Python 决策树算法原理及实现

　　决策树是机器学习常见的算法，最著名的代表是ID3、C4.5和CART算法。也有其他的如：OC1算法、ID4h和ITI。他们如何选择最优划分属性不尽相同，ID3决策树是以信息增益为准则来选择划分属性的，C4.5决策树是以信息增益率来选择最优划分属性的，CART是以基尼指数来选择划分属性的,其他决策树算法还没有弄清楚。

（1）信息论中的相关概念

熵：熵原本是物理学中的一个定义，后来香农将其引申到了信息论领域，用来表示信息量的大小。信息量越大（分类越不“纯净”），对应的熵值就越大，反之亦然。信息熵的计算公式如下：

 

 

 

在实际应用中，会将概率𝑝_𝑘的值用经验概率替换，所以经验信息熵可以表示为：

 

 

 

条件熵：

信息增益：

信息增益率：

基尼指数：

 （二）代码实现

from math import log
import operator

def createDataSet():
    """创建数据集"""
    dataSet = [[1, 1, 'yes'],
               [1, 1, 'yes'],
               [1, 0, 'no'],
               [0, 1, 'no'],
               [0, 1, 'no']]
    labels = ['no surfacing','flippers']  #标签类别
    return dataSet, labels

def calcShannonEnt(dataSet):
    """计算信息熵"""
    numEntries = len(dataSet)
    labelCounts = {}
    for featVec in dataSet: #the the number of unique elements and their occurance，遍历一次得到dataset的一行数据
        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

def splitDataSet(dataSet, axis, value):
    """返回axis特征下==value的数据样本
    ，axis为数据集第几个特征
    ，value为特征下的某个值"""
    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

def chooseBestFeatureToSplit(dataSet):
    """以信息增益度的划分方法来选择最优属性"""
    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

def majorityCnt(classList):
    """用来筛选数据最后一个特征时的划分，即投票表决法"""
    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):
    """生成决策树"""
    classList = [example[-1] for example in dataSet]
    
    """迭代终止条件"""
    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                            

if __name__=='__main__':
    """代码测试"""
    dataset ,labels = createDataSet()
    s = calcShannonEnt(dataset)
    retdataset = splitDataSet(dataSet, 1, 1)
    bestFeature = chooseBestFeatureToSplit(dataSet)
    myTree = createTree(dataSet,labels)
    print(myTree)

 结果：

{'no surfacing': {0: 'no', 1: {'flippers': {0: 'no', 1: 'yes'}}}}

二、生成树可视化

import matplotlib.pyplot as plt

decisionNode = 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()
    firstSides = list(myTree.keys())
    firstStr = firstSides[0]#找到输入的第一个元素
    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

def getTreeDepth(myTree):
    maxDepth = 0
#     firstStr = myTree.keys()[0]
    firstSides = list(myTree.keys())
    firstStr = firstSides[0]#找到输入的第一个元素
    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

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 )
    
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
    lt = list(myTree.keys())
    firstStr = lt[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)

if __name__=='__main__':
    """代码测试"""
    dataset ,labels = createDataSet()
    s = calcShannonEnt(dataset)
    retdataset = splitDataSet(dataset, 1, 1)
    bestFeature = chooseBestFeatureToSplit(dataset)
    myTree = createTree(dataset,labels)
    createPlot(myTree)

结果：

参考图书：机器学习实战【美】Peter Harrington 著  李锐 李鹏 曲亚东 王斌 译