X[rand_indices,:])     # 显示100个数字
    
    Lambda = 0.1    # 正则化系数
    #y = y.reshape(-1,1)
    all_theta = oneVsAll(X, y, num_labels, Lambda)  # 计算所有的theta
    
    p = predict_oneVsAll(all_theta,X)               # 预测
    # 将预测结果和真实结果保存到文件中
    #res = np.hstack((p,y.reshape(-1,1)))
    #np.savetxt("predict.csv", res, delimiter=',')
    
    print(u"预测准确度为：%f%%"%np.mean(np.float64(p == y.reshape(-1,1))*100))
     
# 加载mat文件
def loadmat_data(fileName):
    return spio.loadmat(fileName)
    
# 显示100个数字
def display_data(imgData):
    sum = 0
    '''
    显示100个数（若是一个一个绘制将会非常慢，可以将要画的数字整理好，放到一个矩阵中，显示这个矩阵即可）
    - 初始化一个二维数组
    - 将每行的数据调整成图像的矩阵，放进二维数组
    - 显示即可
    '''
    pad = 1
    display_array = -np.ones((pad+10*(20+pad),pad+10*(20+pad)))
    for i in range(10):
        for j in range(10):
            display_array[pad+i*(20+pad):pad+i*(20+pad)+20,pad+j*(20+pad):pad+j*(20+pad)+20] = (imgData[sum,:].reshape(20,20,order="F"))    # order=F指定以列优先，在matlab中是这样的，python中需要指定，默认以行
            sum += 1
            
    plt.imshow(display_array,cmap='gray')   #显示灰度图像
    plt.axis('off')
    plt.show()

# 求每个分类的theta，最后返回所有的all_theta    
def oneVsAll(X,y,num_labels,Lambda):
    # 初始化变量
    m,n = X.shape
    all_theta = np.zeros((n+1,num_labels))  # 每一列对应相应分类的theta,共10列
    X = np.hstack((np.ones((m,1)),X))       # X前补上一列1的偏置bias
    class_y = np.zeros((m,num_labels))      # 数据的y对应0-9，需要映射为0/1的关系
    initial_theta = np.zeros((n+1,1))       # 初始化一个分类的theta
    
    # 映射y
    for i in range(num_labels):
        class_y[:,i] = np.int32(y==i).reshape(1,-1) # 注意reshape(1,-1)才可以赋值
    
    #np.savetxt("class_y.csv", class_y[0:600,:], delimiter=',')    
    
    '''遍历每个分类，计算对应的theta值'''
    for i in range(num_labels):
        #optimize.fmin_cg
        result = optimize.fmin_bfgs(costFunction, initial_theta, fprime=gradient, args=(X,class_y[:,i],Lambda)) # 调用梯度下降的优化方法
        all_theta[:,i] = result.reshape(1,-1)   # 放入all_theta中
        
    all_theta = np.transpose(all_theta) 
    return all_theta

# 代价函数
def costFunction(initial_theta,X,y,inital_lambda):
    m = len(y)
    J = 0
    
    h = sigmoid(np.dot(X,initial_theta))    # 计算h(z)
    theta1 = initial_theta.copy()           # 因为正则化j=1从1开始，不包含0，所以复制一份，前theta(0)值为0 
    theta1[0] = 0   
    
    temp = np.dot(np.transpose(theta1),theta1)
    J = (-np.dot(np.transpose(y),np.log(h))-np.dot(np.transpose(1-y),np.log(1-h))+temp*inital_lambda/2)/m   # 正则化的代价方程
    return J

# 计算梯度
def gradient(initial_theta,X,y,inital_lambda):
    m = len(y)
    grad = np.zeros((initial_theta.shape[0]))
    
    h = sigmoid(np.dot(X,initial_theta))  # 计算h(z)
    theta1 = initial_theta.copy()
    theta1[0] = 0

    grad = np.dot(np.transpose(X),h-y)/m+inital_lambda/m*theta1 #正则化的梯度
    return grad   
    
# S型函数    
def sigmoid(z):
    h = np.zeros((len(z),1))    # 初始化，与z的长度一致
    
    h = 1.0/(1.0+np.exp(-z))
    return h

# 预测
def predict_oneVsAll(all_theta,X):
    m = X.shape[0]
    num_labels = all_theta.shape[0]
    p = np.zeros((m,1))
    X = np.hstack((np.ones((m,1)),X))   #在X最前面加一列1
    
    h = sigmoid(np.dot(X,np.transpose(all_theta)))  #预测

    '''
    返回h中每一行最大值所在的列号
    - np.max(h, axis=1)返回h中每一行的最大值（是某个数字的最大概率）
    - 最后where找到的最大概率所在的列号（列号即是对应的数字）
    '''
    p = np.array(np.where(h[0,:] == np.max(h, axis=1)[0]))  
    for i in np.arange(1, m):
        t = np.array(np.where(h[i,:] == np.max(h, axis=1)[i]))
        p = np.vstack((p,t))
    return p
        
        
if __name__ == "__main__":
    logisticRegression_OneVsAll()

================================================
FILE: LogisticRegression/LogisticRegression_OneVsAll_scikit-learn.py
================================================
#-*- coding: utf-8 -*-
from __future__ import print_function
from scipy import io as spio
import numpy as np
from sklearn import svm
from sklearn.linear_model import LogisticRegression



def logisticRegression_oneVsAll():
    data = loadmat_data("data_digits.mat") 
    X = data['X']   # 获取X数据，每一行对应一个数字20x20px
    y = data['y']   # 这里读取mat文件y的shape=(5000, 1)
    y = np.ravel(y) # 调用sklearn需要转化成一维的(5000,)
    
    model = LogisticRegression()
    model.fit(X, y) # 拟合
    
    predict = model.predict(X) #预测
    
    print(u"预测准确度为：%f%%"%np.mean(np.float64(predict == y)*100))

# 加载mat文件
def loadmat_data(fileName):
    return spio.loadmat(fileName)

if __name__ == "__main__":
    logisticRegression_oneVsAll()

================================================
FILE: LogisticRegression/LogisticRegression_scikit-learn.py
================================================
# -*- coding: utf-8 -*-

from sklearn.linear_model import LogisticRegression
from sklearn.preprocessing import StandardScaler
# from sklearn.cross_validation import train_test_split  # 0.18版本之后废弃
from sklearn.model_selection import train_test_split
import numpy as np

def logisticRegression():
    data = loadtxtAndcsv_data("data1.txt", ",", np.float64)
    X = data[:,0:-1]
    y = data[:,-1]

    # 划分为训练集和测试集
    x_train,x_test,y_train,y_test = train_test_split(X,y,test_size=0.2)

    # 归一化
    scaler = StandardScaler()
    # scaler.fit(x_train)
    x_train = scaler.fit_transform(x_train)
    x_test = scaler.fit_transform(x_test)

    # 逻辑回归
    model = LogisticRegression()
    model.fit(x_train,y_train)

    # 预测
    predict = model.predict(x_test)
    right = sum(predict == y_test)

    predict = np.hstack((predict.reshape(-1,1),y_test.reshape(-1,1)))   # 将预测值和真实值放在一块，好观察
    print(predict)
    print('测试集准确率：%f%%'%(right*100.0/predict.shape[0]))          # 计算在测试集上的准确度

# 加载txt和csv文件
def loadtxtAndcsv_data(fileName,split,dataType):
    return np.loadtxt(fileName,delimiter=split,dtype=dataType)

# 加载npy文件
def loadnpy_data(fileName):
    return np.load(fileName)


if __name__ == "__main__":
    logisticRegression()


================================================
FILE: LogisticRegression/class_y.csv
================================================
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================================================
FILE: LogisticRegression/data1.txt
================================================
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FILE: LogisticRegression/data2.txt
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FILE: LogisticRegression/predict.csv
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================================================
FILE: NeuralNetwok/NeuralNetwork.py
================================================
#-*- coding: utf-8 -*-
import numpy as np
from scipy import io as spio
from matplotlib import pyplot as plt
from scipy import optimize
from matplotlib.font_manager import FontProperties
font = FontProperties(fname=r"c:\windows\fonts\simsun.ttc", size=14)    # 解决windows环境下画图汉字乱码问题

from sklearn import datasets
from sklearn.preprocessing import StandardScaler
import time

def neuralNetwork(input_layer_size,hidden_layer_size,out_put_layer):
    data_img = loadmat_data("data_digits.mat")
    X = data_img['X']
    y = data_img['y']

    '''scaler = StandardScaler()
    scaler.fit(X)
    X = scaler.transform(X)'''  
    
    m,n = X.shape
    """digits = datasets.load_digits()
    X = digits.data
    y = digits.target
    m,n = X.shape
    
    scaler = StandardScaler()
    scaler.fit(X)
    X = scaler.transform(X)"""
    
    ##　随机显示几行数据
    rand_indices = [t for t in [np.random.randint(x-x, m) for x in range(100)]]  # 生成100个0-m的随机数
    display_data(X[rand_indices,:])     # 显示100个数字    
    
    #nn_params = np.vstack((Theta1.reshape(-1,1),Theta2.reshape(-1,1)))
    
    Lambda = 1
    
    initial_Theta1 = randInitializeWeights(input_layer_size,hidden_layer_size); 
    initial_Theta2 = randInitializeWeights(hidden_layer_size,out_put_layer)
    
    initial_nn_params = np.vstack((initial_Theta1.reshape(-1,1),initial_Theta2.reshape(-1,1)))  #展开theta    
    #np.savetxt("testTheta.csv",initial_nn_params,delimiter=",")
    start = time.time()
    result = optimize.fmin_cg(nnCostFunction, initial_nn_params, fprime=nnGradient, args=(input_layer_size,hidden_layer_size,out_put_layer,X,y,Lambda), maxiter=100)
    print (u'执行时间：',time.time()-start)
    print (result)
    '''可视化 Theta1'''
    length = result.shape[0]
    Theta1 = result[0:hidden_layer_size*(input_layer_size+1)].reshape(hidden_layer_size,input_layer_size+1)
    Theta2 = result[hidden_layer_size*(input_layer_size+1):length].reshape(out_put_layer,hidden_layer_size+1)    
    display_data(Theta1[:,1:length])
    display_data(Theta2[:,1:length])
    '''预测'''
    p = predict(Theta1,Theta2,X)
    print (u"预测准确度为：%f%%"%np.mean(np.float64(p == y.reshape(-1,1))*100))    
    res = np.hstack((p,y.reshape(-1,1)))
    np.savetxt("predict.csv", res, delimiter=',')
    

# 加载mat文件
def loadmat_data(fileName):
    return spio.loadmat(fileName)

# 显示100个数字
def display_data(imgData):
    sum = 0
    '''
    显示100个数（若是一个一个绘制将会非常慢，可以将要画的数字整理好，放到一个矩阵中，显示这个矩阵即可）
    - 初始化一个二维数组
    - 将每行的数据调整成图像的矩阵，放进二维数组
    - 显示即可
    '''
    m,n = imgData.shape
    width = np.int32(np.round(np.sqrt(n)))
    height = np.int32(n/width);
    rows_count = np.int32(np.floor(np.sqrt(m)))
    cols_count = np.int32(np.ceil(m/rows_count))
    pad = 1
    display_array = -np.ones((pad+rows_count*(height+pad),pad+cols_count*(width+pad)))
    for i in range(rows_count):
        for j in range(cols_count):
            if sum >= m: #超过了行数，退出当前循环
                break;
            display_array[pad+i*(height+pad):pad+i*(height+pad)+height,pad+j*(width+pad):pad+j*(width+pad)+width] = imgData[sum,:].reshape(height,width,order="F")    # order=F指定以列优先，在matlab中是这样的，python中需要指定，默认以行
            sum += 1
        if sum >= m:  #超过了行数，退出当前循环
            break;
            
    plt.imshow(display_array,cmap='gray')   #显示灰度图像
    plt.axis('off')
    plt.show()

# 代价函数
def nnCostFunction(nn_params,input_layer_size,hidden_layer_size,num_labels,X,y,Lambda):
    length = nn_params.shape[0] # theta的中长度
    # 还原theta1和theta2
    Theta1 = nn_params[0:hidden_layer_size*(input_layer_size+1)].reshape(hidden_layer_size,input_layer_size+1)
    Theta2 = nn_params[hidden_layer_size*(input_layer_size+1):length].reshape(num_labels,hidden_layer_size+1)
    
    # np.savetxt("Theta1.csv",Theta1,delimiter=',')
    
    m = X.shape[0]
    class_y = np.zeros((m,num_labels))      # 数据的y对应0-9，需要映射为0/1的关系
    # 映射y
    for i in range(num_labels):
        class_y[:,i] = np.int32(y==i).reshape(1,-1) # 注意reshape(1,-1)才可以赋值
     
    '''去掉theta1和theta2的第一列，因为正则化时从1开始'''    
    Theta1_colCount = Theta1.shape[1]    
    Theta1_x = Theta1[:,1:Theta1_colCount]
    Theta2_colCount = Theta2.shape[1]    
    Theta2_x = Theta2[:,1:Theta2_colCount]
    # 正则化向theta^2
    term = np.dot(np.transpose(np.vstack((Theta1_x.reshape(-1,1),Theta2_x.reshape(-1,1)))),np.vstack((Theta1_x.reshape(-1,1),Theta2_x.reshape(-1,1))))
    
    '''正向传播,每次需要补上一列1的偏置bias'''
    a1 = np.hstack((np.ones((m,1)),X))      
    z2 = np.dot(a1,np.transpose(Theta1))    
    a2 = sigmoid(z2)
    a2 = np.hstack((np.ones((m,1)),a2))
    z3 = np.dot(a2,np.transpose(Theta2))
    h  = sigmoid(z3)    
    '''代价'''    
    J = -(np.dot(np.transpose(class_y.reshape(-1,1)),np.log(h.reshape(-1,1)))+np.dot(np.transpose(1-class_y.reshape(-1,1)),np.log(1-h.reshape(-1,1)))-Lambda*term/2)/m   
    #temp1 = (h.reshape(-1,1)-class_y.reshape(-1,1))
    #temp2 = (temp1**2).sum()
    #J = 1/(2*m)*temp2
    return np.ravel(J)

# 梯度
def nnGradient(nn_params,input_layer_size,hidden_layer_size,num_labels,X,y,Lambda):
    length = nn_params.shape[0]
    Theta1 = nn_params[0:hidden_layer_size*(input_layer_size+1)].reshape(hidden_layer_size,input_layer_size+1).copy()   # 这里使用copy函数，否则下面修改Theta的值，nn_params也会一起修改
    Theta2 = nn_params[hidden_layer_size*(input_layer_size+1):length].reshape(num_labels,hidden_layer_size+1).copy()
    m = X.shape[0]
    class_y = np.zeros((m,num_labels))      # 数据的y对应0-9，需要映射为0/1的关系    
    # 映射y
    for i in range(num_labels):
        class_y[:,i] = np.int32(y==i).reshape(1,-1) # 注意reshape(1,-1)才可以赋值
     
    '''去掉theta1和theta2的第一列，因为正则化时从1开始'''
    Theta1_colCount = Theta1.shape[1]    
    Theta1_x = Theta1[:,1:Theta1_colCount]
    Theta2_colCount = Theta2.shape[1]    
    Theta2_x = Theta2[:,1:Theta2_colCount]
    
    Theta1_grad = np.zeros((Theta1.shape))  #第一层到第二层的权重
    Theta2_grad = np.zeros((Theta2.shape))  #第二层到第三层的权重
      
   
    '''正向传播，每次需要补上一列1的偏置bias'''
    a1 = np.hstack((np.ones((m,1)),X))
    z2 = np.dot(a1,np.transpose(Theta1))
    a2 = sigmoid(z2)
    a2 = np.hstack((np.ones((m,1)),a2))
    z3 = np.dot(a2,np.transpose(Theta2))
    h  = sigmoid(z3)
    
    
    '''反向传播，delta为误差，'''
    delta3 = np.zeros((m,num_labels))
    delta2 = np.zeros((m,hidden_layer_size))
    for i in range(m):
        #delta3[i,:] = (h[i,:]-class_y[i,:])*sigmoidGradient(z3[i,:])  # 均方误差的误差率
        delta3[i,:] = h[i,:]-class_y[i,:]                              # 交叉熵误差率
        Theta2_grad = Theta2_grad+np.dot(np.transpose(delta3[i,:].reshape(1,-1)),a2[i,:].reshape(1,-1))
        delta2[i,:] = np.dot(delta3[i,:].reshape(1,-1),Theta2_x)*sigmoidGradient(z2[i,:])
        Theta1_grad = Theta1_grad+np.dot(np.transpose(delta2[i,:].reshape(1,-1)),a1[i,:].reshape(1,-1))
    
    Theta1[:,0] = 0
    Theta2[:,0] = 0          
    '''梯度'''
    grad = (np.vstack((Theta1_grad.reshape(-1,1),Theta2_grad.reshape(-1,1)))+Lambda*np.vstack((Theta1.reshape(-1,1),Theta2.reshape(-1,1))))/m
    return np.ravel(grad)

# S型函数    
def sigmoid(z):
    h = np.zeros((len(z),1))    # 初始化，与z的长度一致
    
    h = 1.0/(1.0+np.exp(-z))
    return h

# S型函数导数
def sigmoidGradient(z):
    g = sigmoid(z)*(1-sigmoid(z))
    return g

# 随机初始化权重theta
def randInitializeWeights(L_in,L_out):
    W = np.zeros((L_out,1+L_in))    # 对应theta的权重
    epsilon_init = (6.0/(L_out+L_in))**0.5
    W = np.random.rand(L_out,1+L_in)*2*epsilon_init-epsilon_init # np.random.rand(L_out,1+L_in)产生L_out*(1+L_in)大小的随机矩阵
    return W


