diff --git a/docs/6.支持向量机.md b/docs/6.支持向量机.md index 11da7481..8b90446d 100644 --- a/docs/6.支持向量机.md +++ b/docs/6.支持向量机.md @@ -151,4 +151,21 @@ SMO伪代码大致如下: > SVM完整版:使用完整 Platt SMO算法加速优化 -* 优化点:选择alpha的方式不同。 \ No newline at end of file +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 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+00000000000001111111111111000000 +00000000000000011111101111100000 +00000000000000001111001111100000 +00000000000000000000001111100000 +00000001000000000000001111100000 +00000011100000000000001111100000 +00000011111000000000000111110000 +00000001111000000000001111100000 +00000001111000000000001111100000 +00000001111000000000001111100000 +00000001111100000000011111000000 +00000001111110000000011111000000 +00000000011111000000011111100000 +00000000011111100000011111100000 +00000000001111111111111111000000 +00000000000111111111111111000000 +00000000000111111111111110000000 +00000000000001111111111110000000 +00000000000000011111111100000000 diff --git a/src/python/6.SVM/svm-complete.py b/src/python/6.SVM/svm-complete.py index e2e0c0cc..4028a6eb 100644 --- a/src/python/6.SVM/svm-complete.py +++ b/src/python/6.SVM/svm-complete.py @@ -66,6 +66,7 @@ def kernelTrans(X, A, kTup): # calc the kernel or transform data to a higher di 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') @@ -360,6 +361,100 @@ def calcWs(alphas, dataArr, classLabels): 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: @@ -400,134 +495,28 @@ def plotfig_SVM(xArr, yArr, ws, b, alphas): 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) + # # 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) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -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)) + # 项目实战 + # 示例:手写识别问题回顾 + testDigits(('rbf', 20))