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python機(jī)器學(xué)習(xí)理論與實(shí)戰(zhàn)（六）支持向量機(jī)

更新時(shí)間：2018年01月19日 11:18:59 作者：marvin521

這篇文章主要介紹了python機(jī)器學(xué)習(xí)理論與實(shí)戰(zhàn)第六篇，支持向量機(jī)的相關(guān)資料，具有一定的參考價(jià)值，感興趣的小伙伴們可以參考一下

上節(jié)基本完成了SVM的理論推倒，尋找最大化間隔的目標(biāo)最終轉(zhuǎn)換成求解拉格朗日乘子變量alpha的求解問(wèn)題，求出了alpha即可求解出SVM的權(quán)重W，有了權(quán)重也就有了最大間隔距離，但是其實(shí)上節(jié)我們有個(gè)假設(shè)：就是訓(xùn)練集是線性可分的，這樣求出的alpha在[0,infinite]。但是如果數(shù)據(jù)不是線性可分的呢？此時(shí)我們就要允許部分的樣本可以越過(guò)分類器，這樣優(yōu)化的目標(biāo)函數(shù)就可以不變，只要引入松弛變量即可，它表示錯(cuò)分類樣本點(diǎn)的代價(jià)，分類正確時(shí)它等于0，當(dāng)分類錯(cuò)誤時(shí)，其中Tn表示樣本的真實(shí)標(biāo)簽-1或者1，回顧上節(jié)中，我們把支持向量到分類器的距離固定為1，因此兩類的支持向量間的距離肯定大于1的，當(dāng)分類錯(cuò)誤時(shí)肯定也大于1，如（圖五）所示（這里公式和圖標(biāo)序號(hào)都接上一節(jié)）。

（圖五）

這樣有了錯(cuò)分類的代價(jià)，我們把上節(jié)（公式四）的目標(biāo)函數(shù)上添加上這一項(xiàng)錯(cuò)分類代價(jià)，得到如（公式八）的形式：

（公式八）

重復(fù)上節(jié)的拉格朗日乘子法步驟，得到（公式九）：

（公式九）

多了一個(gè)Un乘子，當(dāng)然我們的工作就是繼續(xù)求解此目標(biāo)函數(shù)，繼續(xù)重復(fù)上節(jié)的步驟，求導(dǎo)得到（公式十）：

（公式十）

又因?yàn)閍lpha大于0，而且Un大于0，所以0<alpha<C,為了解釋的清晰一些，我們把（公式九）的KKT條件也發(fā)出來(lái)（上節(jié)中的第三類優(yōu)化問(wèn)題），注意Un是大于等于0：

推導(dǎo)到現(xiàn)在，優(yōu)化函數(shù)的形式基本沒(méi)變，只是多了一項(xiàng)錯(cuò)分類的價(jià)值，但是多了一個(gè)條件，0<alpha<C，C是一個(gè)常數(shù)，它的作用就是在允許有錯(cuò)誤分類的情況下，控制最大化間距，它太大了會(huì)導(dǎo)致過(guò)擬合，太小了會(huì)導(dǎo)致欠擬合。接下來(lái)的步驟貌似大家都應(yīng)該知道了，多了一個(gè)C常量的限制條件，然后繼續(xù)用SMO算法優(yōu)化求解二次規(guī)劃，但是我想繼續(xù)把核函數(shù)也一次說(shuō)了，如果樣本線性不可分，引入核函數(shù)后，把樣本映射到高維空間就可以線性可分，如（圖六）所示的線性不可分的樣本：

（圖六）

在（圖六）中，現(xiàn)有的樣本是很明顯線性不可分，但是加入我們利用現(xiàn)有的樣本X之間作些不同的運(yùn)算，如（圖六）右邊所示的樣子，而讓f作為新的樣本（或者說(shuō)新的特征）是不是更好些？現(xiàn)在把X已經(jīng)投射到高維度上去了，但是f我們不知道，此時(shí)核函數(shù)就該上場(chǎng)了，以高斯核函數(shù)為例，在（圖七）中選幾個(gè)樣本點(diǎn)作為基準(zhǔn)點(diǎn)，來(lái)利用核函數(shù)計(jì)算f,如（圖七）所示：

