前置信息

1、決策樹

決策樹是一種十分常用的分類算法，屬于監(jiān)督學習；也就是給出一批樣本，每個樣本都有一組屬性和一個分類結(jié)果。算法通過學習這些樣本，得到一個決策樹，這個決策樹能夠?qū)π碌臄?shù)據(jù)給出合適的分類

2、樣本數(shù)據(jù)

假設現(xiàn)有用戶14名，其個人屬性及是否購買某一產(chǎn)品的數(shù)據(jù)如下：

決策樹分類算法

1、構建數(shù)據(jù)集

為了方便處理，對模擬數(shù)據(jù)按以下規(guī)則轉(zhuǎn)換為數(shù)值型列表數(shù)據(jù)：
年齡：<30賦值為0；30-40賦值為1；>40賦值為2
收入：低為0；中為1；高為2
工作性質(zhì)：不穩(wěn)定為0；穩(wěn)定為1
信用評級：差為0；好為1

#創(chuàng)建數(shù)據(jù)集
def createdataset():
    dataSet=[[0,2,0,0,'N'],
            [0,2,0,1,'N'],
            [1,2,0,0,'Y'],
            [2,1,0,0,'Y'],
            [2,0,1,0,'Y'],
            [2,0,1,1,'N'],
            [1,0,1,1,'Y'],
            [0,1,0,0,'N'],
            [0,0,1,0,'Y'],
            [2,1,1,0,'Y'],
            [0,1,1,1,'Y'],
            [1,1,0,1,'Y'],
            [1,2,1,0,'Y'],
            [2,1,0,1,'N'],]
    labels=['age','income','job','credit']
    return dataSet,labels

調(diào)用函數(shù)，可獲得數(shù)據(jù)：

ds1,lab = createdataset()
print(ds1)
print(lab)

[[0, 2, 0, 0, ‘N’], [0, 2, 0, 1, ‘N’], [1, 2, 0, 0, ‘Y’], [2, 1, 0, 0, ‘Y’], [2, 0, 1, 0, ‘Y’], [2, 0, 1, 1, ‘N’], [1, 0, 1, 1, ‘Y’], [0, 1, 0, 0, ‘N’], [0, 0, 1, 0, ‘Y’], [2, 1, 1, 0, ‘Y’], [0, 1, 1, 1, ‘Y’], [1, 1, 0, 1, ‘Y’], [1, 2, 1, 0, ‘Y’], [2, 1, 0, 1, ‘N’]]
[‘a(chǎn)ge’, ‘income’, ‘job’, ‘credit’]

2、數(shù)據(jù)集信息熵

信息熵也稱為香農(nóng)熵，是隨機變量的期望。度量信息的不確定程度。信息的熵越大，信息就越不容易搞清楚。處理信息就是為了把信息搞清楚，就是熵減少的過程。

def calcShannonEnt(dataSet):
    numEntries = len(dataSet)
    labelCounts = {}
    for featVec in dataSet:
        currentLabel = featVec[-1]
        if currentLabel not in labelCounts.keys():
            labelCounts[currentLabel] = 0
        
        labelCounts[currentLabel] += 1            
        
    shannonEnt = 0.0
    for key in labelCounts:
        prob = float(labelCounts[key])/numEntries
        shannonEnt -= prob*log(prob,2)
    
    return shannonEnt

樣本數(shù)據(jù)信息熵：

shan = calcShannonEnt(ds1)
print(shan)

0.9402859586706309

3、信息增益

信息增益：用于度量屬性A降低樣本集合X熵的貢獻大小。信息增益越大，越適于對X分類。

def chooseBestFeatureToSplit(dataSet):
    numFeatures = len(dataSet[0])-1
    baseEntropy = calcShannonEnt(dataSet)
    bestInfoGain = 0.0;bestFeature = -1
    for i in range(numFeatures):
        featList = [example[i] for example in dataSet]
        uniqueVals = set(featList)
        newEntroy = 0.0
        for value in uniqueVals:
            subDataSet = splitDataSet(dataSet, i, value)
            prop = len(subDataSet)/float(len(dataSet))
            newEntroy += prop * calcShannonEnt(subDataSet)
        infoGain = baseEntropy - newEntroy
        if(infoGain > bestInfoGain):
            bestInfoGain = infoGain
            bestFeature = i    
    return bestFeature

