diff --git a/docs/6.支持向量机.md b/docs/6.支持向量机.md index 277c79e6..8b90446d 100644 --- a/docs/6.支持向量机.md +++ b/docs/6.支持向量机.md @@ -115,3 +115,57 @@ SVM的一般流程 测试算法：十分简单的计算过程就可以实现。 使用算法：几乎所有分类问题都可以使用SVM，值得一提的是，SVM本身是一个二类分类器，对多类问题应用SVM需要对代码做一些修改。 ``` +* 到目前为止，我们已经了解了一些理论知识，现在我们通过`Code`来实现我们的算法吧。 + +## SMO高效优化算法 + +> 序列最小优化(Sequential Minimal Optimization, SMO) + +* 创建作者：John Platt +* 创建时间：1996年 +* SMO用途：用于训练SVM +* SMO目标：求出一系列alpha和b,一旦求出alpha，就很容易计算出权重向量w并得到分隔超平面。 +* SMO思想：是讲大优化问题分解为多个小优化问题来求解的。 +* SMO原理：每次循环选择两个alpha进行优化处理，一旦找出一对合适的alpha，那么就增大一个同时减少一个。 + * 这里指的合适必须要符合一定的条件 + * 1.这两个alpha必须要在间隔边界之外 + * 2.这两个alpha还没有进行过区间化处理或者不在边界上。 + * 之所以要同时改变2个alpha；原因，我们有一个约束条件：`Σ a[i]*label(i)=0`；如果只是修改一个alpha，很可能导致约束条件失效。 + +``` +SMO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 + 同时优化这两个向量 + 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 +``` + +> SVM简化版：应用简化版SMO算法处理小规模数据集 + +代码可参考 svm-simple.py + +> SVM完整版：使用完整 Platt SMO算法加速优化 + +代码可参考 svm-complete_Non-Kernel.py +* 优化点：选择alpha的方式不同。 + +## 在复杂数据上应用核函数 + +* 对于线性可分的情况，效果明显 +* 对于非线性的情况也一样，此时需要用到一种叫`核函数(kernel)`的工具将数据转化为分类器易于理解的形式。 + +> 利用核函数将数据映射到高维空间 + +* 使用核函数：可以将数据从某个特征空间到另一个特征空间的映射。（通常情况下：这种映射会将低维特征空间映射到高维空间。） +* 如果觉得特征空间很装逼、很难理解。 +* 可以把核函数想象成一个包装器(wrapper)或者是接口(interface)，它能将数据从某个很难处理的形式转换成为另一个较容易处理的形式。 +* 经过空间转换后：低维需要解决的非线性问题，就变成了高维需要解决的线性问题。 +* SVM优化特别好的地方，在于所有的运算都可以写成内积(inner product: 是指2个向量相乘，得到单个标量 或者 数值)；内核替换成核函数的方式被称为`核技巧(kernel trick)`或者`核"变电"(kernel substation)` +* 核函数并不仅仅应用于支持向量机，很多其他的机器学习算法也都用到核函数。最流行的核函数：径向基函数(radial basis function) +* 径向基函数的高斯版本，其具体的公式为： +* ![径向基函数的高斯版本](/images/6.SVM/SVM_6_radial-basis-function.jpg) diff --git a/images/6.SVM/SVM_6_radial-basis-function.jpg b/images/6.SVM/SVM_6_radial-basis-function.jpg new file mode 100644 index 00000000..3b7e680e Binary files /dev/null and b/images/6.SVM/SVM_6_radial-basis-function.jpg differ diff --git a/input/6.SVM/testDigits/1_0.txt b/input/6.SVM/testDigits/1_0.txt new file mode 100755 index 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b/input/6.SVM/trainingDigits/9_99.txt @@ -0,0 +1,32 @@ +00000000000000111111000000000000 +00000000000001111111100000000000 +00000000000111111111110000000000 +00000000000111111111111100000000 +00000000000111111111111100000000 +00000000001111111111111110000000 +00000000001111111011111110000000 +00000000001111100001111100000000 +00000000000111110001111111000000 +00000000000111110011111111000000 +00000000000111111111111111000000 +00000000000111111111111111000000 +00000000000011111111111111000000 +00000000000001111111111111000000 +00000000000000011111101111100000 +00000000000000001111001111100000 +00000000000000000000001111100000 +00000001000000000000001111100000 +00000011100000000000001111100000 +00000011111000000000000111110000 +00000001111000000000001111100000 +00000001111000000000001111100000 +00000001111000000000001111100000 +00000001111100000000011111000000 +00000001111110000000011111000000 +00000000011111000000011111100000 +00000000011111100000011111100000 +00000000001111111111111111000000 +00000000000111111111111111000000 +00000000000111111111111110000000 +00000000000001111111111110000000 +00000000000000011111111100000000 diff --git a/src/python/1.MLFoundation/NumPy.py b/src/python/1.MLFoundation/NumPy.py index 4a4721fe..8aa3c9cd 100644 --- a/src/python/1.MLFoundation/NumPy.py +++ b/src/python/1.MLFoundation/NumPy.py @@ -27,6 +27,7 @@ randMat = mat(randArray) https://www.zhihu.com/question/33258489 http://blog.csdn.net/vernice/article/details/48506027 .T 表示对矩阵转置(行列颠倒) + * 等同于: .transpose() .A 返回矩阵基于的数组 参考案例链接： http://blog.csdn.net/qq403977698/article/details/47254539 diff --git a/src/python/6.SVM/svm-complete.py b/src/python/6.SVM/svm-complete.py new file mode 100644 index 00000000..4028a6eb --- /dev/null +++ b/src/python/6.SVM/svm-complete.py @@ -0,0 +1,522 @@ +#!