SVM（python实现）

看《机器学习（西瓜书）》可以理解SVM的推导过程，重点是看附录理解“对偶问题”，以及核函数的定义。

SVM的代码主要是SMO算法的实现，主要参考《统计学习方法》，即如何选择pair进行优化，收敛后即可得到α、w、b

代码：

# _*_ coding:utf-8 _*_
from numpy import *

def loadDataSet(filename): #读取数据
    dataMat=[]
    labelMat=[]
    fr=open(filename)
    for line in fr.readlines():
        lineArr=line.strip().split()
        dataMat.append([float(lineArr[0]), float(lineArr[1])])
        labelMat.append(float(lineArr[2]))
    return dataMat,labelMat #返回数据特征和数据类别

def selectJrand(i,m): #在0-m中随机选择一个不是i的整数
    j=i
    while (j==i):
        j=int(random.uniform(0,m))
    return j

def clipAlpha(aj,H,L):  #保证a在L和H范围内（L <= a <= H）
    if aj>H:
        aj=H
    if L>aj:
        aj=L
    return aj

def kernelTrans(X, A, kTup): #核函数，输入参数,X:支持向量的特征树；A：某一行特征数据；kTup：('lin',k1)核函数的类型和参数
    m,n = shape(X)
    K = mat(zeros((m,1)))
    if kTup[0]=='lin': #线性函数
        K = X * A.T
    elif kTup[0]=='rbf': # 径向基函数(radial bias function)
        for j in range(m):
            deltaRow = X[j,:] - A
            K[j] = deltaRow*deltaRow.T
        K = exp(K/(-1*kTup[1]**2)) #返回生成的结果
    else:
        raise NameError('Houston We Have a Problem -- That Kernel is not recognized')
    return K


#定义类，方便存储数据
class optStruct:
    def __init__(self, dataMatIn, classLabels, C, toler, kTup):  # 存储各类参数
        self.X = dataMatIn  #数据特征
        self.labelMat = classLabels #数据类别
        self.C = C #软间隔参数C，参数越大，非线性拟合能力越强
        self.tol = toler #停止阀值
        self.m = shape(dataMatIn)[0] #数据行数
        self.alphas = mat(zeros((self.m, 1)))
        self.b = 0 #初始设为0
        self.eCache = mat(zeros((self.m, 2)))  # 缓存
        self.K = mat(zeros((self.m, self.m)))  # 核函数的计算结果
        for i in range(self.m):
            self.K[:, i] = kernelTrans(self.X, self.X[i, :], kTup)


def calcEk(oS, k): #计算Ek（参考《统计学习方法》p127公式7.105）
    fXk = float(multiply(oS.alphas, oS.labelMat).T*oS.K[:, k] + oS.b)
    Ek = fXk - float(oS.labelMat[k])
    return Ek

#随机选取aj，并返回其E值
def selectJ(i, oS, Ei):
    maxK = -1
    maxDeltaE = 0
    Ej = 0
    oS.eCache[i] = [1, Ei]
    validEcacheList = nonzero(oS.eCache[:,0].A)[0]  #返回矩阵中的非零位置的行数
    if (len(validEcacheList)) > 1:
        for k in validEcacheList:
            if k == i:
                continue
            Ek = calcEk(oS, k)
            deltaE = abs(Ei - Ek)
            if (deltaE > maxDeltaE): #返回步长最大的aj
                maxK = k
                maxDeltaE = deltaE
                Ej = Ek
        return maxK, Ej
    else:
        j = selectJrand(i, oS.m)
        Ej = calcEk(oS, j)
    return j, Ej


def updateEk(oS, k): #更新os数据
    Ek = calcEk(oS, k)
    oS.eCache[k] = [1, Ek]