以上代碼實現(xiàn)了基于信息熵增益的ID3決策樹學習算法。其核心邏輯原理是：依次選取屬性集中的每一個屬性，將樣本集按照此屬性的取值分割為若干個子集；對這些子集計算信息熵，其與樣本的信息熵的差，即為按照此屬性分割的信息熵增益；找出所有增益中最大的那一個對應的屬性，就是用于分割樣本集的屬性。

計算樣本最佳的分割樣本屬性，結(jié)果顯示為第0列，即age屬性：

col = chooseBestFeatureToSplit(ds1)
col

0

4、構造決策樹

def majorityCnt(classList):
    classCount = {}
    for vote in classList:
        if vote not in classCount.keys():classCount[vote] = 0
        classCount[vote] += 1
    sortedClassCount = sorted(classList.iteritems(),key=operator.itemgetter(1),reverse=True)#利用operator操作鍵值排序字典
    return sortedClassCount[0][0]

#創(chuàng)建樹的函數(shù)    
def createTree(dataSet,labels):
    classList = [example[-1] for example in dataSet]
    if classList.count(classList[0]) == len(classList):
        return classList[0]
    if len(dataSet[0]) == 1:
        return majorityCnt(classList)
    bestFeat = chooseBestFeatureToSplit(dataSet)
    bestFeatLabel = labels[bestFeat]
    myTree = {bestFeatLabel:{}}
    del(labels[bestFeat])
    featValues = [example[bestFeat] for example in dataSet]
    uniqueVals = set(featValues)
    for value in uniqueVals:
        subLabels = labels[:]
        myTree[bestFeatLabel][value] = createTree(splitDataSet(dataSet, bestFeat, value), subLabels)
        
    return myTree

majorityCnt函數(shù)用于處理一下情況：最終的理想決策樹應該沿著決策分支到達最底端時，所有的樣本應該都是相同的分類結(jié)果。但是真實樣本中難免會出現(xiàn)所有屬性一致但分類結(jié)果不一樣的情況，此時majorityCnt將這類樣本的分類標簽都調(diào)整為出現(xiàn)次數(shù)最多的那一個分類結(jié)果。

createTree是核心任務函數(shù)，它對所有的屬性依次調(diào)用ID3信息熵增益算法進行計算處理，最終生成決策樹。

5、實例化構造決策樹

利用樣本數(shù)據(jù)構造決策樹：

Tree = createTree(ds1, lab)
print("樣本數(shù)據(jù)決策樹：")
print(Tree)

樣本數(shù)據(jù)決策樹：
{‘a(chǎn)ge’: {0: {‘job’: {0: ‘N’, 1: ‘Y’}},
1: ‘Y’,
2: {‘credit’: {0: ‘Y’, 1: ‘N’}}}}

6、測試樣本分類

給出一個新的用戶信息，判斷ta是否購買某一產(chǎn)品：

def classify(inputtree,featlabels,testvec):
    firststr = list(inputtree.keys())[0]
    seconddict = inputtree[firststr]
    featindex = featlabels.index(firststr)
    for key in seconddict.keys():
        if testvec[featindex]==key:
            if type(seconddict[key]).__name__=='dict':
                classlabel=classify(seconddict[key],featlabels,testvec)
            else:
                classlabel=seconddict[key]
    return classlabel

labels=['age','income','job','credit']
tsvec=[0,0,1,1]
print('result:',classify(Tree,labels,tsvec))
tsvec1=[0,2,0,1]
print('result1:',classify(Tree,labels,tsvec1))

result: Y
result1: N

后置信息：繪制決策樹代碼

以下代碼用于繪制決策樹圖形，非決策樹算法重點，有興趣可參考學習文章來源地址http://www.zghlxwxcb.cn/news/detail-417153.html

import matplotlib.pyplot as plt

decisionNode = dict(boxstyle="sawtooth", fc="0.8")
leafNode = dict(boxstyle="round4", fc="0.8")
arrow_args = dict(arrowstyle="<-")

