数据挖掘学习札记：ID3算法

原文地址：http://blog.csdn.net/zhaoyl03/article/details/8665663

参考：

1. Wiki上的 

2. 百度文库里的一个PPT，有算例， 3. 百度文库，PPT，很多算例，开始有信息理论，极力推荐阅读，4.用Python实现ID3和C4.5 决策树ID3和C4.5算法Python实现源码

下面是整理的学习笔记。

用途：

The ID3 algorithm is used by training on a dataset  to produce a which is stored in memory. At runtime, this decision tree is used to classify new unseen test cases by working down the decision tree using the values of this test case to arrive at a terminal node that tells you what class this test case belongs to.

基本思想：

角度1.  越是小型的决策树越优于大的决策树（尽管如此该算法也不是总是生成最小的树形结构）

角度2. 引入信息论中互信息（信息增益），作为判别因素的度量，即：以信息熵的下降速度作为选取测试属性的标准，所选的测试属性是从根到当前节点的路径上尚未被考虑的具有最高信息增益的属性

算法描述：

1. ID3 (Examples, Target_Attribute, Attributes)  
2.     Create a root node for the tree  
3.     If all examples are positive, Return the single-node tree Root, with label = +.  
4.     If all examples are negative, Return the single-node tree Root, with label = -.  
5.     If number of predicting attributes is empty, then Return the single node tree Root,  
6.     with label = most common value of the target attribute in the examples.  
7.     Otherwise Begin  
8.         A ← The Attribute that best classifies examples.  
9.         Decision Tree attribute for Root = A.  
10.         For each possible value, v_i, of A,  
11.             Add a new tree branch below Root, corresponding to the test A = v_i.  
12.             Let Examples(v_i) be the subset of examples that have the value v_i for A  
13.             If Examples(v_i) is empty  
14.                 Then below this new branch add a leaf node with label = most common target value in the examples  
15.             Else below this new branch add the subtree ID3 (Examples(v_i), Target_Attribute, Attributes – {A})  
16.     End  
17.     Return Root  

算例：

If S is a collection of 14 examples with 9 YES and 5 NO examples then

Entropy(S) = - (9/14) Log2 (9/14) - (5/14) Log2 (5/14)= 0.940

Notice entropy is 0 if all members of S belong to the same class(the data is perfectly classified). The range of entropy is 0("perfectly classified") to 1 ("totally random").

Gain(S, A) is information gain of example set S on attribute A is defined as

Gain(S, A) = Entropy(S) - S((|S_v| / |S|) * Entropy(S_v))

Where:

S is each value v of all possible values of attribute A

S_v = subset of S for which attribute A has value v

|S_v| = number of elements in S_v

|S| = number of elements in S

Suppose S is a set of 14 examples in which one of the attributes is wind speed. The values of Wind can be Weak or Strong.The classification of these 14 examples are 9 YES and 5 NO. For attribute Wind, suppose there are 8 occurrences of Wind = Weak and 6 occurrences of Wind = Strong. For Wind = Weak, 6 of the examples are YES and 2 are NO. For Wind = Strong, 3 are YES and3 are NO. Therefore

Gain(S,Wind)=Entropy(S)-(8/14)*Entropy(S_weak)-(6/14)*Entropy(S_strong)

= 0.940 - (8/14)*0.811 - (6/14)*1.00

= 0.048

Entropy(S_weak) = - (6/8)*log2(6/8) - (2/8)*log2(2/8)= 0.811

Entropy(S_strong) = - (3/6)*log2(3/6) - (3/6)*log2(3/6)= 1.00

For each attribute, the gain is calculated and the highest gain is used in the decision node.

selects the attribute which has the smallest entropy (or largest information gain) value.