SVM推导帖子收藏

本文主要是介绍SVM推导帖子收藏，希望对大家解决编程问题提供一定的参考价值，需要的开发者们随着小编来一起学习吧！

SVM推导里看过的不错的两个帖子，还有就是《机器学习实战》中的SVM那一章的SMO的简单实现的python代码，学习SVM的可以看一看，比《统计学习》书里的部分，细节要详细些。也可以看看周志华老师的《机器学习》，svm那一章从margin到对偶求解，kkt条件，以及SMO，核函数，正则化惩罚这些都说的很透测。以后会更新一篇自己整理好的整个流程的帖子，估计是手写吧。

'''
Created on Nov 4, 2010
Chapter 5 source file for Machine Learing in Action
@author: Peter
'''
from numpy import *
from time import sleepdef 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,labelMatdef selectJrand(i,m):j=i #we want to select any J not equal to iwhile (j==i):j = int(random.uniform(0,m))return jdef clipAlpha(aj,H,L):if aj > H: aj = Hif L > aj:aj = Lreturn ajdef 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 = 0while (iter < maxIter):alphaPairsChanged = 0for i in range(m):fXi = float(multiply(alphas,labelMat).T*(dataMatrix*dataMatrix[i,:].T)) + bEi = fXi - float(labelMat[i])#if checks if an example violates KKT conditionsif ((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)) + bEj = 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"; continueeta = 2.0 * dataMatrix[i,:]*dataMatrix[j,:].T - dataMatrix[i,:]*dataMatrix[i,:].T - dataMatrix[j,:]*dataMatrix[j,:].Tif eta >= 0: print "eta>=0"; continuealphas[j] -= labelMat[j]*(Ei - Ej)/etaalphas[j] = clipAlpha(alphas[j],H,L)if (abs(alphas[j] - alphaJold) < 0.00001): print "j not moving enough"; continuealphas[i] += labelMat[j]*labelMat[i]*(alphaJold - alphas[j])#update i by the same amount as j#the update is in the oppostie directionb1 = b - Ei- labelMat[i]*(alphas[i]-alphaIold)*dataMatrix[i,:]*dataMatrix[i,:].T - labelMat[j]*(alphas[j]-alphaJold)*dataMatrix[i,:]*dataMatrix[j,:].Tb2 = b - Ej- labelMat[i]*(alphas[i]-alphaIold)*dataMatrix[i,:]*dataMatrix[j,:].T - labelMat[j]*(alphas[j]-alphaJold)*dataMatrix[j,:]*dataMatrix[j,:].Tif (0 < alphas[i]) and (C > alphas[i]): b = b1elif (0 < alphas[j]) and (C > alphas[j]): b = b2else: b = (b1 + b2)/2.0alphaPairsChanged += 1print "iter: %d i:%d, pairs changed %d" % (iter,i,alphaPairsChanged)if (alphaPairsChanged == 0): iter += 1else: iter = 0print "iteration number: %d" % iterreturn b,alphasdef kernelTrans(X, A, kTup): #calc the kernel or transform data to a higher dimensional spacem,n = shape(X)K = mat(zeros((m,1)))if kTup[0]=='lin': K = X * A.T   #linear kernelelif kTup[0]=='rbf':for j in range(m):deltaRow = X[j,:] - AK[j] = deltaRow*deltaRow.TK = exp(K/(-1*kTup[1]**2)) #divide in NumPy is element-wise not matrix like Matlabelse: raise NameError('Houston We Have a Problem -- \That Kernel is not recognized')return Kclass optStruct:def __init__(self,dataMatIn, classLabels, C, toler, kTup):  # Initialize the structure with the parameters self.X = dataMatInself.labelMat = classLabelsself.C = Cself.tol = tolerself.m = shape(dataMatIn)[0]self.alphas = mat(zeros((self.m,1)))self.b = 0self.eCache = mat(zeros((self.m,2))) #first column is valid flagself.