简化版本的SMO
相关函数
from numpy import *
from time import sleep
def loadDataSet(fileName): #得到类标签和数据矩阵
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): #i是第一个alpha的下标，m是所有alpha的数量,确保i和j是不一样的
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): #用于调整alpha的值
if aj > H:
aj = H
if L > aj:
aj = L
return aj
def smoSimple(dataMatIn, classLabels, C, toler, maxIter): #SMO的一个有效的简化版本
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 #预测类别
Ei = fXi - float(labelMat[i])#if checks if an example violates KKT condition 计算误差
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 #alpha的最优修改量
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
测试代码
import svmMLiA
from numpy import *
dataArr, labelArr = svmMLiA.loadDataSet('testSet.txt')
# print labelArr
b, alphas = svmMLiA.smoSimple(dataArr, labelArr, 0.6, 0.001, 40)
print b
print alphas[alphas>0]
print shape(alphas[alphas>0])
for i in range(100):
if alphas[i]>0.0: print dataArr[i], labelArr[i]结果
j not moving enough
iteration number: 39
j not moving enough
j not moving enough
iteration number: 40
[[-3.78661581]]---------------------------------------b的值
[[ 0.14237008 0.19622369 0.02570131 0.36429508]]---------------alpha的值
(1L, 4L)
[4.658191, 3.507396] -1.0 ----------------------------------4个支持向量
[3.457096, -0.082216] -1.0
[2.893743, -1.643468] -1.0
[6.080573, 0.418886] 1.0
利用完整platt SMO算法加速优化
相关函数
class optStruct: #建立一个数据结构
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): #计算E值
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): #完整SMO算法中的优化例程
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): #根据alphas计算W 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
测试代码
import svmMLiA
from numpy import *
dataArr, labelArr = svmMLiA.loadDataSet('testSet.txt')
# print labelArr
# b, alphas = svmMLiA.smoSimple(dataArr, labelArr, 0.6, 0.001, 40)
# print b
# print alphas[alphas>0]
# print shape(alphas[alphas>0])
# for i in range(100):
# if alphas[i]>0.0: print dataArr[i], labelArr[i]
#测试完整版的smo算法
b, alphas = svmMLiA.smoP(dataArr, labelArr, 0.6, 0.001, 40)
print b
print alphas[alphas>0]
print shape(alphas[alphas>0])
for i in range(100):
if alphas[i]>0.0: print dataArr[i], labelArr[i]
#测试完整版的smo算法 b, alphas = svmMLiA.smoP(dataArr, labelArr, 0.6, 0.001, 40) print b print alphas[alphas>0] print shape(alphas[alphas>0]) for i in range(100): if alphas[i]>0.0: print dataArr[i], labelArr[i] ws = svmMLiA.calcWs(alphas,dataArr,labelArr) print ws #现在进行分类 dataMat = mat(dataArr) print dataMat[0]*mat(ws)+b print labelArr[0] print dataMat[1]*mat(ws)+b print labelArr[1] print dataMat[2]*mat(ws)+b print labelArr[2]
结果：
fullSet, iter: 2 i:91, pairs changed 0
fullSet, iter: 2 i:92, pairs changed 0
fullSet, iter: 2 i:93, pairs changed 0
L==H
fullSet, iter: 2 i:94, pairs changed 0
j not moving enough
fullSet, iter: 2 i:95, pairs changed 0
L==H
fullSet, iter: 2 i:96, pairs changed 0
L==H
fullSet, iter: 2 i:97, pairs changed 0
fullSet, iter: 2 i:98, pairs changed 0
fullSet, iter: 2 i:99, pairs changed 0
iteration number: 3
[[-2.3194761]]
[[ 0.09390371 0.05876873 0.02162722 0.00526142 0.01900464 0.01900464
0.01876918 0.02336781 0.02336781]]
(1L, 9L)
[3.542485, 1.977398] -1.0
[8.127113, 1.274372] 1.0
[7.108772, -0.986906] 1.0
[3.634009, 1.730537] -1.0
[3.223038, -0.552392] -1.0
[7.40786, -0.121961] 1.0
[5.286862, -2.358286] 1.0
[6.080573, 0.418886] 1.0
[1.966279, -1.840439] -1.0
[[ 0.55449368]
[-0.12452835]]
[[-0.60143267]]
-1.0
[[-0.96386361]]
-1.0
[[ 2.06454697]]
1.0
核函数
相关函数
def testRbf(k1=1.3): #在测试中使用核函数
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)
测试代码
#在复杂数据上应用核函数 svmMLiA.testRbf()
结果
fullSet, iter: 3 i:94, pairs changed 0
fullSet, iter: 3 i:95, pairs changed 0
fullSet, iter: 3 i:96, pairs changed 0
fullSet, iter: 3 i:97, pairs changed 0
fullSet, iter: 3 i:98, pairs changed 0
fullSet, iter: 3 i:99, pairs changed 0
iteration number: 4
there are 20 Support Vectors
the training error rate is: 0.040000
the test error rate is: 0.040000
手写识别问题
相关函数
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)
测试代码
#手写识别
svmMLiA.testDigits(('rbf',50))
结果
fullSet, iter: 5 i:398, pairs changed 0
L==H
fullSet, iter: 5 i:399, pairs changed 0
fullSet, iter: 5 i:400, pairs changed 0
j not moving enough
fullSet, iter: 5 i:401, pairs changed 0
iteration number: 6
there are 402 samples,there are 34 Support Vectors
the training error rate is: 0.014925
the test error rate is: 0.010753