source: src/main/java/weka/estimators/UnivariateKernelEstimator.java @ 22

/*
 *    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 2 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, write to the Free Software
 *    Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
 */

/*
 *    UnivariateKernelEstimator.java
 *    Copyright (C) 2009 University of Waikato, Hamilton, New Zealand
 *
 */

package weka.estimators;

import java.util.Random;
import java.util.Collection;
import java.util.Set;
import java.util.Map;
import java.util.Iterator;
import java.util.TreeMap;
import java.util.ArrayList;

import weka.core.Statistics;
import weka.core.Utils;

/**
 * Simple weighted kernel density estimator.
 *
 * @author Eibe Frank (eibe@cs.waikato.ac.nz)
 * @version $Revision: 5680 $
 */
public class UnivariateKernelEstimator implements UnivariateDensityEstimator,
                                                  UnivariateIntervalEstimator {

  /** The collection used to store the weighted values. */
  protected TreeMap<Double, Double> m_TM = new TreeMap<Double, Double>();

  /** The weighted sum of values */
  protected double m_WeightedSum = 0;

  /** The weighted sum of squared values */
  protected double m_WeightedSumSquared = 0;

  /** The weight of the values collected so far */
  protected double m_SumOfWeights = 0;

  /** The current bandwidth (only computed when needed) */
  protected double m_Width = Double.MAX_VALUE;

  /** The exponent to use in computation of bandwidth (default: -0.25) */
  protected double m_Exponent = -0.25;

  /** The minimum allowed value of the kernel width (default: 1.0E-6) */
  protected double m_MinWidth = 1.0E-6;

  /** Constant for Gaussian density. */
  public static final double CONST = - 0.5 * Math.log(2 * Math.PI);

  /** Threshold at which further kernels are no longer added to sum. */
  protected double m_Threshold = 1.0E-6;

  /** The number of intervals used to approximate prediction interval. */
  protected int m_NumIntervals = 1000;

  /**
   * Adds a value to the density estimator.
   *
   * @param value the value to add
   * @param weight the weight of the value
   */
  public void addValue(double value, double weight) {

    m_WeightedSum += value * weight;
    m_WeightedSumSquared += value * value * weight;
    m_SumOfWeights += weight;
    if (m_TM.get(value) == null) {
      m_TM.put(value, weight);
    } else {
      m_TM.put(value, m_TM.get(value) + weight);
    }
  }

  /**
   * Updates bandwidth: the sample standard deviation is multiplied by
   * the total weight to the power of the given exponent.
   *
   * If the total weight is not greater than zero, the width is set to
   * Double.MAX_VALUE. If that is not the case, but the width becomes
   * smaller than m_MinWidth, the width is set to the value of
   * m_MinWidth.
   */
  public void updateWidth() {

    // OK, need to do some work
    if (m_SumOfWeights > 0) {

      // Compute variance for scaling
      double mean = m_WeightedSum / m_SumOfWeights;
      double variance = m_WeightedSumSquared / m_SumOfWeights - mean * mean;
      if (variance < 0) {
        variance = 0;
      }

      // Compute kernel bandwidth
      m_Width = Math.sqrt(variance) * Math.pow(m_SumOfWeights, m_Exponent);

      if (m_Width <= m_MinWidth) {
        m_Width = m_MinWidth;
      }
    } else {
      m_Width = Double.MAX_VALUE;
    }
  }
   

  /**
   * Returns the interval for the given confidence value. 
   * 
   * @param conf the confidence value in the interval [0, 1]
   * @return the interval
   */
  public double[][] predictIntervals(double conf) {

    // Update the bandwidth
    updateWidth();

    // Compute minimum and maximum value, and delta
    double val = Statistics.normalInverse(1.0 - (1.0 - conf) / 2);
    double min = m_TM.firstKey() - val * m_Width;
    double max = m_TM.lastKey() + val * m_Width;
    double delta = (max - min) / m_NumIntervals;

    // Create array with estimated probabilities
    double[] probabilities = new double[m_NumIntervals];
    double leftVal = Math.exp(logDensity(min));
    for (int i = 0; i < m_NumIntervals; i++) {
      double rightVal = Math.exp(logDensity(min + (i + 1) * delta));
      probabilities[i] = 0.5 * (leftVal + rightVal) * delta;
      leftVal = rightVal;
    }

    // Sort array based on area of bin estimates
    int[] sortedIndices = Utils.sort(probabilities);

    // Mark the intervals to use
    double sum = 0;
    boolean[] toUse = new boolean[probabilities.length];
    int k = 0;
    while ((sum < conf) && (k < toUse.length)){
      toUse[sortedIndices[toUse.length - (k + 1)]] = true;
      sum += probabilities[sortedIndices[toUse.length - (k + 1)]];
      k++;
    }

