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Finds算法和ID3算法

作业要求

  1. 实现FINDS算法
  2. 实现ID3算法
  • 不要调库自己写。如果有能力可以继续用课件里的数据集测试两个算法(用天气的4条记录测试FINDS,用贷款的15条记录测试ID3)给出训练误差测试误差等;
  • 再有能力可以使用更大的数据集测试算法。

算法实现

FINDS算法

  1. 目标:寻找极大特殊假设。
  2. 从假设集合H中最特殊的假设开始。在该假设不能正确地划分一个正例的时候将其进行一般化。算法如下:
    算法流程
  3. FINDS算法是一种利用more-general-than的偏序结构来搜索假设空间的方法,这一搜索沿着偏序链,从较特殊的假设逐渐演变为较一般的假设。
  4. 算法Python实现:

"""
-*- coding: utf-8 -*-
Created on 2019/10/21 21:02
FINDS
@Author : Zhouy
@Blog : www.crocodilezs.top

"""

# create dataset
def CreateDataset():
dataset = [['Sunny', 'Warm', 'Normal', 'Strong', 'Warm', 'Same', 'Yes'],
['Sunny', 'Warm', 'High', 'Strong', 'Warm', 'Same', 'Yes'],
['Rainy', 'Cold', 'High', 'Strong', 'Warm', 'Change', 'No'],
['Sunny', 'Warm', 'High', 'Strong', 'Cold', 'Change', 'Yes']]
labels = ['Sky', 'Temp', 'Humidity', 'Wind', 'Water', 'Forest', 'OutdoorSport']
return dataset, labels

# Find one version space by using FINDS
# '/' means null, and '*' means generalization
def FINDS(dataset):
constraint = ['/', '/', '/', '/', '/', '/']
for item in dataset:
if item[-1] == 'Yes':
# only go through positive instances
for i in range(len(item)-1):
if(item[i] != constraint[i] and constraint[i] != '*'):
if(constraint[i] == '/'):
constraint[i] = item[i]
else:
constraint[i] = '*'
return constraint

def main():
dataset, labels = CreateDataset()
constraint = FINDS(dataset)
print(constraint)

if __name__ == "__main__":
main()

ID3算法

  1. 决策树:决策树是一种常用的分类与回归方法。决策树的模型为树形结构,在针对分类问题时,实际上就是针对输入数据的各个特征对实例进行分类的过程,即通过树形结构的模型,在每一层级上对特征值进行判断,进而到达决策树叶子节点,即完成分类过程。
    决策树的本质是概念学习。
  2. 信息熵(香浓熵)、条件熵和信息增益的概念

    • 信息量:一件事发生的概率越小,我们说它所蕴含的信息量越大。
      信息量
    • 信息熵:信息熵就是所有可能发生的事件的信息量的期望
      信息熵
    • 条件熵:表示在X给定条件下,Y的条件概率分布的熵对X的数学期望。
      ![条件熵(https://s2.ax1x.com/2019/10/22/K36FBj.jpg)
    • 信息增益:当我们用另一个变量X对原变量Y分类后,原变量Y的不确定性就会减小了(即熵值减小)。而熵就是不确定性,不确定程度减少了多少其实就是信息增益。这就是信息增益的由来,所以信息增益定义如下:
      信息增益
  3. 算法’python’实现:
    (用课件上的贷款数据集一直没法成功分类,于是参考了csdn博客的另一个数据集合代码)

myTrees.py

"""
-*- coding: utf-8 -*-
Created on 2019/10/22 11:59
myTrees
@Author : Zhouy
@Blog : www.crocodilezs.top

"""
from math import log
import operator


# 原始数据
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

# 多数表决器
# 列中相同值数量最多为结果
def majorityCnt(classList):
classCounts = {}
for value in classList:
if (value not in classCounts.keys()):
classCounts[value] = 0
classCounts[value] += 1
sortedClassCount = sorted(classCounts.iteritems(), key=operator.itemgetter(1), reverse=True)
return sortedClassCount[0][0]


# 划分数据集
# dataSet:原始数据集
# axis:进行分割的指定列索引
# value:指定列中的值
def splitDataSet(dataSet, axis, value):
retDataSet = []
for featDataVal in dataSet:
if featDataVal[axis] == value:
# 下面两行去除某一项指定列的值,很巧妙有没有
reducedFeatVal = featDataVal[:axis]
reducedFeatVal.extend(featDataVal[axis + 1:])
retDataSet.append(reducedFeatVal)
return retDataSet


