1.決策樹的相關知識

在之前的接觸中決策樹直觀印象應該就是if-else的迴圈，if會怎麼樣，else之後再繼續if-else直至最終的結果。在上節講的kNN它其實已經可以完成很多工，但是它最大的缺點就是無法給資料集的內在含義，決策樹的主要優勢在於資料形式非常容易理解。

我們來對決策樹進行一個總的概括，然後對程式進行解釋。優點：計算複雜度不高，輸出結果易於理解，對中間值的缺失不敏感，可以處理不相關特徵資料。缺點：可能會產生過度匹配的問題。適用資料型別：數值型和標稱型。

標稱型：標稱型目標變數的結果只在有限目標集中取值，如真與假(標稱型目標變數主要用於分類)

數值型：數值型目標變數則可以從無限的數值集合中取值，如0.100，42.001等 (數值型目標變數主要用於迴歸)

2.資訊增益

講到資訊增益，或者資訊理論也總會提到一個名人克勞德 · 夏農，集合資訊度量方式稱為夏農熵或者簡稱為熵，這個名字就是來源於資訊之父夏農。（其實，我這裡會有一個經驗的分享啦，就是在學習過程中，經常會遇到一些之前從來沒有聽過的名詞或者專業術語，文人之間的咬文嚼字真的我是很無感的，但是理工科是以嚴謹和細緻著稱的，所以看論文或書籍的時候經常看到沒聽過的詞，這時候可千萬不要放棄或者煩操，看一下公式或者解釋，也許會豁然開朗，不過是名字罷了，追根究底有可能只是一行公式）

熵定義為資訊的期望值，在明晰這個概念之前，我們必須明白資訊的定義。如果待分類的事物可能劃分在多個分類之中，則符號xi的資訊定義為如下所示：

l(xi)=−log2p(xi)

其中p(xi)是選擇該分類的概率。

為了計算熵，我們需要計算所有類別可能值包含的資訊期望值，通過下面的公式得到:

H=−∑ni=1p(xi)log2p(xi)

是不是看到熵，心裡就一顫，莫名的煩躁呢。。。也有可能只有LZ是這樣的，但是如果靜下心來，會發現只是個代號，內容還是比較平易近人的哈。

3.決策樹的程式碼實現
這裡主要有兩個.py檔案。
第一個是trees.py，程式碼如下：

#coding:utf-8
#匯入一些操作符

from math import log
import operator

#建立一個簡單的資料集
def createDataSet
 
():
    dataSet = [[1, 1, 'yes'],
               [1, 1, 'yes'],
               [1, 0, 'no'],
               [0, 1, 'no'],
               [0, 1, 'no']]
    labels = ['no surfacing','flippers']
    #change to discrete values
    return dataSet, labels

#計算給定資料集的夏農熵
def calcShannonEnt(dataSet):
    numEntries = len(dataSet)
    labelCounts = {}
    for featVec in dataSet: #the the number of unique elements and their occurance
        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) #log base 2
    return shannonEnt

#按照給定特徵劃分資料集    
def splitDataSet(dataSet, axis, value):
    retDataSet = []
    for featVec in dataSet:
        if featVec[axis] == value:
            reducedFeatVec = featVec[:axis]     #chop out axis used for splitting
            reducedFeatVec.extend(featVec[axis+1:])
            retDataSet.append(reducedFeatVec)
    return retDataSet

#選擇最好的資料集劃分方式    
def chooseBestFeatureToSplit(dataSet):
    numFeatures = len(dataSet[0]) - 1      #the last column is used for the labels
    baseEntropy = calcShannonEnt(dataSet)
    bestInfoGain = 0.0; bestFeature = -1
    for i in range(numFeatures):        #iterate over all the features
        featList = [example[i] for example in dataSet]#create a list of all the examples of this feature
        uniqueVals = set(featList)       #get a set of unique values
        newEntropy = 0.0
        for value in uniqueVals:
            subDataSet = splitDataSet(dataSet, i, value)
            prob = len(subDataSet)/float(len(dataSet))
            newEntropy += prob * calcShannonEnt(subDataSet)     
        infoGain = baseEntropy - newEntropy     #calculate the info gain; ie reduction in entropy
        if (infoGain > bestInfoGain):       #compare this to the best gain so far
            bestInfoGain = infoGain         #if better than current best, set to best
            bestFeature = i
    return bestFeature                      #returns an integer

