原理

• 寻找一个分割超平面来作为分类边界，找到离分割超平面最近的点，确保它们离分割超平面的距离尽可能远。
• 支持向量就是离分割超平面最近的那些点

优点：

• 泛化错误率低，计算开销不大，结果易解释。

缺点：

• 对参数调节和核函数的选择敏感，原始分类器不加修改仅适用于处理二类问题。

适用数据类型：

• 数值型和标称型数据。

简化版SMO算法

#加载数据
def loadData(path):
    #新建数据和标签列表
    dataList = [];labelList= []
    #获得文件指针
    fr = open(path)
    #一行一行读取
    for line in fr.readlines():
        #分割返回list
        lineList = line.strip().split()
        #取前1，2列作为训练数据
        dataList.append([float(lineList[0]),float(lineList[1])])
        #最后一列最为标签数据
        labelList.append(float(lineList[-1]))
    return dataList,labelList
dataList,labelList = loadData('../../Reference Code/Ch06/testSet.txt')

#随机选择alpha
import random
def selectJrand(i,m):     
    #这里可能随机取值会取到和i相等的值，为了j!=i，所以才先赋值j=1，再while循环
    j = i 
    while(j==i):
        j = int(random.uniform(0,m)) 
    return j

j = selectJrand(1,6)
# def selectJrand(m):
#     j = int(random.uniform(0,m))
#     return j
# j = selectJrand(6)
def clipAlpha(aj,H,L):
    if aj>H:
        aj = H
    if L>aj:
        aj = L
    return aj

import sys
print(sys.executable)

/home/ubuntu/anaconda3/bin/python

#简化版SMO
import numpy as np
def smoSimple(dataMatIn,classLabels,C,toler,maxIter):
    '''
    参数：
        dataMatIn:输入数据
        classLabels：标签
        C：惩罚项
        toler：错误容忍度
        maxIter：最大迭代次数
    返回：
        b:偏置项
        alphas：拉格朗日乘子
    '''
    #数据和标签转化为ndarray
    #等价于dataMatrix = mat(dataMatIn); labelMat = mat(classLabels).T
    dataArray = np.array(dataMatIn); labelArray = np.array(classLabels).reshape(-1,1)
    #初始化b，获得数据矩阵的行列
    b = 0; m,n = shape(dataArray)
    #初始化alphas全为0向量
    alphas = np.zeros((m,1))