# 检验梯度是否计算正确
def checkGradient(Lambda = 0):
    '''构造一个小型的神经网络验证，因为数值法计算梯度很浪费时间，而且验证正确后之后就不再需要验证了'''
    input_layer_size = 3
    hidden_layer_size = 5
    num_labels = 3
    m = 5
    initial_Theta1 = debugInitializeWeights(input_layer_size,hidden_layer_size); 
    initial_Theta2 = debugInitializeWeights(hidden_layer_size,num_labels)
    X = debugInitializeWeights(input_layer_size-1,m)
    y = np.transpose(np.mod(np.arange(1,m+1), num_labels))# 初始化y
    
    y = y.reshape(-1,1)
    nn_params = np.vstack((initial_Theta1.reshape(-1,1),initial_Theta2.reshape(-1,1)))  #展开theta 
    '''BP求出梯度'''
    grad = nnGradient(nn_params, input_layer_size, hidden_layer_size, 
                     num_labels, X, y, Lambda)  
    '''使用数值法计算梯度'''
    num_grad = np.zeros((nn_params.shape[0]))
    step = np.zeros((nn_params.shape[0]))
    e = 1e-4
    for i in range(nn_params.shape[0]):
        step[i] = e
        loss1 = nnCostFunction(nn_params-step.reshape(-1,1), input_layer_size, hidden_layer_size, 
                              num_labels, X, y, 
                              Lambda)
        loss2 = nnCostFunction(nn_params+step.reshape(-1,1), input_layer_size, hidden_layer_size, 
                              num_labels, X, y, 
                              Lambda)
        num_grad[i] = (loss2-loss1)/(2*e)
        step[i]=0
    # 显示两列比较
    res = np.hstack((num_grad.reshape(-1,1),grad.reshape(-1,1)))
    print("检查梯度的结果，第一列为数值法计算得到的，第二列为BP得到的:")
    print (res)

# 初始化调试的theta权重
def debugInitializeWeights(fan_in,fan_out):
    W = np.zeros((fan_out,fan_in+1))
    x = np.arange(1,fan_out*(fan_in+1)+1)
    W = np.sin(x).reshape(W.shape)/10
    return W

# 预测
def predict(Theta1,Theta2,X):
    m = X.shape[0]
    num_labels = Theta2.shape[0]
    #p = np.zeros((m,1))
    '''正向传播，预测结果'''
    X = np.hstack((np.ones((m,1)),X))
    h1 = sigmoid(np.dot(X,np.transpose(Theta1)))
    h1 = np.hstack((np.ones((m,1)),h1))
    h2 = sigmoid(np.dot(h1,np.transpose(Theta2)))
    
    '''
    返回h中每一行最大值所在的列号
    - np.max(h, axis=1)返回h中每一行的最大值（是某个数字的最大概率）
    - 最后where找到的最大概率所在的列号（列号即是对应的数字）
    '''
    #np.savetxt("h2.csv",h2,delimiter=',')
    p = np.array(np.where(h2[0,:] == np.max(h2, axis=1)[0]))  
    for i in np.arange(1, m):
        t = np.array(np.where(h2[i,:] == np.max(h2, axis=1)[i]))
        p = np.vstack((p,t))
    return p    

if __name__ == "__main__":
    checkGradient()
    neuralNetwork(400, 25, 10)

================================================
FILE: NeuralNetwok/predict.csv
================================================
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================================================
FILE: PCA/PCA.py
================================================
#-*- coding: utf-8 -*-
# Author: Bob
# Date:   2016.11.24
import numpy as np
from matplotlib import pyplot as plt
from scipy import io as spio
from sklearn.decomposition import pca

'''
主成分分析_2维数据降维1维演示函数
'''
def PCA_2D():
    data_2d = spio.loadmat("data.mat")
    X = data_2d['X']
    m = X.shape[0]
    plt = plot_data_2d(X,'bo') # 显示二维的数据
    plt.show()
    
    X_copy = X.copy()
    X_norm,mu,sigma = featureNormalize(X_copy)    # 归一化数据
    #plot_data_2d(X_norm)    # 显示归一化后的数据
    #plt.show()
    
    Sigma = np.dot(np.transpose(X_norm),X_norm)/m  # 求Sigma
    U,S,V = np.linalg.svd(Sigma)       # 求Sigma的奇异值分解
    
    plt = plot_data_2d(X,'bo') # 显示原本数据
    drawline(plt, mu, mu+S[0]*(U[:,0]), 'r-')  # 线，为投影的方向

    plt.axis('square')
    plt.show()
    
    K = 1  # 定义降维多少维（本来是2维的，这里降维1维）
    '''投影之后数据（降维之后）'''
    Z = projectData(X_norm,U,K)   # 投影
    '''恢复数据'''
    X_rec = recoverData(Z,U,K)    # 恢复
    '''作图-----原数据与恢复的数据'''
    plt = plot_data_2d(X_norm,'bo')
    plot_data_2d(X_rec,'ro')
    for i in range(X_norm.shape[0]):
        drawline(plt, X_norm[i,:], X_rec[i,:], '--k')
    plt.axis('square')
    plt.show()


'''主成分分析_PCA图像数据降维'''    
def PCA_faceImage():
    print (u'加载图像数据.....')
    data_image = spio.loadmat('data_faces.mat')
    X = data_image['X']
    display_imageData(X[0:100,:])
    m = X.shape[0]  # 数据条数
    
    print (u'运行PCA....')
    X_norm,mu,sigma = featureNormalize(X)  # 归一化
    
    Sigma = np.dot(np.transpose(X_norm),X_norm)/m  # 求Sigma
    U,S,V = np.linalg.svd(Sigma)                   # 奇异值分解
    display_imageData(np.transpose(U[:,0:36]))     # 显示U的数据
    
    print (u'对face数据降维.....')
    K = 100     # 降维100维(原先是32*32=1024维的)
    Z = projectData(X_norm, U, K)
    print (u'投影之后Z向量的大小：%d %d' %Z.shape)
    
    print (u'显示降维之后的数据......')
    X_rec = recoverData(Z, U, K) # 恢复数据
    display_imageData(X_rec[0:100,:])
    
    
      
    
# 可视化二维数据
def plot_data_2d(X,marker):
    plt.plot(X[:,0],X[:,1],marker) 
    return plt

# 归一化数据
def featureNormalize(X):
    '''（每一个数据-当前列的均值）/当前列的标准差'''
    n = X.shape[1]
    mu = np.zeros((1,n));
    sigma = np.zeros((1,n))
    
    mu = np.mean(X,axis=0)   # axis=0表示列
    sigma = np.std(X,axis=0)
    for i in range(n):
        X[:,i] = (X[:,i]-mu[i])/sigma[i]
    return X,mu,sigma


# 映射数据
def projectData(X_norm,U,K):
    Z = np.zeros((X_norm.shape[0],K))
    
    U_reduce = U[:,0:K]          # 取前K个
    Z = np.dot(X_norm,U_reduce) 
    return Z

# 画一条线
def drawline(plt,p1,p2,line_type):
    plt.plot(np.array([p1[0],p2[0]]),np.array([p1[1],p2[1]]),line_type)  

  

# 恢复数据
def recoverData(Z,U,K):
    X_rec = np.zeros((Z.shape[0],U.shape[0]))
    U_recude = U[:,0:K]
    X_rec = np.dot(Z,np.transpose(U_recude))  # 还原数据（近似）
    return X_rec

# 显示图片
def display_imageData(imgData):
    sum = 0
    '''
    显示100个数（若是一个一个绘制将会非常慢，可以将要画的图片整理好，放到一个矩阵中，显示这个矩阵即可）
    - 初始化一个二维数组
    - 将每行的数据调整成图像的矩阵，放进二维数组
    - 显示即可
    '''
    m,n = imgData.shape
    width = np.int32(np.round(np.sqrt(n)))
    height = np.int32(n/width);
    rows_count = np.int32(np.floor(np.sqrt(m)))
    cols_count = np.int32(np.ceil(m/rows_count))
    pad = 1
    display_array = -np.ones((pad+rows_count*(height+pad),pad+cols_count*(width+pad)))
    for i in range(rows_count):
        for j in range(cols_count):
            max_val = np.max(np.abs(imgData[sum,:]))
            display_array[pad+i*(height+pad):pad+i*(height+pad)+height,pad+j*(width+pad):pad+j*(width+pad)+width] = imgData[sum,:].reshape(height,width,order="F")/max_val    # order=F指定以列优先，在matlab中是这样的，python中需要指定，默认以行
            sum += 1
            
    plt.imshow(display_array,cmap='gray')   #显示灰度图像
    plt.axis('off')
    plt.show()

if __name__ == "__main__":
    PCA_2D()
    PCA_faceImage()

================================================
FILE: PCA/PCA_scikit-learn.py
================================================
#-*- coding: utf-8 -*-
# Author:bob
# Date:2016.12.22
import numpy as np
from matplotlib import pyplot as plt
from scipy import io as spio
from sklearn.decomposition import pca
from sklearn.preprocessing import StandardScaler

'''二维降为一维主运行函数'''
def PCA_2d_example():
    '''加载数据并作图'''
    data = spio.loadmat('data.mat')
    X = data['X']
    plt = plot_data_2d(X,'bo')
    plt.axis('square')
    plt.title('original data')
    plt.show()
    '''归一化数据并作图'''
    scaler = StandardScaler()
    scaler.fit(X)
    x_train = scaler.transform(X)
    
    plot_data_2d(x_train, 'bo')
    plt.axis('square')
    plt.title('scaler data')
    plt.show()
    
    '''拟合数据'''
    K=1 # 要降的维度
    model = pca.PCA(n_components=K).fit(x_train)   # 拟合数据，n_components定义要降的维度
    Z = model.transform(x_train)    # transform就会执行降维操作
    
    '''数据恢复并作图'''
    Ureduce = model.components_     # 得到降维用的Ureduce
    x_rec = np.dot(Z,Ureduce)       # 数据恢复
    
    plot_data_2d(x_rec,'bo')
    plt.plot()
    plt.axis('square')
    plt.title('recover data')
    plt.show()

def PCA_face_example():
    '''加载数据并显示'''
    image_data = spio.loadmat('data_faces.mat')
    X = image_data['X']
    display_imageData(X[0:100,:])  # 显示100个最初图像
    
    '''归一化数据'''
    scaler = StandardScaler()
    scaler.fit(X)
    x_train = scaler.transform(X)
    
    '''拟合模型'''
    K=100
    model = pca.PCA(n_components=K).fit(x_train)
    Z = model.transform(x_train)
    Ureduce = model.components_
    
    display_imageData(Ureduce[0:36,:])  # 可视化部分U数据
    x_rec = np.dot(Z,Ureduce)
    
    display_imageData(x_rec[0:100,:])  # 显示恢复的数据
    

    
    
# 可视化二维数据
def plot_data_2d(X,marker):
    plt.plot(X[:,0],X[:,1],marker) 
    return plt

# 显示图片
def display_imageData(imgData):
    sum = 0
    '''
    显示100个数（若是一个一个绘制将会非常慢，可以将要画的图片整理好，放到一个矩阵中，显示这个矩阵即可）
    - 初始化一个二维数组
    - 将每行的数据调整成图像的矩阵，放进二维数组
    - 显示即可
    '''
    m,n = imgData.shape
    width = np.int32(np.round(np.sqrt(n)))
    height = np.int32(n/width);
    rows_count = np.int32(np.floor(np.sqrt(m)))
    cols_count = np.int32(np.ceil(m/rows_count))
    pad = 1
    display_array = -np.ones((pad+rows_count*(height+pad),pad+cols_count*(width+pad)))
    for i in range(rows_count):
        for j in range(cols_count):
            max_val = np.max(np.abs(imgData[sum,:]))
            display_array[pad+i*(height+pad):pad+i*(height+pad)+height,pad+j*(width+pad):pad+j*(width+pad)+width] = imgData[sum,:].reshape(height,width,order="F")/max_val    # order=F指定以列优先，在matlab中是这样的，python中需要指定，默认以行
            sum += 1
            
    plt.imshow(display_array,cmap='gray')   #显示灰度图像
    plt.axis('off')
    plt.show()


if __name__ == '__main__':
    #PCA_2d_example()
    PCA_face_example()

================================================
FILE: PCA/data_faces.mat
================================================
[File too large to display: 10.5 MB]

================================================
FILE: SVM/SVM_scikit-learn.py
================================================
#-*- coding: utf-8 -*-
import numpy as np
from scipy import io as spio
from matplotlib import pyplot as plt
from sklearn import svm


def SVM():
    '''data1——线性分类'''
    data1 = spio.loadmat('data1.mat')
    X = data1['X']
    y = data1['y']
    y = np.ravel(y)
    plot_data(X, y)

    model = svm.SVC(C=1.0, kernel='linear').fit(X, y)  # 指定核函数为线性核函数
    plot_decisionBoundary(X, y, model)  # 画决策边界
    '''data2——非线性分类'''
    data2 = spio.loadmat('data2.mat')
    X = data2['X']
    y = data2['y']
    y = np.ravel(y)
    plt = plot_data(X, y)
    plt.show()

    model = svm.SVC(gamma=100).fit(X, y)  # gamma为核函数的系数，值越大拟合的越好
    plot_decisionBoundary(X, y, model, class_='notLinear')  # 画决策边界


# 作图
def plot_data(X, y):
    plt.figure(figsize=(10, 8))
    pos = np.where(y == 1)  # 找到y=1的位置
    neg = np.where(y == 0)  # 找到y=0的位置
    p1, = plt.plot(np.ravel(X[pos, 0]), np.ravel(X[pos, 1]), 'ro', markersize=8)
    p2, = plt.plot(np.ravel(X[neg, 0]), np.ravel(X[neg, 1]), 'g^', markersize=8)
    plt.xlabel("X1")
    plt.ylabel("X2")
    plt.legend([p1, p2], ["y==1", "y==0"])
    return plt


# 画决策边界
def plot_decisionBoundary(X, y, model, class_='linear'):
    plt = plot_data(X, y)

    # 线性边界        
    if class_ == 'linear':
        w = model.coef_
        b = model.intercept_
        xp = np.linspace(np.min(X[:, 0]), np.max(X[:, 0]), 100)
        yp = -(w[0, 0] * xp + b) / w[0, 1]
        plt.plot(xp, yp, 'b-', linewidth=2.0)
        plt.show()
    else:  # 非线性边界
        x_1 = np.transpose(np.linspace(np.min(X[:, 0]), np.max(X[:, 0]), 100).reshape(1, -1))
        x_2 = np.transpose(np.linspace(np.min(X[:, 1]), np.max(X[:, 1]), 100).reshape(1, -1))
        X1, X2 = np.meshgrid(x_1, x_2)
        vals = np.zeros(X1.shape)
        for i in range(X1.shape[1]):
            this_X = np.hstack((X1[:, i].reshape(-1, 1), X2[:, i].reshape(-1, 1)))
            vals[:, i] = model.predict(this_X)

        plt.contour(X1, X2, vals, [0, 1], color='blue')
        plt.show()


if __name__ == "__main__":
    SVM()

================================================
FILE: SVM/data.txt
================================================
0.051267,0.69956,1
-0.092742,0.68494,1
-0.21371,0.69225,1
-0.375,0.50219,1
-0.51325,0.46564,1
-0.52477,0.2098,1
-0.39804,0.034357,1
-0.30588,-0.19225,1
0.016705,-0.40424,1
0.13191,-0.51389,1
0.38537,-0.56506,1
0.52938,-0.5212,1
0.63882,-0.24342,1
0.73675,-0.18494,1
0.54666,0.48757,1
0.322,0.5826,1
0.16647,0.53874,1
-0.046659,0.81652,1
-0.17339,0.69956,1
-0.47869,0.63377,1
-0.60541,0.59722,1
-0.62846,0.33406,1
-0.59389,0.005117,1
-0.42108,-0.27266,1
-0.11578,-0.39693,1
0.20104,-0.60161,1
0.46601,-0.53582,1
0.67339,-0.53582,1
-0.13882,0.54605,1
-0.29435,0.77997,1
-0.26555,0.96272,1
-0.16187,0.8019,1
-0.17339,0.64839,1
-0.28283,0.47295,1
-0.36348,0.31213,1
-0.30012,0.027047,1
-0.23675,-0.21418,1
-0.06394,-0.18494,1
0.062788,-0.16301,1
0.22984,-0.41155,1
0.2932,-0.2288,1
0.48329,-0.18494,1
0.64459,-0.14108,1
0.46025,0.012427,1
0.6273,0.15863,1
0.57546,0.26827,1
0.72523,0.44371,1
0.22408,0.52412,1
0.44297,0.67032,1
0.322,0.69225,1
0.13767,0.57529,1
-0.0063364,0.39985,1
-0.092742,0.55336,1
-0.20795,0.35599,1
-0.20795,0.17325,1
-0.43836,0.21711,1
-0.21947,-0.016813,1
-0.13882,-0.27266,1
0.18376,0.93348,0
0.22408,0.77997,0
0.29896,0.61915,0
0.50634,0.75804,0
0.61578,0.7288,0
0.60426,0.59722,0
0.76555,0.50219,0
0.92684,0.3633,0
0.82316,0.27558,0
0.96141,0.085526,0
0.93836,0.012427,0
0.86348,-0.082602,0
0.89804,-0.20687,0
0.85196,-0.36769,0
0.82892,-0.5212,0
0.79435,-0.55775,0
0.59274,-0.7405,0
0.51786,-0.5943,0
0.46601,-0.41886,0
0.35081,-0.57968,0
0.28744,-0.76974,0
0.085829,-0.75512,0
0.14919,-0.57968,0
-0.13306,-0.4481,0
-0.40956,-0.41155,0
-0.39228,-0.25804,0
-0.74366,-0.25804,0
-0.69758,0.041667,0
-0.75518,0.2902,0
-0.69758,0.68494,0
-0.4038,0.70687,0
-0.38076,0.91886,0
-0.50749,0.90424,0
-0.54781,0.70687,0
0.10311,0.77997,0
0.057028,0.91886,0
-0.10426,0.99196,0
-0.081221,1.1089,0
0.28744,1.087,0
0.39689,0.82383,0
0.63882,0.88962,0
0.82316,0.66301,0
0.67339,0.64108,0
1.0709,0.10015,0
-0.046659,-0.57968,0
-0.23675,-0.63816,0
-0.15035,-0.36769,0
-0.49021,-0.3019,0
-0.46717,-0.13377,0
-0.28859,-0.060673,0
-0.61118,-0.067982,0
-0.66302,-0.21418,0
-0.59965,-0.41886,0
-0.72638,-0.082602,0
-0.83007,0.31213,0
-0.72062,0.53874,0
-0.59389,0.49488,0
-0.48445,0.99927,0
-0.0063364,0.99927,0
0.63265,-0.030612,0


