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此篇文章主要介绍逻辑回归的推导和实现。

LR推导

LR推导

上面的推导可以分为几个步骤：
1.给出映射函数。
2.写出单样本，多样本的似然函数，然后给出-log似然函数。
3.使用梯度上升法进行求解参数更新。

最后的参数迭代式子中，i表示当前为第几个样本，j表示当前样本的第几个特征。参数迭代式子就是当前参数加上学习率(当前标签值-当前样本拟合值）当前样本i的第j个特征值。

LR实现

from numpy import *
#—————————————————————数据预处理过程——————————————————————#
def loadDataSet():
    dataMat = []; labelMat = []
    fr = open('./Ch05/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

#————————————————————sigmoid函数定义———————————————————————#
def sigmoid(inX):
    return 1.0/(1+exp(-inX))

#———————————————————梯度上升法实现函数————————————————————-——#
def gradAscent(dataMatIn, classLabels):
    dataMatrix = mat(dataMatIn)             #convert to NumPy matrix
    labelMat = mat(classLabels).transpose() #convert to NumPy matrix
    m,n = shape(dataMatrix)
    alpha = 0.001            #学习率，步长
    maxCycles = 500          #迭代次数
    weights = ones((n,1))
    for k in range(maxCycles):              #heavy on matrix operations
        h = sigmoid(dataMatrix*weights)     #matrix mult
        error = (labelMat - h)              #vector subtraction
        weights = weights + alpha * dataMatrix.transpose()* error #matrix mult
    return weights

#——————————————————画出数据集和决策直线————————————————————————#
def plotBestFit(weights):
    import matplotlib.pyplot as plt
    dataMat,labelMat=loadDataSet()
    dataArr = array(dataMat)
    n = shape(dataArr)[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]    #z=0,则simod=0.5,是分类点
    ax.plot(x, y)
    plt.xlabel('X1'); plt.ylabel('X2');
    plt.show()
#——————————————————————随机梯度上升法实现函数——————————————————#
def stocGradAscent0(dataMatrix, classLabels):
    m,n = shape(dataMatrix)
    alpha = 0.01
    weights = ones(n)   #initialize to all ones
    for i in range(m):
        h = sigmoid(sum(dataMatrix[i]*weights))
        error = classLabels[i] - h
        weights = weights + alpha * error * dataMatrix[i]
    return weights

#———————————————————改进的随机梯度上升法实现函数——————————————————#
def stocGradAscent1(dataMatrix, classLabels, numIter=150):  #迭代次数默认为150次
    m,n = shape(dataMatrix)
    weights = ones(n)   #initialize to all ones
    for j in range(numIter):
        dataIndex = range(m)
        for i in range(m):
            alpha = 4/(1.0+j+i)+0.0001    #apha 随着迭代次数逐渐减小，
            randIndex = int(random.uniform(0,len(dataIndex)))   #随机选取样本来跟新回归系数，这样能减少周期性波动
            h = sigmoid(sum(dataMatrix[randIndex]*weights))
            error = classLabels[randIndex] - h
            weights = weights + alpha * error * dataMatrix[randIndex]
            del(list(dataIndex)[randIndex])
    return weights

#————————————————————回归结果输出函数——————————————————————#
def classifyVector(inX, weights):
    prob = sigmoid(sum(inX*weights))
    if prob > 0.8: return 1.0
    else: return 0.0

#—————————————————————测试函数————————————————————————#
def colicTest():
    frTrain = open(r'E:\Data1\train.txt')
    frTest = open(r'E:\Data1\test.txt')
    trainingSet = []; trainingLabels = []
    for line in frTrain.readlines():
        currLine = line.strip().split('\t')
        lineArr =[]
        for i in range(13):
            lineArr.append(int(currLine[i+3]))
        trainingSet.append(lineArr)
        trainingLabels.append(float(currLine[16]))
    trainWeights = stocGradAscent1(array(trainingSet), trainingLabels, 10)    #traingSet必须由列表变为数组
    errorCount = 0; numTestVec = 0.0
    for line in frTest.readlines():
        numTestVec += 1.0         #测试次数记录
        currLine = line.strip().split('\t')
        user_id = int(currLine[0]);item_id = int(currLine[2])
        lineArr =[]
        for i in range(13):
            lineArr.append(int(currLine[i+3]))
        a = int(classifyVector(array(lineArr), trainWeights))
        print(user_id,item_id,a)
        #if int(classifyVector(array(lineArr), trainWeights))!= int(currLine[16]):
        #   errorCount += 1
    #errorRate = (float(errorCount)/numTestVec)
    #print("the error rate of this test is: %f" % errorRate)
    #return errorRate

#————————————————————调用10次coliTest函数并求结果的平均值——————————————————#
def multiTest():
    numTests = 10; errorSum=0.0
    for k in range(numTests):
        errorSum += colicTest()
    print("after %d iterations the average error rate is: %f" % (numTests, errorSum/float(numTests)))

完整代码见git