ID3算法缺點
它一般會優先選擇有較多屬性值的Feature，因為屬性值多的特征會有相對較大的信息增益，信息增益反映的是，在給定一個條件以后，不確定性減少的程度，
這必然是分得越細的數據集確定性更高，也就是條件熵越小，信息增益越大。為了解決這個問題，C4.5就應運而生，它采用信息增益率來作為選擇分支的准則。
C4.5算法原理
信息增益率定義為：
其中，分子為信息增益（信息增益計算可參考上一節ID3的算法原理），分母為屬性X的熵。
需要注意的是，增益率准則對可取值數目較少的屬性有所偏好。
所以一般這樣選取划分屬性：選擇增益率最高的特征列作為划分屬性的依據。
代碼實現
與ID3代碼實現不同的是：只改變計算香農熵的函數calcShannonEnt，以及選擇最優特征索引函數chooseBestFeatureToSplit，具體代碼如下：
1 # -*- coding: utf-8 -*- 2 """ 3 Created on Thu Aug 2 17:09:34 2018 4 決策樹ID3,C4.5的實現 5 @author: weixw 6 """ 7 from math import log 8 import operator 9 #原始數據 10 def createDataSet(): 11 dataSet = [[1, 1, 'yes'], 12 [1, 1, 'yes'], 13 [1, 0, 'no'], 14 [0, 1, 'no'], 15 [0, 1, 'no']] 16 labels = ['no surfacing','flippers'] 17 return dataSet, labels 18 19 #多數表決器 20 #列中相同值數量最多為結果 21 def majorityCnt(classList): 22 classCounts = {} 23 for value in classList: 24 if(value not in classCounts.keys()): 25 classCounts[value] = 0 26 classCounts[value] +=1 27 sortedClassCount = sorted(classCounts.iteritems(),key = operator.itemgetter(1),reverse =True) 28 return sortedClassCount[0][0] 29 30 31 #划分數據集 32 #dataSet:原始數據集 33 #axis:進行分割的指定列索引 34 #value:指定列中的值 35 def splitDataSet(dataSet,axis,value): 36 retDataSet= [] 37 for featDataVal in dataSet: 38 if featDataVal[axis] == value: 39 #下面兩行去除某一項指定列的值，很巧妙有沒有 40 reducedFeatVal = featDataVal[:axis] 41 reducedFeatVal.extend(featDataVal[axis+1:]) 42 retDataSet.append(reducedFeatVal) 43 return retDataSet 44 45 #計算香農熵 46 #columnIndex = -1表示獲取數據集每一項的最后一列的標簽值 47 #其他表示獲取特征列 48 def calcShannonEnt(columnIndex, dataSet): 49 #數據集總項數 50 numEntries = len(dataSet) 51 #標簽計數對象初始化 52 labelCounts = {} 53 for featDataVal in dataSet: 54 #獲取數據集每一項的最后一列的標簽值 55 currentLabel = featDataVal[columnIndex] 56 #如果當前標簽不在標簽存儲對象里，則初始化，然后計數 57 if currentLabel not in labelCounts.keys(): 58 labelCounts[currentLabel] = 0 59 labelCounts[currentLabel] += 1 60 #熵初始化 61 shannonEnt = 0.0 62 #遍歷標簽對象，求概率，計算熵 63 for key in labelCounts.keys(): 64 prop = labelCounts[key]/float(numEntries) 65 shannonEnt -= prop*log(prop,2) 66 return shannonEnt 67 68 69 #通過信息增益，選出最優特征列索引(ID3) 70 def chooseBestFeatureToSplit(dataSet): 71 #計算特征個數，dataSet最后一列是標簽屬性，不是特征量 72 numFeatures = len(dataSet[0])-1 73 #計算初始數據香農熵 74 baseEntropy = calcShannonEnt(-1, dataSet) 75 #初始化信息增益，最優划分特征列索引 76 bestInfoGain = 0.0 77 bestFeatureIndex = -1 78 for i in range(numFeatures): 79 #獲取每一列數據 80 featList = [example[i] for example in dataSet] 81 #將每一列數據去重 82 uniqueVals = set(featList) 83 newEntropy = 0.0 84 for value in uniqueVals: 85 subDataSet = splitDataSet(dataSet,i,value) 86 #計算條件概率 87 prob = len(subDataSet)/float(len(dataSet)) 88 #計算條件熵 89 newEntropy +=prob*calcShannonEnt(-1, subDataSet) 90 #計算信息增益 91 infoGain = baseEntropy - newEntropy 92 if(infoGain > bestInfoGain): 93 bestInfoGain = infoGain 94 