================================================
FILE: readme.md
================================================
机器学习算法Python实现
=========

[![MIT license](https://img.shields.io/dub/l/vibe-d.svg)](https://github.com/lawlite19/MachineLearning_Python/blob/master/LICENSE)

## 目录
* [机器学习算法Python实现](#机器学习算法python实现)
	* [一、线性回归](#一线性回归)
		* [1、代价函数](#1代价函数)
		* [2、梯度下降算法](#2梯度下降算法)
		* [3、均值归一化](#3均值归一化)
		* [4、最终运行结果](#4最终运行结果)
		* [5、使用scikit-learn库中的线性模型实现](#5使用scikit-learn库中的线性模型实现)
	* [二、逻辑回归](#二逻辑回归)
		* [1、代价函数](#1代价函数)
		* [2、梯度](#2梯度)
		* [3、正则化](#3正则化)
		* [4、S型函数（即）](#4s型函数即)
		* [5、映射为多项式](#5映射为多项式)
		* [6、使用的优化方法](#6使用scipy的优化方法)
		* [7、运行结果](#7运行结果)
		* [8、使用scikit-learn库中的逻辑回归模型实现](#8使用scikit-learn库中的逻辑回归模型实现)
	* [逻辑回归_手写数字识别_OneVsAll](#逻辑回归_手写数字识别_onevsall)
		* [1、随机显示100个数字](#1随机显示100个数字)
		* [2、OneVsAll](#2onevsall)
		* [3、手写数字识别](#3手写数字识别)
		* [4、预测](#4预测)
		* [5、运行结果](#5运行结果)
		* [6、使用scikit-learn库中的逻辑回归模型实现](#6使用scikit-learn库中的逻辑回归模型实现)
	* [三、BP神经网络](#三bp神经网络)
		* [1、神经网络model](#1神经网络model)
		* [2、代价函数](#2代价函数)
		* [3、正则化](#3正则化)
		* [4、反向传播BP](#4反向传播bp)
		* [5、BP可以求梯度的原因](#5bp可以求梯度的原因)
		* [6、梯度检查](#6梯度检查)
		* [7、权重的随机初始化](#7权重的随机初始化)
		* [8、预测](#8预测)
		* [9、输出结果](#9输出结果)
	* [四、SVM支持向量机](#四svm支持向量机)
		* [1、代价函数](#1代价函数)
		* [2、Large Margin](#2large-margin)
		* [3、SVM Kernel（核函数）](#3svm-kernel核函数)
		* [4、使用中的模型代码](#4使用scikit-learn中的svm模型代码)
		* [5、运行结果](#5运行结果)
	* [五、K-Means聚类算法](#五k-means聚类算法)
		* [1、聚类过程](#1聚类过程)
		* [2、目标函数](#2目标函数)
		* [3、聚类中心的选择](#3聚类中心的选择)
		* [4、聚类个数K的选择](#4聚类个数k的选择)
		* [5、应用——图片压缩](#5应用图片压缩)
		* [6、使用scikit-learn库中的线性模型实现聚类](#6使用scikit-learn库中的线性模型实现聚类)
		* [7、运行结果](#7运行结果)
	* [六、PCA主成分分析（降维）](#六pca主成分分析降维)
		* [1、用处](#1用处)
		* [2、2D-->1D，nD-->kD](#22d--1dnd--kd)
		* [3、主成分分析PCA与线性回归的区别](#3主成分分析pca与线性回归的区别)
		* [4、PCA降维过程](#4pca降维过程)
		* [5、数据恢复](#5数据恢复)
		* [6、主成分个数的选择（即要降的维度）](#6主成分个数的选择即要降的维度)
		* [7、使用建议](#7使用建议)
		* [8、运行结果](#8运行结果)
		* [9、使用scikit-learn库中的PCA实现降维](#9使用scikit-learn库中的pca实现降维)
	* [七、异常检测 Anomaly Detection](#七异常检测-anomaly-detection)
		* [1、高斯分布（正态分布）](#1高斯分布正态分布gaussian-distribution)
		* [2、异常检测算法](#2异常检测算法)
		* [3、评价的好坏，以及的选取](#3评价px的好坏以及ε的选取)
		* [4、选择使用什么样的feature（单元高斯分布）](#4选择使用什么样的feature单元高斯分布)
		* [5、多元高斯分布](#5多元高斯分布)
		* [6、单元和多元高斯分布特点](#6单元和多元高斯分布特点)
		* [7、程序运行结果](#7程序运行结果)

## 一、[线性回归](/LinearRegression)
- [全部代码](/LinearRegression/LinearRegression.py)

### 1、代价函数
- ![J(\theta ) = \frac{1}{{2{\text{m}}}}\sum\limits_{i = 1}^m {{{({h_\theta }({x^{(i)}}) - {y^{(i)}})}^2}} ](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=J%28%5Ctheta%20%29%20%3D%20%5Cfrac%7B1%7D%7B%7B2%7B%5Ctext%7Bm%7D%7D%7D%7D%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%7B%7B%28%7Bh_%5Ctheta%20%7D%28%7Bx%5E%7B%28i%29%7D%7D%29%20-%20%7By%5E%7B%28i%29%7D%7D%29%7D%5E2%7D%7D%20)
- 其中：
![{h_\theta }(x) = {\theta _0} + {\theta _1}{x_1} + {\theta _2}{x_2} + ...](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bh_%5Ctheta%20%7D%28x%29%20%3D%20%7B%5Ctheta%20_0%7D%20%2B%20%7B%5Ctheta%20_1%7D%7Bx_1%7D%20%2B%20%7B%5Ctheta%20_2%7D%7Bx_2%7D%20%2B%20...)

- 下面就是要求出theta，使代价最小，即代表我们拟合出来的方程距离真实值最近
- 共有m条数据，其中![{{{({h_\theta }({x^{(i)}}) - {y^{(i)}})}^2}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%7B%7B%28%7Bh_%5Ctheta%20%7D%28%7Bx%5E%7B%28i%29%7D%7D%29%20-%20%7By%5E%7B%28i%29%7D%7D%29%7D%5E2%7D%7D)代表我们要拟合出来的方程到真实值距离的平方，平方的原因是因为可能有负值，正负可能会抵消
- 前面有系数`2`的原因是下面求梯度是对每个变量求偏导，`2`可以消去

- 实现代码：
```
# 计算代价函数
def computerCost(X,y,theta):
    m = len(y)
    J = 0
    
    J = (np.transpose(X*theta-y))*(X*theta-y)/(2*m) #计算代价J
    return J
```
 - 注意这里的X是真实数据前加了一列1，因为有theta(0)

### 2、梯度下降算法
- 代价函数对![{{\theta _j}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%7B%5Ctheta%20_j%7D%7D)求偏导得到：   
![\frac{{\partial J(\theta )}}{{\partial {\theta _j}}} = \frac{1}{m}\sum\limits_{i = 1}^m {[({h_\theta }({x^{(i)}}) - {y^{(i)}})x_j^{(i)}]} ](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cfrac%7B%7B%5Cpartial%20J%28%5Ctheta%20%29%7D%7D%7B%7B%5Cpartial%20%7B%5Ctheta%20_j%7D%7D%7D%20%3D%20%5Cfrac%7B1%7D%7Bm%7D%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%5B%28%7Bh_%5Ctheta%20%7D%28%7Bx%5E%7B%28i%29%7D%7D%29%20-%20%7By%5E%7B%28i%29%7D%7D%29x_j%5E%7B%28i%29%7D%5D%7D%20)
- 所以对theta的更新可以写为：   
![{\theta _j} = {\theta _j} - \alpha \frac{1}{m}\sum\limits_{i = 1}^m {[({h_\theta }({x^{(i)}}) - {y^{(i)}})x_j^{(i)}]} ](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Ctheta%20_j%7D%20%3D%20%7B%5Ctheta%20_j%7D%20-%20%5Calpha%20%5Cfrac%7B1%7D%7Bm%7D%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%5B%28%7Bh_%5Ctheta%20%7D%28%7Bx%5E%7B%28i%29%7D%7D%29%20-%20%7By%5E%7B%28i%29%7D%7D%29x_j%5E%7B%28i%29%7D%5D%7D%20)
- 其中![\alpha ](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Calpha%20)为学习速率，控制梯度下降的速度，一般取**0.01,0.03,0.1,0.3.....**
- 为什么梯度下降可以逐步减小代价函数
 - 假设函数`f(x)`
 - 泰勒展开：`f(x+△x)=f(x)+f'(x)*△x+o(△x)`
 - 令：`△x=-α*f'(x)`   ,即负梯度方向乘以一个很小的步长`α`
 - 将`△x`代入泰勒展开式中：`f(x+△x)=f(x)-α*[f'(x)]²+o(△x)`
 - 可以看出，`α`是取得很小的正数，`[f'(x)]²`也是正数，所以可以得出：`f(x+△x)<=f(x)`
 - 所以沿着**负梯度**放下，函数在减小，多维情况一样。
- 实现代码
```
# 梯度下降算法
def gradientDescent(X,y,theta,alpha,num_iters):
    m = len(y)      
    n = len(theta)
    
    temp = np.matrix(np.zeros((n,num_iters)))   # 暂存每次迭代计算的theta，转化为矩阵形式
    
    
    J_history = np.zeros((num_iters,1)) #记录每次迭代计算的代价值
    
    for i in range(num_iters):  # 遍历迭代次数    
        h = np.dot(X,theta)     # 计算内积，matrix可以直接乘
        temp[:,i] = theta - ((alpha/m)*(np.dot(np.transpose(X),h-y)))   #梯度的计算
        theta = temp[:,i]
        J_history[i] = computerCost(X,y,theta)      #调用计算代价函数
        print '.',      
    return theta,J_history  
```

### 3、均值归一化
- 目的是使数据都缩放到一个范围内，便于使用梯度下降算法
- ![{x_i} = \frac{{{x_i} - {\mu _i}}}{{{s_i}}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bx_i%7D%20%3D%20%5Cfrac%7B%7B%7Bx_i%7D%20-%20%7B%5Cmu%20_i%7D%7D%7D%7B%7B%7Bs_i%7D%7D%7D)
- 其中 ![{{\mu _i}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%7B%5Cmu%20_i%7D%7D) 为所有此feture数据的平均值
- ![{{s_i}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%7Bs_i%7D%7D)可以是**最大值-最小值**，也可以是这个feature对应的数据的**标准差**
- 实现代码：
```
# 归一化feature
def featureNormaliza(X):
    X_norm = np.array(X)            #将X转化为numpy数组对象，才可以进行矩阵的运算
    #定义所需变量
    mu = np.zeros((1,X.shape[1]))   
    sigma = np.zeros((1,X.shape[1]))
    
    mu = np.mean(X_norm,0)          # 求每一列的平均值（0指定为列，1代表行）
    sigma = np.std(X_norm,0)        # 求每一列的标准差
    for i in range(X.shape[1]):     # 遍历列
        X_norm[:,i] = (X_norm[:,i]-mu[i])/sigma[i]  # 归一化
    
    return X_norm,mu,sigma
```
- 注意预测的时候也需要均值归一化数据

### 4、最终运行结果
- 代价随迭代次数的变化   
![enter description here][1]


### 5、[使用scikit-learn库中的线性模型实现](/LinearRegression/LinearRegression_scikit-learn.py)
- 导入包
```
from sklearn import linear_model
from sklearn.preprocessing import StandardScaler    #引入缩放的包
```
- 归一化
```
    # 归一化操作
    scaler = StandardScaler()   
    scaler.fit(X)
    x_train = scaler.transform(X)
    x_test = scaler.transform(np.array([1650,3]))
```
- 线性模型拟合
```
    # 线性模型拟合
    model = linear_model.LinearRegression()
    model.fit(x_train, y)
``` 
- 预测
```
    #预测结果
    result = model.predict(x_test)
```

-------------------

  
## 二、[逻辑回归](/LogisticRegression)
- [全部代码](/LogisticRegression/LogisticRegression.py)

### 1、代价函数
- ![\left\{ \begin{gathered}
  J(\theta ) = \frac{1}{m}\sum\limits_{i = 1}^m {\cos t({h_\theta }({x^{(i)}}),{y^{(i)}})}  \hfill \\
  \cos t({h_\theta }(x),y) = \left\{ {\begin{array}{c}    { - \log ({h_\theta }(x))} \\    { - \log (1 - {h_\theta }(x))}  \end{array} \begin{array}{c}    {y = 1} \\    {y = 0}  \end{array} } \right. \hfill \\ 
\end{gathered}  \right.](http://latex.codecogs.com/gif.latex?%5Clarge%20%5Cleft%5C%7B%20%5Cbegin%7Bgathered%7D%20J%28%5Ctheta%20%29%20%3D%20%5Cfrac%7B1%7D%7Bm%7D%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%5Ccos%20t%28%7Bh_%5Ctheta%20%7D%28%7Bx%5E%7B%28i%29%7D%7D%29%2C%7By%5E%7B%28i%29%7D%7D%29%7D%20%5Chfill%20%5C%5C%20%5Ccos%20t%28%7Bh_%5Ctheta%20%7D%28x%29%2Cy%29%20%3D%20%5Cleft%5C%7B%20%7B%5Cbegin%7Barray%7D%7Bc%7D%20%7B%20-%20%5Clog%20%28%7Bh_%5Ctheta%20%7D%28x%29%29%7D%20%5C%5C%20%7B%20-%20%5Clog%20%281%20-%20%7Bh_%5Ctheta%20%7D%28x%29%29%7D%20%5Cend%7Barray%7D%20%5Cbegin%7Barray%7D%7Bc%7D%20%7By%20%3D%201%7D%20%5C%5C%20%7By%20%3D%200%7D%20%5Cend%7Barray%7D%20%7D%20%5Cright.%20%5Chfill%20%5C%5C%20%5Cend%7Bgathered%7D%20%5Cright.)
- 可以综合起来为：
![J(\theta ) =  - \frac{1}{m}\sum\limits_{i = 1}^m {[{y^{(i)}}\log ({h_\theta }({x^{(i)}}) + (1 - } {y^{(i)}})\log (1 - {h_\theta }({x^{(i)}})]](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=J%28%5Ctheta%20%29%20%3D%20%20-%20%5Cfrac%7B1%7D%7Bm%7D%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%5B%7By%5E%7B%28i%29%7D%7D%5Clog%20%28%7Bh_%5Ctheta%20%7D%28%7Bx%5E%7B%28i%29%7D%7D%29%20%2B%20%281%20-%20%7D%20%7By%5E%7B%28i%29%7D%7D%29%5Clog%20%281%20-%20%7Bh_%5Ctheta%20%7D%28%7Bx%5E%7B%28i%29%7D%7D%29%5D)
其中：
![{h_\theta }(x) = \frac{1}{{1 + {e^{ - x}}}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bh_%5Ctheta%20%7D%28x%29%20%3D%20%5Cfrac%7B1%7D%7B%7B1%20%2B%20%7Be%5E%7B%20-%20x%7D%7D%7D%7D)
- 为什么不用线性回归的代价函数表示，因为线性回归的代价函数可能是非凸的，对于分类问题，使用梯度下降很难得到最小值，上面的代价函数是凸函数
- ![{ - \log ({h_\theta }(x))}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%20-%20%5Clog%20%28%7Bh_%5Ctheta%20%7D%28x%29%29%7D)的图像如下，即`y=1`时：
![enter description here][2]

可以看出，当![{{h_\theta }(x)}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%7Bh_%5Ctheta%20%7D%28x%29%7D)趋于`1`，`y=1`,与预测值一致，此时付出的代价`cost`趋于`0`，若![{{h_\theta }(x)}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%7Bh_%5Ctheta%20%7D%28x%29%7D)趋于`0`，`y=1`,此时的代价`cost`值非常大，我们最终的目的是最小化代价值
- 同理![{ - \log (1 - {h_\theta }(x))}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%20-%20%5Clog%20%281%20-%20%7Bh_%5Ctheta%20%7D%28x%29%29%7D)的图像如下（`y=0`）：   
![enter description here][3]

### 2、梯度
- 同样对代价函数求偏导：
![\frac{{\partial J(\theta )}}{{\partial {\theta _j}}} = \frac{1}{m}\sum\limits_{i = 1}^m {[({h_\theta }({x^{(i)}}) - {y^{(i)}})x_j^{(i)}]} ](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cfrac%7B%7B%5Cpartial%20J%28%5Ctheta%20%29%7D%7D%7B%7B%5Cpartial%20%7B%5Ctheta%20_j%7D%7D%7D%20%3D%20%5Cfrac%7B1%7D%7Bm%7D%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%5B%28%7Bh_%5Ctheta%20%7D%28%7Bx%5E%7B%28i%29%7D%7D%29%20-%20%7By%5E%7B%28i%29%7D%7D%29x_j%5E%7B%28i%29%7D%5D%7D%20)   
可以看出与线性回归的偏导数一致
- 推到过程
![enter description here][4]

