《机器学习实战》 第三章【决策树】

决策树(Decision Tree）是在已知各种情况发生概率的基础上，通过构成决策树来求取净现值的期望值大于等于零的概率，评价项目风险，判断其可行性的决策分析方法，是直观运用概率分析的一种图解法。
——百度百科
按照我个人的理解来解释就是，决策树是一种能够理解数据集中内含的知识信息并进行递归划分，直到划分为不可再分的标签为止，它是一种用于分类的递归算法。

算法描述

算法简述

def createBranch():
检测数据集中的每一个子项是否属于同一分类：
if yes:
	return 类标签
else:
	寻找划分数据集的最好特征 #利用熵进行选择
	划分数据集
	创建分支节点
		for每一个划分的子集
			调用createBranch()	#这里就是自身递归调用的位置
	return 分支节点

所以，算法的重点就在于如何寻找划分数据集的最好特征和如何划分数据集

信息增益

• 划分数据集的大原则：将无序数据变得更加有序
• 一种方法是使用“信息论”度量信息
• 在划分数据集之前之后信息发生的变化称为信息增益
• 使用香农公式计算将信息增益（混乱程度）

熵：一种描述混乱度的量

熵定义为信息的期望值，如果待分类的食物可能划分在多个分类中，则符号xi的信息定义为：

p(xi)是描述该分了的概率，而H描述了所以类别所有可能值的信息期望值：

优缺点

• 优点：计算复杂度不高，输出结果易于理解，对中间值的缺失不敏感，可以处理不相关特征数据
• 缺点：可能会产生过度匹配问题
• 适用数据类型：数值型和标称型

一般流程

• 收集数据：任何方法
• 准备数据：树构造算法只适用于标称型数据，因此数值型数据必须离散化
• 分析数据：任何方法，构造树完成后，我们应该检查图形是否符合预期
• 训练算法：构造树的数据结构
• 测试算法：使用经验树计算错误率
• 使用算法：可以适用于任何监督学习算法

一个栗子

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

#计算香农值
def calcShannonEnt(dataSet):
    numEntries = len(dataSet)
    labelCounts = {}
    for featVec in dataSet: #集合中所有的数据
        currentLabel = featVec[-1] #当前的标签
        if currentLabel not in labelCounts.keys(): labelCounts[currentLabel] = 0 #如果没有，字典自动补齐1个
        labelCounts[currentLabel] += 1 #加1个
    shannonEnt = 0.0 #香农值
    #print(labelCounts)
    for key in labelCounts:
        prob = float(labelCounts[key])/numEntries #计算式（归一化）
        shannonEnt -= prob * log(prob,2) #香农公式
    return shannonEnt

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

#按照给定特征划分数据集
#parameter1：待划分的数据集
#paramenter2:第axis列
#paramenter3:返回值为value的部分
#顺便去掉第axis列
def splitDataSet(dataSet,axis,value):
    retDataSet = []     #由于列表是引用传递，所以这里使用临时变量
    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
def chooseBestFeatureToSplit(dataSet):
    numFeatures = len(dataSet[0]) - 1      #得到特征数，去掉最后一列的标签
    baseEntropy = calcShannonEnt(dataSet)   #初始化
    bestInfoGain = 0.0; bestFeature = -1
    for i in range(numFeatures):        #在所有特征中进行选择
        featList = [example[i] for example in dataSet] #创建唯一的分类标签列表
        uniqueVals = set(featList)
        newEntropy = 0.0
        for value in uniqueVals:    #遍历当前特征中的所有属性值，对每个唯一属性值进行划分一次数据集
            subDataSet = splitDataSet(dataSet, i, value)
            prob = len(subDataSet)/float(len(dataSet))
            newEntropy += prob * calcShannonEnt(subDataSet)
        infoGain = baseEntropy - newEntropy     #比较哪个更好，如果新的熵更小，则选用它
        if (infoGain > bestInfoGain):
            bestInfoGain = infoGain
            bestFeature = i
    return bestFeature                      #返回最优特征

def majorityCnt(classList):
    classCount={}
    for vote in classList:
        if vote not in classCount.keys(): classCount[vote]=0
        classCount[vote]+=1
        sortedclassCount=sorted(classCount.items(),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]  # 当所有分类都唯一的时候就停止
    if len(dataSet[0]) == 1:  # 当属性就只剩下一个的时候也停止
        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


myDat,labels=createDataSet()
myTree=createTree(myDat,labels)
print(myTree)

结果：得到一棵树

预测隐形眼镜类型

#coding:utf-8
from math import log
import operator
import treePlotter
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

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
    
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
    
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

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]#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                            
    
def classify(inputTree,featLabels,testVec):
    firstStr = inputTree.keys()[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

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)

fr=open('lenses.txt')
lenses=[inst.strip().split('\t') for inst in fr.readlines()]
lensesLabel=['age','prescript','astigmatic','tearRate']
lensesTree=createTree(lenses,lensesLabel)
treePlotter.createPlot(lensesTree)

结果：