#首先检验ai是否满足KKT条件，如果不满足，随机选择aj进行优化，更新ai,aj,b值
def innerL(i, oS): #输入参数i和所有参数数据
    Ei = calcEk(oS, i) #计算E值（预测值与真实值之间的差距）
    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)):
        #检验这行数据是否符合KKT条件 参考《统计学习方法》p128公式7.111-113
        j,Ej = selectJ(i, oS, Ei) #随机选取aj，并返回其E值
        alphaIold = oS.alphas[i].copy()
        alphaJold = oS.alphas[j].copy()
        if (oS.labelMat[i] != oS.labelMat[j]): #以下代码的公式参考《统计学习方法》p126
            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] #参考《统计学习方法》p127公式7.107
        if eta >= 0:
            print("eta>=0")
            return 0
        oS.alphas[j] -= oS.labelMat[j]*(Ei - Ej)/eta #参考《统计学习方法》p127公式7.106
        oS.alphas[j] = clipAlpha(oS.alphas[j],H,L) #参考《统计学习方法》p127公式7.108
        updateEk(oS, j)
        if (abs(oS.alphas[j] - alphaJold) < oS.tol): #alpha变化大小阀值（自己设定）
            print("j not moving enough")
            return 0
        oS.alphas[i] += oS.labelMat[j]*oS.labelMat[i]*(alphaJold - oS.alphas[j])#参考《统计学习方法》p127公式7.109
        updateEk(oS, i) #更新数据
        #以下求解b的过程，参考《统计学习方法》p129公式7.114-7.116
        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] 0) or (entireSet)):
        alphaPairsChanged = 0
        if entireSet:
            for i in range(oS.m):  # 遍历所有数据
                alphaPairsChanged += innerL(i, oS)
                print("fullSet, iter: %d i:%d, pairs changed %d" % (iter, i, alphaPairsChanged))
                #  显示第多少次迭代，那行特征数据使alpha发生了改变，这次改变了多少次alpha
            iter += 1
        else:
            nonBoundIs = nonzero((oS.alphas.A > 0) * (oS.alphas.A < C))[0]
            for i in nonBoundIs: #遍历非边界的数据
                alphaPairsChanged += innerL(i, oS)
                print("non-bound, iter: %d i:%d, pairs changed %d" % (iter, i, alphaPairsChanged))
            iter += 1
        if entireSet:
            entireSet = False
        elif (alphaPairsChanged == 0):
            entireSet = True
        print("iteration number: %d" % iter)
    return oS.b,oS.alphas

def testRbf(data_train, data_test):
    dataArr, labelArr = loadDataSet(data_train) #读取训练数据
    b, alphas = smoP(dataArr, labelArr, 200, 0.0001, 10000, ('rbf', 1.3))  # 通过SMO算法得到b和alpha
    datMat = mat(dataArr)
    labelMat = mat(labelArr).transpose()
    svInd = nonzero(alphas)[0]  # 选取不为0数据的行数（也就是支持向量）
    sVs = datMat[svInd]  # 支持向量的特征数据
    labelSV = labelMat[svInd]  # 支持向量的类别（1或-1）
    print("there are %d Support Vectors" % shape(sVs)[0])  # 打印出共有多少的支持向量
    m, n = shape(datMat) #训练数据的行列数
    errorCount = 0
    for i in range(m):
        kernelEval = kernelTrans(sVs, datMat[i, :], ('rbf', 1.3)) #将支持向量转化为核函数
        predict = kernelEval.T * multiply(labelSV, alphas[svInd]) + b
        #这一行的预测结果（代码来源于《统计学习方法》p133里面最后用于预测的公式）注意最后确定的分离平面只有那些支持向量决定。
        if sign(predict) != sign(labelArr[i]):  # sign函数 -1 if x < 0, 0 if x==0, 1 if x > 0
            errorCount += 1
    print("the training error rate is: %f" % (float(errorCount)/m)) #打印出错误率
    dataArr_test, labelArr_test = loadDataSet(data_test) #读取测试数据
    errorCount_test = 0
    datMat_test=mat(dataArr_test)
    labelMat = mat(labelArr_test).transpose()
    m,n = shape(datMat_test)
    for i in range(m): #在测试数据上检验错误率
        kernelEval = kernelTrans(sVs,datMat_test[i,:],('rbf', 1.3))
        predict=kernelEval.T * multiply(labelSV,alphas[svInd]) + b
        if sign(predict)!=sign(labelArr_test[i]):
            errorCount_test += 1
    print("the test error rate is: %f" % (float(errorCount_test)/m))

#主程序
def main():
    filename_traindata='D:/MLCode/SVM/traindata.txt'
    filename_testdata='D:/MLCode/SVM/testdata.txt'
    testRbf(filename_traindata,filename_testdata)

if __name__=='__main__':
    main()