/usr/bin/python +# coding:utf8 + +""" +Created on Nov 4, 2010 +Update on 2017-03-21 +Chapter 5 source file for Machine Learing in Action +@author: Peter/geekidentity/片刻 +""" +from numpy import * +import matplotlib.pyplot as plt + + +class optStruct: + """ + 建立的数据结构来保存所有的重要值 + """ + def __init__(self, dataMatIn, classLabels, C, toler, kTup): + """ + Args: + dataMatIn 数据集 + classLabels 类别标签 + C 松弛变量(常量值)，允许有些数据点可以处于分隔面的错误一侧。 + 控制最大化间隔和保证大部分的函数间隔小于1.0这两个目标的权重。 + 可以通过调节该参数达到不同的结果。 + toler 容错率 + kTup 包含核函数信息的元组 + """ + + self.X = dataMatIn + self.labelMat = classLabels + self.C = C + self.tol = toler + + # 数据的行数 + self.m = shape(dataMatIn)[0] + self.alphas = mat(zeros((self.m, 1))) + self.b = 0 + + # 误差缓存，第一列给出的是eCache是否有效的标志位，第二列给出的是实际的E值。 + self.eCache = mat(zeros((self.m, 2))) + + # m行m列的矩阵 + 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 kernelTrans(X, A, kTup): # calc the kernel or transform data to a higher dimensional space + """ + 核转换函数 + Args: + X dataMatIn数据集 + A dataMatIn数据集的第i行的数据 + kTup 核函数的信息 + + Returns: + + """ + m, n = shape(X) + K = mat(zeros((m, 1))) + if kTup[0] == 'lin': + # linear kernel: m*n * n*1 = m*1, 求y值 + K = X * A.T + elif kTup[0] == 'rbf': + for j in range(m): + deltaRow = X[j, :] - A + K[j] = deltaRow * deltaRow.T + # 径向基函数的高斯版本 + K = 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 + + +def loadDataSet(fileName): + """loadDataSet（对文件进行逐行解析，从而得到第行的类标签和整个数据矩阵） + + Args: + fileName 文件名 + Returns: + dataMat 数据矩阵 + labelMat 类标签 + """ + 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 calcEk(oS, k): + """calcEk（求 Ek误差：预测值-真实值的差） + + 该过程在完整版的SMO算法中陪出现次数较多，因此将其单独作为一个方法 + Args: + oS optStruct对象 + k 具体的某一行 + + Returns: + + """ + fXk = float(multiply(oS.alphas, oS.labelMat).T * oS.K[:, k] + oS.b) + Ek = fXk - float(oS.labelMat[k]) + return Ek + + +def selectJrand(i, m): + """ + 随机选择一个整数 + Args: + i 第一个alpha的下标 + m 所有alpha的数目 + Returns: + j 返回一个不为i的随机数，在0~m之间的整数值 + """ + j = i + while j == i: + j = int(random.uniform(0, m)) + return j + + +def selectJ(i, oS, Ei): # this is the second choice -heurstic, and calcs Ej + """selectJ（返回最优的j和Ej） + + 内循环的启发式方法。 + 选择第二个(内循环)alpha的alpha值 + 这里的目标是选择合适的第二个alpha值以保证每次优化中采用最大步长。 + 该函数的误差与第一个alpha值Ei和下标i有关。 + Args: + i 具体的第i一行 + oS optStruct对象 + Ei 预测结果与真实结果比对，计算误差Ei + + Returns: + j 随机选出的第j一行 + Ej 预测结果与真实结果比对，计算误差Ej + """ + maxK = -1 + maxDeltaE = 0 + Ej = 0 + # # 首先将输入值Ei在缓存中设置成为有效的。这里的有效意味着它已经计算好了。 + oS.eCache[i] = [1, Ei] + + # print 'oS.eCache[%s]=%s' % (i, oS.eCache[i]) + # print 'oS.eCache[:, 0].A=%s' % oS.eCache[:, 0].A.T + # """ + # # 返回非0的：行列值 + # nonzero(oS.eCache[:, 0].A)= ( + # 行： array([ 0, 2, 4, 5, 8, 10, 17, 18, 20, 21, 23, 25, 26, 29, 30, 39, 46,52, 54, 55, 62, 69, 70, 76, 79, 82, 94, 97]), + # 列： array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,0, 0, 0, 0, 0]) + # ) + # """ + # print 'nonzero(oS.eCache[:, 0].A)=', nonzero(oS.eCache[:, 0].A) + # # 取行的list + # print 'nonzero(oS.eCache[:, 0].A)[0]=', nonzero(oS.eCache[:, 0].A)[0] + # 非零E值的行的list列表，所对应的alpha值 + validEcacheList = nonzero(oS.eCache[:, 0].A)[0] + if (len(validEcacheList)) > 1: + for k in validEcacheList: # 在所有的值上进行循环，并选择其中使得改变最大的那个值 + if k == i: + continue # don't calc for i, waste of time + + # 求 Ek误差：预测值-真实值的差 + Ek = calcEk(oS, k) + deltaE = abs(Ei - Ek) + if (deltaE > maxDeltaE): + # 选择具有最大步长的j + maxK = k + maxDeltaE = deltaE + Ej = Ek + return maxK, Ej + else: # 如果是第一次循环，则随机选择一个alpha值 + j = selectJrand(i, oS.m) + + # 求 Ek误差：预测值-真实值的差 + Ej = calcEk(oS, j) + return j, Ej + + +def updateEk(oS, k): + """updateEk（计算误差值并存入缓存中。） + + 在对alpha值进行优化之后会用到这个值。 + Args: + oS optStruct对象 + k 某一列的行号 + """ + + # 求 误差：预测值-真实值的差 + Ek = calcEk(oS, k) + oS.eCache[k] = [1, Ek] + + +def clipAlpha(aj, H, L): + """clipAlpha(调整aj的值，使aj处于 L<=aj<=H) + Args: + aj 目标值 + H 最大值 + L 最小值 + Returns: + aj 目标值 + """ + if aj > H: + aj = H + if L > aj: + aj = L + return aj + + +def innerL(i, oS): + """innerL + 内循环代码 + Args: + i 具体的某一行 + oS optStruct对象 + + Returns: + 0 找不到最优的值 + 1 找到了最优的值，并且oS.Cache到缓存中 + """ + + # 求 Ek误差：预测值-真实值的差 + Ei = calcEk(oS, i) + + # 约束条件 (KKT条件是解决最优化问题的时用到的一种方法。我们这里提到的最优化问题通常是指对于给定的某一函数，求其在指定作用域上的全局最小值。) + # 0<=alphas[i]<=C，但由于0和C是边界值，我们无法进行优化，因为需要增加一个alphas和降低一个alphas。 + # 表示发生错误的概率：labelMat[i]*Ei 如果超出了 toler， 才需要优化。至于正负号，我们考虑绝对值就对了。 + 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)): + # 选择最大的误差对应的j进行优化。效果更明显 + j, Ej = selectJ(i, oS, Ei) + alphaIold = oS.alphas[i].copy() + alphaJold = oS.alphas[j].copy() + + # L和H用于将alphas[j]调整到0-C之间。如果L==H，就不做任何改变，直接return 0 + 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是alphas[j]的最优修改量，如果eta==0，需要退出for循环的当前迭代过程 + # 如果ETA为0，那么计算新的alphas[j]就比较麻烦了, 为什么呢？ 因为2个值一样。 + # 2ab <= a^2 + b^2 + 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 + + # 计算出一个新的alphas[j]值 + oS.alphas[j] -= oS.labelMat[j] * (Ei - Ej) / eta + # 并使用辅助函数，以及L和H对其进行调整 + oS.alphas[j] = clipAlpha(oS.alphas[j], H, L) + # 更新误差缓存 + updateEk(oS, j) + + # 检查alpha[j]是否只是轻微的改变，如果是的话，就退出for循环。 + if (abs(oS.alphas[j] - alphaJold) < 0.00001): + print("j not moving enough") + return 0 + + # 然后alphas[i]和alphas[j]同样进行改变，虽然改变的大小一样，但是改变的方向正好相反 + oS.alphas[i] += oS.labelMat[j] * oS.labelMat[i] * (alphaJold - oS.alphas[j]) + # 更新误差缓存 + updateEk(oS, i) + + # 在对alpha[i], alpha[j] 进行优化之后，给这两个alpha值设置一个常数b。 + # w= Σ[1~n] ai*yi*xi => b = yi- Σ[1~n] ai*yi(xi*xj) + # 所以： b1 - b = (y1-y) - Σ[1~n] yi*(a1-a)*(xi*x1) + 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)): + """ + 完整SMO算法外循环，与smoSimple有些类似，但这里的循环退出条件更多一些 + Args: + dataMatIn 数据集 + classLabels 类别标签 + C 松弛变量(常量值)，允许有些数据点可以处于分隔面的错误一侧。 + 控制最大化间隔和保证大部分的函数间隔小于1.0这两个目标的权重。 + 可以通过调节该参数达到不同的结果。 + toler 容错率 + maxIter 退出前最大的循环次数 + kTup 包含核函数信息的元组 + Returns: + b 模型的常量值 + alphas 拉格朗日乘子 + """ + + # 创建一个 optStruct 对象 + oS = optStruct(mat(dataMatIn), mat(classLabels).transpose(), C, toler, kTup) + iter = 0 + entireSet = True + alphaPairsChanged = 0 + + # 循环遍历：循环maxIter次 并且 （alphaPairsChanged存在可以改变 or 所有行遍历一遍） + while (iter < maxIter) and ((alphaPairsChanged > 0) or (entireSet)): + alphaPairsChanged = 0 + + # 当entireSet=true or 非边界alpha对没有了；就开始寻找 alpha对，然后决定是否要进行else。 + if entireSet: + # 在数据集上遍历所有可能的alpha + for i in range(oS.m): + # 是否存在alpha对，存在就+1 + alphaPairsChanged += innerL(i, oS) + print("fullSet, iter: %d i:%d, pairs changed %d" % (iter, i, alphaPairsChanged)) + iter += 1 + + # 对已存在 alpha对，选出非边界的alpha值，进行优化。 + else: + # 遍历所有的非边界alpha值，也就是不在边界0或C上的值。 + 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 + + # 如果找到alpha对，就优化非边界alpha值，否则，就重新进行寻找，如果寻找一遍 遍历所有的行还是没找到，就退出循环。 + 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): + """ + 基于alpha计算w值 + Args: + alphas 拉格朗日乘子 + dataArr feature数据集 + classLabels 目标变量数据集 + + Returns: + wc 回归系数 + """ + 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 + + +def testRbf(k1=1.3): + dataArr, labelArr = loadDataSet('input/6.SVM/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)) + + # 和这个svm-simple类似： fXi = float(multiply(alphas, labelMat).T*(dataMatrix*dataMatrix[i, :].T)) + b + 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('input/6.SVM/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)) + + +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 = [] + print(dirName) + 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)): + + # 1. 