本篇內容為《機器學習實戰》第 7 章利用 AdaBoost 元演算法提高分類效能程式清單。所用程式碼為 python3。

AdaBoost優點：泛化錯誤率低，易編碼，可以應用在大部分分類器上，無引數調整。缺點：對離群點敏感。適用資料型別：數值型和標稱型資料。

boosting 方法擁有多個版本，這裡將只關注其中一個最流行的版本 AdaBoost。

在構造 AdaBoost 的程式碼時，我們將首先通過一個簡單資料集來確保在演算法實現上一切就緒。使用如下的資料集：

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
 

在 python 提示符下，執行程式碼載入資料集：

>>> import adaboost
>>> datMat, classLabels=adaboost.loadSimpData()

我們先給出函式buildStump()的虛擬碼：

程式清單 7-1 單層決策樹生成函式

'''
Created on Sep 20, 2018

@author: yufei
Adaboost is short for Adaptive Boosting
'''

"""
測試是否有某個值小於或大於我們正在測試的閾值
"""
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


"""
在一個加權資料集中迴圈
buildStump()將會遍歷stumpClassify()函式所有的可能輸入值
並找到具有最低錯誤率的單層決策樹
"""
def buildStump(dataArr,classLabels,D):
    dataMatrix = mat(dataArr); labelMat = mat(classLabels).T
    m,n = shape(dataMatrix)
    # 變數 numSteps 用於在特徵的所有可能值上進行遍歷
    numSteps = 10.0
    # 建立一個空字典，用於儲存給定權重向量 D 時所得到的最佳單層決策樹的相關資訊
    bestStump = {}; bestClasEst = mat(zeros((m,1)))
    # 初始化為正無窮大，之後用於尋找可能的最小錯誤率
    minError = inf

    # 第一層迴圈在資料集的所有特徵上遍歷
    for i in range(n):#loop over all dimensions
        rangeMin = dataMatrix[:,i].min(); rangeMax = dataMatrix[:,i].max();
        # 計算步長
        stepSize = (rangeMax-rangeMin)/numSteps
        # 第二層迴圈是瞭解步長後再在這些值上遍歷
        for j in range(-1,int(numSteps)+1):#loop over all range in current dimension
            # 第三個迴圈是在大於和小於之間切換不等式
            for inequal in ['lt', 'gt']: #go over less than and greater than
                threshVal = (rangeMin + float(j) * stepSize)
                # 呼叫 stumpClassify() 函式，返回分類預測結果
                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))
                
                # 將當前錯誤率與已有的最小錯誤率進行比較
                if weightedError < minError:
                    minError = weightedError
                    bestClasEst = predictedVals.copy()
                    bestStump['dim'] = i
                    bestStump['thresh'] = threshVal
                    bestStump['ineq'] = inequal
    return bestStump,minError,bestClasEst
 