### 3、正则化
- 目的是为了防止过拟合
- 在代价函数中加上一项![J(\theta ) =  - \frac{1}{m}\sum\limits_{i = 1}^m {[{y^{(i)}}\log ({h_\theta }({x^{(i)}}) + (1 - } {y^{(i)}})\log (1 - {h_\theta }({x^{(i)}})] + \frac{\lambda }{{2m}}\sum\limits_{j = 1}^n {\theta _j^2} ](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=J%28%5Ctheta%20%29%20%3D%20%20-%20%5Cfrac%7B1%7D%7Bm%7D%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%5B%7By%5E%7B%28i%29%7D%7D%5Clog%20%28%7Bh_%5Ctheta%20%7D%28%7Bx%5E%7B%28i%29%7D%7D%29%20%2B%20%281%20-%20%7D%20%7By%5E%7B%28i%29%7D%7D%29%5Clog%20%281%20-%20%7Bh_%5Ctheta%20%7D%28%7Bx%5E%7B%28i%29%7D%7D%29%5D%20%2B%20%5Cfrac%7B%5Clambda%20%7D%7B%7B2m%7D%7D%5Csum%5Climits_%7Bj%20%3D%201%7D%5En%20%7B%5Ctheta%20_j%5E2%7D%20)
- 注意j是重1开始的，因为theta(0)为一个常数项，X中最前面一列会加上1列1，所以乘积还是theta(0),feature没有关系，没有必要正则化
- 正则化后的代价：
```
# 代价函数
def costFunction(initial_theta,X,y,inital_lambda):
    m = len(y)
    J = 0
    
    h = sigmoid(np.dot(X,initial_theta))    # 计算h(z)
    theta1 = initial_theta.copy()           # 因为正则化j=1从1开始，不包含0，所以复制一份，前theta(0)值为0 
    theta1[0] = 0   
    
    temp = np.dot(np.transpose(theta1),theta1)
    J = (-np.dot(np.transpose(y),np.log(h))-np.dot(np.transpose(1-y),np.log(1-h))+temp*inital_lambda/2)/m   # 正则化的代价方程
    return J
```
- 正则化后的代价的梯度
```
# 计算梯度
def gradient(initial_theta,X,y,inital_lambda):
    m = len(y)
    grad = np.zeros((initial_theta.shape[0]))
    
    h = sigmoid(np.dot(X,initial_theta))# 计算h(z)
    theta1 = initial_theta.copy()
    theta1[0] = 0

    grad = np.dot(np.transpose(X),h-y)/m+inital_lambda/m*theta1 #正则化的梯度
    return grad  
```

### 4、S型函数（即![{{h_\theta }(x)}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%7Bh_%5Ctheta%20%7D%28x%29%7D)）
- 实现代码：
```
# S型函数    
def sigmoid(z):
    h = np.zeros((len(z),1))    # 初始化，与z的长度一置
    
    h = 1.0/(1.0+np.exp(-z))
    return h
```

### 5、映射为多项式
- 因为数据的feture可能很少，导致偏差大，所以创造出一些feture结合
- eg:映射为2次方的形式:![1 + {x_1} + {x_2} + x_1^2 + {x_1}{x_2} + x_2^2](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=1%20%2B%20%7Bx_1%7D%20%2B%20%7Bx_2%7D%20%2B%20x_1%5E2%20%2B%20%7Bx_1%7D%7Bx_2%7D%20%2B%20x_2%5E2)
- 实现代码：
```
# 映射为多项式 
def mapFeature(X1,X2):
    degree = 3;                     # 映射的最高次方
    out = np.ones((X1.shape[0],1))  # 映射后的结果数组（取代X）
    '''
    这里以degree=2为例，映射为1,x1,x2,x1^2,x1,x2,x2^2
    '''
    for i in np.arange(1,degree+1): 
        for j in range(i+1):
            temp = X1**(i-j)*(X2**j)    #矩阵直接乘相当于matlab中的点乘.*
            out = np.hstack((out, temp.reshape(-1,1)))
    return out
```

### 6、使用`scipy`的优化方法
- 梯度下降使用`scipy`中`optimize`中的`fmin_bfgs`函数
- 调用scipy中的优化算法fmin_bfgs（拟牛顿法Broyden-Fletcher-Goldfarb-Shanno
 - costFunction是自己实现的一个求代价的函数，
 - initial_theta表示初始化的值,
 - fprime指定costFunction的梯度
 - args是其余测参数，以元组的形式传入，最后会将最小化costFunction的theta返回 
```
    result = optimize.fmin_bfgs(costFunction, initial_theta, fprime=gradient, args=(X,y,initial_lambda))    
```   

### 7、运行结果
- data1决策边界和准确度  
![enter description here][5]
![enter description here][6]
- data2决策边界和准确度  
![enter description here][7]
![enter description here][8]

### 8、[使用scikit-learn库中的逻辑回归模型实现](/LogisticRegression/LogisticRegression_scikit-learn.py)
- 导入包
```
from sklearn.linear_model import LogisticRegression
from sklearn.preprocessing import StandardScaler
from sklearn.cross_validation import train_test_split
import numpy as np
```
- 划分训练集和测试集
```
    # 划分为训练集和测试集
    x_train,x_test,y_train,y_test = train_test_split(X,y,test_size=0.2)
```
- 归一化
```
    # 归一化
    scaler = StandardScaler()
    x_train = scaler.fit_transform(x_train)
    x_test = scaler.fit_transform(x_test)
```
- 逻辑回归
```
    #逻辑回归
    model = LogisticRegression()
    model.fit(x_train,y_train)
``` 
- 预测
```
    # 预测
    predict = model.predict(x_test)
    right = sum(predict == y_test)
    
    predict = np.hstack((predict.reshape(-1,1),y_test.reshape(-1,1)))   # 将预测值和真实值放在一块，好观察
    print predict
    print ('测试集准确率：%f%%'%(right*100.0/predict.shape[0]))          #计算在测试集上的准确度
```


-------------

## [逻辑回归_手写数字识别_OneVsAll](/LogisticRegression)
- [全部代码](/LogisticRegression/LogisticRegression_OneVsAll.py)

### 1、随机显示100个数字
- 我没有使用scikit-learn中的数据集，像素是20*20px，彩色图如下
![enter description here][9]
灰度图：
![enter description here][10]
- 实现代码：
```
# 显示100个数字
def display_data(imgData):
    sum = 0
    '''
    显示100个数（若是一个一个绘制将会非常慢，可以将要画的数字整理好，放到一个矩阵中，显示这个矩阵即可）
    - 初始化一个二维数组
    - 将每行的数据调整成图像的矩阵，放进二维数组
    - 显示即可
    '''
    pad = 1
    display_array = -np.ones((pad+10*(20+pad),pad+10*(20+pad)))
    for i in range(10):
        for j in range(10):
            display_array[pad+i*(20+pad):pad+i*(20+pad)+20,pad+j*(20+pad):pad+j*(20+pad)+20] = (imgData[sum,:].reshape(20,20,order="F"))    # order=F指定以列优先，在matlab中是这样的，python中需要指定，默认以行
            sum += 1
            
    plt.imshow(display_array,cmap='gray')   #显示灰度图像
    plt.axis('off')
    plt.show()
```

### 2、OneVsAll
- 如何利用逻辑回归解决多分类的问题，OneVsAll就是把当前某一类看成一类，其他所有类别看作一类，这样有成了二分类的问题了
- 如下图，把途中的数据分成三类，先把红色的看成一类，把其他的看作另外一类，进行逻辑回归，然后把蓝色的看成一类，其他的再看成一类，以此类推...
![enter description here][11]
- 可以看出大于2类的情况下，有多少类就要进行多少次的逻辑回归分类

### 3、手写数字识别
- 共有0-9，10个数字，需要10次分类
- 由于**数据集y**给出的是`0,1,2...9`的数字，而进行逻辑回归需要`0/1`的label标记，所以需要对y处理
- 说一下数据集，前`500`个是`0`,`500-1000`是`1`,`...`,所以如下图，处理后的`y`，**前500行的第一列是1，其余都是0,500-1000行第二列是1，其余都是0....**
![enter description here][12]
- 然后调用**梯度下降算法**求解`theta`
- 实现代码：
```
# 求每个分类的theta，最后返回所有的all_theta    
def oneVsAll(X,y,num_labels,Lambda):
    # 初始化变量
    m,n = X.shape
    all_theta = np.zeros((n+1,num_labels))  # 每一列对应相应分类的theta,共10列
    X = np.hstack((np.ones((m,1)),X))       # X前补上一列1的偏置bias
    class_y = np.zeros((m,num_labels))      # 数据的y对应0-9，需要映射为0/1的关系
    initial_theta = np.zeros((n+1,1))       # 初始化一个分类的theta
    
    # 映射y
    for i in range(num_labels):
        class_y[:,i] = np.int32(y==i).reshape(1,-1) # 注意reshape(1,-1)才可以赋值
    
    #np.savetxt("class_y.csv", class_y[0:600,:], delimiter=',')    
    
    '''遍历每个分类，计算对应的theta值'''
    for i in range(num_labels):
        result = optimize.fmin_bfgs(costFunction, initial_theta, fprime=gradient, args=(X,class_y[:,i],Lambda)) # 调用梯度下降的优化方法
        all_theta[:,i] = result.reshape(1,-1)   # 放入all_theta中
        
    all_theta = np.transpose(all_theta) 
    return all_theta
```

### 4、预测
- 之前说过，预测的结果是一个**概率值**，利用学习出来的`theta`代入预测的**S型函数**中，每行的最大值就是是某个数字的最大概率，所在的**列号**就是预测的数字的真实值,因为在分类时，所有为`0`的将`y`映射在第一列，为1的映射在第二列，依次类推
- 实现代码：
```
# 预测
def predict_oneVsAll(all_theta,X):
    m = X.shape[0]
    num_labels = all_theta.shape[0]
    p = np.zeros((m,1))
    X = np.hstack((np.ones((m,1)),X))   #在X最前面加一列1
    
    h = sigmoid(np.dot(X,np.transpose(all_theta)))  #预测

    '''
    返回h中每一行最大值所在的列号
    - np.max(h, axis=1)返回h中每一行的最大值（是某个数字的最大概率）
    - 最后where找到的最大概率所在的列号（列号即是对应的数字）
    '''
    p = np.array(np.where(h[0,:] == np.max(h, axis=1)[0]))  
    for i in np.arange(1, m):
        t = np.array(np.where(h[i,:] == np.max(h, axis=1)[i]))
        p = np.vstack((p,t))
    return p
```

### 5、运行结果
- 10次分类，在训练集上的准确度：   
![enter description here][13]

### 6、[使用scikit-learn库中的逻辑回归模型实现](/LogisticRegression/LogisticRegression_OneVsAll_scikit-learn.py)
- 1、导入包
```
from scipy import io as spio
import numpy as np
from sklearn import svm
from sklearn.linear_model import LogisticRegression
```
- 2、加载数据
```
    data = loadmat_data("data_digits.mat") 
    X = data['X']   # 获取X数据，每一行对应一个数字20x20px
    y = data['y']   # 这里读取mat文件y的shape=(5000, 1)
    y = np.ravel(y) # 调用sklearn需要转化成一维的(5000,)
```
- 3、拟合模型
```
    model = LogisticRegression()
    model.fit(X, y) # 拟合
```
- 4、预测
```
    predict = model.predict(X) #预测
    
    print u"预测准确度为：%f%%"%np.mean(np.float64(predict == y)*100)
```
- 5、输出结果（在训练集上的准确度）
![enter description here][14]


----------

## 三、BP神经网络
- [全部代码](/NeuralNetwok/NeuralNetwork.py)

### 1、神经网络model
- 先介绍个三层的神经网络，如下图所示
 - 输入层（input layer）有三个units（![{x_0}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bx_0%7D)为补上的bias，通常设为`1`）
 - ![a_i^{(j)}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=a_i%5E%7B%28j%29%7D)表示第`j`层的第`i`个激励，也称为为单元unit
 - ![{\theta ^{(j)}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Ctheta%20%5E%7B%28j%29%7D%7D)为第`j`层到第`j+1`层映射的权重矩阵，就是每条边的权重
![enter description here][15]

- 所以可以得到：
 - 隐含层：  
![a_1^{(2)} = g(\theta _{10}^{(1)}{x_0} + \theta _{11}^{(1)}{x_1} + \theta _{12}^{(1)}{x_2} + \theta _{13}^{(1)}{x_3})](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=a_1%5E%7B%282%29%7D%20%3D%20g%28%5Ctheta%20_%7B10%7D%5E%7B%281%29%7D%7Bx_0%7D%20%2B%20%5Ctheta%20_%7B11%7D%5E%7B%281%29%7D%7Bx_1%7D%20%2B%20%5Ctheta%20_%7B12%7D%5E%7B%281%29%7D%7Bx_2%7D%20%2B%20%5Ctheta%20_%7B13%7D%5E%7B%281%29%7D%7Bx_3%7D%29)   
![a_2^{(2)} = g(\theta _{20}^{(1)}{x_0} + \theta _{21}^{(1)}{x_1} + \theta _{22}^{(1)}{x_2} + \theta _{23}^{(1)}{x_3})](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=a_2%5E%7B%282%29%7D%20%3D%20g%28%5Ctheta%20_%7B20%7D%5E%7B%281%29%7D%7Bx_0%7D%20%2B%20%5Ctheta%20_%7B21%7D%5E%7B%281%29%7D%7Bx_1%7D%20%2B%20%5Ctheta%20_%7B22%7D%5E%7B%281%29%7D%7Bx_2%7D%20%2B%20%5Ctheta%20_%7B23%7D%5E%7B%281%29%7D%7Bx_3%7D%29)   
![a_3^{(2)} = g(\theta _{30}^{(1)}{x_0} + \theta _{31}^{(1)}{x_1} + \theta _{32}^{(1)}{x_2} + \theta _{33}^{(1)}{x_3})](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=a_3%5E%7B%282%29%7D%20%3D%20g%28%5Ctheta%20_%7B30%7D%5E%7B%281%29%7D%7Bx_0%7D%20%2B%20%5Ctheta%20_%7B31%7D%5E%7B%281%29%7D%7Bx_1%7D%20%2B%20%5Ctheta%20_%7B32%7D%5E%7B%281%29%7D%7Bx_2%7D%20%2B%20%5Ctheta%20_%7B33%7D%5E%7B%281%29%7D%7Bx_3%7D%29)
 - 输出层   
![{h_\theta }(x) = a_1^{(3)} = g(\theta _{10}^{(2)}a_0^{(2)} + \theta _{11}^{(2)}a_1^{(2)} + \theta _{12}^{(2)}a_2^{(2)} + \theta _{13}^{(2)}a_3^{(2)})](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bh_%5Ctheta%20%7D%28x%29%20%3D%20a_1%5E%7B%283%29%7D%20%3D%20g%28%5Ctheta%20_%7B10%7D%5E%7B%282%29%7Da_0%5E%7B%282%29%7D%20%2B%20%5Ctheta%20_%7B11%7D%5E%7B%282%29%7Da_1%5E%7B%282%29%7D%20%2B%20%5Ctheta%20_%7B12%7D%5E%7B%282%29%7Da_2%5E%7B%282%29%7D%20%2B%20%5Ctheta%20_%7B13%7D%5E%7B%282%29%7Da_3%5E%7B%282%29%7D%29) 其中，**S型函数**![g(z) = \frac{1}{{1 + {e^{ - z}}}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=g%28z%29%20%3D%20%5Cfrac%7B1%7D%7B%7B1%20%2B%20%7Be%5E%7B%20-%20z%7D%7D%7D%7D)，也成为**激励函数**
- 可以看出![{\theta ^{(1)}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Ctheta%20%5E%7B%281%29%7D%7D) 为3x4的矩阵，![{\theta ^{(2)}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Ctheta%20%5E%7B%282%29%7D%7D)为1x4的矩阵
 - ![{\theta ^{(j)}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Ctheta%20%5E%7B%28j%29%7D%7D) ==》`j+1`的单元数x（`j`层的单元数+1）

### 2、代价函数
- 假设最后输出的![{h_\Theta }(x) \in {R^K}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bh_%5CTheta%20%7D%28x%29%20%5Cin%20%7BR%5EK%7D)，即代表输出层有K个单元
- ![J(\Theta ) =  - \frac{1}{m}\sum\limits_{i = 1}^m {\sum\limits_{k = 1}^K {[y_k^{(i)}\log {{({h_\Theta }({x^{(i)}}))}_k}} }  + (1 - y_k^{(i)})\log {(1 - {h_\Theta }({x^{(i)}}))_k}]](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=J%28%5CTheta%20%29%20%3D%20%20-%20%5Cfrac%7B1%7D%7Bm%7D%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%5Csum%5Climits_%7Bk%20%3D%201%7D%5EK%20%7B%5By_k%5E%7B%28i%29%7D%5Clog%20%7B%7B%28%7Bh_%5CTheta%20%7D%28%7Bx%5E%7B%28i%29%7D%7D%29%29%7D_k%7D%7D%20%7D%20%20%2B%20%281%20-%20y_k%5E%7B%28i%29%7D%29%5Clog%20%7B%281%20-%20%7Bh_%5CTheta%20%7D%28%7Bx%5E%7B%28i%29%7D%7D%29%29_k%7D%5D) 其中，![{({h_\Theta }(x))_i}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%28%7Bh_%5CTheta%20%7D%28x%29%29_i%7D)代表第`i`个单元输出
- 与逻辑回归的代价函数![J(\theta ) =  - \frac{1}{m}\sum\limits_{i = 1}^m {[{y^{(i)}}\log ({h_\theta }({x^{(i)}}) + (1 - } {y^{(i)}})\log (1 - {h_\theta }({x^{(i)}})]](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=J%28%5Ctheta%20%29%20%3D%20%20-%20%5Cfrac%7B1%7D%7Bm%7D%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%5B%7By%5E%7B%28i%29%7D%7D%5Clog%20%28%7Bh_%5Ctheta%20%7D%28%7Bx%5E%7B%28i%29%7D%7D%29%20%2B%20%281%20-%20%7D%20%7By%5E%7B%28i%29%7D%7D%29%5Clog%20%281%20-%20%7Bh_%5Ctheta%20%7D%28%7Bx%5E%7B%28i%29%7D%7D%29%5D)差不多，就是累加上每个输出（共有K个输出）



