支持向量机
优点：泛化错误率低，计算开销不大，结果易于理解
缺点：对参数调节和核函数的选择敏感，原始分类器不加修改仅适用于处理二类问题
适用数据类型：数值型和标称型数据
如果数据集是1024维，需使用一个1023维的某对象来对数据进行分隔，改对象称作超平面。
#6-1 SMO算法辅助函数
def loadDataSet(fileName):#解析
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):#防止alpha下标和alpha的数目相同
j=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
实验这部分代码
import svmMLiA
dataArr,labelArr = svmMLiA.loadDataSet("testSet.txt")
labelArr
Out[7]:
[-1.0,
-1.0,
#略过若干结果...
-1.0,
-1.0]
下面开始SMO算法的第一个版本，伪代码大致如下：
创建一个alpha向量并将其初始化为0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
同事优化这两个向量
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
#6-2 简化版SMO算法
def smoSimple(dataMatIn,classLabels,C,toler,maxIter):#数据集，类别标签，常数C，容错率，退出前最大的循环次数
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 ((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();#深度拷贝
if (labelMat[i]!=labelMat[j]):#保证alpha在0和C之间
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是alpha[j]的最优修改量
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])#修改方向相反
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#设置常数项B
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
#测试实际效果
import svmMLiA
dataArr,labelArr = svmMLiA.loadDataSet("testSet.txt")
b,alphas = svmMLiA.smoSimple(dataArr,labelArr,0.6,0.001,40)
#略过部分
iteration number: 28
iter: 28 i:29, pairs changed 1
iteration number: 0
j not moving enough
iteration number: 1
j not moving enough
j not moving enough
iteration number: 40
对结果进行观察
In [27]: b
Out[27]: matrix([[-3.83810926]])
我们可以直接观察alpha本身，但是其中的0元素过多，为了观察大于0的元素的数量，可以
alphas[alphas>0]#适用于NumPy类型
Out[28]: matrix([[ 0.12749752, 0.24132585, 0.36882337]])
由于SMO算法的随机性，读者运行后的结果可能不同。
#获得支持向量的格式
In [30]: shape(alphas[alphas>0])
Out[30]: (1L, 3L)
#了解哪些数据点是支持向量
In [32]: for i in range(100):
...: if alphas[i]>0.0:print dataArr[i],labelArr[i]
...:
[4.658191, 3.507396] -1.0
[3.457096, -0.082216] -1.0
[6.080573, 0.418886] 1.0
下面开始讨论完整版Platt SMO算法。他通过一个外循环来选择第一个alpha值，并且选择过程会在两种方式之间进行交替：一种是在所有数据集是进行单遍扫描，另一种方式则是非边界alpha中实现单遍扫描。
class optStruct:
def __init__(self,dataMatIn,classLabels,C,toler):
self.X = dataMatIn
self.labelMat = classLabels
self.C = C
self.tol = toler
self.m = shape(dataMatIn)[0]
self.alphas = mat(zeros((self.m,1)))#是否有效的标志位和实际的E值
self.b = 0
self.eCache = mat(zeros((self.m,2)))
#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.X*oS.X[k,:].T)) + oS.b
Ek = fXk - float(oS.labelMat[k])
return Ek
def selectJ(i,oS,Ei):
maxK = -1; maxDeltaE=0; Ej=0
oS.eCache[i] = [1,Ei]
validEcacheList=nonzero(oS.eCache[:,0].A)[0]
if(len(validEcacheList))>1:
for k in validEcacheList:
if k==i: continue
Ek = calcEk(oS,k)
deltaE=abs(Ei-Ek)
if(deltaE>maxDeltaE):#选择最大步长
maxK=k;maxDeltaE=deltaE;Ej=Ek
return maxK,Ej
else:
j=selectJrand(i,oS.m)
Ej=calcEk(oS,j)
return j,Ej
def updateEk(oS, k):
Ek = calcEk(oS, k)
oS.eCache[k] = [1,Ek]
#寻找决策边界的优化例程
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)
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)#更新缓存误差
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])
updateEk(oS, i)
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 smoP(dataMatIn, classLabels, C, toler, maxIter):
oS = optStruct(mat(dataMatIn),mat(classLabels).transpose(),C,toler)#构建数据结构容纳所有数据
iter = 0
entireSet = True; alphaPairsChanged = 0
while (iter < maxIter) and ((alphaPairsChanged > 0) or (entireSet)):
alphaPairsChanged = 0
if entireSet:
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:
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
elif (alphaPairsChanged == 0): entireSet = True
print "iteration number: %d" % iter
return oS.b,oS.alphas
下面观察上述执行结果：
import svmMLiAli
dataArr,labelArr = svmMLiAli.loadDataSet("testSet.txt")
b,alphas = svmMLiAli.smoP(dataArr,labelArr,0.6,0.001,40)
fullSet, iter: 0 i:0, pairs changed 1
fullSet, iter: 0 i:1, pairs changed 1
fullSet, iter: 0 i:2, pairs changed 2
fullSet, iter: 0 i:3, pairs changed 3
L==H
fullSet, iter: 0 i:4, pairs changed 3
L==H
fullSet, iter: 0 i:5, pairs changed 3
L==H
fullSet, iter: 0 i:6, pairs changed 3
fullSet, iter: 0 i:7, pairs changed 3
