决策树-使用ID3算法划分数据集
1. 信息增益
from math import log# 计算给定数据集的香农信息熵def calcShannonEnt(dataSet): numEntries = len(dataSet) labelCounts = {} for featVec in dataSet: 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) return shannonEnt def createDataSet(): dataSet = [[1,1,'yes'], [1,1,'yes'], [1,0,'no'], [0,1,'no'], [0,1,'no']] labels = ['no surfacing', 'filppers'] return dataSet, labels
myDat, labels = createDataSet()
myDat
[[1, 1, 'yes'], [1, 1, 'yes'], [1, 0, 'no'], [0, 1, 'no'], [0, 1, 'no']]
calcShannonEnt(myDat)
0.9709505944546686
myDat[0][-1]='maybe'myDat
[[1, 1, 'maybe'], [1, 1, 'yes'], [1, 0, 'no'], [0, 1, 'no'], [0, 1, 'no']]
calcShannonEnt(myDat)
1.3709505944546687
2. 划分数据集
# 按照给定特征featvec[axis]划分数据集，返回featVec[axis]==value的集合def splitDataSet(dataSet, axis, value): retDataSet = [] for featVec in dataSet: if featVec[axis] == value: reducedFeatVec = featVec[:axis] reducedFeatVec.extend(featVec[axis+1:]) # 抽取 retDataSet.append(reducedFeatVec) return retDataSet
myDat, labels = createDataSet()
myDat
[[1, 1, 'yes'], [1, 1, 'yes'], [1, 0, 'no'], [0, 1, 'no'], [0, 1, 'no']]
splitDataSet(myDat,0,1)
[[1, 'yes'], [1, 'yes'], [0, 'no']]
splitDataSet(myDat,0,0)
[[1, 'no'], [1, 'no']]
# 选择最好的数据集划分方式def chooseBestFeatureToSplit(dataSet): numFeatures = len(dataSet[0]) - 1 baseEntropy = calcShannonEnt(dataSet) bestInfoGain = 0.0 bestFeature = -1 for i in range(numFeatures): featList = [example[i] for example in dataSet] uniqueVals = set(featList) 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 if (infoGain > bestInfoGain): bestInfoGain = infoGain bestFeature = i return bestFeature
myDat, labels = createDataSet()
chooseBestFeatureToSplit(myDat)
0
myDat
[[1, 1, 'yes'], [1, 1, 'yes'], [1, 0, 'no'], [0, 1, 'no'], [0, 1, 'no']]
3. 递归构造决策树
import operator# 如果数据集已经处理了所有属性，凡是类标签依然不是唯一，此时可以通过多数表决的方式定义该叶子节点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] #创建树def createTree(dataSet, labels): classList = [example[-1] for example in dataSet] if classList.count(classList[0]) == len(classList): # 类别完全相同则停止继续划分 return classList[0] if len(dataSet[0]) == 1: # 已经遍历了所有特征，返回多数表决的结果 return majorityCnt(classList) bestFeat = chooseBestFeatureToSplit(dataSet) bestFeatLabel = labels[bestFeat] myTree = {bestFeatLabel:{}} # 使用字典类型存储树 del(labels[bestFeat]) featValues = [example[bestFeat] for example in dataSet] uniqueVals = set(featValues) # 得到列表包含的所有属性值 for value in uniqueVals: subLabels = labels[:] # ！！复制了类标签，因为labels是列表类型的，参数按照引用的方式传递需要保证不改变原始列表的内容 myTree[bestFeatLabel][value] = createTree(splitDataSet(dataSet, bestFeat, value), subLabels) return myTree myDat, labels = createDataSet()myTree = createTree(myDat, labels)myTree
{'no surfacing': {0: 'no', 1: {'filppers': {0: 'no', 1: 'yes'}}}} 4. 构造注解树
# 使用文本注解工具（annotations）绘制树节点import matplotlib.pyplot as plt%matplotlib inline# 定义文本框和箭头格式decisionNode = dict(boxstyle='sawtooth', fc='0.8')leafNode = dict(boxstyle='round4', fc='0.8')arrow_args = dict(arrowstyle='<-')# 绘制带箭头的注解def plotNode(nodeTxt, centerPt, parentPt, nodeType): # centerPt是文本框的位置(xytext)， parentPt是箭头起始位置 createPlot.ax1.annotate(nodeTxt, xy=parentPt, xycoords='axes fraction', xytext=centerPt, textcoords='axes fraction',\ va='center', ha='center', bbox=nodeType, arrowprops=arrow_args)
def createPlot(): fig = plt.figure(1, facecolor='white') fig.clf() # 清空绘图区 createPlot.ax1 = plt.subplot(111, frameon=False) plotNode('decisionNode', (0.5,0.1), (0.1,0.5), decisionNode) plotNode('leafNode', (0.8,0.1), (0.3,0.8), leafNode) plt.show() createPlot()