### 3、正则化
- `L`-->所有层的个数
- ![{S_l}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7BS_l%7D)-->第`l`层unit的个数
- 正则化后的**代价函数**为  
![enter description here][16]
 - ![\theta ](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Ctheta%20)共有`L-1`层，
 - 然后是累加对应每一层的theta矩阵，注意不包含加上偏置项对应的theta(0)
- 正则化后的代价函数实现代码：
```
# 代价函数
def nnCostFunction(nn_params,input_layer_size,hidden_layer_size,num_labels,X,y,Lambda):
    length = nn_params.shape[0] # theta的中长度
    # 还原theta1和theta2
    Theta1 = nn_params[0:hidden_layer_size*(input_layer_size+1)].reshape(hidden_layer_size,input_layer_size+1)
    Theta2 = nn_params[hidden_layer_size*(input_layer_size+1):length].reshape(num_labels,hidden_layer_size+1)
    
    # np.savetxt("Theta1.csv",Theta1,delimiter=',')
    
    m = X.shape[0]
    class_y = np.zeros((m,num_labels))      # 数据的y对应0-9，需要映射为0/1的关系
    # 映射y
    for i in range(num_labels):
        class_y[:,i] = np.int32(y==i).reshape(1,-1) # 注意reshape(1,-1)才可以赋值
     
    '''去掉theta1和theta2的第一列，因为正则化时从1开始'''    
    Theta1_colCount = Theta1.shape[1]    
    Theta1_x = Theta1[:,1:Theta1_colCount]
    Theta2_colCount = Theta2.shape[1]    
    Theta2_x = Theta2[:,1:Theta2_colCount]
    # 正则化向theta^2
    term = np.dot(np.transpose(np.vstack((Theta1_x.reshape(-1,1),Theta2_x.reshape(-1,1)))),np.vstack((Theta1_x.reshape(-1,1),Theta2_x.reshape(-1,1))))
    
    '''正向传播,每次需要补上一列1的偏置bias'''
    a1 = np.hstack((np.ones((m,1)),X))      
    z2 = np.dot(a1,np.transpose(Theta1))    
    a2 = sigmoid(z2)
    a2 = np.hstack((np.ones((m,1)),a2))
    z3 = np.dot(a2,np.transpose(Theta2))
    h  = sigmoid(z3)    
    '''代价'''    
    J = -(np.dot(np.transpose(class_y.reshape(-1,1)),np.log(h.reshape(-1,1)))+np.dot(np.transpose(1-class_y.reshape(-1,1)),np.log(1-h.reshape(-1,1)))-Lambda*term/2)/m   
    
    return np.ravel(J)
```

### 4、反向传播BP
- 上面正向传播可以计算得到`J(θ)`,使用梯度下降法还需要求它的梯度
- BP反向传播的目的就是求代价函数的梯度
- 假设4层的神经网络,![\delta _{\text{j}}^{(l)}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cdelta%20_%7B%5Ctext%7Bj%7D%7D%5E%7B%28l%29%7D)记为-->`l`层第`j`个单元的误差
 - ![\delta _{\text{j}}^{(4)} = a_j^{(4)} - {y_i}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cdelta%20_%7B%5Ctext%7Bj%7D%7D%5E%7B%284%29%7D%20%3D%20a_j%5E%7B%284%29%7D%20-%20%7By_i%7D)《===》![{\delta ^{(4)}} = {a^{(4)}} - y](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Cdelta%20%5E%7B%284%29%7D%7D%20%3D%20%7Ba%5E%7B%284%29%7D%7D%20-%20y)（向量化）
 - ![{\delta ^{(3)}} = {({\theta ^{(3)}})^T}{\delta ^{(4)}}.*{g^}({a^{(3)}})](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Cdelta%20%5E%7B%283%29%7D%7D%20%3D%20%7B%28%7B%5Ctheta%20%5E%7B%283%29%7D%7D%29%5ET%7D%7B%5Cdelta%20%5E%7B%284%29%7D%7D.%2A%7Bg%5E%7D%28%7Ba%5E%7B%283%29%7D%7D%29)
 - ![{\delta ^{(2)}} = {({\theta ^{(2)}})^T}{\delta ^{(3)}}.*{g^}({a^{(2)}})](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Cdelta%20%5E%7B%282%29%7D%7D%20%3D%20%7B%28%7B%5Ctheta%20%5E%7B%282%29%7D%7D%29%5ET%7D%7B%5Cdelta%20%5E%7B%283%29%7D%7D.%2A%7Bg%5E%7D%28%7Ba%5E%7B%282%29%7D%7D%29)
 - 没有![{\delta ^{(1)}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Cdelta%20%5E%7B%281%29%7D%7D)，因为对于输入没有误差
- 因为S型函数![{\text{g(z)}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Ctext%7Bg%28z%29%7D%7D)的导数为：![{g^}(z){\text{ = g(z)(1 - g(z))}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bg%5E%7D%28z%29%7B%5Ctext%7B%20%3D%20g%28z%29%281%20-%20g%28z%29%29%7D%7D)，所以上面的![{g^}({a^{(3)}})](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bg%5E%7D%28%7Ba%5E%7B%283%29%7D%7D%29)和![{g^}({a^{(2)}})](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bg%5E%7D%28%7Ba%5E%7B%282%29%7D%7D%29)可以在前向传播中计算出来

- 反向传播计算梯度的过程为：
 - ![\Delta _{ij}^{(l)} = 0](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5CDelta%20_%7Bij%7D%5E%7B%28l%29%7D%20%3D%200)（![\Delta ](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5CDelta%20)是大写的![\delta ](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cdelta%20)）
 - for i=1-m:     
 -![{a^{(1)}} = {x^{(i)}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Ba%5E%7B%281%29%7D%7D%20%3D%20%7Bx%5E%7B%28i%29%7D%7D)       
-正向传播计算![{a^{(l)}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Ba%5E%7B%28l%29%7D%7D)（l=2,3,4...L）      
-反向计算![{\delta ^{(L)}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Cdelta%20%5E%7B%28L%29%7D%7D)、![{\delta ^{(L - 1)}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Cdelta%20%5E%7B%28L%20-%201%29%7D%7D)...![{\delta ^{(2)}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Cdelta%20%5E%7B%282%29%7D%7D)；       
-![\Delta _{ij}^{(l)} = \Delta _{ij}^{(l)} + a_j^{(l)}{\delta ^{(l + 1)}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5CDelta%20_%7Bij%7D%5E%7B%28l%29%7D%20%3D%20%5CDelta%20_%7Bij%7D%5E%7B%28l%29%7D%20%2B%20a_j%5E%7B%28l%29%7D%7B%5Cdelta%20%5E%7B%28l%20%2B%201%29%7D%7D)          
-![D_{ij}^{(l)} = \frac{1}{m}\Delta _{ij}^{(l)} + \lambda \theta _{ij}^l\begin{array}{c}    {}&amp; {(j \ne 0)}  \end{array} ](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=D_%7Bij%7D%5E%7B%28l%29%7D%20%3D%20%5Cfrac%7B1%7D%7Bm%7D%5CDelta%20_%7Bij%7D%5E%7B%28l%29%7D%20%2B%20%5Clambda%20%5Ctheta%20_%7Bij%7D%5El%5Cbegin%7Barray%7D%7Bc%7D%20%20%20%20%7B%7D%26%20%7B%28j%20%5Cne%200%29%7D%20%20%5Cend%7Barray%7D%20)      
![D_{ij}^{(l)} = \frac{1}{m}\Delta _{ij}^{(l)} + \lambda \theta _{ij}^lj = 0\begin{array}{c}    {}&amp; {j = 0}  \end{array} ](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=D_%7Bij%7D%5E%7B%28l%29%7D%20%3D%20%5Cfrac%7B1%7D%7Bm%7D%5CDelta%20_%7Bij%7D%5E%7B%28l%29%7D%20%2B%20%5Clambda%20%5Ctheta%20_%7Bij%7D%5Elj%20%3D%200%5Cbegin%7Barray%7D%7Bc%7D%20%20%20%20%7B%7D%26%20%7Bj%20%3D%200%7D%20%20%5Cend%7Barray%7D%20)     

- 最后![\frac{{\partial J(\Theta )}}{{\partial \Theta _{ij}^{(l)}}} = D_{ij}^{(l)}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cfrac%7B%7B%5Cpartial%20J%28%5CTheta%20%29%7D%7D%7B%7B%5Cpartial%20%5CTheta%20_%7Bij%7D%5E%7B%28l%29%7D%7D%7D%20%3D%20D_%7Bij%7D%5E%7B%28l%29%7D)，即得到代价函数的梯度
- 实现代码：
```
# 梯度
def nnGradient(nn_params,input_layer_size,hidden_layer_size,num_labels,X,y,Lambda):
    length = nn_params.shape[0]
    Theta1 = nn_params[0:hidden_layer_size*(input_layer_size+1)].reshape(hidden_layer_size,input_layer_size+1).copy()   # 这里使用copy函数，否则下面修改Theta的值，nn_params也会一起修改
    Theta2 = nn_params[hidden_layer_size*(input_layer_size+1):length].reshape(num_labels,hidden_layer_size+1).copy()
    m = X.shape[0]
    class_y = np.zeros((m,num_labels))      # 数据的y对应0-9，需要映射为0/1的关系    
    # 映射y
    for i in range(num_labels):
        class_y[:,i] = np.int32(y==i).reshape(1,-1) # 注意reshape(1,-1)才可以赋值
     
    '''去掉theta1和theta2的第一列，因为正则化时从1开始'''
    Theta1_colCount = Theta1.shape[1]    
    Theta1_x = Theta1[:,1:Theta1_colCount]
    Theta2_colCount = Theta2.shape[1]    
    Theta2_x = Theta2[:,1:Theta2_colCount]
    
    Theta1_grad = np.zeros((Theta1.shape))  #第一层到第二层的权重
    Theta2_grad = np.zeros((Theta2.shape))  #第二层到第三层的权重
      
   
    '''正向传播，每次需要补上一列1的偏置bias'''
    a1 = np.hstack((np.ones((m,1)),X))
    z2 = np.dot(a1,np.transpose(Theta1))
    a2 = sigmoid(z2)
    a2 = np.hstack((np.ones((m,1)),a2))
    z3 = np.dot(a2,np.transpose(Theta2))
    h  = sigmoid(z3)
    
    
    '''反向传播，delta为误差，'''
    delta3 = np.zeros((m,num_labels))
    delta2 = np.zeros((m,hidden_layer_size))
    for i in range(m):
        #delta3[i,:] = (h[i,:]-class_y[i,:])*sigmoidGradient(z3[i,:])  # 均方误差的误差率
        delta3[i,:] = h[i,:]-class_y[i,:]                              # 交叉熵误差率
        Theta2_grad = Theta2_grad+np.dot(np.transpose(delta3[i,:].reshape(1,-1)),a2[i,:].reshape(1,-1))
        delta2[i,:] = np.dot(delta3[i,:].reshape(1,-1),Theta2_x)*sigmoidGradient(z2[i,:])
        Theta1_grad = Theta1_grad+np.dot(np.transpose(delta2[i,:].reshape(1,-1)),a1[i,:].reshape(1,-1))
    
    Theta1[:,0] = 0
    Theta2[:,0] = 0          
    '''梯度'''
    grad = (np.vstack((Theta1_grad.reshape(-1,1),Theta2_grad.reshape(-1,1)))+Lambda*np.vstack((Theta1.reshape(-1,1),Theta2.reshape(-1,1))))/m
    return np.ravel(grad)
```

### 5、BP可以求梯度的原因
- 实际是利用了`链式求导`法则
- 因为下一层的单元利用上一层的单元作为输入进行计算
- 大体的推导过程如下，最终我们是想预测函数与已知的`y`非常接近，求均方差的梯度沿着此梯度方向可使代价函数最小化。可对照上面求梯度的过程。
![enter description here][17]
- 求误差更详细的推导过程：
![enter description here][18]

### 6、梯度检查
- 检查利用`BP`求的梯度是否正确
- 利用导数的定义验证：
![\frac{{dJ(\theta )}}{{d\theta }} \approx \frac{{J(\theta  + \varepsilon ) - J(\theta  - \varepsilon )}}{{2\varepsilon }}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Cfrac%7B%7BdJ%28%5Ctheta%20%29%7D%7D%7B%7Bd%5Ctheta%20%7D%7D%20%5Capprox%20%5Cfrac%7B%7BJ%28%5Ctheta%20%20%2B%20%5Cvarepsilon%20%29%20-%20J%28%5Ctheta%20%20-%20%5Cvarepsilon%20%29%7D%7D%7B%7B2%5Cvarepsilon%20%7D%7D)
- 求出来的数值梯度应该与BP求出的梯度非常接近
- 验证BP正确后就不需要再执行验证梯度的算法了
- 实现代码：
```
# 检验梯度是否计算正确
# 检验梯度是否计算正确
def checkGradient(Lambda = 0):
    '''构造一个小型的神经网络验证，因为数值法计算梯度很浪费时间，而且验证正确后之后就不再需要验证了'''
    input_layer_size = 3
    hidden_layer_size = 5
    num_labels = 3
    m = 5
    initial_Theta1 = debugInitializeWeights(input_layer_size,hidden_layer_size); 
    initial_Theta2 = debugInitializeWeights(hidden_layer_size,num_labels)
    X = debugInitializeWeights(input_layer_size-1,m)
    y = 1+np.transpose(np.mod(np.arange(1,m+1), num_labels))# 初始化y
    
    y = y.reshape(-1,1)
    nn_params = np.vstack((initial_Theta1.reshape(-1,1),initial_Theta2.reshape(-1,1)))  #展开theta 
    '''BP求出梯度'''
    grad = nnGradient(nn_params, input_layer_size, hidden_layer_size, 
                     num_labels, X, y, Lambda)  
    '''使用数值法计算梯度'''
    num_grad = np.zeros((nn_params.shape[0]))
    step = np.zeros((nn_params.shape[0]))
    e = 1e-4
    for i in range(nn_params.shape[0]):
        step[i] = e
        loss1 = nnCostFunction(nn_params-step.reshape(-1,1), input_layer_size, hidden_layer_size, 
                              num_labels, X, y, 
                              Lambda)
        loss2 = nnCostFunction(nn_params+step.reshape(-1,1), input_layer_size, hidden_layer_size, 
                              num_labels, X, y, 
                              Lambda)
        num_grad[i] = (loss2-loss1)/(2*e)
        step[i]=0
    # 显示两列比较
    res = np.hstack((num_grad.reshape(-1,1),grad.reshape(-1,1)))
    print res
```

### 7、权重的随机初始化
- 神经网络不能像逻辑回归那样初始化`theta`为`0`,因为若是每条边的权重都为0，每个神经元都是相同的输出，在反向传播中也会得到同样的梯度，最终只会预测一种结果。
- 所以应该初始化为接近0的数
- 实现代码
```
# 随机初始化权重theta
def randInitializeWeights(L_in,L_out):
    W = np.zeros((L_out,1+L_in))    # 对应theta的权重
    epsilon_init = (6.0/(L_out+L_in))**0.5
    W = np.random.rand(L_out,1+L_in)*2*epsilon_init-epsilon_init # np.random.rand(L_out,1+L_in)产生L_out*(1+L_in)大小的随机矩阵
    return W
```

### 8、预测
- 正向传播预测结果
- 实现代码
```
# 预测
def predict(Theta1,Theta2,X):
    m = X.shape[0]
    num_labels = Theta2.shape[0]
    #p = np.zeros((m,1))
    '''正向传播，预测结果'''
    X = np.hstack((np.ones((m,1)),X))
    h1 = sigmoid(np.dot(X,np.transpose(Theta1)))
    h1 = np.hstack((np.ones((m,1)),h1))
    h2 = sigmoid(np.dot(h1,np.transpose(Theta2)))
    
    '''
    返回h中每一行最大值所在的列号
    - np.max(h, axis=1)返回h中每一行的最大值（是某个数字的最大概率）
    - 最后where找到的最大概率所在的列号（列号即是对应的数字）
    '''
    #np.savetxt("h2.csv",h2,delimiter=',')
    p = np.array(np.where(h2[0,:] == np.max(h2, axis=1)[0]))  
    for i in np.arange(1, m):
        t = np.array(np.where(h2[i,:] == np.max(h2, axis=1)[i]))
        p = np.vstack((p,t))
    return p 
```

### 9、输出结果
- 梯度检查：     
![enter description here][19]
- 随机显示100个手写数字     
![enter description here][20]
- 显示theta1权重     
![enter description here][21]
- 训练集预测准确度     
![enter description here][22]
- 归一化后训练集预测准确度     
![enter description here][23]