為了解實際執行過程，在 python 提示符下，執行程式碼並得到結果：

>>> D=mat(ones((5,1))/5)
>>> adaboost.buildStump(datMat, classLabels, D)
split: dim 0, thresh 0.90, thresh ineqal: lt, the weighted error is 0.400
split: dim 0, thresh 0.90, thresh ineqal: gt, the weighted error is 0.600
split: dim 0, thresh 1.00, thresh ineqal: lt, the weighted error is 0.400
split: dim 0, thresh 1.00, thresh ineqal: gt, the weighted error is 0.600
split: dim 0, thresh 1.10, thresh ineqal: lt, the weighted error is 0.400
split: dim 0, thresh 1.10, thresh ineqal: gt, the weighted error is 0.600
split: dim 0, thresh 1.20, thresh ineqal: lt, the weighted error is 0.400
split: dim 0, thresh 1.20, thresh ineqal: gt, the weighted error is 0.600
split: dim 0, thresh 1.30, thresh ineqal: lt, the weighted error is 0.200
split: dim 0, thresh 1.30, thresh ineqal: gt, the weighted error is 0.800
split: dim 0, thresh 1.40, thresh ineqal: lt, the weighted error is 0.200
split: dim 0, thresh 1.40, thresh ineqal: gt, the weighted error is 0.800
split: dim 0, thresh 1.50, thresh ineqal: lt, the weighted error is 0.200
split: dim 0, thresh 1.50, thresh ineqal: gt, the weighted error is 0.800
split: dim 0, thresh 1.60, thresh ineqal: lt, the weighted error is 0.200
split: dim 0, thresh 1.60, thresh ineqal: gt, the weighted error is 0.800
split: dim 0, thresh 1.70, thresh ineqal: lt, the weighted error is 0.200
split: dim 0, thresh 1.70, thresh ineqal: gt, the weighted error is 0.800
split: dim 0, thresh 1.80, thresh ineqal: lt, the weighted error is 0.200
split: dim 0, thresh 1.80, thresh ineqal: gt, the weighted error is 0.800
split: dim 0, thresh 1.90, thresh ineqal: lt, the weighted error is 0.200
split: dim 0, thresh 1.90, thresh ineqal: gt, the weighted error is 0.800
split: dim 0, thresh 2.00, thresh ineqal: lt, the weighted error is 0.600
split: dim 0, thresh 2.00, thresh ineqal: gt, the weighted error is 0.400
split: dim 1, thresh 0.89, thresh ineqal: lt, the weighted error is 0.400
split: dim 1, thresh 0.89, thresh ineqal: gt, the weighted error is 0.600
split: dim 1, thresh 1.00, thresh ineqal: lt, the weighted error is 0.200
split: dim 1, thresh 1.00, thresh ineqal: gt, the weighted error is 0.800
split: dim 1, thresh 1.11, thresh ineqal: lt, the weighted error is 0.400
split: dim 1, thresh 1.11, thresh ineqal: gt, the weighted error is 0.600
split: dim 1, thresh 1.22, thresh ineqal: lt, the weighted error is 0.400
split: dim 1, thresh 1.22, thresh ineqal: gt, the weighted error is 0.600
split: dim 1, thresh 1.33, thresh ineqal: lt, the weighted error is 0.400
split: dim 1, thresh 1.33, thresh ineqal: gt, the weighted error is 0.600
split: dim 1, thresh 1.44, thresh ineqal: lt, the weighted error is 0.400
split: dim 1, thresh 1.44, thresh ineqal: gt, the weighted error is 0.600
split: dim 1, thresh 1.55, thresh ineqal: lt, the weighted error is 0.400
split: dim 1, thresh 1.55, thresh ineqal: gt, the weighted error is 0.600
split: dim 1, thresh 1.66, thresh ineqal: lt, the weighted error is 0.400
split: dim 1, thresh 1.66, thresh ineqal: gt, the weighted error is 0.600
split: dim 1, thresh 1.77, thresh ineqal: lt, the weighted error is 0.400
split: dim 1, thresh 1.77, thresh ineqal: gt, the weighted error is 0.600
split: dim 1, thresh 1.88, thresh ineqal: lt, the weighted error is 0.400
split: dim 1, thresh 1.88, thresh ineqal: gt, the weighted error is 0.600
split: dim 1, thresh 1.99, thresh ineqal: lt, the weighted error is 0.400
split: dim 1, thresh 1.99, thresh ineqal: gt, the weighted error is 0.600
split: dim 1, thresh 2.10, thresh ineqal: lt, the weighted error is 0.600
split: dim 1, thresh 2.10, thresh ineqal: gt, the weighted error is 0.400
({'dim': 0, 'thresh': 1.3, 'ineq': 'lt'}, matrix([[0.2]]), array([[-1.],
       [ 1.],
       [-1.],
       [-1.],
       [ 1.]]))
 

