决策树及实现

本文基于python逐步实现Decision Tree(决策树)，分为以下几个步骤：

• 加载数据集
• 熵的计算
• 根据最佳分割feature进行数据分割
• 根据最大信息增益选择最佳分割feature
• 递归构建决策树
• 样本分类

关于决策树的理论方面本文几乎不讲，详情请google keywords:“决策树 信息增益  熵”

将分别体现于代码。

本文只建一个.py文件，所有代码都在这个py里

1.加载数据集

我们选用UCI经典Iris为例

Brief of IRIS:

Data Set Characteristics:  

Multivariate

Number of Instances:

150

Area:

Life

Attribute Characteristics:

Real

Number of Attributes:

4

Date Donated

1988-07-01

Associated Tasks:

Classification

Missing Values?

No

Number of Web Hits:

533125

Code：

[python] view plaincopyprint?

1. from numpy import *  
2. #load "iris.data" to workspace   
3. traindata = loadtxt("D:\ZJU_Projects\machine learning\ML_Action\Dataset\Iris.data",delimiter = ',',usecols = (0,1,2,3),dtype = float)  
4. trainlabel = loadtxt("D:\ZJU_Projects\machine learning\ML_Action\Dataset\Iris.data",delimiter = ',',usecols = (range(4,5)),dtype = str)  
5. feaname = ["#0","#1","#2","#3"] # feature names of the 4 attributes (features)  

from numpy import *#load "iris.data" to workspacetraindata = loadtxt("D:\ZJU_Projects\machine learning\ML_Action\Dataset\Iris.data",delimiter = ',',usecols = (0,1,2,3),dtype = float)trainlabel = loadtxt("D:\ZJU_Projects\machine learning\ML_Action\Dataset\Iris.data",delimiter = ',',usecols = (range(4,5)),dtype = str)feaname = ["#0","#1","#2","#3"] # feature names of the 4 attributes (features)

Result:

           

左图为实际数据集，四个离散型feature，一个label表示类别（有Iris-setosa, Iris-versicolor，Iris-virginica 三个类）

2. 熵的计算

entropy是香农提出来的（信息论大牛），定义见wiki

注意这里的entropy是H(C|X=xi)而非H(C|X), H（C|X）的计算见第下一个点，还要乘以概率加和

Code：

[python] view plaincopyprint?

1. from math import log  
2. def calentropy(label):  
3.     n = label.size # the number of samples  
4.     #print n   
5.     count = {} #create dictionary "count"  
6.     for curlabel in label:  
7.         if curlabel not in count.keys():  
8.             count[curlabel] = 0  
9.         count[curlabel] += 1  
10.     entropy = 0  
11.     #print count   
12.     for key in count:  
13.         pxi = float(count[key])/n #notice transfering to float first  
14.         entropy -= pxi*log(pxi,2)  
15.     return entropy  
16.   
17. #testcode:   
18. #x = calentropy(trainlabel)  

from math import logdef calentropy(label):    n = label.size # the number of samples    #print n    count = {} #create dictionary "count"    for curlabel in label:        if curlabel not in count.keys():            count[curlabel] = 0        count[curlabel] += 1    entropy = 0    #print count    for key in count:        pxi = float(count[key])/n #notice transfering to float first        entropy -= pxi*log(pxi,2)    return entropy#testcode:#x = calentropy(trainlabel)

Result：

3. 根据最佳分割feature进行数据分割

假定我们已经得到了最佳分割feature，在这里进行分割（最佳feature为splitfea_idx）

第二个函数idx2data是根据splitdata得到的分割数据的两个index集合返回datal (samples less than pivot), datag(samples greater than pivot), labell, labelg。 这里我们根据所选特征的平均值作为pivot

[python] view plaincopyprint?

1. #split the dataset according to label "splitfea_idx"  
2. def splitdata(oridata,splitfea_idx):  
3.     arg = args[splitfea_idx] #get the average over all dimensions  
4.     idx_less = [] #create new list including data with feature less than pivot  
5.     idx_greater = [] #includes entries with feature greater than pivot  
6.     n = len(oridata)  
7.     for idx in range(n):  
8.         d = oridata[idx]  
9.         if d[splitfea_idx] < arg:  
10.             #add the newentry into newdata_less set  
11.             idx_less.append(idx)  
12.         else:  
13.             idx_greater.append(idx)  
14.     return idx_less,idx_greater  
15.   
16. #testcode:2   
17. #idx_less,idx_greater = splitdata(traindata,2)  
18.   
19.   
20. #give the data and labels according to index  
21. def idx2data(oridata,label,splitidx,fea_idx):  
22.     idxl = splitidx[0] #split_less_indices  
23.     idxg = splitidx[1] #split_greater_indices  
24.     datal = []  
25.     datag = []  
26.     labell = []  
27.     labelg = []  
28.     for i in idxl:  
29.         datal.append(append(oridata[i][:fea_idx],oridata[i][fea_idx+1:]))  
30.     for i in idxg:  
31.         datag.append(append(oridata[i][:fea_idx],oridata[i][fea_idx+1:]))  
32.     labell = label[idxl]  
33.     labelg = label[idxg]  
34.     return datal,datag,labell,labelg  