    #初始化迭代次数
    Iter =0
    #当迭代次数小于最大迭代次数
    while (Iter<maxIter):
        #alpha对
        alphaPairsChanged = 0
        #遍历每一个实例
        for i in range(m):
            #计算g(xi)
            '''
            两个array的相乘*指的是对应元素的相乘；两个array的dot表示矩阵的相乘。
            a=np.array([1,2])
            b=np.array([1,2])
            print(a*b) # [1 4]
            print(np.dot(a,b)) #5
            '''
            fXi = float(np.dot((alphas*labelArray).T,np.dot(dataArray,dataArray[i:i+1,:].T))) + b
            #计算Ei
            Ei = fXi - float(labelArray[i])
            #找到违反KKT条件的实例
            if ((labelArray[i]*Ei < toler) and (alphas[i] < C)) \
            or ((labelArray[i] *Ei > toler) and (alphas[i] > 0)):
                #随机选择j，j!=i
                j = selectJrand(i,m)
                #计算g(xj)
                '''
                注意区别
                shape(dataArray[1,:]) #(2, )
                shape(dataArray[1,:].T)# (2, )
                shape(dataArray[1:2,:].T)(2, 1)
                '''
                fXj = float(np.dot((alphas*labelArray).T,np.dot(dataArray,dataArray[j:j+1,:].T))) + b
                #计算Ej
                Ej = fXj - float(labelArray[j])
                #赋值旧的alphai和alphaj
                alphaIold = alphas[i].copy()
                alphaJold = alphas[j].copy()
                #若yi!=yj
                if (labelArray[i] != labelArray[j]):
                    L = max(0, alphas[j] - alphas[i])
                    H = min(C, C + alphas[j] - alphas[i])
                #若yi=yj
                else:
                    L = max(0, alphas[j] + alphas[i] - C)
                    H = min(C, alphas[j] - alphas[i])
                if L==H:print('L==H');continue
                #计算eta，参考李航PP127
                eta = 2.0*np.dot(dataArray[i],dataArray[j]) \
                - np.dot(dataArray[i],dataArray[i])\
                - np.dot(dataArray[j],dataArray[j])
                if eta >= 0:print('eta>=0');continue
                #跟新alphaj
                alphas[j] -= labelArray[j]*(Ei - Ej)/eta
                #若alphaj>H，则取H，若alphaj<L，则取L，若L<alphaj<H,则取alphaj.
                alphas[j] = clipAlpha(alphas[j],H,L)
                #判断alphaj是否有足够大的变化
                if (abs(alphas[j] - alphaJold) < 0.00001):print('j not moving enough');continue
                #跟新alphai
                alphas[i] += labelArray[j]*labelArray[i]*(alphaJold - alphas[j])
                #重新计算阈值b
                b1 = b- Ei - labelArray[i]*(alphas[i] - alphaIold)*np.dot(dataArray[i],dataArray[i,:]) \
                - labelArray[j]*(alphas[j] - alphaJold) * np.dot(dataArray[i],dataArray[j])
                b2 = b- Ei - labelArray[i]*(alphas[i] - alphaIold)*np.dot(dataArray[i],dataArray[j]) \
                - labelArray[j]*(alphas[j] - alphaJold) * np.dot(dataArray[j],dataArray[j])
                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
                #alpha对加一
                alphaPairsChanged += 1
                print("循环次数: {} alpha:{}, alpha对修改了 {} 次".format(Iter,i,alphaPairsChanged))
        if(alphaPairsChanged == 0): Iter += 1
        else: Iter = 0
        print("迭代次数: {}".format(Iter))
    return b,alphas

dataList,labelList = loadData('../../Reference Code/Ch06/testSet.txt')
b, alphas = smoSimple(dataList, labelList, 0.6, 0.001, 40)
print('b= {}'.format(b))
print('（支持向量对应的alpha>0）alpha>0\n{}'.format(alphas[alphas>0]))

循环次数: 0 alpha:0, alpha对修改了 1 次
循环次数: 0 alpha:2, alpha对修改了 2 次
L==H
j not moving enough
循环次数: 0 alpha:6, alpha对修改了 3 次
j not moving enough
j not moving enough
j not moving enough
循环次数: 0 alpha:22, alpha对修改了 4 次
j not moving enough
L==H
j not moving enough
L==H
j not moving enough
j not moving enough
L==H
L==H
循环次数: 0 alpha:54, alpha对修改了 5 次
循环次数: 0 alpha:55, alpha对修改了 6 次
j not moving enough
L==H
j not moving enough
L==H
L==H
迭代次数: 0
j not moving enough
j not moving enough
L==H
L==H
L==H
j not moving enough
j not moving enough
L==H
j not moving enough
j not moving enough
L==H
L==H
L==H
L==H
j not moving enough
j not moving enough
j not moving enough
j not moving enough
L==H
j not moving enough
j not moving enough
L==H
循环次数: 0 alpha:97, alpha对修改了 1 次
迭代次数: 0
j not moving enough
j not moving enough
j not moving enough
L==H
j not moving enough
j not moving enough
L==H
j not moving enough
j not moving enough
j not moving enough
j not moving enough
j not moving enough
j not moving enough
j not moving enough
循环次数: 0 alpha:54, alpha对修改了 1 次
j not moving enough
j not moving enough
循环次数: 0 alpha:97, alpha对修改了 2 次
迭代次数: 0
j not moving enough
循环次数: 0 alpha:13, alpha对修改了 1 次

