一、應用簡化版SMO算法來處理小規模數據集

該SMO函數的僞代碼：

創建一個alpha向量並將其初始化爲O 向量
當迭代次數小於最大迭代次數時（外循環）
對數據集中的每個數據向量（內循環）：

如果該數據向量可以被優化：
隨機選擇另外一個數據向量
同時優化這兩個向量
如果兩個向量都不能被優化，退出內循環

如果所有向量都沒被優化，增加迭代數目，繼續下一次循環

Python源碼：

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): #選擇兩個不同的alpha值，如果一個選爲第alpha[i]，則另一個alpha值選擇除了i隨機的一個
    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): #由於aj的取值範圍的限制，H爲上限，L爲下限
    if aj > H: 
        aj = H
    if L > aj:
        aj = L
    return aj
'''
功能：簡單版的SMO
輸入參數：
dataMatIn:數據集
classLabels:類別標籤
C:控制參數（懲罰參數）
toler:容錯率
maxIter:最大的循環次數
輸出參數：
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) #將多個列表和輸人蔘數轉換成爲numpy矩陣，這樣就可以簡化很多數學處理操作
    alphas = mat(zeros((m,1)))
    iter = 0 #初始化遍歷次數 
    while (iter < maxIter): #只有在所有數據集上遍歷maxIter次，且不再發生任何alpha修改之後，程序纔會停止並退出while循環
        alphaPairsChanged = 0                       #記錄alpha是否巳經進行優化
        for i in range(m):
            fXi = float(multiply(alphas,labelMat).T*(dataMatrix*dataMatrix[i,:].T)) + b #預測的類別
            Ei = fXi - float(labelMat[i])           #Ei爲計算誤差；if checks if an example violates KKT conditions
            #一旦alphas等於0或C，那麼它們就巳經在“邊界”上了，因而不再能夠減小或增大，因此也就不值得再對它們進行優化了
            if ((labelMat[i]*Ei < -toler) and (alphas[i] < C)) or ((labelMat[i]*Ei > toler) and (alphas[i] > 0)):
                j = selectJrand(i,m)                #隨機選擇第二個alpha值
                fXj = float(multiply(alphas,labelMat).T*(dataMatrix*dataMatrix[j,:].T)) + b
                Ej = fXj - float(labelMat[j])       #計算第二個alpha值的誤差
                alphaIold = alphas[i].copy(); alphaJold = alphas[j].copy();
                if (labelMat[i] != labelMat[j]):    #當y1和y2異號，計算alpha的取值範圍
                    L = max(0, alphas[j] - alphas[i])
                    H = min(C, C + alphas[j] - alphas[i])
                else:                               #當y1和y2同號，計算alpha的取值範圍                       
                    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)的二階導數
                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給原來的舊值b，爲了後續的循環
                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 #檢査alpha值是否做了更新，如果有更新則將iter爲0後繼續運行程序
        else: iter = 0
        print "iteration number: %d" % iter
    return b,alphas

運行以下命令：

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

加上求解w的函數（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

加上畫圖函數：

def plotfig_SVM(xMat,yMat,ws,b,alphas):
    xMat = mat(xMat)
    yMat = mat(yMat)
    b = array(b)[0] #b原來是矩陣，先轉爲數組類型後其數組大小爲（1,1），所以後面加[0]，變爲(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，最小值根據原數據集dataArr[:,0]的大小而定
    y =(-b-ws[0][0]*x)/ws[1][0] #根據x.w + b = 0 得到，其式子展開爲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，並在圖中標紅
        if alphas[i]>0.0:
            ax.plot(xMat[i,0],xMat[i,1],'ro')
    plt.show()

運行命令：

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

畫圖：（其中紅色的點爲支持向量）

二、利用完整Platt SMO算法加速優化

Platt SMO 算法是通過一個外循環來選擇第一個alpha值的，並且其選擇過程會在兩種方式之間進行交替：一種方式是在所有數據集上進行單遍掃描，另一種方式則是在非邊界alpha中實現單遍掃描。而所謂非邊界alpha是指的就是那些不等於邊界0或C的alpha值。對整個數據集的掃描相當容易，而實現非邊界alpha值的掃描時，首先需要建立這些alpha值的列表，然後再對這個表進行遍歷。同時，該步驟會跳過那些已知的不會改變的alpha值。

Python代碼：

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: #建立一個數據結構來保存所有的重要值
    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（是否有效的標誌位），第二列是實際的E值（誤差值）
        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
    #選擇第二個alhpa值以保證在每次優化中採用最大步長
    oS.eCache[i] = [1,Ei]  #set valid（將即將使用的Ei值設爲有效） #choose the alpha that gives the maximum delta E
    validEcacheList = nonzero(oS.eCache[:,0].A)[0]
    if (len(validEcacheList)) > 1: #選擇其中使改變最大的那個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函數一樣（包括選擇第二個乘子）
'''
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]): #當y1和y2異號，計算alpha[j]的取值範圍
            L = max(0, oS.alphas[j] - oS.alphas[i])
            H = min(oS.C, oS.C + oS.alphas[j] - oS.alphas[i])
        else:   #當y1和y2同號，計算alpha[j]的取值範圍
            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：外循環（選擇第一個乘子）
'''
def smoP(dataMatIn, classLabels, C, toler, maxIter,kTup=('lin', 0)):    #full Platt SMO
    oS = optStruct(mat(dataMatIn),mat(classLabels).transpose(),C,toler, kTup) #建立一個數據結構來保存所有的重要值
    iter = 0
    entireSet = True; alphaPairsChanged = 0 #alpha改變標誌位
    while (iter < maxIter) and ((alphaPairsChanged > 0) or (entireSet)):
        alphaPairsChanged = 0 
        if entireSet:   #go over all 遍歷整個數據集
            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

運行以下代碼：

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)

畫圖：

三、在複雜數據上利用核函數

添加函數：

'''
功能：利用核函數進行分類
'''
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，從而得到了所需要的支持向量
    sVs=datMat[svInd] #get matrix of only support vectors
    labelSV = labelMat[svInd];#得到了支持向量的類別標籤，
    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進行分類
        predict = array(kernelEval.T * multiply(labelSV,alphas[svInd]) + b) #計算預測值
        if sign(predict) == -1: #如果預測的標籤是-1，則保存一個列表中
            predict_label0_index.append(i)
        elif sign(predict) == 1: #如果預測的標籤是-1，則保存另一個列表中
            predict_label1_index.append(i)
        if sign(predict)!=sign(labelArr[i]): errorCount += 1 #利用sign函數，判斷預測是否正確
    print "the training error rate is: %f" % (float(errorCount)/m)
    plotfig_kernel(dataArr,predict_label0_index,predict_label1_index,alphas)#畫圖
    dataArr,labelArr = loadDataSet('testSetRBF2.txt') #再用另一個數據集進行測試，後面代碼和前面一樣
    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)

其中畫圖函數換爲：

'''
功能：畫利用核函數進行分類的圖
'''
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: #預測標籤爲-1的標爲紅色三角形
        ax.plot(xMat[i][0],xMat[i][1],'r^')
    for i in predict_label1_index: #預測標籤爲1的標爲藍色正方形
        ax.plot(xMat[i][0],xMat[i][1],'bs')
    plt.show()

運行命令：