# 计算香农熵
def calcShannonEnt(dataSet):
# 数据集总项数
numEntries = len(dataSet)
# 标签计数对象初始化
labelCounts = {}
for featDataVal in dataSet:
# 获取数据集每一项的最后一列的标签值
currentLabel = featDataVal[-1]
# 如果当前标签不在标签存储对象里,则初始化,然后计数
if currentLabel not in labelCounts.keys():
labelCounts[currentLabel] = 0
labelCounts[currentLabel] += 1
# 熵初始化
shannonEnt = 0.0
# 遍历标签对象,求概率,计算熵
for key in labelCounts.keys():
prop = labelCounts[key] / float(numEntries)
shannonEnt -= prop * log(prop, 2)
return shannonEnt


# 选出最优特征列索引
def chooseBestFeatureToSplit(dataSet):
# 计算特征个数,dataSet最后一列是标签属性,不是特征量
numFeatures = len(dataSet[0]) - 1
# 计算初始数据香农熵
baseEntropy = calcShannonEnt(dataSet)
# 初始化信息增益,最优划分特征列索引
bestInfoGain = 0.0
bestFeatureIndex = -1
for i in range(numFeatures):
# 获取每一列数据
featList = [example[i] for example in dataSet]
# 将每一列数据去重
uniqueVals = set(featList)
newEntropy = 0.0
for value in uniqueVals:
subDataSet = splitDataSet(dataSet, i, value)
# 计算条件概率
prob = len(subDataSet) / float(len(dataSet))
# 计算条件熵
newEntropy += prob * calcShannonEnt(subDataSet)
# 计算信息增益
infoGain = baseEntropy - newEntropy
if (infoGain > bestInfoGain):
bestInfoGain = infoGain
bestFeatureIndex = i
return bestFeatureIndex


# 决策树创建
def createTree(dataSet, labels):
# 获取标签属性,dataSet最后一列,区别于labels标签名称
classList = [example[-1] for example in dataSet]
# 树极端终止条件判断
# 标签属性值全部相同,返回标签属性第一项值
if classList.count(classList[0]) == len(classList):
return classList[0]
# 只有一个特征(1列)
if len(dataSet[0]) == 1:
return majorityCnt(classList)
# 获取最优特征列索引
bestFeatureIndex = chooseBestFeatureToSplit(dataSet)
# 获取最优索引对应的标签名称
bestFeatureLabel = labels[bestFeatureIndex]
# 创建根节点
myTree = {bestFeatureLabel: {}}
# 去除最优索引对应的标签名,使labels标签能正确遍历
del (labels[bestFeatureIndex])
# 获取最优列
bestFeature = [example[bestFeatureIndex] for example in dataSet]
uniquesVals = set(bestFeature)
for value in uniquesVals:
# 子标签名称集合
subLabels = labels[:]
# 递归
myTree[bestFeatureLabel][value] = createTree(splitDataSet(dataSet, bestFeatureIndex, value), subLabels)
return myTree


# 获取分类结果
# inputTree:决策树字典
# featLabels:标签列表
# testVec:测试向量 例如:简单实例下某一路径 [1,1] => yes(树干值组合,从根结点到叶子节点)
def classify(inputTree, featLabels, testVec):
# 获取根结点名称,将dict转化为list
firstSide = list(inputTree.keys())
# 根结点名称String类型
firstStr = firstSide[0]
# 获取根结点对应的子节点
secondDict = inputTree[firstStr]
# 获取根结点名称在标签列表中对应的索引
featIndex = featLabels.index(firstStr)
# 由索引获取向量表中的对应值
key = testVec[featIndex]
# 获取树干向量后的对象
valueOfFeat = secondDict[key]
# 判断是子结点还是叶子节点:子结点就回调分类函数,叶子结点就是分类结果
# if type(valueOfFeat).__name__=='dict': 等价 if isinstance(valueOfFeat, dict):
if isinstance(valueOfFeat, dict):
classLabel = classify(valueOfFeat, featLabels, testVec)
else:
classLabel = valueOfFeat
return classLabel


# 将决策树分类器存储在磁盘中,filename一般保存为txt格式
def storeTree(inputTree, filename):
import pickle
fw = open(filename, 'wb+')
pickle.dump(inputTree, fw)
fw.close()


# 将瓷盘中的对象加载出来,这里的filename就是上面函数中的txt文件
def grabTree(filename):
import pickle
fr = open(filename, 'rb')
return pickle.load(fr)

treePlotter.py

"""
-*- coding: utf-8 -*-
Created on 2019/10/22 12:00
treePlotter
@Author : Zhouy
@Blog : www.crocodilezs.top

