python机器学习理论与实战（六）支持向量机

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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