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《machine learning in action》机器学习算法学习笔记支持向量机

2022-02-06 09:04:29 作者：互联网

支持向量机（Support Vector Machine）

前置知识：拉格朗日数乘法、对偶问题、核技巧

拉格朗日数乘法

针对的是约束优化问题：

例题：

已知x>0,y>0,x+2y+2xy=8,则x+2y的最小值__。

解：

引入参数 λ \lambda λ 构造新函数L： x + 2 y + λ ( x + 2 y + 2 x y − 8 ) x+2y+\lambda(x+2y+2xy-8) x+2y+λ(x+2y+2xy−8)

分别对x,y, λ \lambda λ求偏导：
L x = 1 + λ ( 1 + 2 y ) = 0 L y = 2 + λ ( 2 + 2 x ) = 0 L λ = x + 2 y + 2 x y − 8 = 0 L_x = 1+\lambda(1+2y)=0\\ L_y = 2+\lambda(2+2x)=0\\ \ \ \ \ \ \ L_\lambda = x+2y+2xy-8=0\\ Lx=1+λ(1+2y)=0Ly=2+λ(2+2x)=0 Lλ=x+2y+2xy−8=0
三个方程三个未知数，可以求得x=2,y=1。

即当x=2,y=1时x+2y的最小值为4。

对偶问题

用于对优化问题的转换

例如：maxmin -> minmax

默认约束优化问题是弱对偶关系，当满足KNN条件时，具有强对偶关系，即二者等价。

核技巧

用于对高维特征的扩展，当样本数或超平面维数过大时，可以利用核技巧优化，将问题转化为有限维问题。

Machine Learning action

本章学习的是支持向量机中最流行的一种实现–序列最小优化（Sequential Minimal Optimization）算法。

优点：泛化错误率低，计算开销不大，结果容易理解
缺点：对参数调节和和函数的选择敏感，原始分类器不加修改仅适用于处理二分类问题。
适用数据类型：数值型和标称型数据

原理性部分

分割超平面集将不同类别的数据点分割的平面，支持向量机就是由这些分割超平面组成的分类器。

**支持向量(support vector)是指离分割超平面(separating byperplane)**最近的那些点，令支持向量的间隔最大化，就是构造支持向量机的优化目标

在这里插入图片描述

图中a,b,c都可以认为是一个分割超平面，显然，超分割平面b的鲁棒性即泛化能力要优于a,c。

如何建立数理模型来寻找到优质超分割平面？

那么就需要定义一个优化目标，即距离超分割面最近的那些的间隔最大化。
a r g m a x w , b { m i n ( w T ∗ x + b ) } arg\ max_{w,b}\ \{ min(w^T*x+b) \} arg maxw,b {min(wT∗x+b)}
其中w,b就是待优化的参数，假如直接求解该模型是十分困难的一件事，于是开始了一系列的模型转换，感兴趣的可以结合文章开头的视频以及西瓜书进一步学习，这里只将流程梳理一遍。

利用拉格朗朗日乘子法，得到拉格朗日方程组
将拉格朗日函数进行转换，转化为 a r g m i n m a x arg\ min max arg minmax问题
利用对偶问题的强对偶条件(KKT条件)将 a r g m i n m a x arg\ minmax arg minmax转化为 a r g m a x m i n arg\ maxmin arg maxmin问题
由于样本空间的非必线性可分，因此增加松弛变量

同时问题的参数也变为 α \alpha α,接下来便是对该问题进行求解

机器学习实战中使用的方法时SMO算法。

SMO算法伪代码：

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
	如果所有向量都没有被优化，增加迭代数目，继续下一次循环

实际上SMO算法是一个较为有效的贪心算法。

code

'''
Created on Nov 4, 2010
Chapter 5 source file for Machine Learing in Action
@author: Peter
'''
from numpy import *
from time import sleep


def 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, labelMat


def selectJrand(i, m):
    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):
    if aj > H:
        aj = H
    if L > aj:
        aj = L
    return aj


def 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 = 0
    while (iter < maxIter):
        alphaPairsChanged = 0
        for i in range(m):
            fXi = float(multiply(alphas, labelMat).T * (dataMatrix * dataMatrix[i, :].T)) + b
            Ei = fXi - float(labelMat[i])  # if checks if an example violates KKT conditions
            if ((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)) + b
                Ej = 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")
                    continue
                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
                alphas[j] -= labelMat[j] * (Ei - Ej) / eta
                alphas[j] = clipAlpha(alphas[j], H, L)
                if (abs(alphas[j] - alphaJold) < 0.00001):
                    print("j not moving enough")
                    continue
                alphas[i] += labelMat[j] * labelMat[i] * (alphaJold - alphas[j])  # update i by the same amount as j
                # the update is in the oppostie direction
                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))
        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
    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:
    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 = 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
        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 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.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)  # 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.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)):  # full Platt SMO
    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:  # 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 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


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 = []
    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

import svm as svmMLiA
import numpy as np
import matplotlib.pyplot as plt

dataArr,labelArr =svmMLiA.loadDataSet('testSet.txt')
# 数据集、类别标签、常数C、容错率、退出最大循环数
b,alphas= svmMLiA.smoSimple(dataArr,labelArr,0.6,0.001,40)

plt.rcParams['font.sans-serif']=['SimHei']
plt.rcParams['axes.unicode_minus'] = False
# matplotlib画图中中文显示会有问题，需要这两行设置默认字体
plt.xlabel('X')
plt.ylabel('Y')
plt.xlim(xmax=12,xmin=-2.5)
plt.ylim(ymax=10,ymin=-10)
# 画两条（0-9）的坐标轴并设置轴标签x，y
x1=list();x2=list()
y1=list();y2=list()
for j in range(len(labelArr)):
    if labelArr[j]==1:
        x1.append(dataArr[j][0])
        y1.append(dataArr[j][1])
    else :
        x2.append(dataArr[j][0])
        y2.append(dataArr[j][1])
print(x1)
print(y1)
colors1 = '#00CED1'
colors2 = '#DC143C'
area = np.pi*4**2

plt.scatter(x1,y1,s=area,c=colors1,alpha=0.4,label='类别A')
plt.scatter(x2,y2,s=area,c=colors2,alpha=0.4,label='类别B')
plt.plot()
plt.legend()
plt.savefig('1.png',dpi=300)
plt.show()

结语

SVM是经过较严格的数理证明的产物。
其特征提取方式教为单一，当数据集的分布过于类似时难以划分。

标签：alphas,return,self,iter,machine,labelMat,learning,action,oS
来源： https://blog.csdn.net/m0_52186223/article/details/122795635

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《machine learning in action》机器学习算法学习笔记支持向量机