导入训练数据 + dataArr, labelArr = loadImages('input/6.SVM/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)) + + # 2. 导入测试数据 + dataArr, labelArr = loadImages('input/6.SVM/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)) + + +def plotfig_SVM(xArr, yArr, ws, b, alphas): + """ + 参考地址： + http://blog.csdn.net/maoersong/article/details/24315633 + http://www.cnblogs.com/JustForCS/p/5283489.html + http://blog.csdn.net/kkxgx/article/details/6951959 + """ + + xMat = mat(xArr) + yMat = mat(yArr) + + # b原来是矩阵，先转为数组类型后其数组大小为（1,1），所以后面加[0]，变为(1,) + b = array(b)[0] + fig = plt.figure() + ax = fig.add_subplot(111) + + # 注意flatten的用法 + ax.scatter(xMat[:, 0].flatten().A[0], xMat[:, 1].flatten().A[0]) + + # x最大值，最小值根据原数据集dataArr[:, 0]的大小而定 + x = arange(-1.0, 10.0, 0.1) + + # 根据x.w + b = 0 得到，其式子展开为w0.x1 + w1.x2 + b = 0, x2就是y值 + y = (-b-ws[0, 0]*x)/ws[1, 0] + ax.plot(x, y) + + for i in range(shape(yMat[0, :])[1]): + if yMat[0, i] > 0: + ax.plot(xMat[i, 0], xMat[i, 1], 'cx') + else: + ax.plot(xMat[i, 0], xMat[i, 1], 'kp') + + # 找到支持向量，并在图中标红 + for i in range(100): + if alphas[i] > 0.0: + ax.plot(xMat[i, 0], xMat[i, 1], 'ro') + plt.show() + + +if __name__ == "__main__": + + # # 无核函数的测试 + # # 获取特征和目标变量 + # dataArr, labelArr = loadDataSet('input/6.SVM/testSet.txt') + # # print labelArr + + # # b是常量值， alphas是拉格朗日乘子 + # b, alphas = smoP(dataArr, labelArr, 0.6, 0.001, 40) + # print '/n/n/n' + # print 'b=', b + # print 'alphas[alphas>0]=', alphas[alphas > 0] + # print 'shape(alphas[alphas > 0])=', shape(alphas[alphas > 0]) + # for i in range(100): + # if alphas[i] > 0: + # print dataArr[i], labelArr[i] + # # 画图 + # ws = calcWs(alphas, dataArr, labelArr) + # plotfig_SVM(dataArr, labelArr, ws, b, alphas) + + # # 有核函数的测试 + # testRbf(1) + + # 项目实战 + # 示例：手写识别问题回顾 + testDigits(('rbf', 20)) diff --git a/src/python/6.SVM/svm-complete_Non-Kernel.py b/src/python/6.SVM/svm-complete_Non-Kernel.py new file mode 100644 index 00000000..25dfdaad --- /dev/null +++ b/src/python/6.SVM/svm-complete_Non-Kernel.py @@ -0,0 +1,259 @@ +#!/usr/bin/python +# coding:utf8 + +""" +Created on Nov 4, 2010 +Update on 2017-03-21 +Chapter 5 source file for Machine Learing in Action +@author: Peter/geekidentity/片刻 +""" +from numpy import * +import matplotlib.pyplot as plt + + +def loadDataSet(fileName): + """loadDataSet（对文件进行逐行解析，从而得到第行的类标签和整个数据矩阵） + + Args: + fileName 文件名 + Returns: + dataMat 数据矩阵 + labelMat 类标签 + """ + 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): + """ + 随机选择一个整数 + Args: + i 第一个alpha的下标 + m 所有alpha的数目 + Returns: + j 返回一个不为i的随机数，在0~m之间的整数值 + """ + j = i + while j == i: + j = int(random.uniform(0, m)) + return j + + +def clipAlpha(aj, H, L): + """clipAlpha(调整aj的值，使aj处于 L<=aj<=H) + Args: + aj 目标值 + H 最大值 + L 最小值 + Returns: + aj 目标值 + """ + if aj > H: + aj = H + if L > aj: + aj = L + return aj + + +def calcWs(alphas, dataArr, classLabels): + """ + 基于alpha计算w值 + Args: + alphas 拉格朗日乘子 + dataArr feature数据集 + classLabels 目标变量数据集 + + Returns: + wc 回归系数 + """ + 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 + + +''' +#######******************************** +Non-Kernel VErsions below +#######******************************** +''' + +class optStruct: + def __init__(self, dataMatIn, classLabels, C, toler): # 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 + + +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): # 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): + 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.