bestFeatureIndex = i 95 return bestFeatureIndex 96 97 #通過信息增益率，選出最優特征列索引(C4.5) 98 def chooseBestFeatureToSplitOfFurther(dataSet): 99 #計算特征個數，dataSet最后一列是標簽屬性，不是特征量 100 numFeatures = len(dataSet[0])-1 101 #計算初始數據香農熵H(Y) 102 baseEntropy = calcShannonEnt(-1, dataSet) 103 #初始化信息增益，最優划分特征列索引 104 bestInfoGainRatio = 0.0 105 bestFeatureIndex = -1 106 for i in range(numFeatures): 107 #獲取每一特征列香農熵H(X) 108 featEntropy = calcShannonEnt(i, dataSet) 109 #獲取每一列數據 110 featList = [example[i] for example in dataSet] 111 #將每一列數據去重 112 uniqueVals = set(featList) 113 newEntropy = 0.0 114 for value in uniqueVals: 115 subDataSet = splitDataSet(dataSet,i,value) 116 #計算條件概率 117 prob = len(subDataSet)/float(len(dataSet)) 118 #計算條件熵 119 newEntropy +=prob*calcShannonEnt(-1, subDataSet) 120 #計算信息增益 121 infoGain = baseEntropy - newEntropy 122 #計算信息增益率 123 infoGainRatio = infoGain/float(featEntropy) 124 if(infoGainRatio > bestInfoGainRatio): 125 bestInfoGainRatio = infoGainRatio 126 bestFeatureIndex = i 127 return bestFeatureIndex 128 129 #決策樹創建 130 def createTree(dataSet,labels): 131 #獲取標簽屬性，dataSet最后一列，區別於labels標簽名稱 132 classList = [example[-1] for example in dataSet] 133 #樹極端終止條件判斷 134 #標簽屬性值全部相同，返回標簽屬性第一項值 135 if classList.count(classList[0]) == len(classList): 136 return classList[0] 137 #沒有特征，只有標簽列（1列） 138 if len(dataSet[0]) == 1: 139 #返回實例數最大的類 140 return majorityCnt(classList) 141 # #獲取最優特征列索引ID3 142 # bestFeatureIndex = chooseBestFeatureToSplit(dataSet) 143 #獲取最優特征列索引C4.5 144 bestFeatureIndex = chooseBestFeatureToSplitOfFurther(dataSet) 145 #獲取最優索引對應的標簽名稱 146 bestFeatureLabel = labels[bestFeatureIndex] 147 #創建根節點 148 myTree = {bestFeatureLabel:{}} 149 #去除最優索引對應的標簽名，使labels標簽能正確遍歷 150 del(labels[bestFeatureIndex]) 151 #獲取最優列 152 bestFeature = [example[bestFeatureIndex] for example in dataSet] 153 uniquesVals = set(bestFeature) 154 for value in uniquesVals: 155 #子標簽名稱集合 156 subLabels = labels[:] 157 #遞歸 158 myTree[bestFeatureLabel][value] = createTree(splitDataSet(dataSet,bestFeatureIndex,value),subLabels) 159 return myTree 160 161 #獲取分類結果 162 #inputTree:決策樹字典 163 #featLabels:標簽列表 164 #testVec:測試向量 例如：簡單實例下某一路徑 [1,1] => yes（樹干值組合，從根結點到葉子節點） 165 def classify(inputTree,featLabels,testVec): 166 #獲取根結點名稱，將dict轉化為list 167 firstSide = list(inputTree.keys()) 168 #根結點名稱String類型 169 firstStr = firstSide[0] 170 #獲取根結點對應的子節點 171 secondDict = inputTree[firstStr] 172 #獲取根結點名稱在標簽列表中對應的索引 173 featIndex = featLabels.index(firstStr) 174 #由索引獲取向量表中的對應值 175 key = testVec[featIndex] 176 #獲取樹干向量后的對象 177 valueOfFeat = secondDict[key] 178 #判斷是子結點還是葉子節點：子結點就回調分類函數，葉子結點就是分類結果 179 #if type(valueOfFeat).__name__=='dict': 等價 if isinstance(valueOfFeat, dict): 180 if isinstance(valueOfFeat, dict): 181 classLabel = classify(valueOfFeat,featLabels,testVec) 182 else: 183 classLabel = valueOfFeat 184 return classLabel 185 186 187 #將決策樹分類器存儲在磁盤中，filename一般保存為txt格式 188 def storeTree(inputTree,filename): 189 import pickle 190 fw = open(filename,'wb+') 191 pickle.dump(inputTree,fw) 192 fw.close() 193 #將瓷盤中的對象加載出來，這里的filename就是上面函數中的txt文件 194 def grabTree(filename): 195 import pickle 196 fr = open(filename,'rb') 197 return pickle.load(fr) 198 199 200
