Logistic回归及python实现

1、logistics回归原理及sigmoid函数

1.1、sigmoid函数

logistics回归函数是能接受所有的输人然后预测出类别。例如，在两个类的清况下，logistics回归函数输出0或1。或许你之前接触过具有这种性质的函数，该函数称为海维寡德阶跃函数 (Heaviside step function), 或者直接称为单位阶跃函数。然而，海维寒德阶跃函数的问题在于：该函数在跳跃点上从0瞬间跳跃到I,这个瞬间跳跃过程有时很难处理。幸好，另一个函数也有类似的性质。，且数学上更易处理，这就是Sigmoid函数。Sigmoid函数具体的计算公式如下：这里写图片描述，不难发现，sigmoid函数有很好的性质：当x为0时，Sigmoid函数值为0.5。随着x的增大，对应的Sigmoid值将逼近于1; 而随着x的减小，Sigmoid值将逼近于0。如果横坐标
刻度足够大，Sigmoid函数看起来很像一个阶跃函数。

1.2、logistics 回归

知道了sigmoid函数，现在介绍Logistic回归分类器：我们可以在每个特征上都乘以一个回归系数，然后把所有的结果值相加，将这个总和代人Sigmoid函数中，进而得到一个范围在0…..1之间的数值。任何大于0.5的数据被分入1类，小于0.5 即被归入0类。所以， Logistic回归也可以被看成是一种概率估计。

2、logistics回归到实现和解释

2.1、纯python实现logistics回归

'''
Created on Oct 27, 2010
Logistic Regression Working Module
@author: Peter
'''
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)  # convert to NumPy matrix
    labelMat = mat(classLabels).transpose()  # convert to NumPy matrix
    m, n = shape(dataMatrix)
    alpha = 0.001
    maxCycles = 500
    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


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]
    ax.plot(x, y)
    plt.xlabel('X1');
    plt.ylabel('X2');
    plt.show()


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


def stocGradAscent1(dataMatrix, classLabels, numIter=150):
    m, n = shape(dataMatrix)
    weights = ones(n)  # initialize to all ones
    for j in range(numIter):
        dataIndex = list(range(m))
        for i in range(m):
            alpha = 4 / (1.0 + j + i) + 0.0001  # apha decreases with iteration, does not
            randIndex = int(random.uniform(0, len(dataIndex)))  # go to 0 because of the constant
            h = sigmoid(sum(dataMatrix[randIndex] * weights))
            error = classLabels[randIndex] - h
            weights = weights + alpha * error * dataMatrix[randIndex]
            del (dataIndex[randIndex])
    return weights


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.readlines():
        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 = stocGradAscent1(array(trainingSet), trainingLabels, 1000)
    errorCount = 0;
    numTestVec = 0.0
    for line in frTest.readlines():
        numTestVec += 1.0
        currLine = line.strip().split('\t')
        lineArr = []
        for i in range(21):
            lineArr.append(float(currLine[i]))
        if int(classifyVector(array(lineArr), trainWeights)) != 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 iterations the average error rate is: %f" % (numTests, errorSum / float(numTests)))

multiTest()

注：本篇博客参考自机器学习实战

logistics regression