 

运行结果：

 

 

traindata.txt

-0.214824   0.662756    -1.000000
-0.061569   -0.091875   1.000000
0.406933    0.648055    -1.000000
0.223650    0.130142    1.000000
0.231317    0.766906    -1.000000
-0.748800   -0.531637   -1.000000
-0.557789   0.375797    -1.000000
0.207123    -0.019463   1.000000
0.286462    0.719470    -1.000000
0.195300    -0.179039   1.000000
-0.152696   -0.153030   1.000000
0.384471    0.653336    -1.000000
-0.117280   -0.153217   1.000000
-0.238076   0.000583    1.000000
-0.413576   0.145681    1.000000
0.490767    -0.680029   -1.000000
0.199894    -0.199381   1.000000
-0.356048   0.537960    -1.000000
-0.392868   -0.125261   1.000000
0.353588    -0.070617   1.000000
0.020984    0.925720    -1.000000
-0.475167   -0.346247   -1.000000
0.074952    0.042783    1.000000
0.394164    -0.058217   1.000000
0.663418    0.436525    -1.000000
0.402158    0.577744    -1.000000
-0.449349   -0.038074   1.000000
0.619080    -0.088188   -1.000000
0.268066    -0.071621   1.000000
-0.015165   0.359326    1.000000
0.539368    -0.374972   -1.000000
-0.319153   0.629673    -1.000000
0.694424    0.641180    -1.000000
0.079522    0.193198    1.000000
0.253289    -0.285861   1.000000
-0.035558   -0.010086   1.000000
-0.403483   0.474466    -1.000000
-0.034312   0.995685    -1.000000
-0.590657   0.438051    -1.000000
-0.098871   -0.023953   1.000000
-0.250001   0.141621    1.000000
-0.012998   0.525985    -1.000000
0.153738    0.491531    -1.000000
0.388215    -0.656567   -1.000000
0.049008    0.013499    1.000000
0.068286    0.392741    1.000000
0.747800    -0.066630   -1.000000
0.004621    -0.042932   1.000000
-0.701600   0.190983    -1.000000
0.055413    -0.024380   1.000000
0.035398    -0.333682   1.000000
0.211795    0.024689    1.000000
-0.045677   0.172907    1.000000
0.595222    0.209570    -1.000000
0.229465    0.250409    1.000000
-0.089293   0.068198    1.000000
0.384300    -0.176570   1.000000
0.834912    -0.110321   -1.000000
-0.307768   0.503038    -1.000000
-0.777063   -0.348066   -1.000000
0.017390    0.152441    1.000000
-0.293382   -0.139778   1.000000
-0.203272   0.286855    1.000000
0.957812    -0.152444   -1.000000
0.004609    -0.070617   1.000000
-0.755431   0.096711    -1.000000
-0.526487   0.547282    -1.000000
-0.246873   0.833713    -1.000000
0.185639    -0.066162   1.000000
0.851934    0.456603    -1.000000
-0.827912   0.117122    -1.000000
0.233512    -0.106274   1.000000
0.583671    -0.709033   -1.000000
-0.487023   0.625140    -1.000000
-0.448939   0.176725    1.000000
0.155907    -0.166371   1.000000
0.334204    0.381237    -1.000000
0.081536    -0.106212   1.000000
0.227222    0.527437    -1.000000
0.759290    0.330720    -1.000000
0.204177    -0.023516   1.000000
0.577939    0.403784    -1.000000
-0.568534   0.442948    -1.000000
-0.011520   0.021165    1.000000
0.875720    0.422476    -1.000000
0.297885    -0.632874   -1.000000
-0.015821   0.031226    1.000000
0.541359    -0.205969   -1.000000
-0.689946   -0.508674   -1.000000
-0.343049   0.841653    -1.000000
0.523902    -0.436156   -1.000000
0.249281    -0.711840   -1.000000
0.193449    0.574598    -1.000000
-0.257542   -0.753885   -1.000000
-0.021605   0.158080    1.000000
0.601559    -0.727041   -1.000000
-0.791603   0.095651    -1.000000
-0.908298   -0.053376   -1.000000
0.122020    0.850966    -1.000000
-0.725568   -0.292022   -1.000000

