机器学习（非线性回归）

1.概率：
1.1定义：概率：对一件事情发生的可能性的衡量

1.2范围：0<=P<=1

1.3计算方法：
1.3.1根据各人之置信
1.3.2根据历史数据
1.3.3根据模拟数据

1.4条件概率
P(A|B) = P(A&&B)/P(B)

2.逻辑回归

2.1基本模型
测试数据为X（x0,x1,x2,……xn）
要学习的参数为Θ（θ1,θ2,θ3……θn）

Z = θ0*x0+θ1*x1+θ2*x2+…..+θn*xn
向量表示：
Z = Θ转置*X

在处理二值数据时，使用Sigmoid函数使曲线平滑化

设置目标函数cost，学习参数θ使得目标函数最小

2.2解法：梯度下降
更新法则：
θ = θ - α(loss加和)(X的转置)

α为学习率，同时对所有的θ进行更新，重复更新直到收敛

代码实现：

# -*- coding:utf-8 -*-import numpy as npimport random#定义梯度下降算法函数，传入参数为自变量x，因变量y，学习率alpha，样本数m，训练次数numIterationdef gradientDescent(x,y,theta,alpha,m,numIterations):    xTrans = x.transpose()  #将自变量转换为转置矩阵    for i in range(0,numIterations):        hypothesis = np.dot(x,theta)  #定义h(函数)        loss = hypothesis - y  #定义估计值与实际值之间的误差        cost = np.sum(loss**2)/(2*m)   #定义目标函数，使得目标函数最小        print("Iteration %d | cost:%f"%(i,cost))   #输出均方误差        gradient = np.dot(xTrans,loss)/m   #dot()函数为点乘函数，利用梯度下降计算每一步的更新量        theta = theta - alpha*gradient    #更新theta，使得每一步更新theta之后的估计值与实际值之间累加和的平方和即目标函数变小    return theta#创建x,y数据，产生实例数据，数据类型为numpyArray类型，多元函数def genData(numPoints,bias,variance):  #numPoints为实例数量，bias为生成y时所加的偏值，variance为方差    x = np.zeros(shape=(numPoints,2))  #定义x为numPoints*2的矩阵，初始值都是0    y = np.zeros(shape=numPoints)  #定义y为numPoints*1的矩阵，初始值为0    #对x进行赋值，两个x,二元函数    for i in range(0,numPoints):        x[i][0] = 1  #初始化矩阵x的第一列为1        x[i][1] = i  #初始化矩阵x的第二列为2        #根据自变量x设置y值，x+偏向值+随机值，保证y值的不同        y[i] = (i + bias) + random.uniform(0,1)*variance  #在自变量x的基础的基础上增加一些随机因素产生y值    return x,yx,y = genData(100, 25, 10)print("x:")print(x)print("y:")print(y)m,n = np.shape(x)  #获取矩阵x的行、列数n_y = np.shape(y)  #获取矩阵y的行数print("x shape:",str(m)," ",str(n))   #树池print("y length:",str(n_y))numIterations = 100000alpha = 0.0005theta = np.ones(n)  #theta的 个数与矩阵x的列数把持已知，即有几元自变量，其自变量的系数就为几个theta = gradientDescent(x, y, theta, alpha, m, numIterations)print(theta)    