（圖七）

這樣就有了f,而核函數(shù)此時(shí)相當(dāng)于對(duì)樣本的X和基準(zhǔn)點(diǎn)一個(gè)度量，做權(quán)重衰減，形成依賴于x的新的特征f,把f放在上面說(shuō)的SVM中繼續(xù)求解alpha，然后得出權(quán)重就行了，原理很簡(jiǎn)單吧，為了顯得有點(diǎn)學(xué)術(shù)味道，把核函數(shù)也做個(gè)樣子加入目標(biāo)函數(shù)中去吧，如（公式十一）所示：

（公式十一）

其中K(Xn,Xm)是核函數(shù)，和上面目標(biāo)函數(shù)比沒(méi)有多大的變化，用SMO優(yōu)化求解就行了，代碼如下：

def smoPK(dataMatIn, classLabels, C, toler, maxIter): #full Platt SMO 
 oS = optStruct(mat(dataMatIn),mat(classLabels).transpose(),C,toler) 
 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

下面演示一個(gè)小例子，手寫識(shí)別。

(1)收集數(shù)據(jù)：提供文本文件

(2)準(zhǔn)備數(shù)據(jù)：基于二值圖像構(gòu)造向量

(3)分析數(shù)據(jù)：對(duì)圖像向量進(jìn)行目測(cè)

(4)訓(xùn)練算法：采用兩種不同的核函數(shù)，并對(duì)徑向基函數(shù)采用不同的設(shè)置來(lái)運(yùn)行SMO算法。

(5)測(cè)試算法：編寫一個(gè)函數(shù)來(lái)測(cè)試不同的核函數(shù)，并計(jì)算錯(cuò)誤率

(6)使用算法：一個(gè)圖像識(shí)別的完整應(yīng)用還需要一些圖像處理的只是，此demo略。

完整代碼如下：

from numpy import * 
from time import sleep 
 
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): 
 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): 
 if aj > H: 
  aj = H 
 if L > aj: 
  aj = L 
 return aj 
 
def 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 
   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)): 
    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 
    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 
 
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: 
 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, 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): 
 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 0 
 
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): 
 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) 
 
 
'''''#######******************************** 
Non-Kernel VErsions below 
'''#######******************************** 
 
class optStructK: 
 def __init__(self,dataMatIn, classLabels, C, toler): # 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 
   
def calcEkK(oS, k): 
 fXk = float(multiply(oS.alphas,oS.labelMat).T*(oS.X*oS.X[k,:].T)) + oS.b 
 Ek = fXk - float(oS.labelMat[k]) 
 return Ek 
   
def selectJK(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, Ej 
 
def updateEkK(oS, k):#after any alpha has changed update the new value in the cache 
 Ek = calcEk(oS, k) 
 oS.eCache[k] = [1,Ek] 
   
def innerLK(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.X[i,:]*oS.X[j,:].T - oS.X[i,:]*oS.X[i,:].T - oS.X[j,:]*oS.X[j,:].T 
  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.X[i,:]*oS.X[i,:].T - oS.labelMat[j]*(oS.alphas[j]-alphaJold)*oS.X[i,:]*oS.X[j,:].T 
  b2 = oS.b - Ej- oS.labelMat[i]*(oS.alphas[i]-alphaIold)*oS.X[i,:]*oS.X[j,:].T - oS.labelMat[j]*(oS.alphas[j]-alphaJold)*oS.X[j,:]*oS.X[j,:].T 
  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 
 
def smoPK(dataMatIn, classLabels, C, toler, maxIter): #full Platt SMO 
 oS = optStruct(mat(dataMatIn),mat(classLabels).transpose(),C,toler) 
 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

運(yùn)行結(jié)果如（圖八）所示：