#split the dataset according to label "splitfea_idx"def splitdata(oridata,splitfea_idx):    arg = args[splitfea_idx] #get the average over all dimensions    idx_less = [] #create new list including data with feature less than pivot    idx_greater = [] #includes entries with feature greater than pivot    n = len(oridata)    for idx in range(n):        d = oridata[idx]        if d[splitfea_idx] < arg:            #add the newentry into newdata_less set            idx_less.append(idx)        else:            idx_greater.append(idx)    return idx_less,idx_greater#testcode:2#idx_less,idx_greater = splitdata(traindata,2)#give the data and labels according to indexdef idx2data(oridata,label,splitidx,fea_idx):    idxl = splitidx[0] #split_less_indices    idxg = splitidx[1] #split_greater_indices    datal = []    datag = []    labell = []    labelg = []    for i in idxl:        datal.append(append(oridata[i][:fea_idx],oridata[i][fea_idx+1:]))    for i in idxg:        datag.append(append(oridata[i][:fea_idx],oridata[i][fea_idx+1:]))    labell = label[idxl]    labelg = label[idxg]    return datal,datag,labell,labelg

这里args是参数，决定分裂节点的阈值（每个参数对应一个feature，大于该值分到>branch，小于该值分到<branch）,我们可以定义如下：

[python] view plaincopyprint?

1. args = mean(traindata,axis = 0)  

args = mean(traindata,axis = 0)

测试：按特征2进行分类，得到的less和greater set of indices分别为：

也就是按args[2]进行样本集分割，<和>args[2]的branch分别有57和93个样本。

4. 根据最大信息增益选择最佳分割feature

信息增益为代码中的info_gain, 注释中是熵的计算

[python] view plaincopyprint?

1. #select the best branch to split   
2. def choosebest_splitnode(oridata,label):  
3.     n_fea = len(oridata[0])  
4.     n = len(label)  
5.     base_entropy = calentropy(label)  
6.     best_gain = -1  
7.     for fea_i in range(n_fea): #calculate entropy under each splitting feature  
8.         cur_entropy = 0  
9.         idxset_less,idxset_greater = splitdata(oridata,fea_i)  
10.         prob_less = float(len(idxset_less))/n  
11.         prob_greater = float(len(idxset_greater))/n  
12.           
13.         #entropy(value|X) = \sum{p(xi)*entropy(value|X=xi)}  
14.         cur_entropy += prob_less*calentropy(label[idxset_less])  
15.         cur_entropy += prob_greater * calentropy(label[idxset_greater])  
16.           
17.         info_gain = base_entropy - cur_entropy #notice gain is before minus after  
18.         if(info_gain>best_gain):  
19.             best_gain = info_gain  
20.             best_idx = fea_i  
21.     return best_idx    
22.   
23. #testcode:   
24. #x = choosebest_splitnode(traindata,trainlabel)  

#select the best branch to splitdef choosebest_splitnode(oridata,label):    n_fea = len(oridata[0])    n = len(label)    base_entropy = calentropy(label)    best_gain = -1    for fea_i in range(n_fea): #calculate entropy under each splitting feature        cur_entropy = 0        idxset_less,idxset_greater = splitdata(oridata,fea_i)        prob_less = float(len(idxset_less))/n        prob_greater = float(len(idxset_greater))/n                #entropy(value|X) = \sum{p(xi)*entropy(value|X=xi)}        cur_entropy += prob_less*calentropy(label[idxset_less])        cur_entropy += prob_greater * calentropy(label[idxset_greater])                info_gain = base_entropy - cur_entropy #notice gain is before minus after        if(info_gain>best_gain):            best_gain = info_gain            best_idx = fea_i    return best_idx  #testcode:#x = choosebest_splitnode(traindata,trainlabel)

这里的测试针对所有数据，分裂一次选择哪个特征呢？

5. 递归构建决策树

详见code注释，buildtree递归地构建树。

递归终止条件：

①该branch内没有样本（subset为空） or

②分割出的所有样本属于同一类 or 

③由于每次分割消耗一个feature，当没有feature的时候停止递归，返回当前样本集中大多数sample的label

[python] view plaincopyprint?