#獲取葉節(jié)點的數(shù)目
def getNumLeafs(myTree):
    numLeafs = 0
    firstStr = list(myTree.keys())[0]
    secondDict = myTree[firstStr]
    for key in secondDict.keys():
        if type(secondDict[key]).__name__=='dict':#測試節(jié)點的數(shù)據(jù)是否為字典，以此判斷是否為葉節(jié)點
            numLeafs += getNumLeafs(secondDict[key])
        else:   numLeafs +=1
    return numLeafs

#獲取樹的層數(shù)
def getTreeDepth(myTree):
    maxDepth = 0
    firstStr = list(myTree.keys())[0]
    secondDict = myTree[firstStr]
    for key in secondDict.keys():
        if type(secondDict[key]).__name__=='dict':#測試節(jié)點的數(shù)據(jù)是否為字典，以此判斷是否為葉節(jié)點
            thisDepth = 1 + getTreeDepth(secondDict[key])
        else:   thisDepth = 1
        if thisDepth > maxDepth: maxDepth = thisDepth
    return maxDepth

#繪制節(jié)點
def plotNode(nodeTxt, centerPt, parentPt, nodeType):
    createPlot.ax1.annotate(nodeTxt, xy=parentPt,  xycoords='axes fraction',
             xytext=centerPt, textcoords='axes fraction',
             va="center", ha="center", bbox=nodeType, arrowprops=arrow_args )

#繪制連接線  
def plotMidText(cntrPt, parentPt, txtString):
    xMid = (parentPt[0]-cntrPt[0])/2.0 + cntrPt[0]
    yMid = (parentPt[1]-cntrPt[1])/2.0 + cntrPt[1]
    createPlot.ax1.text(xMid, yMid, txtString, va="center", ha="center", rotation=30)

#繪制樹結(jié)構  
def plotTree(myTree, parentPt, nodeTxt):#if the first key tells you what feat was split on
    numLeafs = getNumLeafs(myTree)  #this determines the x width of this tree
    depth = getTreeDepth(myTree)
    firstStr = list(myTree.keys())[0]     #the text label for this node should be this
    cntrPt = (plotTree.xOff + (1.0 + float(numLeafs))/2.0/plotTree.totalW, plotTree.yOff)
    plotMidText(cntrPt, parentPt, nodeTxt)
    plotNode(firstStr, cntrPt, parentPt, decisionNode)
    secondDict = myTree[firstStr]
    plotTree.yOff = plotTree.yOff - 1.0/plotTree.totalD
    for key in secondDict.keys():
        if type(secondDict[key]).__name__=='dict':#test to see if the nodes are dictonaires, if not they are leaf nodes   
            plotTree(secondDict[key],cntrPt,str(key))        #recursion
        else:   #it's a leaf node print the leaf node
            plotTree.xOff = plotTree.xOff + 1.0/plotTree.totalW
            plotNode(secondDict[key], (plotTree.xOff, plotTree.yOff), cntrPt, leafNode)
            plotMidText((plotTree.xOff, plotTree.yOff), cntrPt, str(key))
    plotTree.yOff = plotTree.yOff + 1.0/plotTree.totalD

#創(chuàng)建決策樹圖形    
def createPlot(inTree):
    fig = plt.figure(1, facecolor='white')
    fig.clf()
    axprops = dict(xticks=[], yticks=[])
    createPlot.ax1 = plt.subplot(111, frameon=False, **axprops)    #no ticks
    #createPlot.ax1 = plt.subplot(111, frameon=False) #ticks for demo puropses 
    plotTree.totalW = float(getNumLeafs(inTree))
    plotTree.totalD = float(getTreeDepth(inTree))
    plotTree.xOff = -0.5/plotTree.totalW; plotTree.yOff = 1.0;
    plotTree(inTree, (0.5,1.0), '')
    plt.savefig('決策樹.png',dpi=300,bbox_inches='tight')
    plt.show()

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