决策树
- 1. 决策树的构造
- 1.1 信息增益
- 1.2 划分数据集
- 1.3 递归构建决策树
- 2. 使用 Matplotlib 注解绘制树形图
- 3. 测试和存储分类器
- 3.1 测试算法:使用决策树执行分类
- 3.2 使用算法:决策树的存储
- 4. 实例:使用决策树预测隐形眼镜类型
1. 决策树的构造
- 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
- 适用数据类型:数值型(离散化)和标称型
- 本文使用 ID3 算法(信息增益)划分数据集
1.1 信息增益
- 信息(information):如果待分类的事物可能划分在多个分类之中,则符号 x i x_i xi 的信息定义为: l ( x i ) = − l o g 2 p ( x i ) l(x_i) = -log_2p(x_i) l(xi)=−log2p(xi) 其中 p ( x i ) p(x_i) p(xi) 是选择该分类的概率
- 熵(entropy):定义为信息的期望值,即 H = − ∑ i = 1 n p ( x i ) l o g 2 p ( x i ) H = -\sum_{i=1}^np(x_i)log_2p(x_i) H=−i=1∑np(xi)log2p(xi) 其中 n 是分类的数目
- 计算给定数据集的香农熵:增加类别数,熵也相应地增加
'''计算给定数据集的香农熵'''
from math import log
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
[IN]: test = [[1, 1, 'yes'], [1, 1, 'yes'], [1, 0, 'no'], [0, 1, 'no'], [0, 1, 'no']]
[IN]: calcShannonEnt(test)
[OUT]: 0.9709505944546686
[IN]: test2 = [[1, 1, 'maybe'], [1, 1, 'yes'], [1, 0, 'no'], [0, 1, 'no'], [0, 1, 'no']]
[IN]: calcShannonEnt(test2)
[OUT]: 1.3709505944546687
1.2 划分数据集
- 按照给定特征划分数据集,输出为符合特征要求的数据集(去掉了给定的特征列):
'''按照给定特征划分数据集'''
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]: splitDataSet(test, 0, 1)
[OUT]: [[1, 'yes'], [1, 'yes'], [0, 'no']]
[IN]: splitDataSet(test, 0, 0)
[OUT]: [[1, 'no'], [1, 'no']]
- 选择最好的数据集划分方式:遍历特征计算信息增益,然后进行比较,取信息增益最大的特征
'''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
[IN]: chooseBestFeatureToSplit(test)
[OUT]: 0
1.3 递归构建决策树
- 构建决策树的原理:对于原始数据集,基于最好的属性值划分数据集。第一次划分之后,数据将被向下传递到树分支的下一个节点,在该节点上再次划分数据。递归结束的条件是:程序遍历完所有划分数据集的属性,或者每个分支下的所有实例都具有相同的分类,此时我们得到了所有的叶子节点,任何到达叶子节点的数据必然属于叶子节点的分类。
- 如果数据集已经处理了所有的属性,但是类标签依然不是唯一的,此时我们需要决定如何定义该子节点,一般情况下采用多数表决的方式
- 多数表决函数代码:
'''多数表决'''
from collections import Counter
def majorityCnt(classList):
class_count = Counter(classList).most_common(1)
return class_count[0][0]
- 创建树:
'''创建树'''
def createTree(dataSet, labels):
classList = [example[-1] for example in dataSet]
if classList.count(classList[0]) == len(classList):
return classList[0]
if len(dataSet[0]) == 1:
return majorityCnt(classList)
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]: labels = ['no surfacing', 'flippers']
[IN]: createTree(test, labels)
[OUT]: {'no surfacing': {0: 'no', 1: {'flippers': {0: 'no', 1: 'yes'}}}}
2. 使用 Matplotlib 注解绘制树形图
- 使用文本注解绘制树节点:
'''使用文本注解绘制树节点'''
import matplotlib.pyplot as plt
plt.rcParams['figure.constrained_layout.use'] = True
plt.rcParams['font.sans-serif'] = ['SimHei']
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.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()
createPlot.ax1 = plt.subplot(111, frameon=False)
plotNode('决策节点', (0.5, 0.1), (0.1, 0.5), decisionNode)
plotNode('叶节点', (0.8, 0.1), (0.3, 0.8), leafNode)
plt.show()
2. 获取叶节点的数目和树的层数
'''获取叶节点的数目和树的层数'''
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
- 绘制决策树
'''plotTree 函数'''
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)
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
def 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))
plotTree.totalD = float(getTreeDepth(inTree))
plotTree.xOff = -0.5 / plotTree.totalW
plotTree.yOff = 1.0
plotTree(inTree, (0.5, 1.0), '')
plt.show()
3. 测试和存储分类器
3.1 测试算法:使用决策树执行分类
'''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)
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
[IN]: 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'}}}}]
[IN]: labels = ['no surfacing', 'flippers']
[IN]: classify(listOfTrees[0], labels, [1,0])
[OUT]: 'no'
[IN]: classify(listOfTrees[0], labels, [1,1])
[OUT]: 'yes'
3.2 使用算法:决策树的存储
'''使用 pickle 模块存储/载入决策树'''
def storeTree(inputTree, filename):
import pickle
with open(filename, 'wb') as f:
pickle.dump(inputTree, f)
def grabTree(filename):
import pickle
with open(filename, 'rb') as f:
tree = pickle.load(f)
return tree
4. 实例:使用决策树预测隐形眼镜类型
import pandas as pd
data = pd.read_csv('Ch03/lenses.txt', sep='\t', header=None)
featMat = data.values.tolist()
lenselabels = ['age', 'prescript', 'astigmatic', 'tearRate', 'class']
lenseTree = createTree(featMat, lenselabels)
[IN]: lenseTree
[OUT]: {'tearRate': {'normal': {'astigmatic': {'yes': {'prescript': {'myope': 'hard',
'hyper': {'age': {'young': 'hard',
'pre': 'no lenses',
'presbyopic': 'no lenses'}}}},
'no': {'age': {'young': 'soft',
'pre': 'soft',
'presbyopic': {'prescript': {'myope': 'no lenses', 'hyper': 'soft'}}}}}},
'reduced': 'no lenses'}}
createPlot(lenseTree)