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支持向量机(未完)

开源分享 2022-01-04 阅读 60

系列文章目录

文章目录


提示:参考书籍图灵程序设计丛书《机器学习实战》

一、寻找最大间隔

在这里插入图片描述

二、SMO高效优化算法

1.应用简化版SMO算法处理小规模数据集

from numpy import *
from time import sleep

#SMO算法中的辅助函数
def loadDataSet(fileName):  # 得到类标签和数据矩阵
    dataMat = [];
    labelMat = []
    fr = open(fileName)
    for line in fr.readlines():
        lineArr = line.strip().split('\t')
        dataMat.append([float(lineArr[0]), float(lineArr[1])])
        labelMat.append(float(lineArr[2]))
    return dataMat, labelMat

def selectJrand(i, m):  # i是第一个alpha的下标,m是所有alpha的数量,确保i和j是不一样的
    j = i  # we want to select any J not equal to i
    while (j == i):
        j = int(random.uniform(0, m))
    return j

def clipAlpha(aj, H, L):  # 用于调整alpha的值
    if aj > H:
        aj = H
    if L > aj:
        aj = L
    return aj

#简化版SMO算法
def smoSimple(dataMatIn, classLabels, C, toler, maxIter):  # SMO的一个有效的简化版本
    dataMatrix = mat(dataMatIn);
    labelMat = mat(classLabels).transpose()
    b = 0;
    m, n = shape(dataMatrix)
    alphas = mat(zeros((m, 1)))
    iter = 0
    while (iter < maxIter):
        alphaPairsChanged = 0
        for i in range(m):
            fXi = float(multiply(alphas, labelMat).T * (dataMatrix * dataMatrix[i, :].T)) + b  # 预测类别
            Ei = fXi - float(labelMat[i])  # if checks if an example violates KKT condition 计算误差
            if ((labelMat[i] * Ei < -toler) and (alphas[i] < C)) or ((labelMat[i] * Ei > toler) and (alphas[i] > 0)):
                j = selectJrand(i, m)
                fXj = float(multiply(alphas, labelMat).T * (dataMatrix * dataMatrix[j, :].T)) + b
                Ej = fXj - float(labelMat[j])
                alphaIold = alphas[i].copy();
                alphaJold = alphas[j].copy();
                if (labelMat[i] != labelMat[j]):
                    L = max(0, alphas[j] - alphas[i])
                    H = min(C, C + alphas[j] - alphas[i])
                else:
                    L = max(0, alphas[j] + alphas[i] - C)
                    H = min(C, alphas[j] + alphas[i])
                if L == H: print("L==H");
                continue
                eta = 2.0 * dataMatrix[i, :] * dataMatrix[j, :].T -\
                      dataMatrix[i, :] * dataMatrix[i, :].T - \
                      dataMatrix[j, :] * dataMatrix[j, :].T  # alpha的最优修改量
                if eta >= 0: print ("eta>=0");
                continue
                alphas[j] -= labelMat[j] * (Ei - Ej) / eta
                alphas[j] = clipAlpha(alphas[j], H, L)
                if (abs(alphas[j] - alphaJold) < 0.00001):
                    print ("j not moving enough");
                continue
                alphas[i] += labelMat[j] * labelMat[i] * (alphaJold - alphas[j])  # update i by the same amount as j
                # the update is in the oppostie direction
                b1 = b - Ei - labelMat[i] * (alphas[i] - alphaIold) * dataMatrix[i, :] * dataMatrix[i, :].T - labelMat[
                    j] * (alphas[j] - alphaJold) * dataMatrix[i, :] * dataMatrix[j, :].T
                b2 = b - Ej - labelMat[i] * (alphas[i] - alphaIold) * dataMatrix[i, :] * dataMatrix[j, :].T - labelMat[
                    j] * (alphas[j] - alphaJold) * dataMatrix[j, :] * dataMatrix[j, :].T
                if (0 < alphas[i]) and (C > alphas[i]):
                    b = b1
                elif (0 < alphas[j]) and (C > alphas[j]):
                    b = b2
                else:
                    b = (b1 + b2) / 2.0
                alphaPairsChanged += 1
                print ("iter: %d i:%d, pairs changed %d" % (iter, i, alphaPairsChanged))
        if (alphaPairsChanged == 0):
            iter += 1
        else:
            iter = 0
        print("iteration number: %d" % iter)
    return b, alphas
    
if __name__ == '__main__':
    dataArr, labelArr = loadDataSet('testSet.txt')
     # print labelArr
    b, alphas = smoSimple(dataArr, labelArr, 0.6, 0.001, 40)
    print(b)
    print(alphas[alphas > 0])
    print(shape(alphas[alphas > 0]))
    for i in range(100):
        if alphas[i] > 0.0:
            print (dataArr[i], labelArr[i])