迭代次数: 25
j not moving enough
j not moving enough
j not moving enough
j not moving enough
迭代次数: 26
j not moving enough
j not moving enough
j not moving enough
j not moving enough
迭代次数: 27
j not moving enough
j not moving enough
j not moving enough
j not moving enough
迭代次数: 28
j not moving enough
j not moving enough
j not moving enough
j not moving enough
迭代次数: 29
j not moving enough
j not moving enough
j not moving enough
j not moving enough
迭代次数: 30
j not moving enough
j not moving enough
j not moving enough
j not moving enough
迭代次数: 31
j not moving enough
j not moving enough
j not moving enough
j not moving enough
迭代次数: 32
j not moving enough
j not moving enough
j not moving enough
j not moving enough
迭代次数: 33
j not moving enough
j not moving enough
j not moving enough
j not moving enough
迭代次数: 34
j not moving enough
j not moving enough
j not moving enough
j not moving enough
迭代次数: 35
j not moving enough
j not moving enough
j not moving enough
j not moving enough
迭代次数: 36
j not moving enough
j not moving enough
j not moving enough
j not moving enough
迭代次数: 37
j not moving enough
j not moving enough
j not moving enough
j not moving enough
迭代次数: 38
j not moving enough
j not moving enough
j not moving enough
j not moving enough
迭代次数: 39
j not moving enough
j not moving enough
j not moving enough
j not moving enough
迭代次数: 40
b= [-4.48392829]
（支持向量对应的alpha>0）alpha>0
[0.02485639 0.33967041 0.26285541 0.1016714 ]

#打印支持向量
for i in range(len(dataList)):
    if alphas[i]>0.0:
        print(dataList[i],labelList[i])

[4.658191, 3.507396] -1
[2.893743, -1.643468] -1
[5.286862, -2.358286] 1
[6.080573, 0.418886] 1

import matplotlib.pyplot as plt
import numpy as np
def dataToShow(dataList,labelList,b,alphas):
    #array形式方便处理
    dataArray = np.array(dataList)
    alphasArray = np.array(alphas.tolist())
    #变成一列，-1表示自动计算多少行
    labelArray = np.array(labelList).reshape(-1,1)
    #正类负类分开画图,np.squeeze转化成一维的
    posData = dataArray[np.squeeze(labelArray>0.0)]    
    negData = dataArray[np.squeeze(labelArray<0.0)]
    svData = dataArray[np.squeeze(alphasArray>0.0)]
    plt.figure()
    plt.scatter(posData[:,0],posData[:,1],c='b',s=20)
    plt.scatter(negData[:,0],negData[:,1],c='r',s=20)
    plt.scatter(svData[:,0],svData[:,1],marker='o',c='',s=100,edgecolors='g')
    plt.legend(['Positive Point','Negatibe Poin','Support Vector'])
    #画分割超平面
    w = np.dot((alphasArray * labelArray).T, dataArray)
    x0 = np.array([2, 8])
    #分割线：w0*x0+w1*x1+b=0
    x1 = -(w[0, 0] * x0 + np.squeeze(np.array(b))) / w[0, 1]
    plt.plot(x0, x1, color = 'y')
    plt.ylim((-10,12))
    plt.show()
dataToShow(dataList,labelList,b,alphas)

output_7_0.png

完整版SMO算法

class optStructK:
    def __init__(self,dataMatIn, classLabels, C, toler):
        self.X = dataMatIn
        selef.labelMat = classLabels
        self.C = C
        self.tol = toler
        self.m = shape(dataMatIn)[0]
        self.alphas = np.zeros((self.m,1))
        self.b = 0
        self.eCache = np.zeros((self.m,2)) #误差缓存
#计算Ei
def calcEk(oS, k):
    fXk = float(np.dot((oS.alphas * oS.labelMat).T, np.dot(oS.X, oS.X[k:k+1,:].T))) + oS.b
    #计算Ej
    Ek = fXk - float(oS.labelMat[k])
    return Ek
#内循环选择alpha
def selectJK(i,oS,Ei):
    '''
    内循环选择alpha的启发式算法
    参数：
        i -- 外循环alpha的下标
        oS -- 类
        Ei -- 误差
    返回：
        j -- 选择alpha的下标
        Ej -- 误差
    '''
    #初始化
    maxK = -1;maxDeltaE = 0; Ej = 0
    oS.eCache[i] = [1,Ei]
    #选择合理的集合
    validEcacheList = np.nonzero(oS.eCache[:,0])[0]
    if (len(validEcacheList)) > 1:
        #选择最大步长的alpha
        for k in validEcacheList:
            #不重复计算
            if k == i: continue
            #计算误差
            Ek = calcEk(oS, k)
            #计算步长
            deltaE = abs(Ei - Ek)
            #记录最佳选择
            if (deltaE > maxDeltaE):
                maxK = k;maxDeltaE = deltaE; Ej = Ek
        return maxK, Ej
    #没有合理值
    else:
        #随机选择
        j = select(i,oS.m)
        Ej = calcEk(oS,j)
    return j,Ej