--------------------

## 四、SVM支持向量机

### 1、代价函数
- 在逻辑回归中，我们的代价为：   
![\cos t({h_\theta }(x),y) = \left\{ {\begin{array}{c}    { - \log ({h_\theta }(x))} \\    { - \log (1 - {h_\theta }(x))}  \end{array} \begin{array}{c}    {y = 1} \\    {y = 0}  \end{array} } \right.](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Ccos%20t%28%7Bh_%5Ctheta%20%7D%28x%29%2Cy%29%20%3D%20%5Cleft%5C%7B%20%7B%5Cbegin%7Barray%7D%7Bc%7D%20%20%20%20%7B%20-%20%5Clog%20%28%7Bh_%5Ctheta%20%7D%28x%29%29%7D%20%5C%5C%20%20%20%20%7B%20-%20%5Clog%20%281%20-%20%7Bh_%5Ctheta%20%7D%28x%29%29%7D%20%20%5Cend%7Barray%7D%20%5Cbegin%7Barray%7D%7Bc%7D%20%20%20%20%7By%20%3D%201%7D%20%5C%5C%20%20%20%20%7By%20%3D%200%7D%20%20%5Cend%7Barray%7D%20%7D%20%5Cright.)，    
其中：![{h_\theta }({\text{z}}) = \frac{1}{{1 + {e^{ - z}}}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bh_%5Ctheta%20%7D%28%7B%5Ctext%7Bz%7D%7D%29%20%3D%20%5Cfrac%7B1%7D%7B%7B1%20%2B%20%7Be%5E%7B%20-%20z%7D%7D%7D%7D)，![z = {\theta ^T}x](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=z%20%3D%20%7B%5Ctheta%20%5ET%7Dx)
- 如图所示，如果`y=1`，`cost`代价函数如图所示    
![enter description here][24]    
我们想让![{\theta ^T}x &gt;  &gt; 0](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%5Ctheta%20%5ET%7Dx%20%3E%20%20%3E%200)，即`z>>0`，这样的话`cost`代价函数才会趋于最小（这是我们想要的），所以用途中**红色**的函数![\cos {t_1}(z)](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Ccos%20%7Bt_1%7D%28z%29)代替逻辑回归中的cost
- 当`y=0`时同样，用![\cos {t_0}(z)](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Ccos%20%7Bt_0%7D%28z%29)代替
![enter description here][25]
- 最终得到的代价函数为：    
![J(\theta ) = C\sum\limits_{i = 1}^m {[{y^{(i)}}\cos {t_1}({\theta ^T}{x^{(i)}}) + (1 - {y^{(i)}})\cos {t_0}({\theta ^T}{x^{(i)}})} ] + \frac{1}{2}\sum\limits_{j = 1}^{\text{n}} {\theta _j^2} ](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=J%28%5Ctheta%20%29%20%3D%20C%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%5B%7By%5E%7B%28i%29%7D%7D%5Ccos%20%7Bt_1%7D%28%7B%5Ctheta%20%5ET%7D%7Bx%5E%7B%28i%29%7D%7D%29%20%2B%20%281%20-%20%7By%5E%7B%28i%29%7D%7D%29%5Ccos%20%7Bt_0%7D%28%7B%5Ctheta%20%5ET%7D%7Bx%5E%7B%28i%29%7D%7D%29%7D%20%5D%20%2B%20%5Cfrac%7B1%7D%7B2%7D%5Csum%5Climits_%7Bj%20%3D%201%7D%5E%7B%5Ctext%7Bn%7D%7D%20%7B%5Ctheta%20_j%5E2%7D%20)   
最后我们想要![\mathop {\min }\limits_\theta  J(\theta )](http://latex.codecogs.com/gif.latex?%5Clarge%20%5Cmathop%20%7B%5Cmin%20%7D%5Climits_%5Ctheta%20J%28%5Ctheta%20%29)
- 之前我们逻辑回归中的代价函数为：   
![J(\theta ) =  - \frac{1}{m}\sum\limits_{i = 1}^m {[{y^{(i)}}\log ({h_\theta }({x^{(i)}}) + (1 - } {y^{(i)}})\log (1 - {h_\theta }({x^{(i)}})] + \frac{\lambda }{{2m}}\sum\limits_{j = 1}^n {\theta _j^2} ](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=J%28%5Ctheta%20%29%20%3D%20%20-%20%5Cfrac%7B1%7D%7Bm%7D%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%5B%7By%5E%7B%28i%29%7D%7D%5Clog%20%28%7Bh_%5Ctheta%20%7D%28%7Bx%5E%7B%28i%29%7D%7D%29%20%2B%20%281%20-%20%7D%20%7By%5E%7B%28i%29%7D%7D%29%5Clog%20%281%20-%20%7Bh_%5Ctheta%20%7D%28%7Bx%5E%7B%28i%29%7D%7D%29%5D%20%2B%20%5Cfrac%7B%5Clambda%20%7D%7B%7B2m%7D%7D%5Csum%5Climits_%7Bj%20%3D%201%7D%5En%20%7B%5Ctheta%20_j%5E2%7D%20)   
可以认为这里的![C = \frac{m}{\lambda }](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=C%20%3D%20%5Cfrac%7Bm%7D%7B%5Clambda%20%7D)，只是表达形式问题，这里`C`的值越大，SVM的决策边界的`margin`也越大，下面会说明

### 2、Large Margin
- 如下图所示,SVM分类会使用最大的`margin`将其分开    
![enter description here][26]
- 先说一下向量内积
 - ![u = \left[ {\begin{array}{c}    {{u_1}} \\    {{u_2}}  \end{array} } \right]](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=u%20%3D%20%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7Bc%7D%20%20%20%20%7B%7Bu_1%7D%7D%20%5C%5C%20%20%20%20%7B%7Bu_2%7D%7D%20%20%5Cend%7Barray%7D%20%7D%20%5Cright%5D)，![v = \left[ {\begin{array}{c}    {{v_1}} \\    {{v_2}}  \end{array} } \right]](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=v%20%3D%20%5Cleft%5B%20%7B%5Cbegin%7Barray%7D%7Bc%7D%20%20%20%20%7B%7Bv_1%7D%7D%20%5C%5C%20%20%20%20%7B%7Bv_2%7D%7D%20%20%5Cend%7Barray%7D%20%7D%20%5Cright%5D)    
 - ![||u||](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7C%7Cu%7C%7C)表示`u`的**欧几里得范数**（欧式范数），![||u||{\text{ = }}\sqrt {{\text{u}}_1^2 + u_2^2} ](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7C%7Cu%7C%7C%7B%5Ctext%7B%20%3D%20%7D%7D%5Csqrt%20%7B%7B%5Ctext%7Bu%7D%7D_1%5E2%20%2B%20u_2%5E2%7D%20)
 - `向量V`在`向量u`上的投影的长度记为`p`，则：向量内积：    
 ![{{\text{u}}^T}v = p||u|| = {u_1}{v_1} + {u_2}{v_2}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7B%7B%5Ctext%7Bu%7D%7D%5ET%7Dv%20%3D%20p%7C%7Cu%7C%7C%20%3D%20%7Bu_1%7D%7Bv_1%7D%20%2B%20%7Bu_2%7D%7Bv_2%7D)      
 ![enter description here][27]  
根据向量夹角公式推导一下即可，![\cos \theta  = \frac{{\overrightarrow {\text{u}} \overrightarrow v }}{{|\overrightarrow {\text{u}} ||\overrightarrow v |}}](http://latex.codecogs.com/gif.latex?%5Clarge%20%5Ccos%20%5Ctheta%20%3D%20%5Cfrac%7B%7B%5Coverrightarrow%20%7B%5Ctext%7Bu%7D%7D%20%5Coverrightarrow%20v%20%7D%7D%7B%7B%7C%5Coverrightarrow%20%7B%5Ctext%7Bu%7D%7D%20%7C%7C%5Coverrightarrow%20v%20%7C%7D%7D)

- 前面说过，当`C`越大时，`margin`也就越大，我们的目的是最小化代价函数`J(θ)`,当`margin`最大时，`C`的乘积项![\sum\limits_{i = 1}^m {[{y^{(i)}}\cos {t_1}({\theta ^T}{x^{(i)}}) + (1 - {y^{(i)}})\cos {t_0}({\theta ^T}{x^{(i)}})} ]](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%5B%7By%5E%7B%28i%29%7D%7D%5Ccos%20%7Bt_1%7D%28%7B%5Ctheta%20%5ET%7D%7Bx%5E%7B%28i%29%7D%7D%29%20%2B%20%281%20-%20%7By%5E%7B%28i%29%7D%7D%29%5Ccos%20%7Bt_0%7D%28%7B%5Ctheta%20%5ET%7D%7Bx%5E%7B%28i%29%7D%7D%29%7D%20%5D)要很小，所以近似为：   
![J(\theta ) = C0 + \frac{1}{2}\sum\limits_{j = 1}^{\text{n}} {\theta _j^2}  = \frac{1}{2}\sum\limits_{j = 1}^{\text{n}} {\theta _j^2}  = \frac{1}{2}(\theta _1^2 + \theta _2^2) = \frac{1}{2}{\sqrt {\theta _1^2 + \theta _2^2} ^2}](http://latex.codecogs.com/gif.latex?%5Clarge%20J%28%5Ctheta%20%29%20%3D%20C0%20&plus;%20%5Cfrac%7B1%7D%7B2%7D%5Csum%5Climits_%7Bj%20%3D%201%7D%5E%7B%5Ctext%7Bn%7D%7D%20%7B%5Ctheta%20_j%5E2%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D%5Csum%5Climits_%7Bj%20%3D%201%7D%5E%7B%5Ctext%7Bn%7D%7D%20%7B%5Ctheta%20_j%5E2%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D%28%5Ctheta%20_1%5E2%20&plus;%20%5Ctheta%20_2%5E2%29%20%3D%20%5Cfrac%7B1%7D%7B2%7D%7B%5Csqrt%20%7B%5Ctheta%20_1%5E2%20&plus;%20%5Ctheta%20_2%5E2%7D%20%5E2%7D)，      
我们最后的目的就是求使代价最小的`θ`
- 由   
![\left\{ {\begin{array}{c}    {{\theta ^T}{x^{(i)}} \geqslant 1} \\    {{\theta ^T}{x^{(i)}} \leqslant  - 1}  \end{array} } \right.\begin{array}{c}    {({y^{(i)}} = 1)} \\    {({y^{(i)}} = 0)}  \end{array} ](http://latex.codecogs.com/gif.latex?%5Clarge%20%5Cleft%5C%7B%20%7B%5Cbegin%7Barray%7D%7Bc%7D%20%7B%7B%5Ctheta%20%5ET%7D%7Bx%5E%7B%28i%29%7D%7D%20%5Cgeqslant%201%7D%20%5C%5C%20%7B%7B%5Ctheta%20%5ET%7D%7Bx%5E%7B%28i%29%7D%7D%20%5Cleqslant%20-%201%7D%20%5Cend%7Barray%7D%20%7D%20%5Cright.%5Cbegin%7Barray%7D%7Bc%7D%20%7B%28%7By%5E%7B%28i%29%7D%7D%20%3D%201%29%7D%20%5C%5C%20%7B%28%7By%5E%7B%28i%29%7D%7D%20%3D%200%29%7D%20%5Cend%7Barray%7D)可以得到：    
![\left\{ {\begin{array}{c}    {{p^{(i)}}||\theta || \geqslant 1} \\    {{p^{(i)}}||\theta || \leqslant  - 1}  \end{array} } \right.\begin{array}{c}    {({y^{(i)}} = 1)} \\    {({y^{(i)}} = 0)}  \end{array} ](http://latex.codecogs.com/gif.latex?%5Clarge%20%5Cleft%5C%7B%20%7B%5Cbegin%7Barray%7D%7Bc%7D%20%7B%7Bp%5E%7B%28i%29%7D%7D%7C%7C%5Ctheta%20%7C%7C%20%5Cgeqslant%201%7D%20%5C%5C%20%7B%7Bp%5E%7B%28i%29%7D%7D%7C%7C%5Ctheta%20%7C%7C%20%5Cleqslant%20-%201%7D%20%5Cend%7Barray%7D%20%7D%20%5Cright.%5Cbegin%7Barray%7D%7Bc%7D%20%7B%28%7By%5E%7B%28i%29%7D%7D%20%3D%201%29%7D%20%5C%5C%20%7B%28%7By%5E%7B%28i%29%7D%7D%20%3D%200%29%7D%20%5Cend%7Barray%7D)，`p`即为`x`在`θ`上的投影
- 如下图所示，假设决策边界如图，找其中的一个点，到`θ`上的投影为`p`,则![p||\theta || \geqslant 1](http://latex.codecogs.com/gif.latex?%5Clarge%20p%7C%7C%5Ctheta%20%7C%7C%20%5Cgeqslant%201)或者![p||\theta || \leqslant  - 1](http://latex.codecogs.com/gif.latex?%5Clarge%20p%7C%7C%5Ctheta%20%7C%7C%20%5Cleqslant%20-%201)，若是`p`很小，则需要![||\theta ||](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7C%7C%5Ctheta%20%7C%7C)很大，这与我们要求的`θ`使![||\theta || = \frac{1}{2}\sqrt {\theta _1^2 + \theta _2^2} ](http://latex.codecogs.com/gif.latex?%5Clarge%20%7C%7C%5Ctheta%20%7C%7C%20%3D%20%5Cfrac%7B1%7D%7B2%7D%5Csqrt%20%7B%5Ctheta%20_1%5E2%20&plus;%20%5Ctheta%20_2%5E2%7D)最小相违背，**所以**最后求的是`large margin`   
![enter description here][28]

### 3、SVM Kernel（核函数）
- 对于线性可分的问题，使用**线性核函数**即可
- 对于线性不可分的问题，在逻辑回归中，我们是将`feature`映射为使用多项式的形式![1 + {x_1} + {x_2} + x_1^2 + {x_1}{x_2} + x_2^2](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=1%20%2B%20%7Bx_1%7D%20%2B%20%7Bx_2%7D%20%2B%20x_1%5E2%20%2B%20%7Bx_1%7D%7Bx_2%7D%20%2B%20x_2%5E2)，`SVM`中也有**多项式核函数**，但是更常用的是**高斯核函数**，也称为**RBF核**
- 高斯核函数为：![f(x) = {e^{ - \frac{{||x - u|{|^2}}}{{2{\sigma ^2}}}}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=f%28x%29%20%3D%20%7Be%5E%7B%20-%20%5Cfrac%7B%7B%7C%7Cx%20-%20u%7C%7B%7C%5E2%7D%7D%7D%7B%7B2%7B%5Csigma%20%5E2%7D%7D%7D%7D%7D)     
假设如图几个点，
![enter description here][29]
令：   
![{f_1} = similarity(x,{l^{(1)}}) = {e^{ - \frac{{||x - {l^{(1)}}|{|^2}}}{{2{\sigma ^2}}}}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bf_1%7D%20%3D%20similarity%28x%2C%7Bl%5E%7B%281%29%7D%7D%29%20%3D%20%7Be%5E%7B%20-%20%5Cfrac%7B%7B%7C%7Cx%20-%20%7Bl%5E%7B%281%29%7D%7D%7C%7B%7C%5E2%7D%7D%7D%7B%7B2%7B%5Csigma%20%5E2%7D%7D%7D%7D%7D)   
![{f_2} = similarity(x,{l^{(2)}}) = {e^{ - \frac{{||x - {l^{(2)}}|{|^2}}}{{2{\sigma ^2}}}}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bf_2%7D%20%3D%20similarity%28x%2C%7Bl%5E%7B%282%29%7D%7D%29%20%3D%20%7Be%5E%7B%20-%20%5Cfrac%7B%7B%7C%7Cx%20-%20%7Bl%5E%7B%282%29%7D%7D%7C%7B%7C%5E2%7D%7D%7D%7B%7B2%7B%5Csigma%20%5E2%7D%7D%7D%7D%7D)
.
.
.
- 可以看出，若是`x`与![{l^{(1)}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bl%5E%7B%281%29%7D%7D)距离较近，==》![{f_1} \approx {e^0} = 1](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bf_1%7D%20%5Capprox%20%7Be%5E0%7D%20%3D%201)，（即相似度较大）   
若是`x`与![{l^{(1)}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bl%5E%7B%281%29%7D%7D)距离较远，==》![{f_2} \approx {e^{ - \infty }} = 0](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bf_2%7D%20%5Capprox%20%7Be%5E%7B%20-%20%5Cinfty%20%7D%7D%20%3D%200)，（即相似度较低）
- 高斯核函数的`σ`越小，`f`下降的越快      
![enter description here][30]
![enter description here][31]

- 如何选择初始的![{l^{(1)}}{l^{(2)}}{l^{(3)}}...](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bl%5E%7B%281%29%7D%7D%7Bl%5E%7B%282%29%7D%7D%7Bl%5E%7B%283%29%7D%7D...)
 - 训练集：![(({x^{(1)}},{y^{(1)}}),({x^{(2)}},{y^{(2)}}),...({x^{(m)}},{y^{(m)}}))](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%28%28%7Bx%5E%7B%281%29%7D%7D%2C%7By%5E%7B%281%29%7D%7D%29%2C%28%7Bx%5E%7B%282%29%7D%7D%2C%7By%5E%7B%282%29%7D%7D%29%2C...%28%7Bx%5E%7B%28m%29%7D%7D%2C%7By%5E%7B%28m%29%7D%7D%29%29)
 - 选择：![{l^{(1)}} = {x^{(1)}},{l^{(2)}} = {x^{(2)}}...{l^{(m)}} = {x^{(m)}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bl%5E%7B%281%29%7D%7D%20%3D%20%7Bx%5E%7B%281%29%7D%7D%2C%7Bl%5E%7B%282%29%7D%7D%20%3D%20%7Bx%5E%7B%282%29%7D%7D...%7Bl%5E%7B%28m%29%7D%7D%20%3D%20%7Bx%5E%7B%28m%29%7D%7D)
 - 对于给出的`x`，计算`f`,令：![f_0^{(i)} = 1](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=f_0%5E%7B%28i%29%7D%20%3D%201)所以：![{f^{(i)}} \in {R^{m + 1}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bf%5E%7B%28i%29%7D%7D%20%5Cin%20%7BR%5E%7Bm%20%2B%201%7D%7D)
 - 最小化`J`求出`θ`，          
 ![J(\theta ) = C\sum\limits_{i = 1}^m {[{y^{(i)}}\cos {t_1}({\theta ^T}{f^{(i)}}) + (1 - {y^{(i)}})\cos {t_0}({\theta ^T}{f^{(i)}})} ] + \frac{1}{2}\sum\limits_{j = 1}^{\text{n}} {\theta _j^2} ](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=J%28%5Ctheta%20%29%20%3D%20C%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%5B%7By%5E%7B%28i%29%7D%7D%5Ccos%20%7Bt_1%7D%28%7B%5Ctheta%20%5ET%7D%7Bf%5E%7B%28i%29%7D%7D%29%20%2B%20%281%20-%20%7By%5E%7B%28i%29%7D%7D%29%5Ccos%20%7Bt_0%7D%28%7B%5Ctheta%20%5ET%7D%7Bf%5E%7B%28i%29%7D%7D%29%7D%20%5D%20%2B%20%5Cfrac%7B1%7D%7B2%7D%5Csum%5Climits_%7Bj%20%3D%201%7D%5E%7B%5Ctext%7Bn%7D%7D%20%7B%5Ctheta%20_j%5E2%7D%20)
 - 如果![{\theta ^T}f \geqslant 0](http://latex.codecogs.com/gif.latex?%5Clarge%20%7B%5Ctheta%20%5ET%7Df%20%5Cgeqslant%200)，==》预测`y=1`

