这节和上一节很像，不同的是，上一篇的是通过支持向量和待分类数据內积进行分类的，只是这里不同的是，在计算內积时使用核函数进行代替，这里参考的是机器学习实战中的核函数，如果前面理解的比较深入，读代码还是很简单的，这里的代码建议不要刚开始就去读核函数定义，建议先从测试核函数的代码读，然后搞清楚核函数的参数都代表什么，同时想想作者为什么这样使用，刚开始理解这些可能有点难，多读几遍，在读代码时建议时刻想着前面的理论，因为代码就是按照前面的理论思路进行写的，最后就是尝试自己写一下，下面的代码关键的地方均已注释，上代码：
先把结果贴出来：
fullSet, iter: 8 i:97, pairs changed 0 fullSet, iter: 8 i:98, pairs changed 0 fullSet, iter: 8 i:99, pairs changed 0 iteration number: 9 there are 18 Support Vectors the training error rate is: 0.000000 the test error rate is: 0.050000源码：
#!/usr/bin/env/python # -*- coding: utf-8 -*- # Author: 赵守风 # File name: kernelfunction.py # Time:2018/10/25 # Email:1583769112@qq.com from numpy import * import numpy as np from SMO_simplify import * # 其实这里看定义函数有点不好理解，虽然理论大家都知道，但是实现方式可以由很多种，大家可以先看看这个函数是 # 如何调用的即核函数测试，就知道这个函数的参数的意义，下面我把测试关键代码拿过来便于分析 ''' 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]; 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 ''' # kernelEval = kernelTrans(sVs,datMat[i,:],('rbf', k1)) # 从这一句我们看到传入的第一个参数X就是sVs，往上追溯可知这是参与內积的训练数据点即支持向量 # 第二个参数A，传入的是datMat[i:0]，传入的是一条数据，即待分类的一个数据，原因是每个待分类的数据都需要和支持向量相內积 # kTup[0]第一个元素是选择核函数类型的参数，第二个kTup[1]就更简单了，其实就是径向基的那个参数，大家看这个就懂了 # K = exp(K / (-1 * kTup[1] ** 2)) def kernelTrans(X, A, kTup): # calc the kernel or transform data to a higher dimensional space m, n = np.shape(X) K = np.mat(np.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 = np.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 = np.shape(dataMatIn)[0] self.alphas = np.mat(zeros((self.m, 1))) self.b = 0 self.eCache = np.mat(zeros((self.m, 2))) # first column is valid flag self.K = np.mat(zeros((self.m, self.m))) for i in range(self.m): # 调用核函数处理 self.K[:, i] = kernelTrans(self.X, self.X[i, :], kTup) def calcEk(oS, k): fXk = float(np.multiply(oS.alphas, oS.labelMat).T * oS.K[:, k] + oS.b) Ek = fXk - float(oS.labelMat[k]) return Ek # 内循环调选第二个alpha的策略 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 = np.nonzero(oS.eCache[:, 0].A)[0] # 这个大家还记的啥意思吧，不解释了，不会的翻看前面的看 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 = np.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] # 内循环 def innerL(i, oS): Ei = calcEk(oS, i) # 和前面的一样，Ei为第一个alpha的差值 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) # 调选第二个参数 alphaIold = oS.alphas[i].copy() alphaJold = oS.alphas[j].copy() # 加入约束 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 # 这里就开始使用核函数了，2K12-K11-K22,这些都是核函数，在这里他就是通过调用核函数，而以前的是直接数据內积 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 # 更新alpha值 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 if (abs(oS.alphas[j] - alphaJold) < 0.00001): print("j not moving enough") return 0 # 更新另外一个alpha oS.alphas[i] += oS.labelMat[j] * oS.labelMat[i] * (alphaJold - oS.alphas[j]) # update i by the same amount as j updateEk(oS, i) # 更新缓存 #the update is in the oppostie direction # 计算b值,同时计算b值也使用到了核函数，大家看代码的时候，时刻回忆前面的理论分析过程，每一句代码都和理论保持同步 # 这样多看，多理解几次，思路就顺了 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 # 参数大多数和线性一样的，只是增加了一个kTup即可以选择线性lin也可以选择核函数rbf def smoP(dataMatIn, classLabels, C, toler, maxIter, kTup=('lin', 0)): # full Platt SMO # kTup参数主要是传给自定义的类，后面使用 oS = optStruct(np.mat(dataMatIn), np.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): # 外循环，遍历所有alppha参数即第一个alpha参数 alphaPairsChanged += innerL(i, oS) # 调用内循环，寻找是Ei-Ej差值最大的E，同时在inner使用核函数 print("fullSet, iter: %d i:%d, pairs changed %d" % (iter, i, alphaPairsChanged)) iter += 1 else: # go over non-bound (railed) alphas nonBoundIs = np.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 # 这个和前面是一样的，其实就是计算w def calcWs(alphas, dataArr, classLabels): X = np.mat(dataArr) labelMat = np.mat(classLabels).transpose() m, n = np.shape(X) w = np.zeros((n, 1)) for i in range(m): w += np.multiply(alphas[i] * labelMat[i], X[i, :].T) return w def testRbf(k1=1.3): dataArr, labelArr = loadDataSet('testSetRBF.txt') # 读取数据 # 计算 b和alpha b, alphas = smoP(dataArr, labelArr, 200, 0.0001, 10000, ('rbf', k1)) # C=200 important datMat = np.mat(dataArr); labelMat = np.mat(labelArr).transpose() svInd = np.nonzero(alphas.A > 0)[0] sVs = datMat[svInd] # get matrix of only support vectors labelSV = labelMat[svInd]; print("there are %d Support Vectors" % np.shape(sVs)[0]) m, n = np.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 = np.mat(dataArr); labelMat = np.mat(labelArr).transpose() m, n = np.shape(datMat) for i in range(m): kernelEval = kernelTrans(sVs, datMat[i, :], ('rbf', k1)) predict = kernelEval.T * np.multiply(labelSV, alphas[svInd]) + b if np.sign(predict) != np.sign(labelArr[i]): errorCount += 1 print("the test error rate is: %f" % (float(errorCount) / m))
