机器学习（logistics回归）

这篇中我们将首次接触到最优化算法。

logistics回归进行分类的主要思想是：根据现有数据对分类边界线建立回归公式，以此进行分类。这个分类的边界就是我们所求的回归函数。

回归一词源于最佳拟合，表示要找到最佳拟合参数，使用的是最优化算法。回归函数就是确定最佳回归参数，然后对不同的特征赋予不同的权重

优点：计算代价不高，易于理解和实现

缺点：容易欠拟合，分类精度不高

适用用标称型与数值型数据

算法基础

所采用的的映射函数是Sigmoid函数，Sigmoid函数比0-1函数好的一点是在局部上看是平滑的，而整体上看是近似跳跃的，而0-1函数本身是跳跃的，这个瞬间跳跃过程很难处理，不够平滑，误差较大

为了实现logistics回归分类器，我们可以在每一个特征上都乘以一个回归系数，然后把所有结果的值相加，将这个总和带入Sigmoid函数，结果比0.5大就分入1类，比0.5小就分入0类，因此该分类方法也是一种概率估计

最佳回归系数的确定方法

1.梯度上升法，该法是用来求函数最大值的，常说的梯度下降法是用来求函数最小值

2.所谓的梯度其实是数学意义中的导数，也是数据变化最大的方向，一般用倒三角符号来表示梯度

3.公式为 w= w+ a.tidu(f(w)),其中a是步长，该公式会一直迭代到某一个值，或者达到误差允许的范围

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 = 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))