1 ID3演算法流程

理論知識不必多講，下面講解ID3演算法的工作流程，以西瓜書為例：

2 演算法程式碼講解

2.1 前提必知

2.2 演算法流程

3:程式碼

整個演算法實際就 makeTree和distributionForInstance兩個函式，下面把程式碼附上來，想要執行的話請下在weka平臺，裡面包含依賴的類。

package weka.classifiers.trees;
/*
 *   This program is free software: you can redistribute it and/or modify
 *   it under the terms of the GNU General Public License as published by
 *   the Free Software Foundation, either version 3 of the License, or
 *   (at your option) any later version.
 *
 *   This program is distributed in the hope that it will be useful,
 *   but WITHOUT ANY WARRANTY; without even the implied warranty of
 *   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *   GNU General Public License for more details.
 *
 *   You should have received a copy of the GNU General Public License
 *   along with this program.  If not, see <http://www.gnu.org/licenses/>.
 */

/*
 *    Id3.java
 *    Copyright (C) 1999 University of Waikato, Hamilton, New Zealand
 *
 */
import weka.classifiers.AbstractClassifier;
import weka.classifiers.Sourcable;
import weka.core.Attribute;
import weka.core.Capabilities;
import weka.core.Drawable;
import weka.core.Instance;
import weka.core.Instances;
import weka.core.NoSupportForMissingValuesException;
import weka.core.RevisionUtils;
import weka.core.TechnicalInformation;
import weka.core.TechnicalInformationHandler;
import weka.core.Utils;
import weka.core.Capabilities.Capability;
import weka.core.TechnicalInformation.Field;
import weka.core.TechnicalInformation.Type;

import java.util.Enumeration;

/**
 <!-- globalinfo-start -->
 * Class for constructing an unpruned decision tree based on the ID3 algorithm. Can only deal with nominal attributes. No missing values allowed. Empty leaves may result in unclassified instances. For more information see: <br/>
 * <br/>
 * R. Quinlan (1986). Induction of decision trees. Machine Learning. 1(1):81-106.
 * <p/>
 <!-- globalinfo-end -->
 *
 <!-- technical-bibtex-start -->
 * BibTeX:
 * <pre>
 * &#64;article{Quinlan1986,
 *    author = {R. Quinlan},
 *    journal = {Machine Learning},
 *    number = {1},
 *    pages = {81-106},
 *    title = {Induction of decision trees},
 *    volume = {1},
 *    year = {1986}
 * }
 * </pre>
 * <p/>
 <!-- technical-bibtex-end -->
 *
 <!-- options-start -->
 * Valid options are: <p/>
 * 
 * <pre> -D
 *  If set, classifier is run in debug mode and
 *  may output additional info to the console</pre>
 * 
 <!-- options-end -->
 *
 * @author Eibe Frank (
 
[email protected])
 * @version $Revision: 8109 $ 
 */
public class myID3
  extends AbstractClassifier 
  implements TechnicalInformationHandler, Sourcable,Drawable{

  /** for serialization */
  static final long serialVersionUID = -2693678647096322561L;
  /** the node's id */
  private int m_id;

  /** static count to assign the ids */
  private static int ms_count = 0;
  /** The node's successors. */ 
  private myID3[] m_Successors;

  /** Attribute used for splitting. */
  private Attribute m_Attribute;

  /** Class value if node is leaf. */
  private double m_ClassValue;

  /** Class distribution if node is leaf. */
  private double[] m_Distribution;

  /** Class attribute of dataset. */
  private Attribute m_ClassAttribute;
  public myID3()
  {
      // Assign ids to each node of the id3 tree
      m_id = ms_count++;
  }

  /**
   * Returns a string describing the classifier.
   * @return a description suitable for the GUI.
   */
  public String globalInfo() {

    return  "Class for constructing an unpruned decision tree based on the ID3 "
      + "algorithm. Can only deal with nominal attributes. No missing values "
      + "allowed. Empty leaves may result in unclassified instances. For more "
      + "information see: \n\n"
      + getTechnicalInformation().toString();
  }