测试结果
在这里插入图片描述

2.利用完整Platt SMO算法加快优化

#完整版Platt SMO的支持函数
class optStruct:  # 建立一个数据结构
    def __init__(self, dataMatIn, classLabels, C, toler, kTup):  # Initialize the structure with the parameters
        self.X = dataMatIn
        self.labelMat = classLabels
        self.C = C
        self.tol = toler
        self.m = shape(dataMatIn)[0]
        self.alphas = mat(zeros((self.m, 1)))
        self.b = 0
        self.eCache = mat(zeros((self.m, 2)))  # first column is valid flag
        self.K = mat(zeros((self.m, self.m)))
        for i in range(self.m):
            self.K[:, i] = kernelTrans(self.X, self.X[i, :], kTup)

#误差缓存
def calcEk(oS, k):  # 计算E值
    fXk = float(multiply(oS.alphas, oS.labelMat).T * oS.K[:, k] + oS.b)
    Ek = fXk - float(oS.labelMat[k])
    return Ek

#内循环的启发式方法
def selectJ(i, oS, Ei):  # this is the second choice -heurstic, and calcs Ej
    maxK = -1;
    maxDeltaE = 0;
    Ej = 0
    oS.eCache[i] = [1, Ei]  # set valid #choose the alpha that gives the maximum delta E
    validEcacheList = nonzero(oS.eCache[:, 0].A)[0]
    if (len(validEcacheList)) > 1:
        for k in validEcacheList:  # loop through valid Ecache values and find the one that maximizes delta E
            if k == i: continue  # don't calc for i, waste of time
            Ek = calcEk(oS, k)
            deltaE = abs(Ei - Ek)
            if (deltaE > maxDeltaE):    #选择具有最大步长的j
                maxK = k;
                maxDeltaE = deltaE;
                Ej = Ek
        return maxK, Ej
    else:  # in this case (first time around) we don't have any valid eCache values
        j = selectJrand(i, oS.m)
        Ej = calcEk(oS, j)
    return j, Ej


def updateEk(oS, k):  # after any alpha has changed update the new value in the cache
    Ek = calcEk(oS, k)
    oS.eCache[k] = [1, Ek]

def innerL(i, oS):  # 完整SMO算法中的优化例程
    Ei = calcEk(oS, i)
    if ((oS.labelMat[i] * Ei < -oS.tol) and (oS.alphas[i] < oS.C)) or (
            (oS.labelMat[i] * Ei > oS.tol) and (oS.alphas[i] > 0)):
        j, Ej = selectJ(i, oS, Ei)  # this has been changed from selectJrand 第二个alpha选择中的启发式方法
        alphaIold = oS.alphas[i].copy();
        alphaJold = oS.alphas[j].copy();
        if (oS.labelMat[i] != oS.labelMat[j]):
            L = max(0, oS.alphas[j] - oS.alphas[i])
            H = min(oS.C, oS.C + oS.alphas[j] - oS.alphas[i])
        else:
            L = max(0, oS.alphas[j] + oS.alphas[i] - oS.C)
            H = min(oS.C, oS.alphas[j] + oS.alphas[i])
        if L == H:
            print("L==H");
        return 0
        eta = 2.0 * oS.K[i, j] - oS.K[i, i] - oS.K[j, j]  # changed for kernel
        if eta >= 0:
            print("eta>=0");
        return 0
        oS.alphas[j] -= oS.labelMat[j] * (Ei - Ej) / eta
        oS.alphas[j] = clipAlpha(oS.alphas[j], H, L)
        updateEk(oS, j)  # added this for the Ecache
        if (abs(oS.alphas[j] - alphaJold) < 0.00001):
            print("j not moving enough");
        return 0
        oS.alphas[i] += oS.labelMat[j] * oS.labelMat[i] * (alphaJold - oS.alphas[j])  # update i by the same amount as j
        updateEk(oS, i)  # added this for the Ecache                    #the update is in the oppostie direction
        b1 = oS.b - Ei - oS.labelMat[i] * (oS.alphas[i] - alphaIold) * oS.K[i, i] - oS.labelMat[j] * (
                    oS.alphas[j] - alphaJold) * oS.K[i, j]
        b2 = oS.b - Ej - oS.labelMat[i] * (oS.alphas[i] - alphaIold) * oS.K[i, j] - oS.labelMat[j] * (
                    oS.alphas[j] - alphaJold) * oS.K[j, j]
        if (0 < oS.alphas[i]) and (oS.C > oS.alphas[i]):
            oS.b = b1
        elif (0 < oS.alphas[j]) and (oS.C > oS.alphas[j]):
            oS.b = b2
        else:
            oS.b = (b1 + b2) / 2.0
        return 1
    else:
        return 0


