基本概念
决策树是一种常见的分类算法。
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。
信息增益
这里是使用的ID3算法来划分数据集，每次划分数据集时是只选取一个特征，那么就有一个问题出现了，训练集中会存在多个特征，选取特征的顺序应该如何来界定呢？这里，划分数据集的大原则是：将无序的数据变得更加有序。
信息增益和熵
介绍信息论中的两个概念，信息增益和熵。
信息增益指的是划分数据集之前之后信息发生的变化。依据这一点，那么获得信息增益最高的特征就是最好的选择，而信息增益的一个度量方式称之为熵（entropy）。
单个的信息定义为：
整个熵的公式：
Python实现
计算给定数据集的信息熵
from math import log
def calcShannonEnt(dataSet):
numEntries = len(dataSet)
labelCounts = {}
# 为所有可能分类创建字典
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) # 以2为底求对数
return shannonEnt
创建数据集
def createDataSet():
dataSet = [[1, 1, 'yes'],
[1, 1, 'yes'],
[1, 0, 'no'],
[0, 1, 'no'],
[0, 1, 'no']]
labels = ['no surfacing', 'flippers']
return dataSet, labels
In [12]: myDat,labels = trees.createDataSet()
In [14]: myDat
Out[14]: [[1, 1, 'yes'], [1, 1, 'yes'], [1, 0, 'no'], [0, 1, 'no'], [0, 1, 'no']]
In [15]: trees.calcShannonEnt(myDat)
Out[15]: 0.9709505944546686
划分数据集
def splitDataSet(dataSet, axis, value):
retDataSet = []
for featVec in dataSet:
if featVec[axis] == value:
reducedFeatVec = featVec[:axis]
reducedFeatVec.extend(featVec[axis+1:])
retDataSet.append(reducedFeatVec)
return retDataSet
In [16]: myDat,labels = trees.createDataSet()
In [17]: myDat
Out[17]: [[1, 1, 'yes'], [1, 1, 'yes'], [1, 0, 'no'], [0, 1, 'no'], [0, 1, 'no']]
In [18]: trees.splitDataSet(myDat, 0, 1)
Out[18]: [[1, 'yes'], [1, 'yes'], [0, 'no']]
In [19]: trees.splitDataSet(myDat, 0, 0)
Out[19]: [[1, 'no'], [1, 'no']]
选择最好的数据集划分方式
其中的数据集想要用此函数要满足一定要求：第一，数据必须是一种由列表元素组成的列表，而且所有的列表元素都要具有相同的数据长度；第二，数据的最后一列或者每个实例的最后一个元素是当前实例的类别标签。
def chooseBestFeatureToSplit(dataSet):
numFeatures = len(dataSet[0]) - 1 # 数据集特征个数
baseEntropy = calcShannonEnt(dataSet) # 计算整个数据集的原始信息熵
bestInfoGain = 0.0; bestFeatrue = -1
for i in xrange(numFeatures):
featList = [example[i] for example in dataSet]
uniqueVals = set(featList) # 获取唯一元素值
newEntropy = 0.0
for value in uniqueVals:
subDataSet = splitDataSet(dataSet, i, value)
prob = len(subDataSet) / float(len(dataSet))
newEntropy += prob * calcShannonEnt(subDataSet) # 该特征划分所对应的熵
infoGain = baseEntropy - newEntropy
if (infoGain > bestInfoGain):
bestInfoGain = infoGain
bestFeatrue = i
return bestFeatrue
In [20]: myDat,labels = trees.createDataSet()
In [21]: trees.chooseBestFeatureToSplit(myDat)
Out[21]: 0
In [22]: myDat
Out[22]: [[1, 1, 'yes'], [1, 1, 'yes'], [1, 0, 'no'], [0, 1, 'no'], [0, 1, 'no']]
叶节点的分类
# 多数表决
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), reversed = 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): # 类标签完全相同则停止划分
return classList[0]
if len(dataSet[0]) == 1:
return majorityCnt(classList) # 遍历完所有特征
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
In [23]: myDat,labels = trees.createDataSet()
In [24]: myTree = trees.createTree(myDat, labels)
In [25]: myTree
Out[25]: {'no surfacing': {0: 'no', 1: {'flippers': {0: 'no', 1: 'yes'}}}}
绘制树形图
绘图代码不是重点，就直接贴上来了。
import matplotlib.pyplot as plt
decisionNode = dict(boxstyle = "sawtooth", fc = "0.8")
leafNode = dict(boxstyle = "round4", fc = "0.8")
arrow_args = dict(arrowstyle = "<-")
def plotNode(nodeTxt, centerPt, parentPt, nodeType):
createPlot.axl.annotate(nodeTxt, xy = parentPt, xycoords = 'axes fraction', xytext = centerPt, textcoords = 'axes fraction', va = "center", ha = "center", bbox = nodeType, arrowprops = arrow_args)
def createPlot(inTree):
fig = plt.figure(1, facecolor = 'white')
fig.clf()
axprops = dict(xticks = [], yticks = [])
createPlot.axl = plt.subplot(111, frameon = False, **axprops)
plotTree.totalW = float(getNumLeafs(inTree))
plotTree.totalD = float(getTreeDepth(inTree))
plotTree.xOff = -0.5/plotTree.totalW; plotTree.yOff = 1.0;
plotTree(inTree, (0.5, 1.0), '')
plt.show()
def getNumLeafs(myTree):
numLeafs = 0
firstStr = 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 = 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
def retrieveTree(i):
listOfTrees = [{'no surfacing': {0: 'no', 1: {'flippers': {0: 'no', 1: 'yes'}}}},
{'no surfacing': {0: 'no', 1: {'flippers': {0: {'head': {0: 'no', 1: 'yes'}}, 1: 'no'}}}}
]
return listOfTrees[i]
def plotMidText(cntrPt, parentPt, txtString):
xMid = (parentPt[0] - cntrPt[0])/2.0 + cntrPt[0]
yMid = (parentPt[1] - cntrPt[1])/2.0 + cntrPt[1]
createPlot.axl.text(xMid, yMid, txtString)
def plotTree(myTree, parentPt, nodeTxt):
numLeafs = getNumLeafs(myTree)
depth = getTreeDepth(myTree)
firstStr = myTree.keys()[0]
cntrPt = (plotTree.xOff + (1.0 + float(numLeafs))/2.0/plotTree.totalW, plotTree.yOff)
plotMidText(cntrPt, parentPt, nodeTxt)
plotNode(firstStr, cntrPt, parentPt, decisionNode)
secondDict = myTree[firstStr]
plotTree.yOff = plotTree.yOff - 1.0/plotTree.totalD
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
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
In [27]: myTree = treePlotter.retrieveTree(0)
In [28]: treePlotter.createPlot(myTree)
In [29]: myTree['no surfacing'][3] = 'maybe'
In [30]: myTree
Out[30]: {'no surfacing': {0: 'no', 1: {'flippers': {0: 'no', 1: 'yes'}}, 3: 'maybe'}}
In [31]: treePlotter.createPlot(myTree)
使用决策树的分类函数
def classify(inputTree, featLabels, testVec):
firstStr = inputTree.keys()[0]
secondDict = inputTree[firstStr]
featIndex = featLabels.index(firstStr)
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
决策树的存储
主要是为了解决耗时的问题，引入pickle模块来序列化对象。
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)
参考文献
[1][机器学习实战]