python机器学习理论与实战（六）支持向量机

脚本专栏 2024/11/26 佚名

3 2 1

上节基本完成了SVM的理论推倒，寻找最大化间隔的目标最终转换成求解拉格朗日乘子变量alpha的求解问题，求出了alpha即可求解出SVM的权重W，有了权重也就有了最大间隔距离，但是其实上节我们有个假设：就是训练集是线性可分的，这样求出的alpha在[0,infinite]。但是如果数据不是线性可分的呢？此时我们就要允许部分的样本可以越过分类器，这样优化的目标函数就可以不变，只要引入松弛变量即可，它表示错分类样本点的代价，分类正确时它等于0，当分类错误时，其中Tn表示样本的真实标签-1或者1，回顾上节中，我们把支持向量到分类器的距离固定为1，因此两类的支持向量间的距离肯定大于1的，当分类错误时肯定也大于1，如（图五）所示（这里公式和图标序号都接上一节）。

（图五）

这样有了错分类的代价，我们把上节（公式四）的目标函数上添加上这一项错分类代价，得到如（公式八）的形式：

（公式八）

重复上节的拉格朗日乘子法步骤，得到（公式九）：

（公式九）

多了一个Un乘子，当然我们的工作就是继续求解此目标函数，继续重复上节的步骤，求导得到（公式十）：

（公式十）

又因为alpha大于0，而且Un大于0，所以0<alpha<C,为了解释的清晰一些，我们把（公式九）的KKT条件也发出来（上节中的第三类优化问题），注意Un是大于等于0：

推导到现在，优化函数的形式基本没变，只是多了一项错分类的价值，但是多了一个条件，0<alpha<C，C是一个常数，它的作用就是在允许有错误分类的情况下，控制最大化间距，它太大了会导致过拟合，太小了会导致欠拟合。接下来的步骤貌似大家都应该知道了，多了一个C常量的限制条件，然后继续用SMO算法优化求解二次规划，但是我想继续把核函数也一次说了，如果样本线性不可分，引入核函数后，把样本映射到高维空间就可以线性可分，如（图六）所示的线性不可分的样本：

（图六）

在（图六）中，现有的样本是很明显线性不可分，但是加入我们利用现有的样本X之间作些不同的运算，如（图六）右边所示的样子，而让f作为新的样本（或者说新的特征）是不是更好些？现在把X已经投射到高维度上去了，但是f我们不知道，此时核函数就该上场了，以高斯核函数为例，在（图七）中选几个样本点作为基准点，来利用核函数计算f,如（图七）所示：

（图七）

这样就有了f,而核函数此时相当于对样本的X和基准点一个度量，做权重衰减，形成依赖于x的新的特征f,把f放在上面说的SVM中继续求解alpha，然后得出权重就行了，原理很简单吧，为了显得有点学术味道，把核函数也做个样子加入目标函数中去吧，如（公式十一）所示：

（公式十一）

其中K(Xn,Xm)是核函数，和上面目标函数比没有多大的变化，用SMO优化求解就行了，代码如下：

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

下面演示一个小例子，手写识别。

(1)收集数据：提供文本文件

(2)准备数据：基于二值图像构造向量

(3)分析数据：对图像向量进行目测

(4)训练算法：采用两种不同的核函数，并对径向基函数采用不同的设置来运行SMO算法。

(5)测试算法：编写一个函数来测试不同的核函数，并计算错误率

(6)使用算法：一个图像识别的完整应用还需要一些图像处理的只是，此demo略。

完整代码如下：

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

运行结果如（图八）所示：

（图八）

上面代码有兴趣的可以读读，用的话，建议使用libsvm。

参考文献：

[1]machine learning in action. PeterHarrington

[2] pattern recognition and machinelearning. Christopher M. Bishop

[3]machine learning.Andrew Ng

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python,机器学习,支持向量机

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python,机器学习,支持向量机

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python机器学习理论与实战（六）支持向量机

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