1 ''' 2 Created on Oct 14, 2010 3 4 @author: Peter Harrington 5 ''' 6 import matplotlib.pyplot as plt 7 8 decisionNode = dict(boxstyle="sawtooth", fc="0.8") 9 leafNode = dict(boxstyle="round4", fc="0.8") 10 arrow_args = dict(arrowstyle="<-") 11 12 #獲取樹的葉子節點 13 def getNumLeafs(myTree): 14 numLeafs = 0 15 #dict轉化為list 16 firstSides = list(myTree.keys()) 17 firstStr = firstSides[0] 18 secondDict = myTree[firstStr] 19 for key in secondDict.keys(): 20 #判斷是否是葉子節點（通過類型判斷，子類不存在，則類型為str；子類存在，則為dict） 21 if type(secondDict[key]).__name__=='dict':#test to see if the nodes are dictonaires, if not they are leaf nodes 22 numLeafs += getNumLeafs(secondDict[key]) 23 else: numLeafs +=1 24 return numLeafs 25 26 #獲取樹的層數 27 def getTreeDepth(myTree): 28 maxDepth = 0 29 #dict轉化為list 30 firstSides = list(myTree.keys()) 31 firstStr = firstSides[0] 32 secondDict = myTree[firstStr] 33 for key in secondDict.keys(): 34 if type(secondDict[key]).__name__=='dict':#test to see if the nodes are dictonaires, if not they are leaf nodes 35 thisDepth = 1 + getTreeDepth(secondDict[key]) 36 else: thisDepth = 1 37 if thisDepth > maxDepth: maxDepth = thisDepth 38 return maxDepth 39 40 def plotNode(nodeTxt, centerPt, parentPt, nodeType): 41 createPlot.ax1.annotate(nodeTxt, xy=parentPt, xycoords='axes fraction', 42 xytext=centerPt, textcoords='axes fraction', 43 va="center", ha="center", bbox=nodeType, arrowprops=arrow_args ) 44 45 def plotMidText(cntrPt, parentPt, txtString): 46 xMid = (parentPt[0]-cntrPt[0])/2.0 + cntrPt[0] 47 yMid = (parentPt[1]-cntrPt[1])/2.0 + cntrPt[1] 48 createPlot.ax1.text(xMid, yMid, txtString, va="center", ha="center", rotation=30) 49 50 def plotTree(myTree, parentPt, nodeTxt):#if the first key tells you what feat was split on 51 numLeafs = getNumLeafs(myTree) #this determines the x width of this tree 52 depth = getTreeDepth(myTree) 53 firstSides = list(myTree.keys()) 54 firstStr = firstSides[0] #the text label for this node should be this 55 cntrPt = (plotTree.xOff + (1.0 + float(numLeafs))/2.0/plotTree.totalW, plotTree.yOff) 56 plotMidText(cntrPt, parentPt, nodeTxt) 57 plotNode(firstStr, cntrPt, parentPt, decisionNode) 58 secondDict = myTree[firstStr] 59 plotTree.yOff = plotTree.yOff - 1.0/plotTree.totalD 60 for key in secondDict.keys(): 61 if type(secondDict[key]).