testdata.txt

0.676771    -0.486687   -1.000000
0.008473    0.186070    1.000000
-0.727789   0.594062    -1.000000
0.112367    0.287852    1.000000
0.383633    -0.038068   1.000000
-0.927138   -0.032633   -1.000000
-0.842803   -0.423115   -1.000000
-0.003677   -0.367338   1.000000
0.443211    -0.698469   -1.000000
-0.473835   0.005233    1.000000
0.616741    0.590841    -1.000000
0.557463    -0.373461   -1.000000
-0.498535   -0.223231   -1.000000
-0.246744   0.276413    1.000000
-0.761980   -0.244188   -1.000000
0.641594    -0.479861   -1.000000
-0.659140   0.529830    -1.000000
-0.054873   -0.238900   1.000000
-0.089644   -0.244683   1.000000
-0.431576   -0.481538   -1.000000
-0.099535   0.728679    -1.000000
-0.188428   0.156443    1.000000
0.267051    0.318101    1.000000
0.222114    -0.528887   -1.000000
0.030369    0.113317    1.000000
0.392321    0.026089    1.000000
0.298871    -0.915427   -1.000000
-0.034581   -0.133887   1.000000
0.405956    0.206980    1.000000
0.144902    -0.605762   -1.000000
0.274362    -0.401338   1.000000
0.397998    -0.780144   -1.000000
0.037863    0.155137    1.000000
-0.010363   -0.004170   1.000000
0.506519    0.486619    -1.000000
0.000082    -0.020625   1.000000
0.057761    -0.155140   1.000000
0.027748    -0.553763   -1.000000
-0.413363   -0.746830   -1.000000
0.081500    -0.014264   1.000000
0.047137    -0.491271   1.000000
-0.267459   0.024770    1.000000
-0.148288   -0.532471   -1.000000
-0.225559   -0.201622   1.000000
0.772360    -0.518986   -1.000000
-0.440670   0.688739    -1.000000
0.329064    -0.095349   1.000000
0.970170    -0.010671   -1.000000
-0.689447   -0.318722   -1.000000
-0.465493   -0.227468   -1.000000
-0.049370   0.405711    1.000000
-0.166117   0.274807    1.000000
0.054483    0.012643    1.000000
0.021389    0.076125    1.000000
-0.104404   -0.914042   -1.000000
0.294487    0.440886    -1.000000
0.107915    -0.493703   -1.000000
0.076311    0.438860    1.000000
0.370593    -0.728737   -1.000000
0.409890    0.306851    -1.000000
0.285445    0.474399    -1.000000
-0.870134   -0.161685   -1.000000
-0.654144   -0.675129   -1.000000
0.285278    -0.767310   -1.000000
0.049548    -0.000907   1.000000
0.030014    -0.093265   1.000000
-0.128859   0.278865    1.000000
0.307463    0.085667    1.000000
0.023440    0.298638    1.000000
0.053920    0.235344    1.000000
0.059675    0.533339    -1.000000
0.817125    0.016536    -1.000000
-0.108771   0.477254    1.000000
-0.118106   0.017284    1.000000
0.288339    0.195457    1.000000
0.567309    -0.200203   -1.000000
-0.202446   0.409387    1.000000
-0.330769   -0.240797   1.000000
-0.422377   0.480683    -1.000000
-0.295269   0.326017    1.000000
0.261132    0.046478    1.000000
-0.492244   -0.319998   -1.000000
-0.384419   0.099170    1.000000
0.101882    -0.781145   -1.000000
0.234592    -0.383446   1.000000
-0.020478   -0.901833   -1.000000
0.328449    0.186633    1.000000
-0.150059   -0.409158   1.000000
-0.155876   -0.843413   -1.000000
-0.098134   -0.136786   1.000000
0.110575    -0.197205   1.000000
0.219021    0.054347    1.000000
0.030152    0.251682    1.000000
0.033447    -0.122824   1.000000
-0.686225   -0.020779   -1.000000
-0.911211   -0.262011   -1.000000
0.572557    0.377526    -1.000000
-0.073647   -0.519163   -1.000000
-0.281830   -0.797236   -1.000000
-0.555263   0.126232    -1.000000

参考资料：

https://blog.csdn.net/csqazwsxedc/article/details/71513197

《机器学习》

《统计学习方法》