#完整版Platt SMO的外循环代码
def smoP(dataMatIn, classLabels, C, toler, maxIter, kTup=('lin', 0)):  # full Platt SMO
    oS = optStruct(mat(dataMatIn), mat(classLabels).transpose(), C, toler, kTup)
    iter = 0
    entireSet = True;
    alphaPairsChanged = 0
    while (iter < maxIter) and ((alphaPairsChanged > 0) or (entireSet)):
        alphaPairsChanged = 0
        if entireSet:  # go over all
            for i in range(oS.m):
                alphaPairsChanged += innerL(i, oS)
                print("fullSet, iter: %d i:%d, pairs changed %d" % (iter, i, alphaPairsChanged))
            iter += 1
        else:  # go over non-bound (railed) alphas
            nonBoundIs = nonzero((oS.alphas.A > 0) * (oS.alphas.A < C))[0]
            for i in nonBoundIs:
                alphaPairsChanged += innerL(i, oS)
                print( "non-bound, iter: %d i:%d, pairs changed %d" % (iter, i, alphaPairsChanged))
            iter += 1
        if entireSet:
            entireSet = False  # toggle entire set loop
        elif (alphaPairsChanged == 0):
            entireSet = True
        print("iteration number: %d" % iter)
    return oS.b, oS.alphas

def calcWs(alphas,dataArr,classLabels):  #根据alphas计算W
    X = mat(dataArr); labelMat = mat(classLabels).transpose()
    m,n = shape(X)
    w = zeros((n,1))
    for i in range(m):
        w += multiply(alphas[i]*labelMat[i],X[i,:].T)
    return w

if __name__ == '__main__':
 dataArr, labelArr = loadDataSet('testSet.txt')
  #测试完整版的smo算法
    b, alphas = smoP(dataArr, labelArr, 0.6, 0.001, 40)
    print (b)
    print (alphas[alphas>0])
    print (shape(alphas[alphas>0]))
    for i in range(100):
        if alphas[i]>0.0:
            print (dataArr[i], labelArr[i])

测试结果
在这里插入图片描述

2.在复杂数据上应用核函数


#核转换函数
def kernelTrans(X, A, kTup): #calc the kernel or transform data to a higher dimensional space
    m,n = shape(X)
    K = mat(zeros((m,1)))
    if kTup[0]=='lin': K = X * A.T   #linear kernel
    elif kTup[0]=='rbf':
        for j in range(m):
            deltaRow = X[j,:] - A
            K[j] = deltaRow*deltaRow.T
        K = exp(K/(-1*kTup[1]**2)) #divide in NumPy is element-wise not matrix like Matlab
    else: raise NameError('Houston We Have a Problem -- \
    That Kernel is not recognized')
    return K

#完整版Platt SMO的支持函数
class optStruct:  # 建立一个数据结构
    def __init__(self, dataMatIn, classLabels, C, toler, kTup):  # Initialize the structure with the parameters
        self.X = dataMatIn
        self.labelMat = classLabels
        self.C = C
        self.tol = toler
        self.m = shape(dataMatIn)[0]
        self.alphas = mat(zeros((self.m, 1)))
        self.b = 0
        self.eCache = mat(zeros((self.m, 2)))  # first column is valid flag
        self.K = mat(zeros((self.m, self.m)))
        for i in range(self.m):
            self.K[:, i] = kernelTrans(self.X, self.X[i, :], kTup)

2.在测试中使用核函数

#利用核函数进行分类的径向基测试函数
def testRbf(k1=1.3):    #在测试中使用核函数
    dataArr,labelArr = loadDataSet('testSetRBF.txt')
    b,alphas = smoP(dataArr, labelArr, 200, 0.0001, 10000, ('rbf', k1)) #C=200 important
    datMat=mat(dataArr); labelMat = mat(labelArr).transpose()
    svInd=nonzero(alphas.A>0)[0]
    sVs=datMat[svInd] #get matrix of only support vectors
    labelSV = labelMat[svInd];
    print ("there are %d Support Vectors" % shape(sVs)[0])
    m,n = shape(datMat)
    errorCount = 0
    for i in range(m):
        kernelEval = kernelTrans(sVs,datMat[i,:],('rbf', k1))
        predict=kernelEval.T * multiply(labelSV,alphas[svInd]) + b
        if sign(predict)!=sign(labelArr[i]): errorCount += 1
    print ("the training error rate is: %f" % (float(errorCount)/m))
    dataArr,labelArr = loadDataSet('testSetRBF2.txt')
    errorCount = 0
    datMat=mat(dataArr); labelMat = mat(labelArr).transpose()
    m,n = shape(datMat)
    for i in range(m):
        kernelEval = kernelTrans(sVs,datMat[i,:],('rbf', k1))
        predict=kernelEval.T * multiply(labelSV,alphas[svInd]) + b
        if sign(predict)!=sign(labelArr[i]): errorCount += 1
    print ("the test error rate is: %f" % (float(errorCount)/m))
    
if __name__ == '__main__':
 testRbf()

测试结果
在这里插入图片描述

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