def updateEkK(oS,k):
    #在alpha更新后存储计算得到的误差
    Ek = calcEk(oS,k)
    oS.eCache[k] = [1,Ek]
def innerLK(i, oS):
    #计算误差
    Ei = calcEkK(oS, i)
    #找出不满足KKT条件的alpha
    if ((oS.labelMat[i, 0]*Ei < -oS.tol) and (oS.alphas[i, 0] < oS.C)) or ((oS.labelMat[i, 0]*Ei > oS.tol) and (oS.alphas[i, 0] > 0)):
        #选择j
        j,Ej = selectJK(i, oS, Ei)
        #存储旧的值
        alphaIold = oS.alphas[i, 0].copy(); alphaJold = oS.alphas[j, 0].copy();
        #两种情况求边界值
        if (oS.labelMat[i, 0] != oS.labelMat[j, 0]):
            L = max(0, oS.alphas[j, 0] - oS.alphas[i, 0])
            H = min(oS.C, oS.C + oS.alphas[j, 0] - oS.alphas[i, 0])
        else:
            L = max(0, oS.alphas[j, 0] + oS.alphas[i, 0] - oS.C)
            H = min(oS.C, oS.alphas[j, 0] + oS.alphas[i, 0])
        if L==H: return 0
        #计算变化量
        eta = 2.0 * np.dot(oS.X[i:i+1,:], oS.X[j:j+1,:].T) - np.dot(oS.X[i:i+1,:], oS.X[i:i+1,:].T) - np.dot(oS.X[j:j+1,:], oS.X[j:j+1,:].T)
        if eta >= 0: return 0
        #更新alpha
        oS.alphas[j, 0] -= oS.labelMat[j, 0]*(Ei - Ej)/eta
        #约束alpha
        oS.alphas[j, 0] = clipAlpha(oS.alphas[j, 0],H,L)
        updateEkK(oS, j)
        if (abs(oS.alphas[j, 0] - alphaJold) < 0.00001): return 0
        oS.alphas[i, 0] += oS.labelMat[j, 0]*oS.labelMat[i, 0]*(alphaJold - oS.alphas[j, 0])
        updateEkK(oS, i)
        b1 = oS.b - Ei- oS.labelMat[i, 0]*(oS.alphas[i, 0]-alphaIold)*np.dot(oS.X[i:i+1,:], oS.X[i:i+1,:].T) - oS.labelMat[j, 0]*(oS.alphas[j, 0]-alphaJold)*np.dot(oS.X[i:i+1,:], oS.X[j:j+1,:].T)
        b2 = oS.b - Ej- oS.labelMat[i, 0]*(oS.alphas[i, 0]-alphaIold)*np.dot(oS.X[i:i+1,:], oS.X[j:j+1,:].T) - oS.labelMat[j, 0]*(oS.alphas[j, 0]-alphaJold)*np.dot(oS.X[j:j+1,:], oS.X[j:j+1,:].T)
        if (0 < oS.alphas[i, 0]) and (oS.C > oS.alphas[i, 0]): oS.b = b1
        elif (0 < oS.alphas[j, 0]) and (oS.C > oS.alphas[j, 0]): oS.b = b2
        else: oS.b = (b1 + b2)/2.0
        return 1
    else: return 0
def smoPK(dataMatIn, classLabels, C, toler, maxIter):
    #建立类变量
    oS = optStructK(np.array(dataMatIn),np.array(classLabels).reshape(-1, 1),C,toler)
    iter = 0
    entireSet = True; alphaPairsChanged = 0
    #执行循环
    while (iter < maxIter) and ((alphaPairsChanged > 0) or (entireSet)):
        alphaPairsChanged = 0
        if entireSet:
            #遍历所有
            for i in range(oS.m):        
                alphaPairsChanged += innerLK(i,oS)
                #print("fullSet, iter: {} i:{}, pairs changed {}".format(iter,i,alphaPairsChanged))
            iter += 1
        else:
            #遍历非边界值
            nonBoundIs = np.nonzero((oS.alphas > 0) * (oS.alphas < C))[0]
            for i in nonBoundIs:
                alphaPairsChanged += innerLK(i,oS)
                #print("non-bound, iter: {} i:{}, pairs changed {}".format(iter,i,alphaPairsChanged))
            iter += 1
        if entireSet: entireSet = False 
        elif (alphaPairsChanged == 0): entireSet = True  
        #print("iteration number: {}".format(iter))
    return oS.b,oS.alphas

