本博文的内容主要是在自学《Machine Learning in Action》的中文版《机器学习实战》的小结，原书中对调用的一些模块的函数并没有做出过多的解释，本文进行了总结和补充。

算法原理

根据信息增益的评判准则，选择一个当前最优的特征对数据集进行分割，递归此操作，直到最后被分割的子数据集只含有一种类型的样本或者用完所有的特征，最后选择该子集中多数的类别最为该子集的最终类别（当然也可以有）。

信息增益

熵（Entropy）:指信息的期望值
熵是一个很玄的概念，人类的成长过程其实就是一个降低熵的过程，就像人刚出生事，大脑内的每一个神经元都是互相连接的，然后随着年龄的增长不断的断掉一些相互连接的神经元。这个过程也可以理解为人类将大自然的信息不断的总结，提炼，精简。这个过程就是熵的降低。人之所以感到学习痛苦大抵是这个原因。

熵的定义公式

如果呆分类的事务可能划分在多个分类之中，则符号xi 的信息定义为：

l(xi)=−log2p(xi)



p(xi) 为该类别的概率。


该数据集分熵为所有类别所有可能值包含的信息期望值：



H=∑i=1np(xi)log2p(xi)

Python代码计算熵

def calcShannonEnt(dataSet):
    numEntries = len(dataSet)
    labelCounts = {}
    #计算每个类别出现的次数
    for featVec in dataSet:
        currentLabel = featVec[-1]
        if currentLabel not in labelCounts.keys():
            labelCounts[currentLabel] = 0
        labelCounts[currentLabel] += 1
    #计算香农熵
    shannonEnt = 0.0
    for key in labelCounts:
        prob = float(labelCounts[key])/numEntries
        shannonEnt -= prob * log(prob,2)
    return shannonEnt

上述熵的计算是计算一个集合的熵，而计算根据一个特征分割成的所有数据子集的熵需要将每个子集的熵乘以该子集的概率最后求和。这一部分的Python代码如下：

for value in uniqueEntropy:
            subDataSet =                                         splitDataSet(dataSet,i,value)
            prob = len(subDataSet)/float(len(dataSet))
            newEntropy += prob * calcShannonEnt(subDataSet)

ID3 Python实现

整个ID3 决策树的Python代码实现如下：

#-*- coding:utf-8 -*-
from math import log

def createDataSet():
    dataSet=[[1,1,'yes'],[1,1,'yes'],[1,0,'no'],[0,1,'no'],[0,0,'no']]
    labels=['no surfacing','flippers']
    return dataSet,labels

#计算给定数据集的香农熵
def calcShannonEnt(dataSet):
    numEntries = len(dataSet)
    labelCounts = {}
    #计算每个类别出现的次数
    for featVec in dataSet:
        currentLabel = featVec[-1]
        if currentLabel not in labelCounts.keys():
            labelCounts[currentLabel] = 0
        labelCounts[currentLabel] += 1
    #计算香农熵
    shannonEnt = 0.0
    for key in labelCounts:
        prob = float(labelCounts[key])/numEntries
        shannonEnt -= prob * log(prob,2)
    return shannonEnt
#按照给定特征划分数据集
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
#选择最好的数据集划分方式
def chooseBestFeatureToSplit(dataSet):
    numFeatures = len(dataSet[0])-1
    baseEntroy = calcShannonEnt(dataSet)
    bestInfoGain = 0.0; bestFeature = -1
    for i in range(numFeatures):
        featList = [example[i] for example in dataSet]
        uniqueEntropy = set(featList)
        newEntropy = 0.0
        for value in uniqueEntropy:
            subDataSet = splitDataSet(dataSet,i,value)
            prob = len(subDataSet)/float(len(dataSet))
            newEntropy += prob * calcShannonEnt(subDataSet)
        infoGain = baseEntroy -newEntropy
        if (infoGain>bestInfoGain):
            bestInfoGain = infoGain
            bestFeature = i
    return bestFeature
#如果数据集已经处理了所有属性，但类标签依然不唯一，则返回类标签出现次数最多的分类名称
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=classCount.iteritems(1),reverse=True)
    return sortedClassCount[0][0]

#创建决策树
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:
        subLables = labels[:]
        myTree[bestFeatLabel][value] = createTree(splitDataSet(dataSet,bestFeat,value),subLables)
    return myTree
def classify(inputTree, featLabels, testVec):
    firstStr = 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
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)