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 Ekdef selectJ(i, oS, Ei):         #this is the second choice -heurstic, and calcs EjmaxK = -1; maxDeltaE = 0; Ej = 0oS.eCache[i] = [1,Ei]  #set valid #choose the alpha that gives the maximum delta EvalidEcacheList = 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 Eif k == i: continue #don't calc for i, waste of timeEk = calcEk(oS, k)deltaE = abs(Ei - Ek)if (deltaE > maxDeltaE):maxK = k; maxDeltaE = deltaE; Ej = Ekreturn maxK, Ejelse:   #in this case (first time around) we don't have any valid eCache valuesj = selectJrand(i, oS.m)Ej = calcEk(oS, j)return j, Ejdef updateEk(oS, k):#after any alpha has changed update the new value in the cacheEk = 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 selectJrandalphaIold = 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 0eta = 2.0 * oS.K[i,j] - oS.K[i,i] - oS.K[j,j] #changed for kernelif eta >= 0: print "eta>=0"; return 0oS.alphas[j] -= oS.labelMat[j]*(Ei - Ej)/etaoS.alphas[j] = clipAlpha(oS.alphas[j],H,L)updateEk(oS, j) #added this for the Ecacheif (abs(oS.alphas[j] - alphaJold) < 0.00001): print "j not moving enough"; return 0oS.alphas[i] += oS.labelMat[j]*oS.labelMat[i]*(alphaJold - oS.alphas[j])#update i by the same amount as jupdateEk(oS, i) #added this for the Ecache                    #the update is in the oppostie directionb1 = 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 = b1elif (0 < oS.alphas[j]) and (oS.C > oS.alphas[j]): oS.b = b2else: oS.b = (b1 + b2)/2.0return 1else: return 0def smoP(dataMatIn, classLabels, C, toler, maxIter,kTup=('lin', 0)):    #full Platt SMOoS = optStruct(mat(dataMatIn),mat(classLabels).transpose(),C,toler, kTup)iter = 0entireSet = True; alphaPairsChanged = 0while (iter < maxIter) and ((alphaPairsChanged > 0) or (entireSet)):alphaPairsChanged = 0if entireSet:   #go over allfor i in range(oS.m):        alphaPairsChanged += innerL(i,oS)print "fullSet, iter: %d i:%d, pairs changed %d" % (iter,i,alphaPairsChanged)iter += 1else:#go over non-bound (railed) alphasnonBoundIs = 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 += 1if entireSet: entireSet = False #toggle entire set loopelif (alphaPairsChanged == 0): entireSet = True  print "iteration number: %d" % iterreturn oS.b,oS.alphasdef 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 wdef testRbf(k1=1.3):dataArr,labelArr = loadDataSet('testSetRBF.txt')b,alphas = smoP(dataArr, labelArr, 200, 0.0001, 10000, ('rbf', k1)) #C=200 importantdatMat=mat(dataArr); labelMat = mat(labelArr).transpose()svInd=nonzero(alphas.A>0)[0]sVs=datMat[svInd] #get matrix of only support vectorslabelSV = labelMat[svInd];print "there are %d Support Vectors" % shape(sVs)[0]m,n = shape(datMat)errorCount = 0for i in range(m):kernelEval = kernelTrans(sVs,datMat[i,:],('rbf', k1))predict=kernelEval.T * multiply(labelSV,alphas[svInd]) + bif sign(predict)!=sign(labelArr[i]): errorCount += 1print "the training error rate is: %f" % (float(errorCount)/m)dataArr,labelArr = loadDataSet('testSetRBF2.txt')errorCount = 0datMat=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]) + bif 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 returnVectdef loadImages(dirName):from os import listdirhwLabels = []trainingFileList = listdir(dirName)           #load the training setm = len(trainingFileList)trainingMat = zeros((m,1024))for i in range(m):fileNameStr = trainingFileList[i]fileStr = fileNameStr.split('.')[0]     #take off .txtclassNumStr = 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 = 0for i in range(m):kernelEval = kernelTrans(sVs,datMat[i,:],kTup)predict=kernelEval.T * multiply(labelSV,alphas[svInd]) + bif sign(predict)!