#返回出現次數最多的分類名稱
def majorityCnt(classList):
    classCount={}
    for vote in classList:
        if vote not in classCount.keys(): classCount[vote] = 0
        classCount[vote] += 1
    sortedClassCount = sorted(classCount.iteritems(), key=operator.itemgetter(1), reverse=True)
    return sortedClassCount[0][0]

#建立樹的函式程式碼
def createTree(dataSet,labels):
    classList = [example[-1] for example in dataSet]
    if classList.count(classList[0]) == len(classList): 
        return classList[0]#stop splitting when all of the classes are equal
    if len(dataSet[0]) == 1: #stop splitting when there are no more features in dataSet
        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[:]       #copy all of labels, so trees don't mess up existing labels
        myTree[bestFeatLabel][value] = createTree(splitDataSet(dataSet, bestFeat, value),subLabels)
    return myTree                            

#使用決策樹執行分類    
def classify(inputTree,featLabels,testVec):
    #python2
    # firstStr = inputTree.keys()[0]
    #python3.5
    firstSides = list(inputTree.keys())
    firstStr = firstSides[0]

    secondDict = inputTree[firstStr]
    featIndex = featLabels.index(firstStr)
    key = testVec[featIndex]
    valueOfFeat = secondDict[key]
    if isinstance(valueOfFeat, dict): 
        classLabel = classify(valueOfFeat, featLabels, testVec)
    else: classLabel = valueOfFeat
    return classLabel

#使用pickle模組儲存決策樹
def storeTree(inputTree,filename):
    import pickle
    fw = open(filename,'w')
    pickle.dump(inputTree,fw)
    fw.close()

def grabTree(filename):
    import pickle
    fr = open(filename)
    return pickle.load(fr)

第二個主要是繪製決策樹圖treePlotter.py

#coding:utf-8

#匯入matplotlib.pyplot,如果顯示mo module,請自行安裝
import matplotlib.pyplot as plt

#定義文字框和箭頭格式
decisionNode = dict(boxstyle="sawtooth", fc="0.8")
leafNode = dict(boxstyle="round4", fc="0.8")
arrow_args = dict(arrowstyle="<-")

#獲取葉節點數
def getNumLeafs(myTree):
    numLeafs = 0
    #如果使用的是python2，用註釋掉的一行
    # firstStr = myTree.keys()[0]


    #LZ使用的是python3
    firstSides = list(myTree.keys())
    firstStr = firstSides[0]
    secondDict = myTree[firstStr]
    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
            numLeafs += getNumLeafs(secondDict[key])
        else:   numLeafs +=1
    return numLeafs

#獲取樹的層數
def getTreeDepth(myTree):
    maxDepth = 0
    #python2的用法
    # firstStr = myTree.keys()[0]

    #python3使用下面兩行
    firstSides = list(myTree.keys())
    firstStr = firstSides[0]

    secondDict = myTree[firstStr]
    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
            thisDepth = 1 + getTreeDepth(secondDict[key])
        else:   thisDepth = 1
        if thisDepth > maxDepth: maxDepth = thisDepth
    return maxDepth

#繪製帶箭頭的註解
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 )

#在父子節點填充文字資訊    
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)

#plotTree函式
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)
    #python2
    # firstStr = myTree.keys()[0]     #the text label for this node should be this
    #python3
    firstSides = list(myTree.keys())
    firstStr = firstSides[0]
    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
#if you do get a dictonary you know it's a tree, and the first element will be another dict

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.show()

#def createPlot():
#    fig = plt.figure(1, facecolor='white')
#    fig.clf()
#    createPlot.ax1 = plt.subplot(111, frameon=False) #ticks for demo puropses 
#    plotNode('a decision node', (0.5, 0.1), (0.1, 0.5), decisionNode)
#    plotNode('a leaf node', (0.8, 0.1), (0.3, 0.8), leafNode)
#    plt.show()

def retrieveTree(i):
    listOfTrees =[{'no surfacing': {0: 'no', 1: {'flippers': {0: 'no', 1: 'yes'}}}},
                  {'no surfacing': {0: 'no', 1: {'flippers': {0: {'head': {0: 'no', 1: 'yes'}}, 1: 'no'}}}}
                  ]
    return listOfTrees[i]

#createPlot(thisTree)

這裡貼出程式碼和程式碼配套的資料集，有興趣的小夥伴可以自行下載。連結: https://pan.baidu.com/s/1gf664dP 密碼: 32r5
需要注意的是因為LZ筆記本上裝的是python3.5，為了確保程式碼編譯成功，把原本的python2版本的程式碼改了一部分，註釋裡都有，小夥伴可以根據自己的需求進行選擇O(∩_∩)O