AdaBoost元算法
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boosting：不同的分类器是通过串行训练而获得的，每个新分类器都根据已经训练出的分类器的性能来进行训练。通过集中关注被已有分类器错分的那些样本来获得新的分类器。
权重alpha：弱分类器的线性组合系数，用来构成完整分类器。对每个数据的分类时，其结果是弱分类器结果的线性组合。
权重D：样本的权重向量，每个元素表征对应样本的重要性。m*1阶列向量。
基于单层决策树构建弱分类器：仅基于单个特征来做决策。
单层决策树生成函数：
from numpy import *
def loadSimpData():
datMat = matrix([[ 1. , 2.1],
[ 2. , 1.1],
[ 1.3, 1. ],
[ 1. , 1. ],
[ 2. , 1. ]])
classLabels = [1.0, 1.0, -1.0, -1.0, 1.0]
return datMat,classLabels
def stumpClassify(dataMatrix,dimen,threshVal,threshIneq):
retArray=ones((shape(dataMatrix)[0],1))
if threshIneq==‘lt‘:
retArray[dataMatrix[:,dimen]<=threshVal]=-1.0
else:
retArray[dataMatrix[:,dimen]>threshVal]=-1.0
return retArray
def buildStump(dataArr,classLabels,D):
dataMatrix=mat(dataArr)
labelMat=mat(classLabels).T
m,n=shape(dataMatrix)
numSteps=10.0
bestStump={}
bestClasEst=mat(zeros((m,1)))
minError=inf
for i in range(n):
rangeMin=dataMatrix[:,i].min()
rangeMax=dataMatrix[:,i].max()
stepSize=(rangeMax-rangeMin)/numSteps
for j in range(-1,int(numSteps)+1):
for inequal in [‘lt‘,‘gt‘]:
threshVal=(rangeMin+float(j)*stepSize)
predictVals=stumpClassify(dataMatrix,i,threshVal,inequal)
errArr=mat(ones((m,1)))
errArr[predictVals==labelMat]=0
weightedError=D.T*errArr #1*m m*1 ==>标量
print(‘split:dim %d,thresh:%.2f,thresh inequal:%s,the weighted error is %.3f‘%(i,threshVal,inequal,weightedError))
if weightedError<minError:
minError=weightedError
bestClasEst=predictVals.copy()
bestStump[‘dim‘]=i
bestStump[‘thresh‘]=threshVal
bestStump[‘ineq‘]=inequal
return bestStump,minError,bestClasEst
if __name__==‘__main__‘:
D=mat(ones((5,1))/5)
datMat, classLabels=loadSimpData()
buildStump(datMat,classLabels,D)
stumpClassify(dataMatrix,dimen,threshVal,threshIneq):单层决策树，通过阈值比较对数据分类。所有在阈值一边的数据会分为-1，另一边的数据分为+1.该函数通过数组过滤来实现。分为两种模式：小于等于阈值分为-1，大于阈值分为+1；或者相反。
weightedError=D.T*errArr #1*m m*1 ==>标量 将错误向量errArr和权重向量D的相应元素相乘并求和，得到数值weightedError，这就是AdaBoost与分类器交互的地方。这里基于权重向量D而不是其他错误计算指标来评价分类器。
输出：
split:dim 0,thresh:0.90,thresh inequal:lt,the weighted error is 0.400 split:dim 0,thresh:0.90,thresh inequal:gt,the weighted error is 0.600 split:dim 0,thresh:1.00,thresh inequal:lt,the weighted error is 0.400 split:dim 0,thresh:1.00,thresh inequal:gt,the weighted error is 0.600 split:dim 0,thresh:1.10,thresh inequal:lt,the weighted error is 0.400 split:dim 0,thresh:1.10,thresh inequal:gt,the weighted error is 0.600 split:dim 0,thresh:1.20,thresh inequal:lt,the weighted error is 0.400 split:dim 0,thresh:1.20,thresh inequal:gt,the weighted error is 0.600 split:dim 0,thresh:1.30,thresh inequal:lt,the weighted error is 0.200 split:dim 0,thresh:1.30,thresh inequal:gt,the weighted error is 0.800 split:dim 0,thresh:1.40,thresh inequal:lt,the weighted error is 0.200 split:dim 0,thresh:1.40,thresh inequal:gt,the weighted error is 0.800 split:dim 0,thresh:1.50,thresh inequal:lt,the weighted error is 0.200 split:dim 0,thresh:1.50,thresh inequal:gt,the weighted error is 0.800 split:dim 0,thresh:1.60,thresh inequal:lt,the weighted error is 0.200 split:dim 0,thresh:1.60,thresh inequal:gt,the weighted error is 0.800 split:dim 0,thresh:1.70,thresh inequal:lt,the weighted error is 0.200 split:dim 0,thresh:1.70,thresh inequal:gt,the weighted error is 0.800 split:dim 0,thresh:1.80,thresh inequal:lt,the weighted error is 0.200 split:dim 0,thresh:1.80,thresh inequal:gt,the weighted error is 0.800 split:dim 0,thresh:1.90,thresh inequal:lt,the