  /**
   * Returns an instance of a TechnicalInformation object, containing 
   * detailed information about the technical background of this class,
   * e.g., paper reference or book this class is based on.
   * 
   * @return the technical information about this class
   */
  public TechnicalInformation getTechnicalInformation() {
    TechnicalInformation 	result;
    
    result = new TechnicalInformation(Type.ARTICLE);
    result.setValue(Field.AUTHOR, "R. Quinlan");
    result.setValue(Field.YEAR, "1986");
    result.setValue(Field.TITLE, "Induction of decision trees");
    result.setValue(Field.JOURNAL, "Machine Learning");
    result.setValue(Field.VOLUME, "1");
    result.setValue(Field.NUMBER, "1");
    result.setValue(Field.PAGES, "81-106");
    
    return result;
  }

  /**
   * Returns default capabilities of the classifier.
   *
   * @return      the capabilities of this classifier
   */
  public Capabilities getCapabilities() {
    Capabilities result = super.getCapabilities();
    result.disableAll();

    // attributes
    result.enable(Capability.NOMINAL_ATTRIBUTES);

    // class
    result.enable(Capability.NOMINAL_CLASS);
    result.enable(Capability.MISSING_CLASS_VALUES);

    // instances
    result.setMinimumNumberInstances(0);
    
    return result;
  }

  /**
   * Builds Id3 decision tree classifier.
   *
   * @param data the training data
   * @exception Exception if classifier can't be built successfully
   */
  public void buildClassifier(Instances data) throws Exception {

    // can classifier handle the data?
    getCapabilities().testWithFail(data);

    // remove instances with missing class
    data = new Instances(data);
    data.deleteWithMissingClass();
    
    makeTree(data);
  }

  /**
   * Method for building an Id3 tree.
   *
   * @param data the training data
   * @exception Exception if decision tree can't be built successfully
   */
  private void makeTree(Instances data) throws Exception {

    // Check if no instances have reached this node.
    if (data.numInstances() == 0) {
      m_Attribute = null;
      m_ClassValue = Utils.missingValue();
      m_Distribution = new double[data.numClasses()];
      return;
    }

    // Compute attribute with maximum information gain.
    double[] infoGains = new double[data.numAttributes()];
    Enumeration attEnum = data.enumerateAttributes();
    while (attEnum.hasMoreElements()) {
      Attribute att = (Attribute) attEnum.nextElement();
      infoGains[att.index()] = computeInfoGain(data, att);
    }
    m_Attribute = data.attribute(Utils.maxIndex(infoGains));
    
    // Make leaf if information gain is zero. 
    // Otherwise create successors.
    if (Utils.eq(infoGains[m_Attribute.index()], 0)) {
      m_Attribute = null;
      m_Distribution = new double[data.numClasses()];
      Enumeration instEnum = data.enumerateInstances();
      while (instEnum.hasMoreElements()) {
        Instance inst = (Instance) instEnum.nextElement();
        m_Distribution[(int) inst.classValue()]++;
      }
      Utils.normalize(m_Distribution);
      m_ClassValue = Utils.maxIndex(m_Distribution);
      m_ClassAttribute = data.classAttribute();
    } else {
      Instances[] splitData = splitData(data, m_Attribute);
      m_Successors = new myID3[m_Attribute.numValues()];
      for (int j = 0; j < m_Attribute.numValues(); j++) {
        m_Successors[j] = new myID3();
        m_Successors[j].makeTree(splitData[j]);
      }
    }
  }

  /**
   * Classifies a given test instance using the decision tree.
   *
   * @param instance the instance to be classified
   * @return the classification
   * @throws NoSupportForMissingValuesException if instance has missing values
   */
  public double classifyInstance(Instance instance) 
    throws NoSupportForMissingValuesException {

    if (instance.hasMissingValue()) {
      throw new NoSupportForMissingValuesException("Id3: no missing values, "
                                                   + "please.");
    }
    if (m_Attribute == null) {
      return m_ClassValue;
    } else {
      return m_Successors[(int) instance.value(m_Attribute)].
        classifyInstance(instance);
    }
  }

  /**
   * Computes class distribution for instance using decision tree.
   *
   * @param instance the instance for which distribution is to be computed
   * @return the class distribution for the given instance
   * @throws NoSupportForMissingValuesException if instance has missing values
   */
  public double[] distributionForInstance(Instance instance) 
    throws NoSupportForMissingValuesException {

    if (instance.hasMissingValue()) {
      throw new NoSupportForMissingValuesException("Id3: no missing values, "
                                                   + "please.");
    }
    if (m_Attribute == null) {
      return m_Distribution;
    } else { 
      return m_Successors[(int) instance.value(m_Attribute)].
        distributionForInstance(instance);
    }
  }

