一.Adaboost理論部分

1.1 adaboost運行過程

注釋：算法是利用指數函數降低誤差，運行過程通過迭代進行。其中函數的算法怎么來的，你不用知道！當然你也可以嘗試使用其它的函數代替指數函數，看看效果如何。

1.2 舉例說明算法流程

1.3 算法誤差界的證明

注釋：誤差的上界限由Zm約束，然而Zm又是由Gm(xi)約束，所以選擇適當的Gm(xi)可以加快誤差的減小。

二.代碼實現

2.1程序流程圖

2.2基本程序實現

注釋：真是倒霉玩意，本來代碼全部注釋好了，突然Ubuntu奔潰了，全部程序就GG了。。。下面的代碼就是官網的代碼，部分補上注釋。現在使用Deepin桌面版了，其它方面都比Ubuntu好，但是有點點卡。

from numpy import *

def loadDataSet(fileName): #general function to parse tab -delimited floats

numFeat = len(open(fileName).readline().split(' ')) #get number of fields

dataMat = []; labelMat = []

fr = open(fileName)

for line in fr.readlines():

lineArr =[]

curLine = line.strip().split(' ')

for i in range(numFeat-1):

lineArr.append(float(curLine[i]))

dataMat.append(lineArr)

labelMat.append(float(curLine[-1]))

return dataMat,labelMat

def stumpClassify(dataMatrix,dimen,threshVal,threshIneq):#just classify the data

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 #init error sum, to +infinity

for i in range(n):#loop over all dimensions

rangeMin = dataMatrix[:,i].min(); rangeMax = dataMatrix[:,i].max();

stepSize = (rangeMax-rangeMin)/numSteps

for j in range(-1,int(numSteps)+1):#loop over all range in current dimension

for inequal in ['lt', 'gt']: #go over less than and greater than

threshVal = (rangeMin + float(j) * stepSize)

predictedVals = stumpClassify(dataMatrix,i,threshVal,inequal)#call stump classify with i, j, lessThan

errArr = mat(ones((m,1)))

errArr[predictedVals == labelMat] = 0

weightedError = D.T*errArr #calc total error multiplied by D

#print "split: dim %d, thresh %.2f, thresh ineqal: %s, the weighted error is %.3f" % (i, threshVal, inequal, weightedError)

if weightedError < minError:

minError = weightedError

bestClasEst = predictedVals.copy()

bestStump['dim'] = i

bestStump['thresh'] = threshVal

bestStump['ineq'] = inequal

return bestStump,minError,bestClasEst

def adaBoostTrainDS(dataArr,classLabels,numIt=40):

weakClassArr = []

m = shape(dataArr)[0]

D = mat(ones((m,1))/m) #init D to all equal

aggClassEst = mat(zeros((m,1)))

for i in range(numIt):

bestStump,error,classEst = buildStump(dataArr,classLabels,D)#build Stump

#print "D:",D.T

alpha = float(0.5*log((1.0-error)/max(error,1e-16)))#calc alpha, throw in max(error,eps) to account for error=0

bestStump['alpha'] = alpha

weakClassArr.append(bestStump) #store Stump Params in Array

#print "classEst: ",classEst.T

expon = multiply(-1*alpha*mat(classLabels).T,classEst) #exponent for D calc, getting messy

D = multiply(D,exp(expon)) #Calc New D for next iteration

D = D/D.sum()

#calc training error of all classifiers, if this is 0 quit for loop early (use break)

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,aggClassEst

def adaClassify(datToClass,classifierArr):

dataMatrix = mat(datToClass)#do stuff similar to last aggClassEst in adaBoostTrainDS

m = shape(dataMatrix)[0]

aggClassEst = mat(zeros((m,1)))

for i in range(len(classifierArr)):

classEst = stumpClassify(dataMatrix,classifierArr[i]['dim'],

classifierArr[i]['thresh'],

classifierArr[i]['ineq'])#call stump classify

aggClassEst += classifierArr[i]['alpha']*classEst

#print aggClassEst

return sign(aggClassEst)

def plotROC(predStrengths, classLabels):

import matplotlib.pyplot as plt

cur = (1.0,1.0) #cursor

ySum = 0.0 #variable to calculate AUC

numPosClas = sum(array(classLabels)==1.0)#標簽等于1的和（也等于個數）

yStep = 1/float(numPosClas); xStep = 1/float(len(classLabels)-numPosClas)

sortedIndicies = predStrengths.argsort()#get sorted index, it's reverse

sortData = sorted(predStrengths.tolist()[0])

fig = plt.figure()

fig.clf()

ax = plt.subplot(111)

#loop through all the values, drawing a line segment at each point

for index in sortedIndicies.tolist()[0]:

if classLabels[index] == 1.0:

delX = 0; delY = yStep;

else:

delX = xStep; delY = 0;

ySum += cur[1]

#draw line from cur to (cur[0]-delX,cur[1]-delY)

ax.plot([cur[0],cur[0]-delX],[cur[1],cur[1]-delY], c='b')

cur = (cur[0]-delX,cur[1]-delY)

ax.plot([0,1],[0,1],'b--')

plt.xlabel('False positive rate'); plt.ylabel('True positive rate')

plt.title('ROC curve for AdaBoost horse colic detection system')

ax.axis([0,1,0,1])

plt.show()

print ("the Area Under the Curve is: ",ySum*xStep)

注釋：重點說明一下非均衡分類的圖像繪制問題，想了很久才想明白！

都是相對而言的，其中本文說的曲線在左上方就為好，也是相對而言的，看你怎么定義個理解！