dataList,labelList = loadData('../../Reference Code/Ch06/testSet.txt')
b, alphas = smoSimple(dataList, labelList, 0.6, 0.001, 40)
print('b= {}'.format(b))
print('（支持向量对应的alpha>0）alpha>0\n{}'.format(alphas[alphas>0]))

循环次数: 0 alpha:0, alpha对修改了 1 次
L==H
循环次数: 0 alpha:4, alpha对修改了 2 次
j not moving enough
循环次数: 0 alpha:6, alpha对修改了 3 次
L==H
j not moving enough
L==H
循环次数: 0 alpha:25, alpha对修改了 4 次
L==H
循环次数: 0 alpha:29, alpha对修改了 5 次
L==H
循环次数: 0 alpha:52, alpha对修改了 6 次
j not moving enough
循环次数: 0 alpha:55, alpha对修改了 7 次
L==H
j not moving enough
L==H
j not moving enough
j not moving enough
j not moving enough
j not moving enough
j not moving enough
迭代次数: 0
j not moving enough
j not moving enough
j not moving enough
j not moving enough
L==H
L==H
j not moving enough
j not moving enough
L==H
j not moving enough
j not moving enough
L==H
j not moving enough
j not moving enough
j not moving enough
j not moving enough
L==H
循环次数: 0 alpha:52, alpha对修改了 1 次
j not moving enough
循环次数: 0 alpha:55, alpha对修改了 2 次
j not moving enough
j not moving enough
j not moving enough
j not moving enough
循环次数: 0 alpha:76, alpha对修改了 3 次
j not moving enough
L==H
L==H
迭代次数: 0
循环次数: 0 alpha:0, alpha对修改了 1 次
j not moving enough
j not moving enough
L==H
L==H
j not moving enough
j not moving enough
j not moving enough
j not moving enough
j not moving enough
j not moving enough
j not moving enough
j not moving enough
L==H
j not moving enough
j not moving enough
j not moving enough
j not moving enough
循环次数: 0 alpha:96, alpha对修改了 2 次
j not moving enough
迭代次数: 0
j not moving enough
j not moving enough
循环次数: 0 alpha:8, alpha对修改了 1 次
循环次数: 0 alpha:10, alpha对修改了 2 次
L==H
j not moving enough
j not moving enough
j not moving enough
L==H
L==H
循环次数: 0 alpha:54, alpha对修改了 3 次
j not moving enough
L==H
j not moving enough
j not moving enough
j not moving enough
j not moving enough
迭代次数: 0
j not moving enough
循环次数: 0 alpha:5, alpha对修改了 1 次
j not moving enough
j not moving enough
循环次数: 0 alpha:17, alpha对修改了 2 次
L==H
j not moving enough
j not moving enough
L==H
j not moving enough
L==H
L==H
j not moving enough
L==H
j not moving enough
j not moving enough
j not moving enough
j not moving enough
j not moving enough
j not moving enough
j not moving enough
j not moving enough
L==H
L==H
迭代次数: 0
j not moving enough
循环次数: 0 alpha:5, alpha对修改了 1 次
L==H
j not moving enough
j not moving enough
j not moving enough
j not moving enough
j not moving enough
j not moving enough
j not moving enough
j not moving enough
循环次数: 0 alpha:29, alpha对修改了 2 次
j not moving enough
循环次数: 0 alpha:52, alpha对修改了 3 次
循环次数: 0 alpha:54, alpha对修改了 4 次
j not moving enough
j not moving enough
j not moving enough
j not moving enough
j not moving enough
迭代次数: 0
j not moving enough
j not moving enough
j not moving enough
j not moving enough
L==H
j not moving enough
j not moving enough
L==H
j not moving enough
j not moving enough
循环次数: 0 alpha:54, alpha对修改了 1 次
j not moving enough
j not moving enough
j not moving enough
循环次数: 0 alpha:86, alpha对修改了 2 次
L==H
L==H
j not moving enough
L==H
j not moving enough
迭代次数: 0
j not moving enough
j not moving enough
循环次数: 0 alpha:5, alpha对修改了 1 次
j not moving enough
j not moving enough
L==H
j not moving enough
j not moving enough
j not moving enough
j not moving enough