weighted error is 0.200 split:dim 0,thresh:1.90,thresh inequal:gt,the weighted error is 0.800 split:dim 0,thresh:2.00,thresh inequal:lt,the weighted error is 0.600 split:dim 0,thresh:2.00,thresh inequal:gt,the weighted error is 0.400 split:dim 1,thresh:0.89,thresh inequal:lt,the weighted error is 0.400 split:dim 1,thresh:0.89,thresh inequal:gt,the weighted error is 0.600 split:dim 1,thresh:1.00,thresh inequal:lt,the weighted error is 0.200 split:dim 1,thresh:1.00,thresh inequal:gt,the weighted error is 0.800 split:dim 1,thresh:1.11,thresh inequal:lt,the weighted error is 0.400 split:dim 1,thresh:1.11,thresh inequal:gt,the weighted error is 0.600 split:dim 1,thresh:1.22,thresh inequal:lt,the weighted error is 0.400 split:dim 1,thresh:1.22,thresh inequal:gt,the weighted error is 0.600 split:dim 1,thresh:1.33,thresh inequal:lt,the weighted error is 0.400 split:dim 1,thresh:1.33,thresh inequal:gt,the weighted error is 0.600 split:dim 1,thresh:1.44,thresh inequal:lt,the weighted error is 0.400 split:dim 1,thresh:1.44,thresh inequal:gt,the weighted error is 0.600 split:dim 1,thresh:1.55,thresh inequal:lt,the weighted error is 0.400 split:dim 1,thresh:1.55,thresh inequal:gt,the weighted error is 0.600 split:dim 1,thresh:1.66,thresh inequal:lt,the weighted error is 0.400 split:dim 1,thresh:1.66,thresh inequal:gt,the weighted error is 0.600 split:dim 1,thresh:1.77,thresh inequal:lt,the weighted error is 0.400 split:dim 1,thresh:1.77,thresh inequal:gt,the weighted error is 0.600 split:dim 1,thresh:1.88,thresh inequal:lt,the weighted error is 0.400 split:dim 1,thresh:1.88,thresh inequal:gt,the weighted error is 0.600 split:dim 1,thresh:1.99,thresh inequal:lt,the weighted error is 0.400 split:dim 1,thresh:1.99,thresh inequal:gt,the weighted error is 0.600 split:dim 1,thresh:2.10,thresh inequal:lt,the weighted error is 0.600 split:dim 1,thresh:2.10,thresh inequal:gt,the weighted error is 0.400
基于单层决策树的AdaBoost训练过程：
def adaBoostTrainDS(dataArr,classLabels,numIter=40):
weakClassArr=[]
m=shape(dataArr)[0]
D=mat(ones((m,1))/m) #每个样本的权重均初始化为1/m
aggClassEst=mat(zeros((m,1)))
for i in range(numIter):
beatStump,error,classEst=buildStump(dataArr,classLabels,D)
print(‘D:‘,D.T)
alpha=float(0.5*log((1.0-error)/max(error,e-16)))
print(‘alpha:‘,alpha)
beatStump[‘alpha‘]=alpha
weakClassArr.append(beatStump)
print(‘分类估计：‘,classEst.T)
expon=multiply(-1*alpha*mat(classLabels).T,classEst)
D=multiply(D,exp(expon))
D=D/D.sum()
aggClassEst+=alpha*classEst
print(‘aggClassEst:‘,aggClassEst.T)
aggErrors=multiply(sign(aggClassEst)!=mat(classLabels).T,ones((m,1)))
errorRate=aggErrors.sum()/m
print(‘total error:‘,errorRate,‘
‘)
if errorRate==0.0:
break
return weakClassArr
D是概率分布向量，D中所有元素之和等于1.
首先利用前面的buildStump()函数建立一个单层决策树。该函数的输入为权重向量D，返回的则是利用D得到的具有最小错误率的单层决策树，同时返回的还有最小的错误率以及预测的类别向量。
alpha=float(0.5*log((1.0-error)/max(error,e-16))) 其中的max(error,e-16)是用来防止error很小时发生的除零溢出。
aggClassEst是m*1阶的列向量，用来存储运行时的类别估计值，符号代表预测结果，为正时表示目前此样本的预测类别为1，为负时表示-1.
aggClassEst+=alpha*classEst 用各弱分类器的分类结果与权重alpha的线性组合值作为最终的预测值。迭代一次，就产生一个弱分类器，相当于对最终的结果修正一次。
aggErrors=multiply(sign(aggClassEst)!=mat(classLabels).T,ones((m,1))) 将分类错误的样本对应位置设置为1，方便求出错误分类总数和错误率。
测试AdaBoost：
if __name__==‘__main__‘:
D=mat(ones((5,1))/5)
datMat, classLabels=loadSimpData()
classifyArray=adaBoostTrainDS(datMat,classLabels,9)
print(classifyArray)
输出：
D: [[0.2 0.2 0.2 0.2 0.2]]
alpha: 0.6931471805599453
分类估计： [[-1. 1. -1. -1. 1.]]