### 4、使用`scikit-learn`中的`SVM`模型代码
- [全部代码](/SVM/SVM_scikit-learn.py)
- 线性可分的,指定核函数为`linear`：
```
    '''data1——线性分类'''
    data1 = spio.loadmat('data1.mat')
    X = data1['X']
    y = data1['y']
    y = np.ravel(y)
    plot_data(X,y)
    
    model = svm.SVC(C=1.0,kernel='linear').fit(X,y) # 指定核函数为线性核函数
```
- 非线性可分的，默认核函数为`rbf`
```
    '''data2——非线性分类'''
    data2 = spio.loadmat('data2.mat')
    X = data2['X']
    y = data2['y']
    y = np.ravel(y)
    plt = plot_data(X,y)
    plt.show()
    
    model = svm.SVC(gamma=100).fit(X,y)     # gamma为核函数的系数，值越大拟合的越好
```
### 5、运行结果
- 线性可分的决策边界：    
![enter description here][32]
- 线性不可分的决策边界：   
![enter description here][33]

--------------------------

## 五、K-Means聚类算法
- [全部代码](/K-Means/K-Menas.py)

### 1、聚类过程
- 聚类属于无监督学习，不知道y的标记分为K类
- K-Means算法分为两个步骤
 - 第一步：簇分配，随机选`K`个点作为中心，计算到这`K`个点的距离，分为`K`个簇
 - 第二步：移动聚类中心：重新计算每个**簇**的中心，移动中心，重复以上步骤。
- 如下图所示：
 - 随机分配的聚类中心  
 ![enter description here][34]
 - 重新计算聚类中心，移动一次  
 ![enter description here][35]
 - 最后`10`步之后的聚类中心  
 ![enter description here][36]

- 计算每条数据到哪个中心最近实现代码：
```
# 找到每条数据距离哪个类中心最近    
def findClosestCentroids(X,initial_centroids):
    m = X.shape[0]                  # 数据条数
    K = initial_centroids.shape[0]  # 类的总数
    dis = np.zeros((m,K))           # 存储计算每个点分别到K个类的距离
    idx = np.zeros((m,1))           # 要返回的每条数据属于哪个类
    
    '''计算每个点到每个类中心的距离'''
    for i in range(m):
        for j in range(K):
            dis[i,j] = np.dot((X[i,:]-initial_centroids[j,:]).reshape(1,-1),(X[i,:]-initial_centroids[j,:]).reshape(-1,1))
    
    '''返回dis每一行的最小值对应的列号，即为对应的类别
    - np.min(dis, axis=1)返回每一行的最小值
    - np.where(dis == np.min(dis, axis=1).reshape(-1,1)) 返回对应最小值的坐标
     - 注意：可能最小值对应的坐标有多个，where都会找出来，所以返回时返回前m个需要的即可（因为对于多个最小值，属于哪个类别都可以）
    '''  
    dummy,idx = np.where(dis == np.min(dis, axis=1).reshape(-1,1))
    return idx[0:dis.shape[0]]  # 注意截取一下
```
- 计算类中心实现代码：
```
# 计算类中心
def computerCentroids(X,idx,K):
    n = X.shape[1]
    centroids = np.zeros((K,n))
    for i in range(K):
        centroids[i,:] = np.mean(X[np.ravel(idx==i),:], axis=0).reshape(1,-1)   # 索引要是一维的,axis=0为每一列，idx==i一次找出属于哪一类的，然后计算均值
    return centroids
```

### 2、目标函数
- 也叫做**失真代价函数**
- ![J({c^{(1)}}, \cdots ,{c^{(m)}},{u_1}, \cdots ,{u_k}) = \frac{1}{m}\sum\limits_{i = 1}^m {||{x^{(i)}} - {u_{{c^{(i)}}}}|{|^2}} ](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=J%28%7Bc%5E%7B%281%29%7D%7D%2C%20%5Ccdots%20%2C%7Bc%5E%7B%28m%29%7D%7D%2C%7Bu_1%7D%2C%20%5Ccdots%20%2C%7Bu_k%7D%29%20%3D%20%5Cfrac%7B1%7D%7Bm%7D%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%7C%7C%7Bx%5E%7B%28i%29%7D%7D%20-%20%7Bu_%7B%7Bc%5E%7B%28i%29%7D%7D%7D%7D%7C%7B%7C%5E2%7D%7D%20)
- 最后我们想得到：  
![enter description here][37]
- 其中![{c^{(i)}}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bc%5E%7B%28i%29%7D%7D)表示第`i`条数据距离哪个类中心最近，
- 其中![{u_i}](http://chart.apis.google.com/chart?cht=tx&chs=1x0&chf=bg,s,FFFFFF00&chco=000000&chl=%7Bu_i%7D)即为聚类的中心

### 3、聚类中心的选择
- 随机初始化，从给定的数据中随机抽取K个作为聚类中心
- 随机一次的结果可能不好，可以随机多次，最后取使代价函数最小的作为中心
- 实现代码：(这里随机一次)
```
# 初始化类中心--随机取K个点作为聚类中心
def kMeansInitCentroids(X,K):
    m = X.shape[0]
    m_arr = np.arange(0,m)      # 生成0-m-1
    centroids = np.zeros((K,X.shape[1]))
    np.random.shuffle(m_arr)    # 打乱m_arr顺序    
    rand_indices = m_arr[:K]    # 取前K个
    centroids = X[rand_indices,:]
    return centroids
```

### 4、聚类个数K的选择
- 聚类是不知道y的label的，所以不知道真正的聚类个数
- 肘部法则（Elbow method）
 - 作代价函数`J`和`K`的图，若是出现一个拐点，如下图所示，`K`就取拐点处的值，下图此时`K=3`
 ![enter description here][38]
 - 若是很平滑就不明确，人为选择。
- 第二种就是人为观察选择

### 5、应用——图片压缩
- 将图片的像素分为若干类，然后用这个类代替原来的像素值
- 执行聚类的算法代码：
```
# 聚类算法
def runKMeans(X,initial_centroids,max_iters,plot_process):
    m,n = X.shape                   # 数据条数和维度
    K = initial_centroids.shape[0]  # 类数
    centroids = initial_centroids   # 记录当前类中心
    previous_centroids = centroids  # 记录上一次类中心
    idx = np.zeros((m,1))           # 每条数据属于哪个类
    
    for i in range(max_iters):      # 迭代次数
        print u'迭代计算次数：%d'%(i+1)
        idx = findClosestCentroids(X, centroids)
        if plot_process:    # 如果绘制图像
            plt = plotProcessKMeans(X,centroids,previous_centroids) # 画聚类中心的移动过程
            previous_centroids = centroids  # 重置
        centroids = computerCentroids(X, idx, K)    # 重新计算类中心
    if plot_process:    # 显示最终的绘制结果
        plt.show()
    return centroids,idx    # 返回聚类中心和数据属于哪个类
```

### 6、[使用scikit-learn库中的线性模型实现聚类](/K-Means/K-Means_scikit-learn.py)

- 导入包
```
    from sklearn.cluster import KMeans
```
- 使用模型拟合数据
```
    model = KMeans(n_clusters=3).fit(X) # n_clusters指定3类，拟合数据
```
- 聚类中心
```
    centroids = model.cluster_centers_  # 聚类中心
```

### 7、运行结果
- 二维数据类中心的移动  
![enter description here][39]
- 图片压缩  
![enter description here][40]


----------------------

## 六、PCA主成分分析（降维）
- [全部代码](/PCA/PCA.py)

### 1、用处
- 数据压缩（Data Compression）,使程序运行更快
- 可视化数据，例如`3D-->2D`等
- ......

### 2、2D-->1D，nD-->kD
- 如下图所示，所有数据点可以投影到一条直线，是**投影距离的平方和**（投影误差）最小
![enter description here][41]
- 注意数据需要`归一化`处理
- 思路是找`1`个`向量u`,所有数据投影到上面使投影距离最小
- 那么`nD-->kD`就是找`k`个向量![$${u^{(1)}},{u^{(2)}} \ldots {u^{(k)}}$$](http://latex.codecogs.com/gif.latex?%24%24%7Bu%5E%7B%281%29%7D%7D%2C%7Bu%5E%7B%282%29%7D%7D%20%5Cldots%20%7Bu%5E%7B%28k%29%7D%7D%24%24)，所有数据投影到上面使投影误差最小
 - eg:3D-->2D,2个向量![$${u^{(1)}},{u^{(2)}}$$](http://latex.codecogs.com/gif.latex?%24%24%7Bu%5E%7B%281%29%7D%7D%2C%7Bu%5E%7B%282%29%7D%7D%24%24)就代表一个平面了，所有点投影到这个平面的投影误差最小即可

### 3、主成分分析PCA与线性回归的区别
- 线性回归是找`x`与`y`的关系，然后用于预测`y`
- `PCA`是找一个投影面，最小化data到这个投影面的投影误差

### 4、PCA降维过程
- 数据预处理（均值归一化）
 - 公式：![$${\rm{x}}_j^{(i)} = {{{\rm{x}}_j^{(i)} - {u_j}} \over {{s_j}}}$$](http://latex.codecogs.com/gif.latex?%24%24%7B%5Crm%7Bx%7D%7D_j%5E%7B%28i%29%7D%20%3D%20%7B%7B%7B%5Crm%7Bx%7D%7D_j%5E%7B%28i%29%7D%20-%20%7Bu_j%7D%7D%20%5Cover%20%7B%7Bs_j%7D%7D%7D%24%24)
 - 就是减去对应feature的均值，然后除以对应特征的标准差（也可以是最大值-最小值）
 - 实现代码：
 ```
     # 归一化数据
    def featureNormalize(X):
        '''（每一个数据-当前列的均值）/当前列的标准差'''
        n = X.shape[1]
        mu = np.zeros((1,n));
        sigma = np.zeros((1,n))
        
        mu = np.mean(X,axis=0)
        sigma = np.std(X,axis=0)
        for i in range(n):
            X[:,i] = (X[:,i]-mu[i])/sigma[i]
        return X,mu,sigma
 ```
- 计算`协方差矩阵Σ`（Covariance Matrix）：![$$\Sigma  = {1 \over m}\sum\limits_{i = 1}^n {{x^{(i)}}{{({x^{(i)}})}^T}} $$](http://latex.codecogs.com/gif.latex?%24%24%5CSigma%20%3D%20%7B1%20%5Cover%20m%7D%5Csum%5Climits_%7Bi%20%3D%201%7D%5En%20%7B%7Bx%5E%7B%28i%29%7D%7D%7B%7B%28%7Bx%5E%7B%28i%29%7D%7D%29%7D%5ET%7D%7D%20%24%24)
 - 注意这里的`Σ`和求和符号不同
 - 协方差矩阵`对称正定`（不理解正定的看看线代）
 - 大小为`nxn`,`n`为`feature`的维度
 - 实现代码：
 ```
 Sigma = np.dot(np.transpose(X_norm),X_norm)/m  # 求Sigma
 ```
- 计算`Σ`的特征值和特征向量
 - 可以是用`svd`奇异值分解函数：`U,S,V = svd(Σ)`
 - 返回的是与`Σ`同样大小的对角阵`S`（由`Σ`的特征值组成）[**注意**：`matlab`中函数返回的是对角阵，在`python`中返回的是一个向量，节省空间]
 - 还有两个**酉矩阵**U和V，且![$$\Sigma  = US{V^T}$$](http://latex.codecogs.com/gif.latex?%24%24%5CSigma%20%3D%20US%7BV%5ET%7D%24%24)
 - ![enter description here][42]
 - **注意**：`svd`函数求出的`S`是按特征值降序排列的，若不是使用`svd`,需要按**特征值**大小重新排列`U`
- 降维
 - 选取`U`中的前`K`列（假设要降为`K`维）
 - ![enter description here][43]
 - `Z`就是对应降维之后的数据
 - 实现代码：
 ```
     # 映射数据
    def projectData(X_norm,U,K):
        Z = np.zeros((X_norm.shape[0],K))
        
        U_reduce = U[:,0:K]          # 取前K个
        Z = np.dot(X_norm,U_reduce) 
        return Z
 ```
- 过程总结：
 - `Sigma = X'*X/m`
 - `U,S,V = svd(Sigma)`
 - `Ureduce = U[:,0:k]`
 - `Z = Ureduce'*x`

### 5、数据恢复
 - 因为：![$${Z^{(i)}} = U_{reduce}^T*{X^{(i)}}$$](http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%24%24%7BZ%5E%7B%28i%29%7D%7D%20%3D%20U_%7Breduce%7D%5ET*%7BX%5E%7B%28i%29%7D%7D%24%24)
 - 所以：![$${X_{approx}} = {(U_{reduce}^T)^{ - 1}}Z$$](http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%24%24%7BX_%7Bapprox%7D%7D%20%3D%20%7B%28U_%7Breduce%7D%5ET%29%5E%7B%20-%201%7D%7DZ%24%24)     （注意这里是X的近似值）
 - 又因为`Ureduce`为正定矩阵，【正定矩阵满足：![$$A{A^T} = {A^T}A = E$$](http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%24%24A%7BA%5ET%7D%20%3D%20%7BA%5ET%7DA%20%3D%20E%24%24)，所以：![$${A^{ - 1}} = {A^T}$$](http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%24%24%7BA%5E%7B%20-%201%7D%7D%20%3D%20%7BA%5ET%7D%24%24)】，所以这里：
 - ![$${X_{approx}} = {(U_{reduce}^{ - 1})^{ - 1}}Z = {U_{reduce}}Z$$](http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%24%24%7BX_%7Bapprox%7D%7D%20%3D%20%7B%28U_%7Breduce%7D%5E%7B%20-%201%7D%29%5E%7B%20-%201%7D%7DZ%20%3D%20%7BU_%7Breduce%7D%7DZ%24%24)
 - 实现代码：
```
    # 恢复数据 
    def recoverData(Z,U,K):
        X_rec = np.zeros((Z.shape[0],U.shape[0]))
        U_recude = U[:,0:K]
        X_rec = np.dot(Z,np.transpose(U_recude))  # 还原数据（近似）
        return X_rec
```

### 6、主成分个数的选择（即要降的维度）
- 如何选择
 - **投影误差**（project error）：![$${1 \over m}\sum\limits_{i = 1}^m {||{x^{(i)}} - x_{approx}^{(i)}|{|^2}} $$](http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%24%24%7B1%20%5Cover%20m%7D%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%7C%7C%7Bx%5E%7B%28i%29%7D%7D%20-%20x_%7Bapprox%7D%5E%7B%28i%29%7D%7C%7B%7C%5E2%7D%7D%20%24%24)
 - **总变差**（total variation）:![$${1 \over m}\sum\limits_{i = 1}^m {||{x^{(i)}}|{|^2}} $$](http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%24%24%7B1%20%5Cover%20m%7D%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%7C%7C%7Bx%5E%7B%28i%29%7D%7D%7C%7B%7C%5E2%7D%7D%20%24%24)
 - 若**误差率**（error ratio）：![$${{{1 \over m}\sum\limits_{i = 1}^m {||{x^{(i)}} - x_{approx}^{(i)}|{|^2}} } \over {{1 \over m}\sum\limits_{i = 1}^m {||{x^{(i)}}|{|^2}} }} \le 0.01$$](http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%24%24%7B%7B%7B1%20%5Cover%20m%7D%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%7C%7C%7Bx%5E%7B%28i%29%7D%7D%20-%20x_%7Bapprox%7D%5E%7B%28i%29%7D%7C%7B%7C%5E2%7D%7D%20%7D%20%5Cover%20%7B%7B1%20%5Cover%20m%7D%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%7C%7C%7Bx%5E%7B%28i%29%7D%7D%7C%7B%7C%5E2%7D%7D%20%7D%7D%20%5Cle%200.01%24%24)，则称`99%`保留差异性
 - 误差率一般取`1%，5%，10%`等
- 如何实现
 - 若是一个个试的话代价太大
 - 之前`U,S,V = svd(Sigma)`,我们得到了`S`，这里误差率error ratio:    
 ![$$error{\kern 1pt} \;ratio = 1 - {{\sum\limits_{i = 1}^k {{S_{ii}}} } \over {\sum\limits_{i = 1}^n {{S_{ii}}} }} \le threshold$$](http://latex.codecogs.com/gif.latex?%5Cfn_cm%20%24%24error%7B%5Ckern%201pt%7D%20%5C%3Bratio%20%3D%201%20-%20%7B%7B%5Csum%5Climits_%7Bi%20%3D%201%7D%5Ek%20%7B%7BS_%7Bii%7D%7D%7D%20%7D%20%5Cover%20%7B%5Csum%5Climits_%7Bi%20%3D%201%7D%5En%20%7B%7BS_%7Bii%7D%7D%7D%20%7D%7D%20%5Cle%20threshold%24%24)
 - 可以一点点增加`K`尝试。

### 7、使用建议
- 不要使用PCA去解决过拟合问题`Overfitting`，还是使用正则化的方法（如果保留了很高的差异性还是可以的）
- 只有在原数据上有好的结果，但是运行很慢，才考虑使用PCA

### 8、运行结果
- 2维数据降为1维
 - 要投影的方向     
![enter description here][44]
 - 2D降为1D及对应关系        
![enter description here][45]
- 人脸数据降维
 - 原始数据         
 ![enter description here][46]
 - 可视化部分`U`矩阵信息    
 ![enter description here][47]
 - 恢复数据    
 ![enter description here][48]

