决策树实战：从原理到实现

主要内容：

1. 决策树简介

2. 决策树的构建算法

3. python实现决策树

4. 扩展

5. 代码附录

6 .参考资料

决策树（decision tree) 是一种基本的分类和回归方法，决策树的学习包括3个步骤：特征选择、决策树的生成和决策树的剪枝。本文讨论决策树用作分类的原理和ID3算法的python实现。

1.决策树

决策树模型是一种描述对实例进行分类的树形结构。决策树由节点和有向边组成。节点有内部节点(internal node)和叶子节点(leaf node)，内部节点表示一个特征（feature)，叶子节点(node)表示一个类（label)。有向边表示相应特征的取值。如图所示是判断一个是否能够有偿还债务的能力的决策树：

2. 如何从历史数据构建决策树

如何一步步构建决策树，关键是找出相应的节点和边。我们需要解决的第一个问题就是，当前数据集上哪个特征在划分数据分类时起决定性作用。为了找出决定性的特征，划分出最好的结果，必须评估每个特征。信息增益石一种有效的评估方法。

2.1.信息增益（information gain)

信息增益表示得知特征X的信息而使得类Y的信息的不确定性减少程度

                              g(D,A)=H(D)-H(D|A)

H（D)数据D的经验熵， H(D|A) 特征A给定条件下D的经验条件熵， 也称为互信息，决策树学习中的信息增益等价于训练数据集中类与特征的互信息

基于信息增益准则的特征选择方法是：对训练数据集D，计算其每个特征的信息增益熵， 选择信息增益最大的特征。

信息增益算法：

                 

2.2 决策树的生产

算法流程：

3. 实战决策树

首先获取数据：

def createDataSet():      #dataSet=pd.read_csv(datafile)   # label=[]    dataSet=[[1,1,'yes'],[1,1,'yes'],[1,0,'no'],[0,1,'no'],[0,1,'no']]    labels=['no surfacing','flippers']    return dataSet, labels

根据数据递归生成树：

def createTree(dataSet,labels):    classList=[example[-1] for example in dataSet]    if classList.count(classList[0])==len(classList):        return classList[0]      #如果改子集全部是同一类，返回该类    if len(dataSet[0])==1:        return majorityCnt(classList)   #返回叶子节点的值    bestFeat=chooseBestFeatureToSplit(dataSet)   #选取最佳特征    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\     #递归调用，采用字典方式存储树的结构                                    (dataSet,bestFeat,value),subLabels)    return myTree

计算决策用到的子函数：

计算给定数据集的香农熵：

def calcShannonEnt(dataSet):    numEntries=len(dataSet)    labelCounts={}    for featVec in dataSet:        currentLabel=featVec[-1]        #计算feature值的频数，为所有可能的类创建字典        if currentLabel not in labelCounts.keys():            labelCounts[currentLabel]=0        labelCounts[currentLabel]+=1    shannonEnt=0.0    #计算Shannon entropy    for key in labelCounts:        prob=float(labelCounts[key])/numEntries        shannonEnt-=prob*log(prob,2)    return shannonEnt

分割数据

def splitDataSet(dataSet, axis, value):    retDataSet=[]    for featVec in dataSet:        if featVec[axis]==value:            reducedFeatVec=featVec[:axis]            reducedFeatVec.extend(featVec[axis+1:])            retDataSet.append(reducedFeatVec)    return retDataSet

选择最好的特征进行分割数据

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)        newEntropy=0.0        #计算每一种划分方式的信息熵        for value in uniqueVals:            subDataSet=splitDataSet(dataSet,i,value)            prob=len(subDataSet)/float(len(dataSet))            newEntropy+=prob*calcShannonEnt(subDataSet)        #信息增益熵        infoGain=baseEntropy - newEntropy        #最好信息增益        if(infoGain>bestInfoGain):            bestInfoGain=infoGain            bestFeature=i    return bestFeature    

#统计数据集出现频率最高的label

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]

5.附录代码：.

5.1 trees.py

# -*- coding: utf-8 -*-"""Created on Fri Apr 03 09:13:58 2015@author: beta"""from math import logimport operatorimport  treePlotterdef calcShannonEnt(dataSet):    numEntries=len(dataSet)    labelCounts={}    for featVec in dataSet:        currentLabel=featVec[-1]        #计算feature值的频数，为所有可能的类创建字典        if currentLabel not in labelCounts.keys():            labelCounts[currentLabel]=0        labelCounts[currentLabel]+=1    shannonEnt=0.0    #计算Shannon entropy    for key in labelCounts:        prob=float(labelCounts[key])/numEntries        shannonEnt-=prob*log(prob,2)    return shannonEnt    def createDataSet():      #dataSet=pd.read_csv(datafile)   # label=[]    dataSet=[[1,1,'yes'],[1,1,'yes'],[1,0,'no'],[0,1,'no'],[0,1,'no']]    labels=['no surfacing','flippers']    return dataSet, labelsdef splitDataSet(dataSet, axis, value):    retDataSet=[]    for featVec in dataSet:        if featVec[axis]==value:            reducedFeatVec=featVec[:axis]            reducedFeatVec.extend(featVec[axis+1:])            retDataSet.append(reducedFeatVec)    return retDataSetdef 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)        newEntropy=0.0        #计算每一种划分方式的信息熵        for value in uniqueVals:            subDataSet=splitDataSet(dataSet,i,value)            prob=len(subDataSet)/float(len(dataSet))            newEntropy+=prob*calcShannonEnt(subDataSet)        #信息增益熵        infoGain=baseEntropy - newEntropy        #最好信息增益        if(infoGain>bestInfoGain):            bestInfoGain=infoGain            bestFeature=i    return bestFeature    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]    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    if __name__=='__main__':        myDat, labels=createDataSet()    myTree=createTree(myDat,labels)    treePlotter.createPlot(myTree)    

5.2 treePlotter.py 代码：

'''Created on Oct 14, 2010@author: Peter Harrington'''import matplotlib.pyplot as pltdecisionNode = dict(boxstyle="sawtooth", fc="0.8")leafNode = dict(boxstyle="round4", fc="0.8")arrow_args = dict(arrowstyle="<-")def getNumLeafs(myTree):    numLeafs = 0    firstStr = myTree.keys()[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 numLeafsdef getTreeDepth(myTree):    maxDepth = 0    firstStr = myTree.keys()[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 maxDepthdef 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 )    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)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 = 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#if you do get a dictonary you know it's a tree, and the first element will be another dictdef 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)

6.参考资料：

1. 李航 《统计学习方法》

2.. Peter Harrington 《机器学习实战》