fullSet, iter: 0 i:8, pairs changed 4
fullSet, iter: 0 i:9, pairs changed 4
j not moving enough
fullSet, iter: 0 i:10, pairs changed 4
fullSet, iter: 0 i:11, pairs changed 4
fullSet, iter: 0 i:12, pairs changed 4
fullSet, iter: 0 i:13, pairs changed 4
fullSet, iter: 0 i:14, pairs changed 4
fullSet, iter: 0 i:15, pairs changed 4
fullSet, iter: 0 i:16, pairs changed 4
j not moving enough#省略部分结果
下面开始计算w
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
In [2]: ws = calcWs(alphas,dataArr,labelArr)
In [3]: ws
Out[3]:
array([[ 0.65307162],
[-0.17196128]])
现在对数据进行分类处理，比如对第一个数据分类，可以：
In [4]: datMat = mat(dataArr)
In [5]: datMat[0]*mat(ws) + b
Out[5]: matrix([[-0.92555695]])
如果该数字大于0，则属于1类，小于则属于-1类：
In [6]: labelArr[0]
Out[6]: -1.0
In [7]: datMat[2]*mat(ws) + b
Out[7]: matrix([[ 2.30436336]])
In [8]: labelArr[2]
Out[8]: 1.0
In [9]: datMat[1]*mat(ws) + b
Out[9]: matrix([[-1.36706674]])
In [10]: labelArr[1]
Out[10]: -1.0
下面研究不能线性处理的情况，需要使用核函数（kernel）。我们需要将数据从一个特征空间转换到另外一个特征空间。
#6-6 核转换函数
def kernelTrans(X, A, kTup):#kT是核函数信息，第一个是类型，另外两个是可选参数
m,n = shape(X)
K = mat(zeros((m,1)))
if kTup[0]=="lin":K = X * A.T
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))#元素除，NumPy中指矩阵元素展开计算
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):#增加kTup
self.X = dataMatIn
self.labelMat = classLabels
self.C = C
self.tol = toler
self.m = shape(dataMatIn)[0]
self.alphas = mat(zeros((self.m,1)))#是否有效的标志位和实际的E值
self.b = 0
self.eCache = mat(zeros((self.m,2)))
#更新部分
self.K = mat(zeros((self.m,self.m)))
for i in range(self.m):
self.K[:,i] = kernelTrans(self.X,self.X[i,:],kTup)
另外还需要修改：
def innerL(i,oS):
#eta = 2.0*oS.X[i,:]*oS.X[j,:].T - oS.X[i,:]*oS.X[i,:].T - oS.X[j,:]*oS.X[j,:].T
eta = 2.0*oS.K[i,j] - oS.K[i,i] - oS.K[j,j]
#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
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]
def calcEk(oS,k):
#fXk = float(multiply(oS.alphas,oS.labelMat).T*(oS.X*oS.X[k,:].T)) + oS.b
fXk = float(multiply(oS.alphas,oS.labelMat).T*oS.K[:,k] + oS.b)
#6-8 利用核函数进行分类的径向基测试函数
def testRbf(k1=1.3):
dataArr,labelArr = loadDataSet("testSetRBF.txt")
b,alphas = smoP(dataArr, labelArr, 200, 0.0001, 10000, ("rbf", k1))
datMat = mat(dataArr); labelMat = mat(labelArr).transpose()
svInd = nonzero(alphas.A>0)[0] #返回数组中值不为零的元素的下标
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,:],("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):#测试数据集
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)
测试
In [19]: svmMLiA.testRbf()
#省略部分数据
L==H
fullSet, iter: 4 i:91, pairs changed 0
L==H
fullSet, iter: 4 i:92, pairs changed 0
fullSet, iter: 4 i:93, pairs changed 0
fullSet, iter: 4 i:94, pairs changed 0
fullSet, iter: 4 i:95, pairs changed 0
L==H
fullSet, iter: 4 i:96, pairs changed 0
fullSet, iter: 4 i:97, pairs changed 0
fullSet, iter: 4 i:98, pairs changed 0
fullSet, iter: 4 i:99, pairs changed 0
iteration number: 5
there are 29 Support Vectors
the training error rate is: 0.130000
the test error rate is: 0.150000
支持向量的数目存在一个最优值。SVM的优点在于它能对数据进行高效分类。如果支持向量太少，就可能会得到一个很差的决策边界；如果支持向量太多，也就相当于每次都利用整个数据集进行分类，这种分类情况称作k近邻。
先加入第二章knn算法中的img2vector()函数，然后加入如下代码：
#6-9 基于SVM的手写数字识别
def loadImages(dirName):
from os import listdir
hwLabels = []
trainingFileList = listdir(dirName)
m = len(trainingFileList)#总文件个数
trainingMat = zeros((m,1024))
for i in range(m):
fileNameStr = trainingFileList[i]
fileStr = fileNameStr.split('.')[0]#按“.”分开，取第0行
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)
In [35]: svmMLiA.testDigits(("rbf",20))
#省略部分结果
L==H
fullSet, iter: 3 i:397, pairs changed 0
L==H
fullSet, iter: 3 i:398, pairs changed 0
L==H
fullSet, iter: 3 i:399, pairs changed 0
fullSet, iter: 3 i:400, pairs changed 0
j not moving enough
fullSet, iter: 3 i:401, pairs changed 0
iteration number: 4
there are 51 Support Vectors
the training error rate is: 0.000000
the test error rate is: 0.016129
根据课本，σ取10的时候可以得到最小的错误率。可以观察到一个有趣的现象，即最小的训练错误率并不对应于最小的支持向量数目。另外，线性核函数并不是特别的糟糕，可以以牺牲线性核函数的错误率来换取分类速度的提高。
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