一，预备知识：
- 信息量：
- 单个类别的信息熵：
- 条件信息量：
- 单个类别的条件熵：
- 信息增益：
- 信息熵：
- 条件熵：（表示分类的类，表示属性V的取值，m为属性V的取值个数，n为分类的个数）
二.算法流程:
实质:递归的先根建树,结束条件(当前子集类别一致)，建树量化方法（信息增益）
三.示例代码：
package com.mechinelearn.id3;import java.io.BufferedReader;import java.io.File;import java.io.FileReader;import java.io.FileWriter;import java.io.IOException;import java.util.ArrayList;import java.util.Iterator;import java.util.LinkedList;import java.util.List;import java.util.regex.Matcher;import java.util.regex.Pattern;import org.dom4j.Document;import org.dom4j.DocumentHelper;import org.dom4j.Element;import org.dom4j.io.OutputFormat;import org.dom4j.io.XMLWriter;public class ID3 { private ArrayList attribute = new ArrayList (); // 存储属性的名称 private ArrayList > attributevalue = new ArrayList >(); // 存储每个属性的取值 private ArrayList data = new ArrayList ();; // 原始数据 int decatt; // 决策变量在属性集中的索引 public static final String patternString = "@attribute(.*)[{](.*?)[}]"; Document xmldoc; Element root; public ID3() { xmldoc = DocumentHelper.createDocument(); root = xmldoc.addElement("root"); root.addElement("DecisionTree").addAttribute("value", "null"); } public static void main(String[] args) { ID3 inst = new ID3(); inst.readARFF(new File("data.txt")); inst.setDec("play"); LinkedList ll = new LinkedList (); for (int i = 0; i < inst.attribute.size(); i++) { if (i != inst.decatt) ll.add(i); } ArrayList al = new ArrayList (); for (int i = 0; i < inst.data.size(); i++) { al.add(i); } inst.buildDT("DecisionTree", "null", al, ll); inst.writeXML("dt.xml"); return; } // 读取arff文件，给attribute、attributevalue、data赋值 public void readARFF(File file) { try { FileReader fr = new FileReader(file); BufferedReader br = new BufferedReader(fr); String line; Pattern pattern = Pattern.compile(patternString); while ((line = br.readLine()) != null) { Matcher matcher = pattern.matcher(line); if (matcher.find()) { attribute.add(matcher.group(1).trim());// 增加属性 String[] values = matcher.group(2).split(","); ArrayList al = new ArrayList (values.length); for (String value : values) { al.add(value.trim()); } attributevalue.add(al);// 每个属性对应的属性值 } else if (line.startsWith("@data")) { while ((line = br.readLine()) != null) { if (line == "") continue; String[] row = line.split(","); data.add(row);// 增加训练数据 } } else { continue; } } br.close(); } catch (IOException e1) { e1.printStackTrace(); } } // 设置决策变量 public void setDec(String name) { int n = attribute.indexOf(name); if (n < 0 || n >= attribute.size()) { System.err.println("决策变量指定错误。"); System.exit(2); } decatt = n; } // 计算每一个属性的属性值对应的的熵 public double getEntropy(int[] arr) { double entropy = 0.0; int sum = 0; for (int i = 0; i < arr.length; i++) { entropy -= arr[i] * Math.log(arr[i] + Double.MIN_VALUE) / Math.log(2); sum += arr[i]; } entropy += sum * Math.log(sum + Double.MIN_VALUE) / Math.log(2); entropy /= sum; return entropy; } // 给一个样本数组及样本的算术和，计算它的熵 public double getEntropy(int[] arr, int sum) { double entropy = 0.0; for (int i = 0; i < arr.length; i++) { entropy -= arr[i] * Math.log(arr[i] + Double.MIN_VALUE) / Math.log(2); } entropy += sum * Math.log(sum + Double.MIN_VALUE) / Math.log(2); entropy /= sum; return entropy; } //是否到达叶子节点 public boolean infoPure(ArrayList subset) { String value = data.get(subset.get(0))[decatt]; for (int i = 1; i < subset.size(); i++) { String next = data.get(subset.get(i))[decatt]; // equals表示对象内容相同，==表示两个对象指向的是同一片内存 if (!value.equals(next)) return false; } return true; } // 给定原始数据的子集(subset中存储行号),当以第index个属性为节点时计算它的信息熵 public double calNodeEntropy(ArrayList subset, int index) { int sum = subset.size(); double entropy = 0.0; int[][] info = new int[attributevalue.get(index).size()][];//属性值个数为行 for (int i = 0; i < info.length; i++) info[i] = new int[attributevalue.get(decatt).size()];//分类属性值个数为列 int[] count = new int[attributevalue.get(index).size()];//每个属性值在整个样本中出现的概率 for (int i = 0; i < sum; i++) { int n = subset.get(i); String nodevalue = data.get(n)[index]; int nodeind = attributevalue.get(index).indexOf(nodevalue); count[nodeind]++; String decvalue = data.get(n)[decatt]; int decind = attributevalue.get(decatt).indexOf(decvalue); info[nodeind][decind]++; } for (int i = 0; i < info.length; i++) { entropy += getEntropy(info[i]) * count[i] / sum;// 计算条件熵 } return entropy; } // 构建决策树（递归建树） public void buildDT(String name, String value, ArrayList subset, LinkedList selatt) { Element ele = null; @SuppressWarnings("unchecked") List list = root.selectNodes("//" + name); Iterator iter = list.iterator(); while (iter.hasNext()) { ele = iter.next(); if (ele.attributeValue("value").equals(value)) break; } if (infoPure(subset)) { // 深度优先建树是否结束 ele.setText(data.get(subset.get(0))[decatt]);// 设置决策 return; } int minIndex = -1; double minEntropy = Double.MAX_VALUE; for (int i = 0; i < selatt.size(); i++) { if (i == decatt) continue; double entropy = calNodeEntropy(subset, selatt.get(i)); if (entropy < minEntropy) { minIndex = selatt.get(i); minEntropy = entropy; } } String nodeName = attribute.get(minIndex); selatt.remove(new Integer(minIndex)); ArrayList attvalues = attributevalue.get(minIndex); for (String val : attvalues) { ele.addElement(nodeName).addAttribute("value", val); ArrayList al = new ArrayList (); for (int i = 0; i < subset.size(); i++) { if (data.get(subset.get(i))[minIndex].equals(val)) { al.add(subset.get(i)); } } buildDT(nodeName, val, al, selatt);// 递归建树 } } // 把xml写入文件 public void writeXML(String filename) { try { File file = new File(filename); if (!file.exists()) file.createNewFile(); FileWriter fw = new FileWriter(file); OutputFormat format = OutputFormat.createPrettyPrint(); // 美化格式 XMLWriter output = new XMLWriter(fw, format); output.write(xmldoc); output.close(); } catch (IOException e) { System.out.println(e.getMessage()); } }}