  /**
   * Prints the decision tree using the private toString method from below.
   *
   * @return a textual description of the classifier
   */
  public String toString() {

    if ((m_Distribution == null) && (m_Successors == null)) {
      return "Id3: No model built yet.";
    }
    return "Id3\n\n" + toString(0);
  }

  /**
   * Computes information gain for an attribute.
   *
   * @param data the data for which info gain is to be computed
   * @param att the attribute
   * @return the information gain for the given attribute and data
   * @throws Exception if computation fails
   */
  private double computeInfoGain(Instances data, Attribute att) 
    throws Exception {

    double infoGain = computeEntropy(data);
    Instances[] splitData = splitData(data, att);
    for (int j = 0; j < att.numValues(); j++) {
      if (splitData[j].numInstances() > 0) {
        infoGain -= ((double) splitData[j].numInstances() /
                     (double) data.numInstances()) *
          computeEntropy(splitData[j]);
      }
    }
    return infoGain;
  }

  /**
   * Computes the entropy of a dataset.
   * 
   * @param data the data for which entropy is to be computed
   * @return the entropy of the data's class distribution
   * @throws Exception if computation fails
   */
  private double computeEntropy(Instances data) throws Exception {

    double [] classCounts = new double[data.numClasses()];
    Enumeration instEnum = data.enumerateInstances();
    while (instEnum.hasMoreElements()) {
      Instance inst = (Instance) instEnum.nextElement();
      classCounts[(int) inst.classValue()]++;
    }
    double entropy = 0;
    for (int j = 0; j < data.numClasses(); j++) {
      if (classCounts[j] > 0) {
        entropy -= classCounts[j] * Utils.log2(classCounts[j]);
      }
    }
    entropy /= (double) data.numInstances();
    return entropy + Utils.log2(data.numInstances());
  }

  /**
   * Splits a dataset according to the values of a nominal attribute.
   *
   * @param data the data which is to be split
   * @param att the attribute to be used for splitting
   * @return the sets of instances produced by the split
   */
  private Instances[] splitData(Instances data, Attribute att) {

    Instances[] splitData = new Instances[att.numValues()];
    for (int j = 0; j < att.numValues(); j++) {
      splitData[j] = new Instances(data, data.numInstances());
    }
    Enumeration instEnum = data.enumerateInstances();
    while (instEnum.hasMoreElements()) {
      Instance inst = (Instance) instEnum.nextElement();
      splitData[(int) inst.value(att)].add(inst);
    }
    for (int i = 0; i < splitData.length; i++) {
      splitData[i].compactify();
    }
    return splitData;
  }

  /**
   * Outputs a tree at a certain level.
   *
   * @param level the level at which the tree is to be printed
   * @return the tree as string at the given level
   */
  private String toString(int level) {

    StringBuffer text = new StringBuffer();
    
    if (m_Attribute == null) {
      if (Utils.isMissingValue(m_ClassValue)) {
        text.append(": null");
      } else {
        text.append(": " + m_ClassAttribute.value((int) m_ClassValue));
      } 
    } else {
      for (int j = 0; j < m_Attribute.numValues(); j++) {
        text.append("\n");
        for (int i = 0; i < level; i++) {
          text.append("|  ");
        }
        text.append(m_Attribute.name() + " = " + m_Attribute.value(j));
        text.append(m_Successors[j].toString(level + 1));
      }
    }
    return text.toString();
  }

  /**
   * Adds this tree recursively to the buffer.
   * 
   * @param id          the unqiue id for the method
   * @param buffer      the buffer to add the source code to
   * @return            the last ID being used
   * @throws Exception  if something goes wrong
   */
  protected int toSource(int id, StringBuffer buffer) throws Exception {
    int                 result;
    int                 i;
    int                 newID;
    StringBuffer[]      subBuffers;
    
    buffer.append("\n");
    buffer.append("  protected static double node" + id + "(Object[] i) {\n");
    
    // leaf?
    if (m_Attribute == null) {
      result = id;
      if (Double.isNaN(m_ClassValue)) {
        buffer.append("    return Double.NaN;");
      } else {
        buffer.append("    return " + m_ClassValue + ";");
      }
      if (m_ClassAttribute != null) {
        buffer.append(" // " + m_ClassAttribute.value((int) m_ClassValue));
      }
      buffer.append("\n");
      buffer.append("  }\n");
    } else {
      buffer.append("    checkMissing(i, " + m_Attribute.index() + ");\n\n");
      buffer.append("    // " + m_Attribute.name() + "\n");
      