=sign(labelArr[i]): errorCount += 1print "the training error rate is: %f" % (float(errorCount)/m)dataArr,labelArr = loadImages('testDigits')errorCount = 0datMat=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]) + bif 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 = dataMatInself.labelMat = classLabelsself.C = Cself.tol = tolerself.m = shape(dataMatIn)[0]self.alphas = mat(zeros((self.m,1)))self.b = 0self.eCache = mat(zeros((self.m,2))) #first column is valid flagdef calcEkK(oS, k):fXk = float(multiply(oS.alphas,oS.labelMat).T*(oS.X*oS.X[k,:].T)) + oS.bEk = fXk - float(oS.labelMat[k])return Ekdef selectJK(i, oS, Ei):         #this is the second choice -heurstic, and calcs EjmaxK = -1; maxDeltaE = 0; Ej = 0oS.eCache[i] = [1,Ei]  #set valid #choose the alpha that gives the maximum delta EvalidEcacheList = 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 Eif k == i: continue #don't calc for i, waste of timeEk = calcEk(oS, k)deltaE = abs(Ei - Ek)if (deltaE > maxDeltaE):maxK = k; maxDeltaE = deltaE; Ej = Ekreturn maxK, Ejelse:   #in this case (first time around) we don't have any valid eCache valuesj = selectJrand(i, oS.m)Ej = calcEk(oS, j)return j, Ejdef updateEkK(oS, k):#after any alpha has changed update the new value in the cacheEk = 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 selectJrandalphaIold = 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 0eta = 2.0 * oS.X[i,:]*oS.X[j,:].T - oS.X[i,:]*oS.X[i,:].T - oS.X[j,:]*oS.X[j,:].Tif eta >= 0: print "eta>=0"; return 0oS.alphas[j] -= oS.labelMat[j]*(Ei - Ej)/etaoS.alphas[j] = clipAlpha(oS.alphas[j],H,L)updateEk(oS, j) #added this for the Ecacheif (abs(oS.alphas[j] - alphaJold) < 0.00001): print "j not moving enough"; return 0oS.alphas[i] += oS.labelMat[j]*oS.labelMat[i]*(alphaJold - oS.alphas[j])#update i by the same amount as jupdateEk(oS, i) #added this for the Ecache                    #the update is in the oppostie directionb1 = 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,:].Tb2 = 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,:].Tif (0 < oS.alphas[i]) and (oS.C > oS.alphas[i]): oS.b = b1elif (0 < oS.alphas[j]) and (oS.C > oS.alphas[j]): oS.b = b2else: oS.b = (b1 + b2)/2.0return 1else: return 0def smoPK(dataMatIn, classLabels, C, toler, maxIter):    #full Platt SMOoS = optStruct(mat(dataMatIn),mat(classLabels).transpose(),C,toler)iter = 0entireSet = True; alphaPairsChanged = 0while (iter < maxIter) and ((alphaPairsChanged > 0) or (entireSet)):alphaPairsChanged = 0if entireSet:   #go over allfor i in range(oS.m):        alphaPairsChanged += innerL(i,oS)print "fullSet, iter: %d i:%d, pairs changed %d" % (iter,i,alphaPairsChanged)iter += 1else:#go over non-bound (railed) alphasnonBoundIs = 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 += 1if entireSet: entireSet = False #toggle entire set loopelif (alphaPairsChanged == 0): entireSet = True  print "iteration number: %d" % iterreturn oS.b,oS.alphasif __name__ == '__main__':j=selectJrand(1,10)print jdataArr,labelArr=loadDataSet('testSet.txt')b,alphas=smoSimple(dataArr,labelArr,0.6,0.001,40)print bprint alphas[alphas>0]