SVM、AdaBoost

一、SVM

支持向量机，因其英文名为support vector machine，故一般简称SVM，通俗来讲，它是一种二类分类模型，其基本模型定义为特征空间上的间隔最大的线性分类器，其学习策略便是间隔最大化，最终可转化为一个凸二次规划问题的求解。
SVM数学推论可以看：
https://blog.youkuaiyun.com/v_july_v/article/details/7624837
https://blog.youkuaiyun.com/zouxy09/article/details/17291543
写的特别详细
python代码详解可看：
https://blog.youkuaiyun.com/csqazwsxedc/article/details/71513197

from numpy import * 

def loadDataSet(filename): #读取数据
    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 #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
    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]<oS.C):
            oS.b = b1
        elif (0 < oS.alphas[j]<oS.C):
            oS.b = b2
        else:
            oS.b = (b1 + b2)/2.0
        return 1
    else:
        return 0


#SMO函数，用于快速求解出alpha
def smoP(dataMatIn, classLabels, C, toler, maxIter,kTup=('lin', 0)): #输入参数：数据特征，数据类别，参数C，阀值toler，最大迭代次数，核函数（默认线性核）
    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:
            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='./traindata.txt'
    filename_testdata='./testdata.txt'
    testRbf(filename_traindata,filename_testdata)

if __name__=='__main__':
    main()

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

二、AdaBoost

Adaboost作为一种提升集成算法，核心思想是不断训练弱学习器，来针对性的提升前一轮中预测错误样本的权重，最终通过加权所有弱学习器的训练结果得到最终分类标签。Adaboost是一种加权提升的集成算法，关键在于两个权重系数：
弱学习器权重，影响每个弱学习器的结果对最终集成学习结果的影响程度，与该学习器的错误率有关
样本权重，这也是Adaboost算法的精髓所在，即每轮训练弱学习器时不断优化调整样本间的权重，保证前一轮中学习错误的样本在下一轮训练中受到重点照顾
代码：

class DecisionTreeClassifierWithWeight:
    def __init__(self):
        self.best_err = 1  # 最小的加权错误率
        self.best_fea_id = 0  # 最优特征id
        self.best_thres = 0  # 选定特征的最优阈值
        self.best_op = 1  # 阈值符号，其中 1: >, 0: <

    def fit(self, X, y, sample_weight=None):
        if sample_weight is None:
            sample_weight = np.ones(len(X)) / len(X)
        n = X.shape[1]
        for i in range(n):
            feature = X[:, i]  # 选定特征列
            fea_unique = np.sort(np.unique(feature))  # 将所有特征值从小到大排序
            for j in range(len(fea_unique)-1):
                thres = (fea_unique[j] + fea_unique[j+1]) / 2  # 逐一设定可能阈值
                for op in (0, 1):
                    y_ = 2*(feature >= thres)-1 if op==1 else 2*(feature < thres)-1  # 判断何种符号为最优
                    err = np.sum((y_ != y)*sample_weight)
                    if err < self.best_err:  # 当前参数组合可以获得更低错误率，更新最优参数
                        self.best_err = err
                        self.best_op = op
                        self.best_fea_id = i
                        self.best_thres = thres
        return self
    
    def predict(self, X):
        feature = X[:, self.best_fea_id]
        return 2*(feature >= self.best_thres)-1 if self.best_op==1 else 2*(feature < self.best_thres)-1
    
    def score(self, X, y, sample_weight=None):
        y_pre = self.predict(X)
        if sample_weight is not None:
            return np.sum((y_pre == y)*sample_weight)
        return np.mean(y_pre == y)

这里以sklearn库中自带的乳腺癌二分类数据集为例，以上述实现的决策树桩进行训练和评分，得到最终得分0.867，这对于一个仅有单层决策树的分类器来说效果还是比较好的。

from sklearn.datasets import load_breast_cancer
from sklearn.model_selection import train_test_split

X, y = load_breast_cancer(return_X_y=True)
y = 2*y-1  # 将0/1取值映射为-1/1取值
X_train, X_test, y_train, y_test = train_test_split(X, y)

DecisionTreeClassifierWithWeight().fit(X_train, y_train).score(X_test, y_test)

注：按照Adaboost中的算法约定，二分类模型中标签分别用-1和1代表负类和正类。

03 Adaboost集成分类器

在实现决策树桩作为弱分类器的基础上，实现Adaboost算法就仅需按照算法流程逐层训练即可。简单起见，这里仅设置超参数n_estimators用于选择弱分类器的个数。为区分于sklearn中的Adaboost标准内置库，本文将自定义实现的Adaboost分类算法命名为AdaBoostClassifier_，并设置相同的默认弱学习器数量超参数n_estimators=50，其余不做限制。

实质上，在逐渐调整样本权重的基础上，仅需逐层训练一个最优的决策树桩作为每轮的弱学习器，并保存在一个弱学习器列表中，同步记录每个弱学习器的权重系数。最后，在实现predict接口时，用每个弱学习器逐一完成训练，而后按其权重系数加权即可得到最终结果。完整代码如下：

class AdaBoostClassifier_:
    def __init__(self, n_estimators=50):
        self.n_estimators = n_estimators
        self.estimators = []
        self.alphas = []

    def fit(self, X, y):
        sample_weight = np.ones(len(X)) / len(X)  # 初始化样本权重为 1/N
        for _ in range(self.n_estimators):
            dtc = DecisionTreeClassifierWithWeight().fit(X, y, sample_weight)  # 训练弱学习器
            alpha = 1/2 * np.log((1-dtc.best_err)/dtc.best_err)  # 权重系数
            y_pred = dtc.predict(X)
            sample_weight *= np.exp(-alpha*y_pred*y)  # 更新迭代样本权重
            sample_weight /= np.sum(sample_weight)  # 样本权重归一化
            self.estimators.append(dtc)
            self.alphas.append(alpha)
        return self

    def predict(self, X):
        y_pred = np.empty((len(X), self.n_estimators))  # 预测结果二维数组，其中每一列代表一个弱学习器的预测结果
        for i in range(self.n_estimators):
            y_pred[:, i] = self.estimators[i].predict(X)
        y_pred = y_pred * np.array(self.alphas)  # 将预测结果与训练权重乘积作为集成预测结果
        return 2*(np.sum(y_pred, axis=1)>0)-1  # 以0为阈值，判断并映射为-1和1

    def score(self, X, y):
        y_pred = self.predict(X)
        return np.mean(y_pred==y)

最后，继续以乳腺癌二分类数据集为例，对比测试自定义实现的AdaBoostClassifier_算法与sklearn标准库中的AdaBoostClassifer算法性能，得到如下结果：

from sklearn.ensemble import AdaBoostClassifier
AdaBoostClassifier_().fit(X_train, y_train).score(X_test, y_test)
AdaBoostClassifier().fit(X_train, y_train).score(X_test, y_test)