5、支持向量机SVM

支持向量机的思想在于最小间隔最大化。这篇文章主要关注SMO算法的实现，和核函数的使用问题。

SMO算法的工作原理是每次循环中选择两个alpha值进行优化处理。这两个alpha满足一定的条件：

1、两个alpha必须要在间隔边界之外

2、两个alpha还没经过区间化处理或不在边界上

首先看一个简化版的SMO算法函数。

from numpy import *from time import sleepdef loadDataSet(fileName):    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,labelMatdef selectJrand(i,m):#用来随机选择另一个alpha    j=i #we want to select any J not equal to i    while (j==i):        j = int(random.uniform(0,m))    return jdef clipAlpha(aj,H,L):#用来对alpha进行区间剪辑    if aj > H:         aj = H    if L > aj:        aj = L    return ajdef smoSimple(dataMatIn, classLabels, C, toler, maxIter):    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 #计算一个分类结果            #Ei代表误差率            Ei = fXi - float(labelMat[i])#if checks if an example violates KKT conditions            if ((labelMat[i]*Ei < -toler) and (alphas[i] < C)) or ((labelMat[i]*Ei > toler) and (alphas[i] > 0)):                #alpha不满足KKT条件时                j = selectJrand(i,m)#随机选择j                fXj = float(multiply(alphas,labelMat).T*(dataMatrix*dataMatrix[j,:].T)) + b#计算预测结果和误差率                Ej = fXj - float(labelMat[j])                alphaIold = alphas[i].copy(); alphaJold = alphas[j].copy();                if (labelMat[i] != labelMat[j]):#计算两个alpha的下限和上限                    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#计算最优alpha                if eta >= 0: print "eta>=0"; continue                alphas[j] -= labelMat[j]*(Ei - Ej)/eta#这里得到的alphaj是没经过剪辑的                alphas[j] = clipAlpha(alphas[j],H,L)#剪辑                if (abs(alphas[j] - alphaJold) < 0.00001): print "j not moving enough"; continue#改变不明显                alphas[i] += labelMat[j]*labelMat[i]*(alphaJold - alphas[j])#update i by the same amount as j                                                                        #the update is in the oppostie direction由alphaj计算alphai                b1 = b - Ei- labelMat[i]*(alphas[i]-alphaIold)*dataMatrix[i,:]*dataMatrix[i,:].T - labelMat[j]*(alphas[j]-alphaJold)*dataMatrix[i,:]*dataMatrix[j,:].T#最后计算两个更新的b                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#本次迭代中，更新了alpha                print "iter: %d i:%d, pairs changed %d" % (iter,i,alphaPairsChanged)        if (alphaPairsChanged == 0): iter += 1#如果没更新数据，iter+1，iter达到一定值代表收敛        else: iter = 0        print "iteration number: %d" % iter    return b,alphas

算法说明见注释。

接下来放一个完整版的Platt SMO

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):    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):         #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):                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, Ejdef 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):    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        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 0def 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.alphasdef calcWs(alphas,dataArr,classLabels):    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

接下来看一下核函数的使用，以径向基函数为例。

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 Kclass 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)

当使用线性模型时，kernelTrans返回的为矩阵和向量的内积，当使用径向基函数时，使用的就是采用核函数的变换形式。