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 np
import math
import operator
def createDataSet():
dataSet = [[1,1,'yes'],
[1,1,'yes'],
[1,0,'no'],
[0,1,'no'],
[0,1,'no'],]
label = ['no surfacing','flippers']
return dataSet,label
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
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
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]
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 bestFeature
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]
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
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)
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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