决策树通常在机器学习中用于分类。

优点：计算复杂度不高，输出结果易于理解，对中间值缺失不敏感，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。

1.信息增益

划分数据集的目的是：将无序的数据变得更加有序。组织杂乱无章数据的一种方法就是使用信息论度量信息。通常采用信息增益，信息增益是指数据划分前后信息熵的减少值。信息越无序信息熵越大，获得信息增益最高的特征就是最好的选择。
熵定义为信息的期望，符号xi的信息定义为：

其中p(xi)为该分类的概率。
熵，即信息的期望值为：

计算信息熵的代码如下：

def calcShannonEnt(dataSet): numEntries = len(dataSet) labelCounts = {} for featVec in dataSet: currentLabel = featVec[-1] if currentLabel not in labelCounts: labelCounts[currentLabel] = 0 labelCounts[currentLabel] += 1 shannonEnt = 0 for key in labelCounts: shannonEnt = shannonEnt - (labelCounts[key]/numEntries)*math.log2(labelCounts[key]/numEntries) return shannonEnt

可以根据信息熵，按照获取最大信息增益的方法划分数据集。

2.划分数据集

划分数据集就是将所有符合要求的元素抽出来。

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

3.选择最好的数据集划分方式

信息增益是熵的减少或者是信息无序度的减少。

def chooseBestFeatureToSplit(dataSet): numFeatures = len(dataSet[0]) - 1 bestInfoGain = 0 bestFeature = -1 baseEntropy = calcShannonEnt(dataSet) for i in range(numFeatures): allValue = [example[i] for example in dataSet]#列表推倒，创建新的列表 allValue = set(allValue)#最快得到列表中唯一元素值的方法 newEntropy = 0 for value in allValue: splitset = splitDataSet(dataSet,i,value) newEntropy = newEntropy + len(splitset)/len(dataSet)*calcShannonEnt(splitset) infoGain = baseEntropy - newEntropy if infoGain > bestInfoGain: bestInfoGain = infoGain bestFeature = i return bestFeature

4.递归创建决策树

结束条件为：程序遍历完所有划分数据集的属性，或每个分支下的所有实例都具有相同的分类。
当数据集已经处理了所有属性，但是类标签还不唯一时，采用多数表决的方式决定叶子节点的类型。

def majorityCnt(classList): classCount = {} for value in classList: if value not in classCount: classCount[value] = 0 classCount[value] += 1 classCount = sorted(classCount.items(),key=operator.itemgetter(1),reverse=True) return classCount[0][0]

生成决策树：

def createTree(dataSet,labels): classList = [example[-1] for example in dataSet] labelsCopy = labels[:] if classList.count(classList[0]) == len(classList): return classList[0] if len(dataSet[0]) == 1: return majorityCnt(classList) bestFeature = chooseBestFeatureToSplit(dataSet) bestLabel = labelsCopy[bestFeature] myTree = {bestLabel:{}} featureValues = [example[bestFeature] for example in dataSet] featureValues = set(featureValues) del(labelsCopy[bestFeature]) for value in featureValues: subLabels = labelsCopy[:] myTree[bestLabel][value] = createTree(splitDataSet(dataSet,bestFeature,value),subLabels) return myTree

5.测试算法——使用决策树分类

同样采用递归的方式得到分类结果。

def classify(inputTree,featLabels,testVec): currentFeat = list(inputTree.keys())[0] secondTree = inputTree[currentFeat] try: featureIndex = featLabels.index(currentFeat) except ValueError as err: print('yes') try: for value in secondTree.keys(): if value == testVec[featureIndex]: if type(secondTree[value]).__name__ == 'dict': classLabel = classify(secondTree[value],featLabels,testVec) else: classLabel = secondTree[value] return classLabel except AttributeError: print(secondTree)

6.完整代码如下

import numpy as npimport mathimport operatordef createDataSet(): dataSet = [[1,1,'yes'], [1,1,'yes'], [1,0,'no'], [0,1,'no'], [0,1,'no'],] label = ['no surfacing','flippers'] return dataSet,labeldef calcShannonEnt(dataSet): numEntries = len(dataSet) labelCounts = {} for featVec in dataSet: currentLabel = featVec[-1] if currentLabel not in labelCounts: labelCounts[currentLabel] = 0 labelCounts[currentLabel] += 1 shannonEnt = 0 for key in labelCounts: shannonEnt = shannonEnt - (labelCounts[key]/numEntries)*math.log2(labelCounts[key]/numEntries) return shannonEntdef splitDataSet(dataSet,axis,value): retDataset = [] for featVec in dataSet: if featVec[axis] == value: newVec = featVec[:axis] newVec.extend(featVec[axis+1:]) retDataset.append(newVec) return retDatasetdef chooseBestFeatureToSplit(dataSet): numFeatures = len(dataSet[0]) - 1 bestInfoGain = 0 bestFeature = -1 baseEntropy = calcShannonEnt(dataSet) for i in range(numFeatures): allValue = [example[i] for example in dataSet] allValue = set(allValue) newEntropy = 0 for value in allValue: splitset = splitDataSet(dataSet,i,value) newEntropy = newEntropy + len(splitset)/len(dataSet)*calcShannonEnt(splitset) infoGain = baseEntropy - newEntropy if infoGain > bestInfoGain: bestInfoGain = infoGain bestFeature = i return bestFeaturedef majorityCnt(classList): classCount = {} for value in classList: if value not in classCount: classCount[value] = 0 classCount[value] += 1 classCount = sorted(classCount.items(),key=operator.itemgetter(1),reverse=True) return classCount[0][0] def createTree(dataSet,labels): classList = [example[-1] for example in dataSet] labelsCopy = labels[:] if classList.count(classList[0]) == len(classList): return classList[0] if len(dataSet[0]) == 1: return majorityCnt(classList) bestFeature = chooseBestFeatureToSplit(dataSet) bestLabel = labelsCopy[bestFeature] myTree = {bestLabel:{}} featureValues = [example[bestFeature] for example in dataSet] featureValues = set(featureValues) del(labelsCopy[bestFeature]) for value in featureValues: subLabels = labelsCopy[:] myTree[bestLabel][value] = createTree(splitDataSet(dataSet,bestFeature,value),subLabels) return myTreedef classify(inputTree,featLabels,testVec): currentFeat = list(inputTree.keys())[0] secondTree = inputTree[currentFeat] try: featureIndex = featLabels.index(currentFeat) except ValueError as err: print('yes') try: for value in secondTree.keys(): if value == testVec[featureIndex]: if type(secondTree[value]).__name__ == 'dict': classLabel = classify(secondTree[value],featLabels,testVec) else: classLabel = secondTree[value] return classLabel except AttributeError: print(secondTree)if __name__ == "__main__": dataset,label = createDataSet() myTree = createTree(dataset,label) a = [1,1] print(classify(myTree,label,a))

7.编程技巧

extend与append的区别

newVec.extend(featVec[axis+1:]) retDataset.append(newVec)

extend([])，是将列表中的每个元素依次加入新列表中
append()是将括号中的内容当做一项加入到新列表中

列表推到

创建新列表的方式

allValue = [example[i] for example in dataSet]

提取列表中唯一的元素

allValue = set(allValue)

列表/元组排序,sorted()函数

classCount = sorted(classCount.items(),key=operator.itemgetter(1),reverse=True)

列表的复制

labelsCopy = labels[:]

代码及数据集下载：决策树

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