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) # 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.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): # full Platt SMO + oS = optStruct(mat(dataMatIn), mat(classLabels).transpose(), C, toler) + 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 plotfig_SVM(xArr, yArr, ws, b, alphas): + """ + 参考地址： + http://blog.csdn.net/maoersong/article/details/24315633 + http://www.cnblogs.com/JustForCS/p/5283489.html + http://blog.csdn.net/kkxgx/article/details/6951959 + """ + + xMat = mat(xArr) + yMat = mat(yArr) + + # b原来是矩阵，先转为数组类型后其数组大小为（1,1），所以后面加[0]，变为(1,) + b = array(b)[0] + fig = plt.figure() + ax = fig.add_subplot(111) + + # 注意flatten的用法 + ax.scatter(xMat[:, 0].flatten().A[0], xMat[:, 1].flatten().A[0]) + + # x最大值，最小值根据原数据集dataArr[:, 0]的大小而定 + x = arange(-1.0, 10.0, 0.1) + + # 根据x.w + b = 0 得到，其式子展开为w0.x1 + w1.x2 + b = 0, x2就是y值 + y = (-b-ws[0, 0]*x)/ws[1, 0] + ax.plot(x, y) + + for i in range(shape(yMat[0, :])[1]): + if yMat[0, i] > 0: + ax.plot(xMat[i, 0], xMat[i, 1], 'cx') + else: + ax.plot(xMat[i, 0], xMat[i, 1], 'kp') + + # 找到支持向量，并在图中标红 + for i in range(100): + if alphas[i] > 0.0: + ax.plot(xMat[i, 0], xMat[i, 1], 'ro') + plt.show() + + +if __name__ == "__main__": + # 获取特征和目标变量 + dataArr, labelArr = loadDataSet('input/6.SVM/testSet.txt') + # print labelArr + + # b是常量值， alphas是拉格朗日乘子 + b, alphas = smoP(dataArr, labelArr, 0.6, 0.001, 40) + print '/n/n/n' + print 'b=', b + print 'alphas[alphas>0]=', alphas[alphas > 0] + print 'shape(alphas[alphas > 0])=', shape(alphas[alphas > 0]) + for i in range(100): + if alphas[i] > 0: + print dataArr[i], labelArr[i] + # 画图 + ws = calcWs(alphas, dataArr, labelArr) + plotfig_SVM(dataArr, labelArr, ws, b, alphas) diff --git a/src/python/6.SVM/svm-simple.py b/src/python/6.SVM/svm-simple.py new file mode 100644 index 00000000..e7699c3b --- /dev/null +++ b/src/python/6.SVM/svm-simple.py @@ -0,0 +1,244 @@ +#!/usr/bin/python +# coding:utf8 + +""" +Created on Nov 4, 2010 +Update on 2017-03-21 +Chapter 5 source file for Machine Learing in Action +@author: Peter/geekidentity/片刻 +""" +from numpy import * +import matplotlib.pyplot as plt + + +def loadDataSet(fileName): + """ + 对文件进行逐行解析，从而得到第行的类标签和整个数据矩阵 + Args: + fileName 文件名 + Returns: + dataMat 数据矩阵 + labelMat 类标签 + """ + 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): + """ + 随机选择一个整数 + Args: + i 第一个alpha的下标 + m 所有alpha的数目 + Returns: + j 返回一个不为i的随机数，在0~m之间的整数值 + """ + j = i + while j == i: + j = int(random.uniform(0, m)) + return j + + +def clipAlpha(aj, H, L): + """clipAlpha(调整aj的值，使aj处于 L<=aj<=H) + Args: + aj 目标值 + H 最大值 + L 最小值 + Returns: + aj 目标值 + """ + if aj > H: + aj = H + if L > aj: + aj = L + return aj + + +def smoSimple(dataMatIn, classLabels, C, toler, maxIter): + """smoSimple + + Args: + dataMatIn 数据集 + classLabels 类别标签 + C 松弛变量(常量值)，允许有些数据点可以处于分隔面的错误一侧。 + 控制最大化间隔和保证大部分的函数间隔小于1.0这两个目标的权重。 + 可以通过调节该参数达到不同的结果。 + toler 容错率 + maxIter 退出前最大的循环次数 + Returns: + b 模型的常量值 + alphas 拉格朗日乘子 + """ + dataMatrix = mat(dataMatIn) + # 矩阵转制 和 .T 一样的功能 + labelMat = mat(classLabels).transpose() + m, n = shape(dataMatrix) + + # 初始化 b和alphas(alpha有点类似权重值。) + b = 0 + alphas = mat(zeros((m, 1))) + + # 没有任何alpha改变的情况下遍历数据的次数 + iter = 0 + while (iter < maxIter): + # w = calcWs(alphas, dataMatIn, classLabels) + # print("w:", w) + + # 记录alpha是否已经进行优化，每次循环时设为0，然后再对整个集合顺序遍历 + alphaPairsChanged = 0 + for i in range(m): + # print 'alphas=', alphas + # print 'labelMat=', labelMat + # print 'multiply(alphas, labelMat)=', multiply(alphas, labelMat) + # 我们预测的类别 y = w^Tx[i]+b; 其中因为 w = Σ(1~n) a[n]lable[n]x[n] + fXi = float(multiply(alphas, labelMat).T*(dataMatrix*dataMatrix[i, :].T)) + b + # 预测结果与真实结果比对，计算误差Ei + Ei = fXi - float(labelMat[i]) + + # 约束条件 (KKT条件是解决最优化问题的时用到的一种方法。我们这里提到的最优化问题通常是指对于给定的某一函数，求其在指定作用域上的全局最小值。) + # 0<=alphas[i]<=C，但由于0和C是边界值，我们无法进行优化，因为需要增加一个alphas和降低一个alphas。 + # 表示发生错误的概率：labelMat[i]*Ei 如果超出了 toler， 才需要优化。至于正负号，我们考虑绝对值就对了。 + if ((labelMat[i]*Ei < -toler) and (alphas[i] < C)) or ((labelMat[i]*Ei > toler) and (alphas[i] > 0)): + + # 如果满足优化的条件，我们就随机选取非i的一个点，进行优化比较 + j = selectJrand(i, m) + # 预测j的结果 + fXj = float(multiply(alphas, labelMat).T*(dataMatrix*dataMatrix[j, :].T)) + b + Ej = fXj - float(labelMat[j]) + alphaIold = alphas[i].copy() + alphaJold = alphas[j].copy() + + # L和H用于将alphas[j]调整到0-C之间。如果L==H，就不做任何改变，直接执行continue语句 + 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是alphas[j]的最优修改量，如果eta==0，需要退出for循环的当前迭代过程 + # 如果ETA为0，那么计算新的alphas[j]就比较麻烦了, 为什么呢？ 因为2个值一样。 + # 2ab <= a^2 + b^2 + 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 + + # 计算出一个新的alphas[j]值 + alphas[j] -= labelMat[j]*(Ei - Ej)/eta + # 并使用辅助函数，以及L和H对其进行调整 + alphas[j] = clipAlpha(alphas[j], H, L) + # 检查alpha[j]是否只是轻微的改变，如果是的话，就退出for循环。 + if (abs(alphas[j] - alphaJold) < 0.00001): + print("j not moving enough") + continue + # 然后alphas[i]和alphas[j]同样进行改变，虽然改变的大小一样，但是改变的方向正好相反 + alphas[i] += labelMat[j]*labelMat[i]*(alphaJold - alphas[j]) + # 在对alpha[i], alpha[j] 进行优化之后，给这两个alpha值设置一个常数b。 + # w= Σ[1~n] ai*yi*xi => b = yi- Σ[1~n] ai*yi(xi*xj) + # 所以： b1 - b = (y1-y) - Σ[1~n] yi*(a1-a)*(xi*x1) + 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)) + # 在for循环外，检查alpha值是否做了更新，如果在更新则将iter设为0后继续运行程序 + if (alphaPairsChanged == 0): + iter += 1 + else: + iter = 0 + print("iteration number: %d" % iter) + return b, alphas + + +def calcWs(alphas, dataArr, classLabels): + """ + 基于alpha计算w值 + Args: + alphas 拉格朗日乘子 + dataArr feature数据集 + classLabels 目标变量数据集 + + Returns: + wc 回归系数 + """ + 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 + + +def plotfig_SVM(xMat, yMat, ws, b, alphas): + """ + 参考地址： + http://blog.csdn.net/maoersong/article/details/24315633 + http://www.cnblogs.com/JustForCS/p/5283489.html + http://blog.csdn.net/kkxgx/article/details/6951959 + """ + + xMat = mat(xMat) + yMat = mat(yMat) + + # b原来是矩阵，先转为数组类型后其数组大小为（1,1），所以后面加[0]，变为(1,) + b = array(b)[0] + fig = plt.figure() + ax = fig.add_subplot(111) + + # 注意flatten的用法 + ax.scatter(xMat[:, 0].flatten().A[0], xMat[:, 1].flatten().A[0]) + + # x最大值，最小值根据原数据集dataArr[:, 0]的大小而定 + x = arange(-1.0, 10.0, 0.1) + + # 根据x.w + b = 0 得到，其式子展开为w0.x1 + w1.x2 + b = 0, x2就是y值 + y = (-b-ws[0, 0]*x)/ws[1, 0] + ax.plot(x, y) + + for i in range(shape(yMat[0, :])[1]): + if yMat[0, i] > 0: + ax.plot(xMat[i, 0], xMat[i, 1], 'cx') + else: + ax.plot(xMat[i, 0], xMat[i, 1], 'kp') + + # 找到支持向量，并在图中标红 + for i in range(100): + if alphas[i] > 0.0: + ax.plot(xMat[i, 0], xMat[i, 1], 'ro') + plt.show() + + +if __name__ == "__main__": + # 获取特征和目标变量 + dataArr, labelArr = loadDataSet('input/6.SVM/testSet.txt') + # print labelArr + + # b是常量值， alphas是拉格朗日乘子 + b, alphas = smoSimple(dataArr, labelArr, 0.6, 0.001, 40) + print '/n/n/n' + print 'b=', b + print 'alphas[alphas>0]=', alphas[alphas > 0] + print 'shape(alphas[alphas > 