部分运行结果：

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cost:4.208633Iteration 99848 | cost:4.208633Iteration 99849 | cost:4.208633Iteration 99850 | cost:4.208633Iteration 99851 | cost:4.208633Iteration 99852 | cost:4.208633Iteration 99853 | cost:4.208633Iteration 99854 | cost:4.208633Iteration 99855 | cost:4.208633Iteration 99856 | cost:4.208633Iteration 99857 | cost:4.208633Iteration 99858 | cost:4.208633Iteration 99859 | cost:4.208633Iteration 99860 | cost:4.208633Iteration 99861 | cost:4.208633Iteration 99862 | cost:4.208633Iteration 99863 | cost:4.208633Iteration 99864 | cost:4.208633Iteration 99865 | cost:4.208633Iteration 99866 | cost:4.208633Iteration 99867 | cost:4.208633Iteration 99868 | cost:4.208633Iteration 99869 | cost:4.208633Iteration 99870 | cost:4.208633Iteration 99871 | cost:4.208633Iteration 99872 | cost:4.208633Iteration 99873 | cost:4.208633Iteration 99874 | cost:4.208633Iteration 99875 | cost:4.208633Iteration 99876 | cost:4.208633Iteration 99877 | cost:4.208633Iteration 99878 | cost:4.208633Iteration 99879 | cost:4.208633Iteration 99880 | cost:4.208633Iteration 99881 | cost:4.208633Iteration 99882 | cost:4.208633Iteration 99883 | cost:4.208633Iteration 99884 | cost:4.208633Iteration 99885 | cost:4.208633Iteration 99886 | cost:4.208633Iteration 99887 | cost:4.208633Iteration 99888 | cost:4.208633Iteration 99889 | cost:4.208633Iteration 99890 | cost:4.208633Iteration 99891 | cost:4.208633Iteration 99892 | cost:4.208633Iteration 99893 | cost:4.208633Iteration 99894 | cost:4.208633Iteration 99895 | cost:4.208633Iteration 99896 | cost:4.208633Iteration 99897 | cost:4.208633Iteration 99898 | cost:4.208633Iteration 99899 | cost:4.208633Iteration 99900 | cost:4.208633Iteration 99901 | cost:4.208633Iteration 99902 | cost:4.208633Iteration 99903 | cost:4.208633Iteration 99904 | cost:4.208633Iteration 99905 | cost:4.208633Iteration 99906 | cost:4.208633Iteration 99907 | cost:4.208633Iteration 99908 | cost:4.208633Iteration 99909 | cost:4.208633Iteration 99910 | cost:4.208633Iteration 99911 | cost:4.208633Iteration 99912 | cost:4.208633Iteration 99913 | cost:4.208633Iteration 99914 | cost:4.208633Iteration 99915 | cost:4.208633Iteration 99916 | cost:4.208633Iteration 99917 | cost:4.208633Iteration 99918 | cost:4.208633Iteration 99919 | cost:4.208633Iteration 99920 | cost:4.208633Iteration 99921 | cost:4.208633Iteration 99922 | cost:4.208633Iteration 99923 | cost:4.208633Iteration 99924 | cost:4.208633Iteration 99925 | cost:4.208633Iteration 99926 | cost:4.208633Iteration 99927 | cost:4.208633Iteration 99928 | cost:4.208633Iteration 99929 | cost:4.208633Iteration 99930 | cost:4.208633Iteration 99931 | cost:4.208633Iteration 99932 | cost:4.208633Iteration 99933 | cost:4.208633Iteration 99934 | cost:4.208633Iteration 99935 | cost:4.208633Iteration 99936 | cost:4.208633Iteration 99937 | cost:4.208633Iteration 99938 | cost:4.208633Iteration 99939 | cost:4.208633Iteration 99940 | cost:4.208633Iteration 99941 | cost:4.208633Iteration 99942 | cost:4.208633Iteration 99943 | cost:4.208633Iteration 99944 | cost:4.208633Iteration 99945 | cost:4.208633Iteration 99946 | cost:4.208633Iteration 99947 | cost:4.208633Iteration 99948 | cost:4.208633Iteration 99949 | cost:4.208633Iteration 99950 | cost:4.208633Iteration 99951 | cost:4.208633Iteration 99952 | cost:4.208633Iteration 99953 | cost:4.208633Iteration 99954 | cost:4.208633Iteration 99955 | cost:4.208633Iteration 99956 | cost:4.208633Iteration 99957 | cost:4.208633Iteration 99958 | cost:4.208633Iteration 99959 | cost:4.208633Iteration 99960 | cost:4.208633Iteration 99961 | cost:4.208633Iteration 99962 | cost:4.208633Iteration 99963 | cost:4.208633Iteration 99964 | cost:4.208633Iteration 99965 | cost:4.208633Iteration 99966 | cost:4.208633Iteration 99967 | cost:4.208633Iteration 99968 | cost:4.208633Iteration 99969 | cost:4.208633Iteration 99970 | cost:4.208633Iteration 99971 | cost:4.208633Iteration 99972 | cost:4.208633Iteration 99973 | cost:4.208633Iteration 99974 | cost:4.208633Iteration 99975 | cost:4.208633Iteration 99976 | cost:4.208633Iteration 99977 | cost:4.208633Iteration 99978 | cost:4.208633Iteration 99979 | cost:4.208633Iteration 99980 | cost:4.208633Iteration 99981 | cost:4.208633Iteration 99982 | cost:4.208633Iteration 99983 | cost:4.208633Iteration 99984 | cost:4.208633Iteration 99985 | cost:4.208633Iteration 99986 | cost:4.208633Iteration 99987 | cost:4.208633Iteration 99988 | cost:4.208633Iteration 99989 | cost:4.208633Iteration 99990 | cost:4.208633Iteration 99991 | cost:4.208633Iteration 99992 | cost:4.208633Iteration 99993 | cost:4.208633Iteration 99994 | cost:4.208633Iteration 99995 | cost:4.208633Iteration 99996 | cost:4.208633Iteration 99997 | cost:4.208633Iteration 99998 | cost:4.208633Iteration 99999 | cost:4.208633[ 29.60963368   1.01268588]

从运行结果可以看到目标函数cost的值随着theta参数的更新而逐渐减小，说明theta参数的更新是有效的。