本代碼用的是徑向基核函數，徑向基函數是一個采用向量作為自變量的函數，能夠基于向量距離運算輸出一個標量。
所謂徑向基函數 (Radial Basis Function 簡稱 RBF), 就是某種沿徑向對稱的標量函數。 通常定義為空間中任一點x到某一中心xc之間歐氏距離的單調函數 , 可記作 k(||x?xc||), 其作用往往是局部的 , 即當x遠離xc時函數取值很小。最常用的徑向基函數是高斯核函數 ,形式為k(||x?xc||)=exp(?||X?XC||22σ2) 其中xc為核函數中心,σ為函數的寬度參數 , 控制了函數的徑向作用范圍。——百度

其中σ是用戶定義的用于確定到達率，或者說函數值跌落到0的速度參數。

# -*- coding: utf-8 -*- """ Created on Sun Oct 22 09:44:54 2017@author: LiLong """ 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 while (j==i):j = int(random.uniform(0,m))return jdef clipAlpha(aj,H,L):if aj > H: aj = Hif L > aj:aj = Lreturn aj# 核轉換函數 def kernelTrans(X, A, kTup): # X是所有的支持向量，A是數據集每個樣本向量，KTup是核函數的參數元組m,n = shape(X)K = mat(zeros((m,1))) # 構建一個列向量if kTup[0]=='lin': K = X * A.T # 如果核函數是線性核函數，elif kTup[0]=='rbf': # 如果是徑向基核函數for j in range(m):deltaRow = X[j,:] - AK[j] = deltaRow*deltaRow.TK = exp(K/(-1*kTup[1]**2)) # 計算高斯核函數,注意元素間的除法else: raise NameError('Houston We Have a Problem -- \That Kernel is not recognized')return Kclass optStruct:def __init__(self,dataMatIn, classLabels, C, toler, kTup): 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))) 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 Ekdef selectJ(i, oS, Ei): maxK = -1; maxDeltaE = 0; Ej = 0oS.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):maxK = k; maxDeltaE = deltaE; Ej = Ekreturn maxK, Ejelse: j = selectJrand(i, oS.m)Ej = calcEk(oS, j)return j, Ejdef updateEk(oS, k):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) 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 0eta = 2.0 * oS.K[i,j] - oS.K[i,i] - oS.K[j,j] # 不同于不使用核函數的代碼if 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) if (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])updateEk(oS, i) 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 = 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)): oS = optStruct(mat(dataMatIn),mat(classLabels).transpose(),C,toler, kTup)iter = 0entireSet = True; alphaPairsChanged = 0while (iter < maxIter) and ((alphaPairsChanged > 0) or (entireSet)):alphaPairsChanged = 0if entireSet: for i in range(oS.m): alphaPairsChanged += innerL(i,oS)print "fullSet, iter: %d i:%d, pairs changed %d" % (iter,i,alphaPairsChanged)iter += 1else: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 += 1if entireSet: entireSet = False elif (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 w# 利用核函數進行分類的徑向基函數測試函數 def testRbf(k1=1.3):dataArr,labelArr = loadDataSet('testSetRBF.txt')b,alphas = smoP(dataArr, labelArr, 200, 0.0001, 10000, ('rbf', k1)) # 核函數的類型是高斯核函數datMat=mat(dataArr); labelMat = mat(labelArr).transpose()svInd=nonzero(alphas.A>0)[0]sVs=datMat[svInd] #得到支持向量labelSV = labelMat[svInd]; # 得到支持向量的標簽print "there are %d Support Vectors" % shape(sVs)[0]m,n = shape(datMat)errorCount = 0for i in range(m): # 利用核函數進行分類# svs是所有的支持向量，datmat[i,:]是每個樣本數據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) testRbf()

需要注意的就是核函數的應用，和普通的Platt SMO有幾處不同，還有就是kernelTrans函數是如何利用核函數進行分類的，我手寫了下代碼過程：

和代碼對應起來就是高斯核函數的應用過程。

運行結果：

L==H fullSet, iter: 0 i:0, pairs changed 0 fullSet, iter: 0 i:1, pairs changed 1 fullSet, iter: 0 i:2, pairs changed 2 fullSet, iter: 0 i:3, pairs changed 3 fullSet, iter: 0 i:4, pairs changed 3 ..., fullSet, iter: 5 i:92, pairs changed 0 fullSet, iter: 5 i:93, pairs changed 0 j not moving enough fullSet, iter: 5 i:94, pairs changed 0 fullSet, iter: 5 i:95, pairs changed 0 fullSet, iter: 5 i:96, pairs changed 0 fullSet, iter: 5 i:97, pairs changed 0 fullSet, iter: 5 i:98, pairs changed 0 fullSet, iter: 5 i:99, pairs changed 0 iteration number: 6 there are 26 Support Vectors the training error rate is: 0.090000 the test error rate is: 0.180000