1. #create the decision tree based on information gain  
2. def buildtree(oridata, label):  
3.     if label.size==0: #if no samples belong to this branch  
4.         return "NULL"  
5.     listlabel = label.tolist()  
6.     #stop when all samples in this subset belongs to one class  
7.     if listlabel.count(label[0])==label.size:  
8.         return label[0]  
9.           
10.     #return the majority of samples' label in this subset if no extra features avaliable  
11.     if len(feanamecopy)==0:  
12.         cnt = {}  
13.         for cur_l in label:  
14.             if cur_l not in cnt.keys():  
15.                 cnt[cur_l] = 0  
16.             cnt[cur_l] += 1  
17.         maxx = -1   
18.         for keys in cnt:  
19.             if maxx < cnt[keys]:  
20.                 maxx = cnt[keys]  
21.                 maxkey = keys  
22.         return maxkey  
23.       
24.     bestsplit_fea = choosebest_splitnode(oridata,label) #get the best splitting feature  
25.     print bestsplit_fea,len(oridata[0])  
26.     cur_feaname = feanamecopy[bestsplit_fea] # add the feature name to dictionary  
27.     print cur_feaname  
28.     nodedict = {cur_feaname:{}}   
29.     del(feanamecopy[bestsplit_fea]) #delete current feature from feaname  
30.     split_idx = splitdata(oridata,bestsplit_fea) #split_idx: the split index for both less and greater  
31.     data_less,data_greater,label_less,label_greater = idx2data(oridata,label,split_idx,bestsplit_fea)  
32.       
33.     #build the tree recursively, the left and right tree are the "<" and ">" branch, respectively  
34.     nodedict[cur_feaname]["<"] = buildtree(data_less,label_less)  
35.     nodedict[cur_feaname][">"] = buildtree(data_greater,label_greater)  
36.     return nodedict  
37.       
38. #testcode:   
39. #mytree = buildtree(traindata,trainlabel)  
40. #print mytree  

#create the decision tree based on information gaindef buildtree(oridata, label):    if label.size==0: #if no samples belong to this branch        return "NULL"    listlabel = label.tolist()    #stop when all samples in this subset belongs to one class    if listlabel.count(label[0])==label.size:        return label[0]            #return the majority of samples' label in this subset if no extra features avaliable    if len(feanamecopy)==0:        cnt = {}        for cur_l in label:            if cur_l not in cnt.keys():                cnt[cur_l] = 0            cnt[cur_l] += 1        maxx = -1         for keys in cnt:            if maxx < cnt[keys]:                maxx = cnt[keys]                maxkey = keys        return maxkey        bestsplit_fea = choosebest_splitnode(oridata,label) #get the best splitting feature    print bestsplit_fea,len(oridata[0])    cur_feaname = feanamecopy[bestsplit_fea] # add the feature name to dictionary    print cur_feaname    nodedict = {cur_feaname:{}}     del(feanamecopy[bestsplit_fea]) #delete current feature from feaname    split_idx = splitdata(oridata,bestsplit_fea) #split_idx: the split index for both less and greater    data_less,data_greater,label_less,label_greater = idx2data(oridata,label,split_idx,bestsplit_fea)        #build the tree recursively, the left and right tree are the "<" and ">" branch, respectively    nodedict[cur_feaname]["<"] = buildtree(data_less,label_less)    nodedict[cur_feaname][">"] = buildtree(data_greater,label_greater)    return nodedict    #testcode:#mytree = buildtree(traindata,trainlabel)#print mytree

Result:

mytree就是我们的结果，#1表示当前使用第一个feature做分割，'<'和'>'分别对应less 和 greater的数据。

6. 样本分类

根据构建出的mytree进行分类，递归走分支

[python] view plaincopyprint?

1. #classify a new sample   
2. def classify(mytree,testdata):  
3.     if type(mytree).__name__ != 'dict':  
4.         return mytree  
5.     fea_name = mytree.keys()[0] #get the name of first feature  
6.     fea_idx = feaname.index(fea_name) #the index of feature 'fea_name'  
7.     val = testdata[fea_idx]  
8.     nextbranch = mytree[fea_name]  
9.       
10.     #judge the current value > or < the pivot (average)  
11.     if val>args[fea_idx]:  
12.         nextbranch = nextbranch[">"]  
13.     else:  
14.         nextbranch = nextbranch["<"]  
15.     return classify(nextbranch,testdata)  
16.   
17. #testcode   
18. tt = traindata[0]  
19. x = classify(mytree,tt)  
20. print x  

#classify a new sampledef classify(mytree,testdata):    if type(mytree).__name__ != 'dict':        return mytree    fea_name = mytree.keys()[0] #get the name of first feature    fea_idx = feaname.index(fea_name) #the index of feature 'fea_name'    val = testdata[fea_idx]    nextbranch = mytree[fea_name]        #judge the current value > or < the pivot (average)    if val>args[fea_idx]:        nextbranch = nextbranch[">"]    else:        nextbranch = nextbranch["<"]    return classify(nextbranch,testdata)#testcodett = traindata[0]x = classify(mytree,tt)print x

Result：

为了验证代码准确性，我们换一下args参数，把它们都设成0（很小）

args = [0,0,0,0]

建树和分类的结果如下：

可见没有小于pivot(0)的项，于是dict中每个<的key对应的value都为空。

本文中全部代码下载：决策树python实现

Reference: Machine Learning in Action