      // subtree calls
      subBuffers = new StringBuffer[m_Attribute.numValues()];
      newID = id;
      for (i = 0; i < m_Attribute.numValues(); i++) {
        newID++;

        buffer.append("    ");
        if (i > 0) {
          buffer.append("else ");
        }
        buffer.append("if (((String) i[" + m_Attribute.index() 
            + "]).equals(\"" + m_Attribute.value(i) + "\"))\n");
        buffer.append("      return node" + newID + "(i);\n");

        subBuffers[i] = new StringBuffer();
        newID = m_Successors[i].toSource(newID, subBuffers[i]);
      }
      buffer.append("    else\n");
      buffer.append("      throw new IllegalArgumentException(\"Value '\" + i["
          + m_Attribute.index() + "] + \"' is not allowed!\");\n");
      buffer.append("  }\n");

      // output subtree code
      for (i = 0; i < m_Attribute.numValues(); i++) {
        buffer.append(subBuffers[i].toString());
      }
      subBuffers = null;
      
      result = newID;
    }
    
    return result;
  }
  
  /**
   * Returns a string that describes the classifier as source. The
   * classifier will be contained in a class with the given name (there may
   * be auxiliary classes),
   * and will contain a method with the signature:
   * <pre><code>
   * public static double classify(Object[] i);
   * </code></pre>
   * where the array <code>i</code> contains elements that are either
   * Double, String, with missing values represented as null. The generated
   * code is public domain and comes with no warranty. <br/>
   * Note: works only if class attribute is the last attribute in the dataset.
   *
   * @param className the name that should be given to the source class.
   * @return the object source described by a string
   * @throws Exception if the source can't be computed
   */
  public String toSource(String className) throws Exception {
    StringBuffer        result;
    int                 id;
    
    result = new StringBuffer();

    result.append("class " + className + " {\n");
    result.append("  private static void checkMissing(Object[] i, int index) {\n");
    result.append("    if (i[index] == null)\n");
    result.append("      throw new IllegalArgumentException(\"Null values "
        + "are not allowed!\");\n");
    result.append("  }\n\n");
    result.append("  public static double classify(Object[] i) {\n");
    id = 0;
    result.append("    return node" + id + "(i);\n");
    result.append("  }\n");
    toSource(id, result);
    result.append("}\n");

    return result.toString();
  }
  
  /**
   * Returns the revision string.
   * 
   * @return		the revision
   */
  public String getRevision() {
    return RevisionUtils.extract("$Revision: 8109 $");
  }

  /**
   * Main method.
   *
   * @param args the options for the classifier
   */
  public static void main(String[] args) {
    runClassifier(new myID3(), args);
  }

	  public int graphType()
	  {
	      return Drawable.TREE;
	  }
	
	  /**
	   * Returns a string that describes the ID3 tree.
	   * The output is in dotty format.
	   *
	   * @return the graph described by a string
	   * @throws Exception if the graph can't be computed
	   */
	  public String graph() throws Exception
	  {
	      StringBuffer text = new StringBuffer();
	
	      text.append( "digraph ID3Tree {\n" );
	      text.append( graph( 0 ) );
	
	      return text.toString() +"}\n";
	  }
	
	  /**
	   * Returns a string that describes the ID3 tree.
	   * The output is in dotty format.
	   *
	   * @param level the level at which the tree is to be printed
	   * @return the graph described by a string
	   * @throws Exception if the graph can't be computed
	   */
	  public String graph(int level) throws Exception
	  {
	      StringBuffer text = new StringBuffer();
	      if ( m_Attribute == null ) {
	          if (Utils.isMissingValue(m_ClassValue)) {
	              text.append( "N" + m_id + " [label=\"null\" " +
	                           "shape=box style=filled " );
	          } else {
	              text.append( "N" + m_id + " [label=\"" +
	                           m_ClassAttribute.value( (int) m_ClassValue ) + "\" " +
	                           "shape=box style=filled " );
	          }
	          text.append( "]\n" );
	      } else {
	          text.append( "N" + m_id + " [label=\"" +
	                       m_Attribute.name() + "\" " );
	          text.append( "]\n" );
	          for ( int j = 0; j < m_Attribute.numValues(); j++ ) {
	              text.append( "N" + m_id + "->" + "N" + m_Successors[j].m_id +
	                           " [label=\"= " + m_Attribute.value( j ) +
	                           "\"]\n" );
	              text.append( m_Successors[j].graph( level + 1 ) );
	          }
	  }
	      return text.toString();
  }
}