1.利用逻辑回归进行二分类

1.1 批量梯度上升方法

from math import *
from numpy import *
import matplotlib.pyplot as plt

def loadDataSet():
    dataMat = []; labelMat = []
    fr = open("testSet.txt")
    for line in fr:
        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()
    m, n = shape(dataMatrix)
    alpha = 0.001
    maxCycle = 500
    weights = ones((n, 1))
    for k in range(maxCycle):
        h = sigmoid(dataMatrix * weights)
        error = labelMat - h
        weights = weights + alpha*dataMatrix.transpose()*error
    return weights


def plotBestFit(wei):
    weights = wei.getA()
    dataMat, labelMat = loadDataSet()
    dataArr = array(dataMat)
    n = shape(dataMat)[0]
    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()

dataArr, labelMat = loadDataSet()
weight = gradAscent(dataArr, labelMat)
print(weight)
plotBestFit(weight)

代码中将数据添加第一列变量全部为1，是因为考虑到分割方程的0次项目，因为假设的分割线方程为

所以当时具有函数接近1，否则接近0

所以我们最终求出来的weigt的三个变量对应的是 w_0,w_1,w_2,并且x_2是y轴坐标，所以分割线方程就是源码中的形式了,如图是分类的结果(批量梯度上升300次)：

Figure_1.png

1.2随机梯度上升

def stocGraAscent0(dataMatrix, classLabels):
    dataMatrix = array(dataMatrix)
    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*dataMatrix[i]*error
    return weights

随机梯度上升的记过如图所示，可能没有批量梯度上升的好，但是由于样本数目少，所以体现不出优越性(有三分之一的样本出现问题)，但是当样本数目变大时，随机梯度上升的运算量和速度就明显了。

Figure_2.png

1.3改进随机梯度上升

随机梯度上升的步长固定且随机性没有得到充分发挥，因此有一定的局限性，导致权重值会在一定的范围内徘徊，所以可以对随机梯度上升的方法进行改进，自适应步长,下图3是改进梯度上升法迭代20次的结果，因此可以充分体现优越性

def stocGraAscent1(dataMatrix, classLabels, numIter=150):
    dataMatrix = array(dataMatrix)
    m, n = shape(dataMatrix)
    weights = ones(n)
    for j in range(numIter):
        dataIndex = range(m)
        for i in range(m):
            alpha = 4/(1.0+j+i)+0.01
            ranIndex = int(random.uniform(0, len(dataIndex)))
            h = sigmoid(sum(dataMatrix[ranIndex]*weights))
            error = classLabels[ranIndex] - h
            weights = weights + alpha*error*dataMatrix[ranIndex]
            # del(dataIndex[ranIndex])
    return weights

Figure_3.png

2.1预测病马的死亡率

def classifyVector(inX, weights):
    prob = sigmoid(sum(inX*weights))
    if prob > 0.5: return 1.0
    else: return 0.0


def colicTest():
    frTrain = open('horseColicTraining.txt')
    frTest = open('horseColicTest.txt')
    trainingSet = []; trainingLabels = []
    for line in frTrain:
        currLine = line.strip().split('\t')
        lineArr = []
        for i in range(21):
            lineArr.append(float(currLine[i]))
        trainingSet.append(lineArr)
        trainingLabels.append(float(currLine[21]))
    trainWeights = stocGraAscent1(trainingSet, trainingLabels, 500)
    errorCount = 0; numTestVec = 0.0
    for line in frTest:
        numTestVec += 1.0
        currLine = line.strip().split()
        lineArr = []
        for i in range(21):
            lineArr.append(float(currLine[i]))
        if int(classifyVector(array(lineArr), trainWeights)) !=int(int(currLine[21])):
            errorCount += 1
    errorRate = (float(errorCount)/numTestVec)
    print("The error rate of this test is: %f" % errorRate)
    return errorRate


def multiTest():
    numTests = 10; errorSum = 0.0
    for k in range(numTests):
        errorSum += colicTest()
    print("After %d iterration the average error rate is: %f" % (numTests, errorSum/float(numTests)))

最终预测结果为

The error rate of this test is: 0.373134
The error rate of this test is: 0.328358
The error rate of this test is: 0.253731
The error rate of this test is: 0.343284
The error rate of this test is: 0.417910
The error rate of this test is: 0.313433
The error rate of this test is: 0.507463
The error rate of this test is: 0.298507
The error rate of this test is: 0.223881
The error rate of this test is: 0.522388
After 10 iterration the average error rate is: 0.358209

Process finished with exit code 0

由于样本中信息缺失达到百分之30，所以结果还算可以接受，最低可以调整到百分之20左右，通过调整迭代次数。