迭代次数: 38
j not moving enough
j not moving enough
j not moving enough
j not moving enough
迭代次数: 39
j not moving enough
j not moving enough
L==H
j not moving enough
迭代次数: 40
b= [-4.65074859]
（支持向量对应的alpha>0）alpha>0
[0.11699604 0.29638533 0.41338137]

dataToShow(dataList,labelList,b,alphas)

output_12_0.png

引入核函数

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] = np.dot(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: return 0
        eta = 2.0 * oS.K[i,j] - oS.K[i,i] - oS.K[j,j] #changed for kernel
        if 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): 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

测试

def testRbf(k1=1.3):
    dataArr,labelArr  = loadData('../../Reference Code/Ch06/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 = loadData('../../Reference Code/Ch06/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) )
    return b,alphas

import matplotlib.pyplot as plt
import numpy as np
def dataToShow(dataList,labelList,b,alphas):
    #array形式方便处理
    dataArray = np.array(dataList)
    alphasArray = np.array(alphas.tolist())
    #变成一列，-1表示自动计算多少行
    labelArray = np.array(labelList).reshape(-1,1)
    #正类负类分开画图,np.squeeze转化成一维的
    posData = dataArray[np.squeeze(labelArray>0.0)]    
    negData = dataArray[np.squeeze(labelArray<0.0)]
    svData = dataArray[np.squeeze(alphasArray>0.0)]
    plt.figure()
    plt.scatter(posData[:,0],posData[:,1],c='b',s=20)
    plt.scatter(negData[:,0],negData[:,1],c='r',s=20)
    plt.scatter(svData[:,0],svData[:,1],marker='o',c='',s=100,edgecolors='g')
    plt.legend(['Positive Point','Negatibe Poin','Support Vector'])
#     #画分割超平面
#     w = np.dot((alphasArray * labelArray).T, dataArray)
#     x0 = np.array([2, 8])
#     #分割线：w0*x0+w1*x1+b=0
#     x1 = -(w[0, 0] * x0 + np.squeeze(np.array(b))) / w[0, 1]
#     plt.plot(x0, x1, color = 'y')
#     plt.ylim((-10,12))
    plt.show()

dataList,labelList  = loadData('../../Reference Code/Ch06/testSetRBF.txt')

b,alphas = testRbf(k1=1.3)
dataToShow(dataList,labelList,b,alphas)

there are 29 Support Vectors
the training error rate is: 0.130000
the test error rate is: 0.150000

output_19_1.png

b,alphas = testRbf(k1=0.1)
dataToShow(dataList,labelList,b,alphas)

there are 84 Support Vectors
the training error rate is: 0.000000
the test error rate is: 0.090000

output_20_1.png

手写识别

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('../../../Week1/Reference Code/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('../../../Week1/Reference Code/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)) 

testDigits(('rbf', 20))

there are 204 Support Vectors
the training error rate is: 0.000000
the test error rate is: 0.010571