一、应用简化版SMO算法来处理小规模数据集

该SMO函数的伪代码：

创建一个alpha向量并将其初始化为O 向量
当迭代次数小于最大迭代次数时（外循环）
对数据集中的每个数据向量（内循环）：

如果该数据向量可以被优化：
随机选择另外一个数据向量
同时优化这两个向量
如果两个向量都不能被优化，退出内循环

如果所有向量都没被优化，增加迭代数目，继续下一次循环

Python源码：

from numpy import *

def 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,labelMat

def selectJrand(i,m): #选择两个不同的alpha值，如果一个选为第alpha[i]，则另一个alpha值选择除了i随机的一个
    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): #由于aj的取值范围的限制，H为上限，L为下限
    if aj > H: 
        aj = H
    if L > aj:
        aj = L
    return aj
'''
功能：简单版的SMO
输入参数：
dataMatIn:数据集
classLabels:类别标签
C:控制参数（惩罚参数）
toler:容错率
maxIter:最大的循环次数
输出参数：
b：f(x)中的b值
alpha:拉格朗日乘子
'''
def smoSimple(dataMatIn, classLabels, C, toler, maxIter):
    dataMatrix = mat(dataMatIn); labelMat = mat(classLabels).transpose()
    b = 0; m,n = shape(dataMatrix) #将多个列表和输人参数转换成为numpy矩阵，这样就可以简化很多数学处理操作
    alphas = mat(zeros((m,1)))
    iter = 0 #初始化遍历次数 
    while (iter < maxIter): #只有在所有数据集上遍历maxIter次，且不再发生任何alpha修改之后，程序才会停止并退出while循环
        alphaPairsChanged = 0                       #记录alpha是否巳经进行优化
        for i in range(m):
            fXi = float(multiply(alphas,labelMat).T*(dataMatrix*dataMatrix[i,:].T)) + b #预测的类别
            Ei = fXi - float(labelMat[i])           #Ei为计算误差；if checks if an example violates KKT conditions
            #一旦alphas等于0或C，那么它们就巳经在“边界”上了，因而不再能够减小或增大，因此也就不值得再对它们进行优化了
            if ((labelMat[i]*Ei < -toler) and (alphas[i] < C)) or ((labelMat[i]*Ei > toler) and (alphas[i] > 0)):
                j = selectJrand(i,m)                #随机选择第二个alpha值
                fXj = float(multiply(alphas,labelMat).T*(dataMatrix*dataMatrix[j,:].T)) + b
                Ej = fXj - float(labelMat[j])       #计算第二个alpha值的误差
                alphaIold = alphas[i].copy(); alphaJold = alphas[j].copy();
                if (labelMat[i] != labelMat[j]):    #当y1和y2异号，计算alpha的取值范围
                    L = max(0, alphas[j] - alphas[i])
                    H = min(C, C + alphas[j] - alphas[i])
                else:                               #当y1和y2同号，计算alpha的取值范围                       
                    L = max(0, alphas[j] + alphas[i] - C)
                    H = min(C, alphas[j] + alphas[i])
                if L==H: print "i:%d, L==H" %(i); continue
                #eta = K11+K22-2*K12,也是f(x)的二阶导数
                eta = 2.0 * dataMatrix[i,:]*dataMatrix[j,:].T - dataMatrix[i,:]*dataMatrix[i,:].T - dataMatrix[j,:]*dataMatrix[j,:].T
                if eta >= 0: print "i:%d, eta>=0" %(i); continue
                alphas[j] -= labelMat[j]*(Ei - Ej)/eta #利用公式更新alpha[j]
                alphas[j] = clipAlpha(alphas[j],H,L)
                if (abs(alphas[j] - alphaJold) < 0.00001): print "i:%d, j not moving enough" %(i); 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 #把新值b给原来的旧值b，为了后续的循环
                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)
##            else:
##                print "i: %d" %(i)
        if (alphaPairsChanged == 0): iter += 1 #检查alpha值是否做了更新，如果有更新则将iter为0后继续运行程序
        else: iter = 0
        print "iteration number: %d" % iter
    return b,alphas

运行以下命令：

dataArr,labelArr = loadDataSet('testSet.txt')
b,alphas = smoSimple(dataArr,labelArr,0.6,0.001,40)

加上求解w的函数（利用alphas）

def 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

加上画图函数：

def plotfig_SVM(xMat,yMat,ws,b,alphas):
    xMat = mat(xMat)
    yMat = mat(yMat)
    b = array(b)[0] #b原来是矩阵，先转为数组类型后其数组大小为（1,1），所以后面加[0]，变为(1,)
    fig = plt.figure()
    ax = fig.add_subplot(111)
    ax.scatter(xMat[:,0].flatten().A[0],xMat[:,1].flatten().A[0]) #注意flatten的用法
    x = arange(-1.0,10.0,0.1) #x最大值，最小值根据原数据集dataArr[:,0]的大小而定
    y =(-b-ws[0][0]*x)/ws[1][0] #根据x.w + b = 0 得到，其式子展开为w0.x1 + w1.x2 + b = 0,x2就是y值
    ax.plot(x,y)
    for i in range(100): #找到支持向量，并在图中标红
        if alphas[i]>0.0:
            ax.plot(xMat[i,0],xMat[i,1],'ro')
    plt.show()