0])=', shape(alphas[alphas > 0]) + for i in range(100): + if alphas[i] > 0: + print dataArr[i], labelArr[i] + # 画图 + ws = calcWs(alphas, dataArr, labelArr) + plotfig_SVM(dataArr, labelArr, ws, b, alphas) diff --git a/src/python/6.SVM/svmMLiA.py b/src/python/6.SVM/svmMLiA.py deleted file mode 100644 index 1694d6bd..00000000 --- a/src/python/6.SVM/svmMLiA.py +++ /dev/null @@ -1,570 +0,0 @@ -""" -Created on Nov 4, 2010 -Update on 2017-03-21 -Chapter 5 source file for Machine Learing in Action -@author: Peter/geekidentity -""" -from numpy import * -import pylab -from time import sleep - -def main(): - dataArr, labelArr = loadDataSet('testSet.txt') - smoSimple(dataArr, labelArr, 0.6, 0.001, 40) - -def loadDataSet(fileName): - """ - 对文件进行逐行解析，从而得到第行的类标签和整个数据矩阵 - Args: - fileName: testSet.txt - - Returns: - 数据矩阵, 类标签 - """ - 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): - """ - 随机选择一个整数 - Args: - i: 第一个alpha的下标 - m: 所有alpha的数目 - - Returns: - - """ - 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): - """ - 用于调整大于H或小于L的alpha值 - Args: - aj: - H: - L: - - Returns: - - """ - if aj > H: - aj = H - if L > aj: - aj = L - return aj - -def smoSimple(dataMatIn, classLabels, C, toler, maxIter): - """ - SVM SMO算法的简单实现: - 创建一个alpha向量并将其初始化为0向量 - 当迭代次数据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 - 同时优化这两个向量 - 如果两个向量都不能被优化，退出内循环 - 如果所有向量都没有被优化，增加迭代数目，继续下一次循环 - Args: - dataMatIn: 数据集 - classLabels: 类别标签 - C: 松弛变量，允许有些数据点可以处于分隔面的错误一侧。 - 控制最大化间隔和保证大部分的函数间隔小于1.0这两个目标的权重。 - 可以通过调节该参数达到不同的结果。 - toler: 容错率 - maxIter: 退出前最大的循环次数 - - Returns: - - """ - dataMatrix = mat(dataMatIn); labelMat = mat(classLabels).transpose() - b = 0; m,n = shape(dataMatrix) - alphas = mat(zeros((m,1))) - iter = 0 # 没有任何alpha改变的情况下遍历数据的次数 - while (iter < maxIter): - w = calcWs(alphas, dataMatIn, classLabels) - print("w:", w) - alphaPairsChanged = 0 #记录alpha是否已经进行优化，每次循环时设为0，然后再对整个集合顺序遍历 - for i in range(m): - fXi = float(multiply(alphas,labelMat).T*(dataMatrix*dataMatrix[i,:].T)) + b # 我们预测的类别 - Ei = fXi - float(labelMat[i])#检查是否违反KKT条件 误差：基于这个实例的预测结果和真实结果的比对，计算误差Ei 参考：http://blog.csdn.net/puqutogether/article/details/44587653 - if ((labelMat[i]*Ei < -toler) and (alphas[i] < C)) or ((labelMat[i]*Ei > toler) and (alphas[i] > 0)): #不管是正负间隔都会测试，同时检查alpha值，保证其不能等于0或C - j = selectJrand(i,m) # 误差很大时进行优化 - 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]): # 将alpha调整到0-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 = 2.0 * dataMatrix[i,:]*dataMatrix[j,:].T - dataMatrix[i,:]*dataMatrix[i,:].T - dataMatrix[j,:]*dataMatrix[j,:].T #最优修改量 - if eta >= 0: print("eta>=0"); continue # 如果ETA为0，那么计算新的alphas[j]就比较麻烦了 - alphas[j] -= labelMat[j]*(Ei - Ej)/eta - alphas[j] = clipAlpha(alphas[j],H,L) - # 检查alpha[j]是否有轻微的改变，如果是的话，就退出for循环。 - if (abs(alphas[j] - alphaJold) < 0.00001): print("j not moving enough"); continue - # 对alpha[i], alpha[j]同样进行改变，改变方向一样 - alphas[i] += labelMat[j]*labelMat[i]*(alphaJold - alphas[j])#update i by the same amount as j - #the update is in the oppostie direction - # 在对alpha[i], alpha[j] 进行优化之后，给这两个alpha值设置一个常数b。 - 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)) - # 在for循环外，检查alpha值是否做了更新，如果在更新则将iter设为0后继续运行程序 - if (alphaPairsChanged == 0): iter += 1 - else:iter = 0 - print("iteration number: %d" % iter) - return b,alphas - - -def kernelTrans(X, A, kTup): # calc the kernel or transform data to a higher dimensional space - """ - 核转换函数 - Args: - X: - A: - kTup: 核函数的信息 - - Returns: - - """ - m, n = shape(X) - K = mat(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 = 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: - """ - 建立的数据结构来保存所有的重要值 - """ - def __init__(self, dataMatIn, classLabels, C, toler, kTup): # Initialize the structure with the parameters - """ - - Args: - dataMatIn: - classLabels: - C: - toler: - kTup: 包含核函数信息的元组 - """ - self.X = dataMatIn - self.labelMat = classLabels - self.C = C - self.tol = toler - self.m = shape(dataMatIn)[0] # 数据的行数 - self.alphas = mat(zeros((self.m, 1))) - self.b = 0 - self.eCache = mat(zeros((self.m, 2))) # 误差缓存，第一列给出的是eCache是否有效的标志位，第二列给出的是实际的E值。 - 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值并返回 - 该过程在完整版的SMO算法中陪出现次数较多，因此将其单独作为一个方法 - Args: - oS: - k: - - Returns: - - """ - 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 - """ - 内循环的启发式方法。 - 选择第二个(内循环)alpha的alpha值 - 这里的目标是选择合适的第二个alpha值以保证每次优化中采用最大步长。 - 该函数的误差与第一个alpha值Ei和下标i有关。 - Args: - i: - oS: - Ei: - - Returns: - - """ - maxK = -1 - maxDeltaE = 0 - Ej = 0 - oS.eCache[i] = [1, Ei] # 首先将输入值Ei在缓存中设置成为有效的。这里的有效意味着它已经计算好了。 - validEcacheList = nonzero(oS.eCache[:, 0].A)[0] # 非零E值所对应的alpha值 - if (len(validEcacheList)) > 1: - for k in validEcacheList: # 在所有的值上进行循环，并选择其中使得改变最大的那个值 - if k == i: continue # don't calc for i, waste of time - Ek = calcEk(oS, k) - deltaE = abs(Ei - Ek) - if (deltaE > maxDeltaE): - # 选择具有最大步长的j - maxK = k - maxDeltaE = deltaE - Ej = Ek - return maxK, Ej - else: # 如果是第一次循环，则随机选择一个alpha值 - 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 - """ - 计算误差值并存入缓存中。 - 在对alpha值进行优化之后会用到这个值。 - Args: - oS: - k: - - Returns: - - """ - Ek = calcEk(oS, k) - oS.eCache[k] = [1, Ek] - - -def innerL(i, oS): - """ - 内循环代码 - Args: - i: - oS: - - Returns: - - """ - 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) # 更新误差缓存 - 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) # 更新误差缓存 #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)): - """ - 完整SMO算法外循环，与smoSimple有些类似，但这里的循环退出条件更多一些 - Args: - dataMatIn: - classLabels: - C: - toler: - maxIter: - kTup: - - Returns: - - """ - 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: # 在数据集上遍历所有可能的alpha - 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: # 遍历所有的非边界alpha值，也就是不在边界0或C上的值。 - 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): - """ - 基于alpha计算w值 - Args: - alphas: - dataArr: - classLabels: - - Returns: - - """ - 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 - - -def testRbf(k1=1.3): - 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)) - - -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 = [] - print(dirName) - 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)) - - -'''#######******************************** -Non-Kernel VErsions below -''' #######******************************** - - -class optStructK: - def __init__(self, dataMatIn, classLabels, C, toler): # 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 - - -def calcEkK(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 selectJK(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 updateEkK(oS, k): # after any alpha has changed update the new value in the cache - Ek = 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 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.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) # 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.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 smoPK(dataMatIn, classLabels, C, toler, maxIter): # full Platt SMO - oS = optStruct(mat(dataMatIn), mat(classLabels).transpose(), C, toler) - 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 - -if __name__ == "__main__": - main() \ No newline at end of file