# 获取叶节点的数目和树的层数，来确定x轴的长度和y轴的高度def getNumLeafs(myTree): numLeafs = 0 firstStr = list(myTree.keys())[0] secondDict = myTree[firstStr] for key in secondDict.keys(): if type(secondDict[key]).__name__ == 'dict': numLeafs += getNumLeafs(secondDict[key]) else: numLeafs += 1 return numLeafs
def getTreeDepth(myTree): maxDepth = 0 firstStr = list(myTree.keys())[0] secondDict = myTree[firstStr] for key in secondDict.keys(): if type(secondDict[key]).__name__ == 'dict': thisDepth = 1 + getTreeDepth(secondDict[key]) else: thisDepth = 1 if thisDepth > maxDepth: maxDepth = thisDepth return maxDepth
# 预先存储两个树的信息def retrieveTree(i): listOfTrees = [{'no surfacing': {0: 'no', 1: {'filppers': {0: 'no', 1: 'yes'}}}},\ {'no surfacing': {0: 'no', 1: {'filppers': {0: {'head':{0:'no', 1: 'yes'}}, 1:'no'}}}}] return listOfTrees[i] myTree = retrieveTree(0)
list(myTree.keys())[0]
'no surfacing'
getNumLeafs(myTree)
3
getTreeDepth(myTree)
2
# PlotTree# 在父子节点间填充文本信息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)
def plotTree(myTree, parentPt, nodeTxt): numLeafs = getNumLeafs(myTree) depth = getTreeDepth(myTree) firstStr = list(myTree.keys())[0] cntrPt = (plotTree.xOff + (1.0 + float(numLeafs))/2.0/plotTree.totalW, plotTree.yOff) plotMidText(cntrPt, parentPt, nodeTxt) # 标记子节点属性值 plotNode(firstStr, cntrPt, parentPt, decisionNode) # 绘制带箭头的注解 secondDict = myTree[firstStr] plotTree.yOff = plotTree.yOff - 1.0/plotTree.totalD # 减少y偏移 for key in secondDict.keys(): if type(secondDict[key]).__name__ == 'dict': plotTree(secondDict[key], cntrPt, str(key)) else: plotTree.xOff = plotTree.xOff + 1.0/plotTree.totalW # 右移x坐标 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 # ！！当前分支画完返回上一位置
# 重新写createPlotdef createPlot(inTree): fig = plt.figure(1, facecolor='white') fig.clf() axprops = dict(xticks=[], yticks=[]) createPlot.ax1 = plt.subplot(111, frameon=False, **axprops) plotTree.totalW = float(getNumLeafs(inTree)) # 全局变量存储树的宽和高 plotTree.totalD = float(getTreeDepth(inTree)) plotTree.xOff = -0.5/plotTree.totalW plotTree.yOff = 1.0 plotTree(inTree, (0.5,1.0), '') plt.show()
myTree
{'no surfacing': {0: 'no', 1: {'filppers': {0: 'no', 1: 'yes'}}}} createPlot(myTree)
myTree['no surfacing'][3] = 'maybe'
createPlot(myTree)
5. 使用决策树进行分类
# 使用决策树的分类函数def classify(inputTree, featLabels, testVec): firstStr = list(inputTree.keys())[0] secondDict = inputTree[firstStr] featIndex = featLabels.index(firstStr) # 用index()将标签字符串转换为索引，方便查找 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
myDat, labels = createDataSet()labels
['no surfacing', 'filppers']
myTree = retrieveTree(0)myTree
{'no surfacing': {0: 'no', 1: {'filppers': {0: 'no', 1: 'yes'}}}} classify(myTree, labels, [1,1])
'yes'
classify(myTree, labels, [1,0])
'no'
# 使用pickle模块存储决策树def storeTree(inputTree, filename): import pickle fw = open(filename,'wb') pickle.dump(inputTree, fw) fw.close()
def grabTree(filename): import pickle fr = open(filename,'rb') return pickle.load(fr)
storeTree(myTree, 'classifierStorage.txt')
grabTree('classifierStorage.txt') {'no surfacing': {0: 'no', 1: {'filppers': {0: 'no', 1: 'yes'}}}} 示例：使用决策树预测隐形眼镜类型
fr = open('data/lenses.txt')lenses = [inst.strip().split('\t') for inst in fr.readlines()]lensesLabels = ['age', 'prescript', 'astigmatic', 'tearRate']lensesTree = createTree(lenses, lensesLabels)print(lensesTree) {'tearRate': {'normal': {'astigmatic': {'yes': {'prescript': {'hyper': {'age': {'presbyopic': 'no lenses', 'young': 'hard', 'pre': 'no lenses'}}, 'myope': 'hard'}}, 'no': {'age': {'presbyopic': {'prescript': {'hyper': 'soft', 'myope': 'no lenses'}}, 'young': 'soft', 'pre': 'soft'}}}}, 'reduced': 'no lenses'}} createPlot(lensesTree)
匹配的选项过多导致Overfitting的问题，如果一个叶子节点只能增加少许信息，则可以删除该节点。
ID3算法的缺点：无法直接处理数值型数据。