运行命令：

ws = calcWs(alphas,dataArr,labelArr)
plotfig_SVM(dataArr,labelArr,ws,b,alphas)

画图：（其中红色的点为支持向量）

二、利用完整Platt SMO算法加速优化

Platt SMO 算法是通过一个外循环来选择第一个alpha值的，并且其选择过程会在两种方式之间进行交替：一种方式是在所有数据集上进行单遍扫描，另一种方式则是在非边界alpha中实现单遍扫描。而所谓非边界alpha是指的就是那些不等于边界0或C的alpha值。对整个数据集的扫描相当容易，而实现非边界alpha值的扫描时，首先需要建立这些alpha值的列表，然后再对这个表进行遍历。同时，该步骤会跳过那些已知的不会改变的alpha值。

Python代码：

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

class optStruct: #建立一个数据结构来保存所有的重要值
    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（是否有效的标志位），第二列是实际的E值（误差值）
        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
    #选择第二个alhpa值以保证在每次优化中采用最大步长
    oS.eCache[i] = [1,Ei]  #set valid（将即将使用的Ei值设为有效） #choose the alpha that gives the maximum delta E
    validEcacheList = nonzero(oS.eCache[:,0].A)[0]
    if (len(validEcacheList)) > 1: #选择其中使改变最大的那个E值
        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, 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]
'''
功能：和smoSimple函数一样（包括选择第二个乘子）
'''
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]): #当y1和y2异号，计算alpha[j]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[j]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 = 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
'''
功能：外循环（选择第一个乘子）
'''
def smoP(dataMatIn, classLabels, C, toler, maxIter,kTup=('lin', 0)):    #full Platt SMO
    oS = optStruct(mat(dataMatIn),mat(classLabels).transpose(),C,toler, kTup) #建立一个数据结构来保存所有的重要值
    iter = 0
    entireSet = True; alphaPairsChanged = 0 #alpha改变标志位
    while (iter < maxIter) and ((alphaPairsChanged > 0) or (entireSet)):
        alphaPairsChanged = 0 
        if entireSet:   #go over all 遍历整个数据集
            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

运行以下代码：

dataArr,labelArr = loadDataSet('testSet.txt')
b,alphas = smoP(dataArr,labelArr,0.6,0.001,40)

ws = calcWs(alphas,dataArr,labelArr)
plotfig_SVM(dataArr,labelArr,ws,b,alphas)

画图：

三、在复杂数据上利用核函数

添加函数：

'''
功能：利用核函数进行分类
'''
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] #找到非零的alphas值，从而得到了所需要的支持向量
    sVs=datMat[svInd] #get matrix of only support vectors
    labelSV = labelMat[svInd];#得到了支持向量的类别标签，
    print "there are %d Support Vectors" % shape(sVs)[0]
    m,n = shape(datMat)
    predict_label0_index = []
    predict_label1_index = []
    errorCount = 0
    for i in range(m):
        kernelEval = kernelTrans(sVs,datMat[i,:],('rbf', k1)) #只利用支持向量就可进行分类
        predict = array(kernelEval.T * multiply(labelSV,alphas[svInd]) + b) #计算预测值
        if sign(predict) == -1: #如果预测的标签是-1，则保存一个列表中
            predict_label0_index.append(i)
        elif sign(predict) == 1: #如果预测的标签是-1，则保存另一个列表中
            predict_label1_index.append(i)
        if sign(predict)!=sign(labelArr[i]): errorCount += 1 #利用sign函数，判断预测是否正确
    print "the training error rate is: %f" % (float(errorCount)/m)
    plotfig_kernel(dataArr,predict_label0_index,predict_label1_index,alphas)#画图
    dataArr,labelArr = loadDataSet('testSetRBF2.txt') #再用另一个数据集进行测试，后面代码和前面一样
    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)

其中画图函数换为：

'''
功能：画利用核函数进行分类的图
'''
def plotfig_kernel(xMat,predict_label0_index,predict_label1_index,alphas):
    fig = plt.figure()
    ax = fig.add_subplot(111)
    for i in predict_label0_index: #预测标签为-1的标为红色三角形
        ax.plot(xMat[i][0],xMat[i][1],'r^')
    for i in predict_label1_index: #预测标签为1的标为蓝色正方形
        ax.plot(xMat[i][0],xMat[i][1],'bs')
    plt.show()

运行命令：