可以更換不同的k1參數以觀察測試錯誤率，訓練錯誤率，支持向量個數隨k1的變化情況。
σ太小的話，支持向量變多；σ太大，其支持向量的數目少了很多，并且測試錯誤率也在下降。所以σ值的設置在某處存在著最優值，如果降低σ，那么訓練錯誤率就會降低，但是測試錯誤率卻在上升。
支持向量機的數目存在一個最優值。SVM的優點在于它對數據的高效分類。如果支持向量太少，就會得到一個很差的決策邊界；如果支持向量太多，也就相當于每次都會利用整個數據集進行分類，這種分類方法稱為k近鄰。

手寫識別問題回顧

基于SVM的數字識別

收集數據：提供的文本文件

準備數據：基于二值圖像構造向量

分析數據：對圖像向量進行目測

訓練算法：采用兩種不同的核函數，并對徑向基核函數采用不同的設置來運行SMO算法。

測試算法：編寫一個函數來測試不同的核函數并計算錯誤率

使用算法：一個圖像識別的完整應用還需要一些圖像處理的知識，這里不再介紹。

# -*- coding: utf-8 -*- """ Created on Sun Oct 22 09:44:54 2017""" # -*- coding: utf-8 -*- """ Created on Sun Oct 22 09:44:54 2017@author: LiLong """ 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 while (j==i):j = int(random.uniform(0,m))return jdef clipAlpha(aj,H,L):if aj > H: aj = Hif L > aj:aj = Lreturn aj# 核轉換函數 def kernelTrans(X, A, kTup): # X是所有的支持向量，A是數據集每個樣本向量，KTup是核函數的參數元組m,n = shape(X)K = mat(zeros((m,1))) # 構建一個列向量if kTup[0]=='lin': K = X * A.T # 如果核函數是線性核函數，elif kTup[0]=='rbf': # 如果是徑向基核函數for j in range(m):deltaRow = X[j,:] - AK[j] = deltaRow*deltaRow.TK = exp(K/(-1*kTup[1]**2)) # 計算高斯核函數,注意元素間的除法else: raise NameError('Houston We Have a Problem -- \That Kernel is not recognized')return Kclass optStruct:def __init__(self,dataMatIn, classLabels, C, toler, kTup): 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))) 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 Ekdef selectJ(i, oS, Ei): maxK = -1; maxDeltaE = 0; Ej = 0oS.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):maxK = k; maxDeltaE = deltaE; Ej = Ekreturn maxK, Ejelse: j = selectJrand(i, oS.m)Ej = calcEk(oS, j)return j, Ejdef updateEk(oS, k):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) 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 0eta = 2.0 * oS.K[i,j] - oS.K[i,i] - oS.K[j,j] # 不同于不使用核函數的代碼if 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) if (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])updateEk(oS, i) 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 = 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)): oS = optStruct(mat(dataMatIn),mat(classLabels).transpose(),C,toler, kTup)iter = 0entireSet = True; alphaPairsChanged = 0while (iter < maxIter) and ((alphaPairsChanged > 0) or (entireSet)):alphaPairsChanged = 0if entireSet: for i in range(oS.m): alphaPairsChanged += innerL(i,oS)print "fullSet, iter: %d i:%d, pairs changed %d" % (iter,i,alphaPairsChanged)iter += 1else: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 += 1if entireSet: entireSet = False elif (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 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) m = len(trainingFileList)trainingMat = zeros((m,1024))for i in range(m):fileNameStr = trainingFileList[i]fileStr = fileNameStr.split('.')[0] 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 = 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) testDigits(('rbf', 20))

函數loadImages（）是已經被重構為自身的一個函數，參考：KNN識別手寫體數字
其中有一個很大的區別是。在前面的KNN識別手寫體中直接應用類別標簽，既數字向量對應相應的數字大小，而同支持向量機一起使用時，類別標簽為-1或者+1,因此，一旦碰到數字9，則輸出類別標簽-1，否則輸出+1。因為這里只做二分類，因此除了1和9之外的數字都被去掉了。

運行結果：

. . . 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 L==H fullSet, iter: 3 i:401, pairs changed 0 iteration number: 4 there are 47 Support Vectors the training error rate is: 0.004975 the test error rate is: 0.016129

嘗試不同的σ值，并嘗試線性核函數，得到：

偷了個懶，這里是機器學習實戰中的結果。分析：

在σ取10左右時可以得到最小的測試錯誤率

• 數據的不同，特征數的不同，得到的最佳σ<script type="math/tex" id="MathJax-Element-816">σ</script>值就會不同
• 最小的訓練錯誤率并不對應于最小的支持向量數目
• 線性核函數的效果并不是特別的糟糕。

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