__name__=='dict':#test to see if the nodes are dictonaires, if not they are leaf nodes 62 plotTree(secondDict[key],cntrPt,str(key)) #recursion 63 else: #it's a leaf node print the leaf node 64 plotTree.xOff = plotTree.xOff + 1.0/plotTree.totalW 65 plotNode(secondDict[key], (plotTree.xOff, plotTree.yOff), cntrPt, leafNode) 66 plotMidText((plotTree.xOff, plotTree.yOff), cntrPt, str(key)) 67 plotTree.yOff = plotTree.yOff + 1.0/plotTree.totalD 68 #if you do get a dictonary you know it's a tree, and the first element will be another dict 69 #繪制決策樹 70 def createPlot(inTree): 71 fig = plt.figure(1, facecolor='white') 72 fig.clf() 73 axprops = dict(xticks=[], yticks=[]) 74 createPlot.ax1 = plt.subplot(111, frameon=False, **axprops) #no ticks 75 #createPlot.ax1 = plt.subplot(111, frameon=False) #ticks for demo puropses 76 plotTree.totalW = float(getNumLeafs(inTree)) 77 plotTree.totalD = float(getTreeDepth(inTree)) 78 plotTree.xOff = -0.5/plotTree.totalW; plotTree.yOff = 1.0; 79 plotTree(inTree, (0.5,1.0), '') 80 plt.show() 81 82 #繪制樹的根節點和葉子節點（根節點形狀：長方形，葉子節點：橢圓形） 83 #def createPlot(): 84 # fig = plt.figure(1, facecolor='white') 85 # fig.clf() 86 # createPlot.ax1 = plt.subplot(111, frameon=False) #ticks for demo puropses 87 # plotNode('a decision node', (0.5, 0.1), (0.1, 0.5), decisionNode) 88 # plotNode('a leaf node', (0.8, 0.1), (0.3, 0.8), leafNode) 89 # plt.show() 90 91 def retrieveTree(i): 92 listOfTrees =[{'no surfacing': {0: 'no', 1: {'flippers': {0: 'no', 1: 'yes'}}}}, 93 {'no surfacing': {0: 'no', 1: {'flippers': {0: {'head': {0: 'no', 1: 'yes'}}, 1: 'no'}}}} 94 ] 95 return listOfTrees[i] 96 97 #thisTree = retrieveTree(0) 98 #createPlot(thisTree) 99 #createPlot() 100 #myTree = retrieveTree(0) 101 #numLeafs =getNumLeafs(myTree) 102 #treeDepth =getTreeDepth(myTree) 103 #print(u"葉子節點數目：%d"% numLeafs) 104 #print(u"樹深度：%d"%treeDepth)
1 # -*- coding: utf-8 -*- 2 """ 3 Created on Fri Aug 3 19:52:10 2018 4 5 @author: weixw 6 """ 7 import myTrees as mt 8 import treePlotter as tp 9 #測試 10 dataSet, labels = mt.createDataSet() 11 #copy函數：新開辟一塊內存，然后將list的所有值復制到新開辟的內存中 12 labels1 = labels.copy() 13 #createTree函數中將labels1的值改變了，所以在分類測試時不能用labels1 14 myTree = mt.createTree(dataSet,labels1) 15 #保存樹到本地 16 mt.storeTree(myTree,'myTree.txt') 17 #在本地磁盤獲取樹 18 myTree = mt.grabTree('myTree.txt') 19 print(u"采用C4.5算法的決策樹結果") 20 print (u"決策樹結構：%s"%myTree) 21 #繪制決策樹 22 print(u"繪制決策樹：") 23 tp.createPlot(myTree) 24 numLeafs =tp.getNumLeafs(myTree) 25 treeDepth =tp.getTreeDepth(myTree) 26 print(u"葉子節點數目：%d"% numLeafs) 27 print(u"樹深度：%d"%treeDepth) 28 #測試分類 簡單樣本數據3列 29 labelResult =mt.classify(myTree,labels,[1,1]) 30 print(u"[1,1] 測試結果為：%s"%labelResult) 31 labelResult =mt.classify(myTree,labels,[1,0]) 32 print(u"[1,0] 測試結果為：%s"%labelResult)
運行結果
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