aggClassEst: [[-0.69314718 0.69314718 -0.69314718 -0.69314718 0.69314718]]
total error: 0.2
D: [[0.5 0.125 0.125 0.125 0.125]]
alpha: 0.9729550745276565
分类估计： [[ 1. 1. -1. -1. -1.]]
aggClassEst: [[ 0.27980789 1.66610226 -1.66610226 -1.66610226 -0.27980789]]
total error: 0.2
D: [[0.28571429 0.07142857 0.07142857 0.07142857 0.5 ]]
alpha: 0.8958797346140273
分类估计： [[1. 1. 1. 1. 1.]]
aggClassEst: [[ 1.17568763 2.56198199 -0.77022252 -0.77022252 0.61607184]]
total error: 0.0
[{‘alpha‘: 0.6931471805599453, ‘dim‘: 0, ‘ineq‘: ‘lt‘, ‘thresh‘: 1.3}, {‘alpha‘: 0.9729550745276565, ‘dim‘: 1, ‘ineq‘: ‘lt‘, ‘thresh‘: 1.0}, {‘alpha‘: 0.8958797346140273, ‘dim‘: 0, ‘ineq‘: ‘lt‘, ‘thresh‘: 0.9}]
classifyArray是数组，由三个弱分类器组成，包含了分类所需的所有信息。此时的训练错误率为0，以下讨论其测试错误率。
上述函数的返回值中含有弱分类器及其alpha值，容易进行测试：只需要将弱分类器提取出来作用到待分类数据上，每个弱分类器的结果以其对应的alpha值为权重，所有这些弱分类器的结果加权求和就得到了最后的结果。
if __name__==‘__main__‘:
D=mat(ones((5,1))/5)
datMat, classLabels=loadSimpData()
classifyArray=adaBoostTrainDS(datMat,classLabels,9)
result=adaClassify([[5,5],[0,0]],classifyArray)
print(‘最终分类结果为：‘,result)
输出：
aggClassEst: [[ 0.69314718] [-0.69314718]] aggClassEst: [[ 1.66610226] [-1.66610226]] aggClassEst: [[ 2.56198199] [-2.56198199]] 最终分类结果为： [[ 1.] [-1.]]
由aggClassEst可以看出，随着三个弱分类器的叠加，其预测结果越来越强，即为离分类边界值0的距离越来越远。
在一个难数据集上应用AdaBoost，预测疝病马能否存活。
自适应数据加载函数，不需指定每个文件中的特征数目，并且假定最后一列数据是类别标签。
def loadDataSet(filename):
numFeatures = len(open(filename).readline().split(‘ ‘)) - 1
dataMat = []
labelMat = []
f = open(filename)
for line in f.readlines():
lineArr=[]
curLine=line.strip().split(‘ ‘)
for i in range(0,numFeatures):
lineArr.append(float(curLine[i]))
dataMat.append(lineArr)
labelMat.append(float(curLine[-1]))
return dataMat,labelMat
用疝病马数据集测试元算法：
if __name__==‘__main__‘:
dataArr,labelArr=loadDataSet(‘horseColicTraining2.txt‘)
classifyArray=adaBoostTrainDS(dataArr,labelArr,10)
testArr,testLabelArr=loadDataSet(‘horseColicTest2.txt‘)
prediction10=adaClassify(testArr,classifyArray)
errArr=mat(ones((67,1)))
count=errArr[prediction10!=mat(testLabelArr).T].sum()
print(prediction10)
print(count)
输出：
total error: 0.2842809364548495 total error: 0.2842809364548495 total error: 0.24749163879598662 total error: 0.24749163879598662 total error: 0.25418060200668896 total error: 0.2408026755852843 total error: 0.2408026755852843 total error: 0.22073578595317725 total error: 0.24749163879598662 total error: 0.23076923076923078 [[ 1.] [ 1.] [ 1.] [-1.] [ 1.] [ 1.] [-1.] [ 1.] [ 1.] [-1.] [-1.] [-1.] [-1.] [ 1.] [ 1.] [ 1.] [ 1.] [-1.] [-1.] [-1.] [-1.] [ 1.] [-1.] [-1.] [ 1.] [ 1.] [ 1.] [ 1.] [ 1.] [ 1.] [ 1.] [-1.] [-1.] [-1.] [-1.] [ 1.] [ 1.] [ 1.] [ 1.] [ 1.] [ 1.] [ 1.] [ 1.] [-1.] [-1.] [ 1.] [-1.] [ 1.] [ 1.] [ 1.] [-1.] [ 1.] [ 1.] [ 1.] [ 1.] [ 1.] [ 1.] [-1.] [ 1.] [-1.] [ 1.] [-1.] [-1.] [ 1.] [ 1.] [ 1.] [ 1.]] 16.0
迭代了10次，产生10个弱分类器，训练错误率最终为：total error: 0.23076923076923078
测试数据集上有67个样本，分类结果中有16个错误，错误率为16/67=0.23880597014925373，比起logistic回归预测结果35%的错误率降低很多。
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