"""
import matplotlib.pyplot as plt

decisionNode = dict(boxstyle="sawtooth", fc="0.8")
leafNode = dict(boxstyle="round4", fc="0.8")
arrow_args = dict(arrowstyle="<-")


# 获取树的叶子节点
def getNumLeafs(myTree):
numLeafs = 0
# dict转化为list
firstSides = list(myTree.keys())
firstStr = firstSides[0]
secondDict = myTree[firstStr]
for key in secondDict.keys():
# 判断是否是叶子节点(通过类型判断,子类不存在,则类型为str;子类存在,则为dict)
if type(secondDict[
key]).__name__ == 'dict': # test to see if the nodes are dictonaires, if not they are leaf nodes
numLeafs += getNumLeafs(secondDict[key])
else:
numLeafs += 1
return numLeafs


# 获取树的层数
def getTreeDepth(myTree):
maxDepth = 0
# dict转化为list
firstSides = list(myTree.keys())
firstStr = firstSides[0]
secondDict = myTree[firstStr]
for key in secondDict.keys():
if type(secondDict[
key]).__name__ == 'dict': # test to see if the nodes are dictonaires, if not they are leaf nodes
thisDepth = 1 + getTreeDepth(secondDict[key])
else:
thisDepth = 1
if thisDepth > maxDepth: maxDepth = thisDepth
return maxDepth


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 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, va="center", ha="center", rotation=30)


def plotTree(myTree, parentPt, nodeTxt): # if the first key tells you what feat was split on
numLeafs = getNumLeafs(myTree) # this determines the x width of this tree
depth = getTreeDepth(myTree)
firstSides = list(myTree.keys())
firstStr = firstSides[0] # the text label for this node should be this
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': # test to see if the nodes are dictonaires, if not they are leaf nodes
plotTree(secondDict[key], cntrPt, str(key)) # recursion
else: # it's a leaf node print the leaf node
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


# if you do get a dictonary you know it's a tree, and the first element will be another dict
# 绘制决策树
def createPlot(inTree):
fig = plt.figure(1, facecolor='white')
fig.clf()
axprops = dict(xticks=[], yticks=[])
createPlot.ax1 = plt.subplot(111, frameon=False, **axprops) # no ticks
# createPlot.ax1 = plt.subplot(111, frameon=False) #ticks for demo puropses
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()


# 绘制树的根节点和叶子节点(根节点形状:长方形,叶子节点:椭圆形)
# def createPlot():
# fig = plt.figure(1, facecolor='white')
# fig.clf()
# createPlot.ax1 = plt.subplot(111, frameon=False) #ticks for demo puropses
# plotNode('a decision node', (0.5, 0.1), (0.1, 0.5), decisionNode)
# plotNode('a leaf node', (0.8, 0.1), (0.3, 0.8), leafNode)
# plt.show()

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]


# thisTree = retrieveTree(0)
# createPlot(thisTree)
# createPlot()
# myTree = retrieveTree(0)
# numLeafs =getNumLeafs(myTree)
# treeDepth =getTreeDepth(myTree)
# print(u"叶子节点数目:%d"% numLeafs)
# print(u"树深度:%d"%treeDepth)

testTrees_3.py

"""
-*- coding: utf-8 -*-
Created on 2019/10/22 12:00
testTrees_3
@Author : Zhouy
@Blog : www.crocodilezs.top

"""
import myTrees as mt
import treePlotter as tp
#测试
dataSet, labels = mt.createDataSet()
#copy函数:新开辟一块内存,然后将list的所有值复制到新开辟的内存中
labels1 = labels.copy()
#createTree函数中将labels1的值改变了,所以在分类测试时不能用labels1
myTree = mt.createTree(dataSet,labels1)
#保存树到本地
mt.storeTree(myTree,'myTree.txt')
#在本地磁盘获取树
myTree = mt.grabTree('myTree.txt')
print (u"决策树结构:%s"%myTree)
#绘制决策树
print(u"绘制决策树:")
tp.createPlot(myTree)
numLeafs =tp.getNumLeafs(myTree)
treeDepth =tp.getTreeDepth(myTree)
print(u"叶子节点数目:%d"% numLeafs)
print(u"树深度:%d"%treeDepth)
#测试分类 简单样本数据3列
labelResult =mt.classify(myTree,labels,[1,1])
print(u"[1,1] 测试结果为:%s"%labelResult)
labelResult =mt.classify(myTree,labels,[1,0])
print(u"[1,0] 测试结果为:%s"%labelResult)


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