### 9、[使用scikit-learn库中的PCA实现降维](/PCA/PCA.py_scikit-learn.py)
- 导入需要的包：
```
#-*- coding: utf-8 -*-
# Author:bob
# Date:2016.12.22
import numpy as np
from matplotlib import pyplot as plt
from scipy import io as spio
from sklearn.decomposition import pca
from sklearn.preprocessing import StandardScaler
```
- 归一化数据
```
    '''归一化数据并作图'''
    scaler = StandardScaler()
    scaler.fit(X)
    x_train = scaler.transform(X)
```
- 使用PCA模型拟合数据，并降维
 - `n_components`对应要将的维度
```
    '''拟合数据'''
    K=1 # 要降的维度
    model = pca.PCA(n_components=K).fit(x_train)   # 拟合数据，n_components定义要降的维度
    Z = model.transform(x_train)    # transform就会执行降维操作
```

- 数据恢复
 - `model.components_`会得到降维使用的`U`矩阵 
```
    '''数据恢复并作图'''
    Ureduce = model.components_     # 得到降维用的Ureduce
    x_rec = np.dot(Z,Ureduce)       # 数据恢复
```



---------------------------------------------------------------


## 七、异常检测 Anomaly Detection
- [全部代码](/AnomalyDetection/AnomalyDetection.py)

### 1、高斯分布（正态分布）`Gaussian distribution` 
- 分布函数：![$$p(x) = {1 \over {\sqrt {2\pi } \sigma }}{e^{ - {{{{(x - u)}^2}} \over {2{\sigma ^2}}}}}$$](http://latex.codecogs.com/png.latex?%5Cfn_cm%20%24%24p%28x%29%20%3D%20%7B1%20%5Cover%20%7B%5Csqrt%20%7B2%5Cpi%20%7D%20%5Csigma%20%7D%7D%7Be%5E%7B%20-%20%7B%7B%7B%7B%28x%20-%20u%29%7D%5E2%7D%7D%20%5Cover%20%7B2%7B%5Csigma%20%5E2%7D%7D%7D%7D%7D%24%24)
 - 其中，`u`为数据的**均值**，`σ`为数据的**标准差**
 - `σ`越**小**，对应的图像越**尖**
- 参数估计（`parameter estimation`）
 - ![$$u = {1 \over m}\sum\limits_{i = 1}^m {{x^{(i)}}} $$](http://latex.codecogs.com/png.latex?%5Cfn_cm%20%24%24u%20%3D%20%7B1%20%5Cover%20m%7D%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%7Bx%5E%7B%28i%29%7D%7D%7D%20%24%24)
 - ![$${\sigma ^2} = {1 \over m}\sum\limits_{i = 1}^m {{{({x^{(i)}} - u)}^2}} $$](http://latex.codecogs.com/png.latex?%5Cfn_cm%20%24%24%7B%5Csigma%20%5E2%7D%20%3D%20%7B1%20%5Cover%20m%7D%5Csum%5Climits_%7Bi%20%3D%201%7D%5Em%20%7B%7B%7B%28%7Bx%5E%7B%28i%29%7D%7D%20-%20u%29%7D%5E2%7D%7D%20%24%24)

### 2、异常检测算法
- 例子
 - 训练集：![$$\{ {x^{(1)}},{x^{(2)}}, \cdots {x^{(m)}}\} $$](http://latex.codecogs.com/png.latex?%5Cfn_cm%20%24%24%5C%7B%20%7Bx%5E%7B%281%29%7D%7D%2C%7Bx%5E%7B%282%29%7D%7D%2C%20%5Ccdots%20%7Bx%5E%7B%28m%29%7D%7D%5C%7D%20%24%24),其中![$$x \in {R^n}$$](http://latex.codecogs.com/png.latex?%5Cfn_cm%20%24%24x%20%5Cin%20%7BR%5En%7D%24%24)
 - 假设![$${x_1},{x_2} \cdots {x_n}$$](http://latex.codecogs.com/png.latex?%5Cfn_cm%20%24%24%7Bx_1%7D%2C%7Bx_2%7D%20%5Ccdots%20%7Bx_n%7D%24%24)相互独立，建立model模型：![$$p(x) = p({x_1};{u_1},\sigma _1^2)p({x_2};{u_2},\sigma _2^2) \cdots p({x_n};{u_n},\sigma _n^2) = \prod\limits_{j = 1}^n {p({x_j};{u_j},\sigma _j^2)} $$](http://latex.codecogs.com/png.latex?%5Cfn_cm%20%24%24p%28x%29%20%3D%20p%28%7Bx_1%7D%3B%7Bu_1%7D%2C%5Csigma%20_1%5E2%29p%28%7Bx_2%7D%3B%7Bu_2%7D%2C%5Csigma%20_2%5E2%29%20%5Ccdots%20p%28%7Bx_n%7D%3B%7Bu_n%7D%2C%5Csigma%20_n%5E2%29%20%3D%20%5Cprod%5Climits_%7Bj%20%3D%201%7D%5En%20%7Bp%28%7Bx_j%7D%3B%7Bu_j%7D%2C%5Csigma%20_j%5E2%29%7D%20%24%24)
- 过程
 - 选择具有代表异常的`feature`:xi
 - 参数估计：![$${u_1},{u_2}, \cdots ,{u_n};\sigma _1^2,\sigma _2^2 \cdots ,\sigma _n^2$$](http://latex.codecogs.com/png.latex?%5Cfn_cm%20%24%24%7Bu_1%7D%2C%7Bu_2%7D%2C%20%5Ccdots%20%2C%7Bu_n%7D%3B%5Csigma%20_1%5E2%2C%5Csigma%20_2%5E2%20%5Ccdots%20%2C%5Csigma%20_n%5E2%24%24)
 - 计算`p(x)`,若是`P(x)<ε`则认为异常，其中`ε`为我们要求的概率的临界值`threshold`
- 这里只是**单元高斯分布**，假设了`feature`之间是独立的，下面会讲到**多元高斯分布**，会自动捕捉到`feature`之间的关系
- **参数估计**实现代码
```
# 参数估计函数（就是求均值和方差）
def estimateGaussian(X):
    m,n = X.shape
    mu = np.zeros((n,1))
    sigma2 = np.zeros((n,1))
    
    mu = np.mean(X, axis=0) # axis=0表示列，每列的均值
    sigma2 = np.var(X,axis=0) # 求每列的方差
    return mu,sigma2
```

### 3、评价`p(x)`的好坏，以及`ε`的选取
- 对**偏斜数据**的错误度量
 - 因为数据可能是非常**偏斜**的（就是`y=1`的个数非常少，(`y=1`表示异常)），所以可以使用`Precision/Recall`，计算`F1Score`(在**CV交叉验证集**上)
 - 例如：预测癌症，假设模型可以得到`99%`能够预测正确，`1%`的错误率，但是实际癌症的概率很小，只有`0.5%`，那么我们始终预测没有癌症y=0反而可以得到更小的错误率。使用`error rate`来评估就不科学了。
 - 如下图记录：    
 ![enter description here][49]
 - ![$$\Pr ecision = {{TP} \over {TP + FP}}$$](http://latex.codecogs.com/png.latex?%5Cfn_cm%20%24%24%5CPr%20ecision%20%3D%20%7B%7BTP%7D%20%5Cover%20%7BTP%20&plus;%20FP%7D%7D%24%24) ，即：**正确预测正样本/所有预测正样本**
 - ![$${\mathop{\rm Re}\nolimits} {\rm{call}} = {{TP} \over {TP + FN}}$$](http://latex.codecogs.com/png.latex?%5Cfn_cm%20%24%24%7B%5Cmathop%7B%5Crm%20Re%7D%5Cnolimits%7D%20%7B%5Crm%7Bcall%7D%7D%20%3D%20%7B%7BTP%7D%20%5Cover%20%7BTP%20&plus;%20FN%7D%7D%24%24) ，即：**正确预测正样本/真实值为正样本**
 - 总是让`y=1`(较少的类)，计算`Precision`和`Recall`
 - ![$${F_1}Score = 2{{PR} \over {P + R}}$$](http://latex.codecogs.com/png.latex?%5Cfn_cm%20%24%24%7BF_1%7DScore%20%3D%202%7B%7BPR%7D%20%5Cover%20%7BP%20&plus;%20R%7D%7D%24%24)
 - 还是以癌症预测为例，假设预测都是no-cancer，TN=199，FN=1，TP=0，FP=0，所以：Precision=0/0，Recall=0/1=0，尽管accuracy=199/200=99.5%，但是不可信。

- `ε`的选取
 - 尝试多个`ε`值，使`F1Score`的值高
- 实现代码
```
# 选择最优的epsilon，即：使F1Score最大    
def selectThreshold(yval,pval):
    '''初始化所需变量'''
    bestEpsilon = 0.
    bestF1 = 0.
    F1 = 0.
    step = (np.max(pval)-np.min(pval))/1000
    '''计算'''
    for epsilon in np.arange(np.min(pval),np.max(pval),step):
        cvPrecision = pval<epsilon
        tp = np.sum((cvPrecision == 1) & (yval == 1).ravel()).astype(float)  # sum求和是int型的，需要转为float
        fp = np.sum((cvPrecision == 1) & (yval == 0).ravel()).astype(float)
        fn = np.sum((cvPrecision == 0) & (yval == 1).ravel()).astype(float)
        precision = tp/(tp+fp)  # 精准度
        recision = tp/(tp+fn)   # 召回率
        F1 = (2*precision*recision)/(precision+recision)  # F1Score计算公式
        if F1 > bestF1:  # 修改最优的F1 Score
            bestF1 = F1
            bestEpsilon = epsilon
    return bestEpsilon,bestF1
```

### 4、选择使用什么样的feature（单元高斯分布）
- 如果一些数据不是满足高斯分布的，可以变化一下数据，例如`log(x+C),x^(1/2)`等
- 如果`p(x)`的值无论异常与否都很大，可以尝试组合多个`feature`,(因为feature之间可能是有关系的)

### 5、多元高斯分布
- 单元高斯分布存在的问题
 - 如下图，红色的点为异常点，其他的都是正常点（比如CPU和memory的变化）   
 ![enter description here][50]
 - x1对应的高斯分布如下：   
 ![enter description here][51]
 - x2对应的高斯分布如下：   
 ![enter description here][52]
 - 可以看出对应的p(x1)和p(x2)的值变化并不大，就不会认为异常
 - 因为我们认为feature之间是相互独立的，所以如上图是以**正圆**的方式扩展
- 多元高斯分布
 - ![$$x \in {R^n}$$](http://latex.codecogs.com/png.latex?%5Cfn_cm%20%24%24x%20%5Cin%20%7BR%5En%7D%24%24)，并不是建立`p(x1),p(x2)...p(xn)`，而是统一建立`p(x)`
 - 其中参数：![$$\mu  \in {R^n},\Sigma  \in {R^{n \times {\rm{n}}}}$$](http://latex.codecogs.com/png.latex?%5Cfn_cm%20%24%24%5Cmu%20%5Cin%20%7BR%5En%7D%2C%5CSigma%20%5Cin%20%7BR%5E%7Bn%20%5Ctimes%20%7B%5Crm%7Bn%7D%7D%7D%7D%24%24),`Σ`为**协方差矩阵**
 - ![$$p(x) = {1 \over {{{(2\pi )}^{{n \over 2}}}|\Sigma {|^{{1 \over 2}}}}}{e^{ - {1 \over 2}{{(x - u)}^T}{\Sigma ^{ - 1}}(x - u)}}$$](http://latex.codecogs.com/png.latex?%5Cfn_cm%20%24%24p%28x%29%20%3D%20%7B1%20%5Cover%20%7B%7B%7B%282%5Cpi%20%29%7D%5E%7B%7Bn%20%5Cover%202%7D%7D%7D%7C%5CSigma%20%7B%7C%5E%7B%7B1%20%5Cover%202%7D%7D%7D%7D%7D%7Be%5E%7B%20-%20%7B1%20%5Cover%202%7D%7B%7B%28x%20-%20u%29%7D%5ET%7D%7B%5CSigma%20%5E%7B%20-%201%7D%7D%28x%20-%20u%29%7D%7D%24%24)
 - 同样，`|Σ|`越小，`p(x)`越尖
 - 例如：    
 ![enter description here][53]，  
 表示x1,x2**正相关**，即x1越大，x2也就越大，如下图，也就可以将红色的异常点检查出了
 ![enter description here][54]      
 若：   
  ![enter description here][55]，   
 表示x1,x2**负相关**
- 实现代码：
```
# 多元高斯分布函数    
def multivariateGaussian(X,mu,Sigma2):
    k = len(mu)
    if (Sigma2.shape[0]>1):
        Sigma2 = np.diag(Sigma2)
    '''多元高斯分布函数'''    
    X = X-mu
    argu = (2*np.pi)**(-k/2)*np.linalg.det(Sigma2)**(-0.5)
    p = argu*np.exp(-0.5*np.sum(np.dot(X,np.linalg.inv(Sigma2))*X,axis=1))  # axis表示每行
    return p
```
### 6、单元和多元高斯分布特点
- 单元高斯分布
 - 人为可以捕捉到`feature`之间的关系时可以使用
 - 计算量小
- 多元高斯分布
 - 自动捕捉到相关的feature
 - 计算量大，因为：![$$\Sigma  \in {R^{n \times {\rm{n}}}}$$](http://latex.codecogs.com/png.latex?%5Cfn_cm%20%24%24%5CSigma%20%5Cin%20%7BR%5E%7Bn%20%5Ctimes%20%7B%5Crm%7Bn%7D%7D%7D%7D%24%24)
 - `m>n`或`Σ`可逆时可以使用。（若不可逆，可能有冗余的x，因为线性相关，不可逆，或者就是m<n）

### 7、程序运行结果
- 显示数据     
![enter description here][56]
- 等高线      
![enter description here][57]
- 异常点标注   
![enter description here][58]



----------------------------------


  [1]: ./images/LinearRegression_01.png "LinearRegression_01.png"
  [2]: ./images/LogisticRegression_01.png "LogisticRegression_01.png"
  [3]: ./images/LogisticRegression_02.png "LogisticRegression_02.png"
  [4]: ./images/LogisticRegression_03.jpg "LogisticRegression_03.jpg"
  [5]: ./images/LogisticRegression_04.png "LogisticRegression_04.png"
  [6]: ./images/LogisticRegression_05.png "LogisticRegression_05.png"
  [7]: ./images/LogisticRegression_06.png "LogisticRegression_06.png"
  [8]: ./images/LogisticRegression_07.png "LogisticRegression_07.png"
  [9]: ./images/LogisticRegression_08.png "LogisticRegression_08.png"
  [10]: ./images/LogisticRegression_09.png "LogisticRegression_09.png"
  [11]: ./images/LogisticRegression_11.png "LogisticRegression_11.png"
  [12]: ./images/LogisticRegression_10.png "LogisticRegression_10.png"
  [13]: ./images/LogisticRegression_12.png "LogisticRegression_12.png"
  [14]: ./images/LogisticRegression_13.png "LogisticRegression_13.png"
  [15]: ./images/NeuralNetwork_01.png "NeuralNetwork_01.png"
  [16]: ./images/NeuralNetwork_02.png "NeuralNetwork_02.png"
  [17]: ./images/NeuralNetwork_03.jpg "NeuralNetwork_03.jpg"
  [18]: ./images/NeuralNetwork_04.png "NeuralNetwork_04.png"
  [19]: ./images/NeuralNetwork_05.png "NeuralNetwork_05.png"
  [20]: ./images/NeuralNetwork_06.png "NeuralNetwork_06.png"
  [21]: ./images/NeuralNetwork_07.png "NeuralNetwork_07.png"
  [22]: ./images/NeuralNetwork_08.png "NeuralNetwork_08.png"
  [23]: ./images/NeuralNetwork_09.png "NeuralNetwork_09.png"
  [24]: ./images/SVM_01.png "SVM_01.png"
  [25]: ./images/SVM_02.png "SVM_02.png"
  [26]: ./images/SVM_03.png "SVM_03.png"
  [27]: ./images/SVM_04.png "SVM_04.png"
  [28]: ./images/SVM_05.png "SVM_05.png"
  [29]: ./images/SVM_06.png "SVM_06.png"
  [30]: ./images/SVM_07.png "SVM_07.png"
  [31]: ./images/SVM_08.png "SVM_08.png"
  [32]: ./images/SVM_09.png "SVM_09.png"
  [33]: ./images/SVM_10.png "SVM_10.png"
  [34]: ./images/K-Means_01.png "K-Means_01.png"
  [35]: ./images/K-Means_02.png "K-Means_02.png"
  [36]: ./images/K-Means_03.png "K-Means_03.png"
  [37]: ./images/K-Means_07.png "K-Means_07.png"
  [38]: ./images/K-Means_04.png "K-Means_04.png"
  [39]: ./images/K-Means_05.png "K-Means_05.png"
  [40]: ./images/K-Means_06.png "K-Means_06.png"
  [41]: ./images/PCA_01.png "PCA_01.png"
  [42]: ./images/PCA_02.png "PCA_02.png"
  [43]: ./images/PCA_03.png "PCA_03.png"
  [44]: ./images/PCA_04.png "PCA_04.png"
  [45]: ./images/PCA_05.png "PCA_05.png"
  [46]: ./images/PCA_06.png "PCA_06.png"
  [47]: ./images/PCA_07.png "PCA_07.png"
  [48]: ./images/PCA_08.png "PCA_08.png"
  [49]: ./images/AnomalyDetection_01.png "AnomalyDetection_01.png"
  [50]: ./images/AnomalyDetection_04.png "AnomalyDetection_04.png"
  [51]: ./images/AnomalyDetection_02.png "AnomalyDetection_02.png"
  [52]: ./images/AnomalyDetection_03.png "AnomalyDetection_03.png"
  [53]: ./images/AnomalyDetection_05.png "AnomalyDetection_05.png"
  [54]: ./images/AnomalyDetection_07.png "AnomalyDetection_07.png"
  [55]: ./images/AnomalyDetection_06.png "AnomalyDetection_06.png"
  [56]: ./images/AnomalyDetection_08.png "AnomalyDetection_08.png"
  [57]: ./images/AnomalyDetection_09.png "AnomalyDetection_09.png"
  [58]: ./images/AnomalyDetection_10.png "AnomalyDetection_10.png"