机器学习——支持向量机(SVM)之核函数(kernel)

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以下是有核函数的版本
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class optStruct:
    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)
        
def calcEk(oS, k):			#计算误差
    fXk = float(multiply(oS.alphas,oS.labelMat).T*oS.K[:,k] + oS.b)
    Ek = fXk - float(oS.labelMat[k])
    return Ek
        
def selectJ(i, oS, Ei):         	#用于选择第2个循环（内循环）的alpha值，内循环中的启发式方法
    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 	#跳过本身
            Ek = calcEk(oS, k)
            deltaE = abs(Ei - Ek)
            if (deltaE > maxDeltaE):	#选取具有最大步长的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):			#alpha改变后，更新缓存
    Ek = calcEk(oS, k)
    oS.eCache[k] = [1,Ek]
        
#内部循环的代码和简版的SMO代码很相似
def innerL(i, oS):		
    Ei = calcEk(oS, i)
    #判断每一个alpha是否被优化过，如果误差很大，就对该alpha值进行优化，toler是容错率
    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) 			#使用启发式方法选取第2个alpha，选取使得误差最大的alpha
        alphaIold = oS.alphas[i].copy(); alphaJold = oS.alphas[j].copy();
        #保证alpha在0与C之间
        if (oS.labelMat[i] != oS.labelMat[j]):		#当y1和y2异号，计算alpha的取值范围 
            L = max(0, oS.alphas[j] - oS.alphas[i])
            H = min(oS.C, oS.C + oS.alphas[j] - oS.alphas[i])
        else:						#当y1和y2同号，计算alpha的取值范围
            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是alpha[j]的最优修改量，eta=K11+K22-2*K12,也是f(x)的二阶导数，K表示核函数
        eta = 2.0 * oS.K[i,j] - oS.K[i,i] - oS.K[j,j] 	#changed for kernel
        #如果二阶导数-eta <= 0，说明一阶导数没有最小值，就不做任何改变，本次循环结束直接运行下一次for循环
        if eta >= 0: print "eta>=0"; return 0
        oS.alphas[j] -= oS.labelMat[j]*(Ei - Ej)/eta	#利用公式更新alpha[j]，alpha2new=alpha2-yj(Ei-Ej)/eta
        oS.alphas[j] = clipAlpha(oS.alphas[j],H,L)	#判断alpha的范围是否在0和C之间
        updateEk(oS, j) 				#在alpha改变的时候更新Ecache
        print "j=",j
        print oS.alphas.A[j]
        #如果alphas[j]没有调整，就忽略下面语句，本次循环结束直接运行下一次for循环
        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) 				#在alpha改变的时候更新Ecache                   
        print "i=",i
        print oS.alphas.A[i]     
        #已经计算出了alpha，接下来根据模型的公式计算b   
        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]
        #根据公式确定偏移量b，理论上可选取任意支持向量来求解，但是现实任务中通常使用所有支持向量求解的平均值，这样更加鲁棒
        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		#如果有任意一对alpha发生改变，返回1
    else: return 0

#完整版Platt SMO的外循环
def smoP(dataMatIn, classLabels, C, toler, maxIter,kTup=(‘lin‘, 0)):    
    oS = optStruct(mat(dataMatIn),mat(classLabels).transpose(),C,toler, kTup)
    iter = 0
    entireSet = True; alphaPairsChanged = 0
    while (iter < maxIter) and ((alphaPairsChanged > 0) or (entireSet)):	#有alpha改变同时遍历次数小于最大次数，或者需要遍历整个集合
        alphaPairsChanged = 0
        #首先进行完整遍历，过程和简化版的SMO一样
        if entireSet:   	
            for i in range(oS.m):        
                alphaPairsChanged += innerL(i,oS)		#i是第1个alpha的下标
                print "完整遍历, 迭代次数: %d i:%d, 成对改变的次数 %d" % (iter,i,alphaPairsChanged)
            iter += 1
        #非边界遍历，挑选其中alpha值在0和C之间非边界alpha进行优化 
        else:
            nonBoundIs = nonzero((oS.alphas.A > 0) * (oS.alphas.A < C))[0]	#然后挑选其中值在0和C之间的非边界alpha进行遍历
            for i in nonBoundIs:
                alphaPairsChanged += innerL(i,oS)
                print "非边界, 迭代次数: %d i:%d, 成对改变的次数 %d" % (iter,i,alphaPairsChanged)
            iter += 1
        #如果这次是完整遍历的话，下次不用进行完整遍历
        if entireSet: entireSet = False 			#终止完整循环
        elif (alphaPairsChanged == 0): entireSet = True  	#如果alpha的改变数量为0的话，再次遍历所有的集合一次
        print "iteration number: %d" % iter
    return oS.b,oS.alphas

def calcWs(alphas,dataArr,classLabels):		#计算模型的参数w，即alpha*y*x转置的累加
    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

def testRbf(k1=1.3):
    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)    
    
def img2vector(filename):
    returnVect = zeros((1,1024))
    fr = open(filename)
    for i in range(32):
        lineStr = fr.readline()
        for j in range(32):
            returnVect[0,32*i+j] = int(lineStr[j])
    return returnVect

def loadImages(dirName):
    from os import listdir
    hwLabels = []
    trainingFileList = listdir(dirName)           #load the training set
    m = len(trainingFileList)
    trainingMat = zeros((m,1024))
    for i in range(m):
        fileNameStr = trainingFileList[i]
        fileStr = fileNameStr.split(‘.‘)[0]     #take off .txt
        classNumStr = int(fileStr.split(‘_‘)[0])
        if classNumStr == 9: hwLabels.append(-1)
        else: hwLabels.append(1)
        trainingMat[i,:] = img2vector(‘%s/%s‘ % (dirName, fileNameStr))
    return trainingMat, hwLabels    

def testDigits(kTup=(‘rbf‘, 10)):
    dataArr,labelArr = loadImages(‘trainingDigits‘)
    b,alphas = smoP(dataArr, labelArr, 200, 0.0001, 10000, kTup)
    datMat=mat(dataArr); labelMat = mat(labelArr).transpose()
    svInd=nonzero(alphas.A>0)[0]
    sVs=datMat[svInd] 
    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,:],kTup)
        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 = loadImages(‘testDigits‘)
    errorCount = 0
    datMat=mat(dataArr); labelMat = mat(labelArr).transpose()
    m,n = shape(datMat)
    for i in range(m):
        kernelEval = kernelTrans(sVs,datMat[i,:],kTup)
        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)