机器学习实战 Logistic回归

Sigmoid函数：

$\frac{1}{1+e^{-z}}$
$z=w_0x_0+w_1x_1+w_2x_2+...+w_nx_x$（$z=w^Tx$）
在每个特征上都乘以一个回归系数，然后把所有结果值相加，将这个总和代入Sigmoid函数中，进而得到一个范围在0~1直接的数值。（1类：大于0.5； 0类：小于0.5）

梯度上升法

梯度上升法基于的思想是：要找到函数的最大值，最好是沿着该函数的梯度方向探寻
$\nabla f(x,y)=\left(\begin{matrix} \frac{\partial f(x,y)}{\partial x} \\ \frac{\partial f(x,y)}{\partial y} \end{matrix}\right)$

• 梯度上升的迭代公式：$w:=w+\alpha\nabla _w f(w)$（$\alpha$为步长）
• 梯度下降的迭代公式：$w:=w-\alpha\nabla _w f(w)$（$\alpha$为步长）

Logistic回归梯度上升优化算法：

from numpy import *
def loadDataSet():
    dataMat = []
    labelMat = []
    fr = open('testSet.txt')
    for line in fr.readlines():
        lineArr = line.strip().split()
        dataMat.append([1.0, float(lineArr[0]), float(lineArr[1])])
        labelMat.append(int(lineArr[2]))
    return dataMat,labelMat

def sigmoid(inX):
    return 1.0/(1+exp(-inX))

def gradAscent(dataMatIn, classLabels):
    dataMatrix = mat(dataMatIn)
    labelMat = mat(classLabels).transpose()#转置矩阵
    m,n = shape(dataMatrix)
    alpha = 0.001
    maxCycles = 500#迭代次数上限
    weights = ones((n,1))
    for k in range(maxCycles):
        h = sigmoid(dataMatrix * weights)
        error = (labelMat - h)
        weights = weights + alpha * dataMatrix.transpose() * error
    return weights

def plotBestFit(weights):
    import matplotlib.pyplot as plt
    dataMat,labelMat = loadDataSet()
    dataArr = array(dataMat)
    n = shape(dataArr)[0]#行
    xcord1 = []
    ycord1 = []
    xcord2 = []
    ycord2 = []
    for i in range(n):
        if int(labelMat[i]) == 1:
            xcord1.append(dataArr[i,1])
            ycord1.append(dataArr[i,2])
        else:
            xcord2.append(dataArr[i,1])
            ycord2.append(dataArr[i,2])
    fig = plt.figure()
    ax = fig.add_subplot(111)
    ax.scatter(xcord1, ycord1, s=30, c='red', marker='s')
    ax.scatter(xcord2, ycord2, s=30,c='green')
    x = arange(-3.0, 3.0, 0.1)
    y = (-weights[0] - weights[1] * x) / weights[2]
    ax.plot(x,y)
    plt.xlabel('X1')
    plt.ylabel('X2')
    plt.show()

if __name__ == '__main__':
    dataArr,labelMat = loadDataSet()
    w = gradAscent(dataArr,labelMat)
    plotBestFit(w.getA())

输出结果：

随机梯度上升

随机梯度上升算法：

def stocGradAscent0(dataMatrix, classLabels):
    m,n = shape(dataMatrix)
    alpha = 0.01
    weights = ones(n)
    for i in range(m):
        h = sigmoid(sum(dataMatrix[i]*weights))
        error = classLabels[i] - h
        weights = weights + alpha * error * dataMatrix[i]
    return weights

输出结果：