這一行可以註釋掉，這裡為了理解函式的執行而打印出來。

將當前錯誤率與已有的最小錯誤率進行對比後，如果當前的值較小，那麼就在字典baseStump中儲存該單層決策樹。字典、錯誤率和類別估計值都會返回給 AdaBoost 演算法。

上述，我們已經構建了單層決策樹，得到了弱學習器。接下來，我們將使用多個弱分類器來構建 AdaBoost 程式碼。

首先給出整個實現的虛擬碼如下：

程式清單 7-2 基於單層決策樹的 AdaBoost 訓練過程

'''
輸入引數：資料集、類別標籤、迭代次數（需要使用者指定）
'''
def adaBoostTrainDS(dataArr,classLabels,numIt=40):
    weakClassArr = []
    m = shape(dataArr)[0]
    # 向量 D 包含了每個資料點的權重，初始化為 1/m
    D = mat(ones((m,1))/m)   #init D to all equal
    # 記錄每個資料點的類別估計累計值
    aggClassEst = mat(zeros((m,1)))
    for i in range(numIt):
        # 呼叫 buildStump() 函式建立一個單層決策樹
        bestStump,error,classEst = buildStump(dataArr,classLabels,D)#build Stump

        print ("D:",D.T)

        # 計算 alpha，本次單層決策樹輸出結果的權重
        # 確保沒有錯誤時不會發生除零溢位
        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)

        # 為下一次迭代計算 D
        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)
        # 錯誤率累加計算
        aggClassEst += alpha*classEst
        print("aggClassEst: ",aggClassEst.T)
        # 為了得到二值分類結果呼叫 sign() 函式
        aggErrors = multiply(sign(aggClassEst) != mat(classLabels).T,ones((m,1)))
        errorRate = aggErrors.sum()/m
        print ("total error: ",errorRate)
        # 若總錯誤率為 0，則中止 for 迴圈
        if errorRate == 0.0: break
    return weakClassArr,aggClassEst

在 python 提示符下，執行程式碼並得到結果：

>>> classifierArray = adaboost.adaBoostTrainDS(datMat, classLabels, 9)
D: [[0.2 0.2 0.2 0.2 0.2]]
classEst:  [[-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]]
classEst:  [[ 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       ]]
classEst:  [[1. 1. 1. 1. 1.]]
aggClassEst:  [[ 1.17568763  2.56198199 -0.77022252 -0.77022252  0.61607184]]
total error:  0.0

最後，我們來觀察測試錯誤率。

程式清單 7-3 AdaBoost 分類函式

'''
將弱分類器的訓練過程從程式中抽查來，應用到某個具體的例項上去。

datToClass: 一個或多個待分類樣例
classifierArr: 多個弱分類器組成的陣列

返回 aggClassEst 符號，大於 0 返回1；小於 0 返回 -1
'''
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[0][i]['dim'], classifierArr[0][i]['thresh'],
        classifierArr[0][i]['ineq'])
        aggClassEst += classifierArr[0][i]['alpha']*classEst
        print (aggClassEst)
    return sign(aggClassEst)

在 python 提示符下，執行程式碼並得到結果：

>>> datArr, labelArr = adaboost.loadSimpData()
>>> classifierArr = adaboost.adaBoostTrainDS(datArr, labelArr, 30)
D: [[0.2 0.2 0.2 0.2 0.2]]
classEst:  [[-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]]
classEst:  [[ 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       ]]
classEst:  [[1. 1. 1. 1. 1.]]
aggClassEst:  [[ 1.17568763  2.56198199 -0.77022252 -0.77022252  0.61607184]]
total error:  0.0

輸入以下命令進行分類：

>>> adaboost.adaClassify([0,0], classifierArr)
[[-0.69314718]]
[[-1.66610226]]
matrix([[-1.]])

隨著迭代的進行，資料點 [0,0] 的分類結果越來越強。也可以在其它點上分類：

>>> adaboost.adaClassify([[5,5],[0,0]], classifierArr)
[[ 0.69314718]
 [-0.69314718]]
[[ 1.66610226]
 [-1.66610226]]
matrix([[ 1.],
        [-1.]])

這兩個點的分類結果也會隨著迭代的進行而越來越強。

$$$$

不足之處，歡迎指正。