    // Don't need probabilities anymore
    probabilities = null;

    // Create final list of intervals
    ArrayList<double[]> intervals = new ArrayList<double[]>();

    // The current interval
    double[] interval = null;
    
    // Iterate through kernels
    boolean haveStartedInterval = false;
    for (int i = 0; i < m_NumIntervals; i++) {

      // Should the current bin be used?
      if (toUse[i]) {

        // Do we need to create a new interval?
        if (haveStartedInterval == false) {
          haveStartedInterval = true;
          interval = new double[2];
          interval[0] = min + i * delta;
        }

        // Regardless, we should update the upper boundary
        interval[1] = min + (i + 1) * delta;
      } else {

        // We need to finalize and store the last interval
        // if necessary.
        if (haveStartedInterval) {
          haveStartedInterval = false;
          intervals.add(interval);
        }
      }
    }

    // Add last interval if there is one
    if (haveStartedInterval) {
      intervals.add(interval);
    }

    return intervals.toArray(new double[0][0]);
  }

  /**
   * Computes the logarithm of x and y given the logarithms of x and y.
   *
   * This is based on Tobias P. Mann's description in "Numerically
   * Stable Hidden Markov Implementation" (2006).
   */
  protected double logOfSum(double logOfX, double logOfY) {

    // Check for cases where log of zero is present
    if (Double.isNaN(logOfX)) {
      return logOfY;
    } 
    if (Double.isNaN(logOfY)) {
      return logOfX;
    }

    // Otherwise return proper result, taken care of overflows
    if (logOfX > logOfY) {
      return logOfX + Math.log(1 + Math.exp(logOfY - logOfX));
    } else {
      return logOfY + Math.log(1 + Math.exp(logOfX - logOfY));
    }
  }

  /**
   * Compute running sum of density values and weights.
   */
  protected void runningSum(Set<Map.Entry<Double,Double>> c, double value, 
                            double[] sums) {

    // Auxiliary variables
    double offset = CONST - Math.log(m_Width);
    double logFactor = Math.log(m_Threshold) - Math.log(1 - m_Threshold);
    double logSumOfWeights = Math.log(m_SumOfWeights);

    // Iterate through values
    Iterator<Map.Entry<Double,Double>> itr = c.iterator();
    while(itr.hasNext()) {
      Map.Entry<Double,Double> entry = itr.next();

      // Skip entry if weight is zero because it cannot contribute to sum
      if (entry.getValue() > 0) {
        double diff = (entry.getKey() - value) / m_Width;
        double logDensity = offset - 0.5 * diff * diff;
        double logWeight = Math.log(entry.getValue());
        sums[0] = logOfSum(sums[0], logWeight + logDensity);
        sums[1] = logOfSum(sums[1], logWeight);

        // Can we stop assuming worst case?
        if (logDensity + logSumOfWeights < logOfSum(logFactor + sums[0], logDensity + sums[1])) {
          break;
        }
      }
    }
  }

  /**
   * Returns the natural logarithm of the density estimate at the given
   * point.
   *
   * @param value the value at which to evaluate
   * @return the natural logarithm of the density estimate at the given
   * value
   */
  public double logDensity(double value) {

    // Update the bandwidth
    updateWidth();

    // Array used to keep running sums
    double[] sums = new double[2];
    sums[0] = Double.NaN;
    sums[1] = Double.NaN;

    // Examine right-hand size of value
    runningSum(m_TM.tailMap(value, true).entrySet(), value, sums);

    // Examine left-hand size of value
    runningSum(m_TM.headMap(value, false).descendingMap().entrySet(), value, sums);

    // Need to normalize
    return sums[0] - Math.log(m_SumOfWeights);
  }

  /**
   * Returns textual description of this estimator.
   */
  public String toString() {

    return "Kernel estimator with bandwidth " + m_Width + 
      " and total weight " + m_SumOfWeights +
      " based on\n" + m_TM.toString();
  }

  /**
   * Main method, used for testing this class.
   */
  public static void main(String[] args) {

    // Get random number generator initialized by system
    Random r = new Random();

    // Create density estimator
    UnivariateKernelEstimator e = new UnivariateKernelEstimator();

    // Output the density estimator
    System.out.println(e);
    
    // Monte Carlo integration
    double sum = 0;
    for (int i = 0; i < 1000; i++) {
      sum += Math.exp(e.logDensity(r.nextDouble() * 10.0 - 5.0));
    }
    System.out.println("Approximate integral: " + 10.0 * sum / 1000);
    
    // Add Gaussian values into it
    for (int i = 0; i < 1000; i++) {
      e.addValue(0.1 * r.nextGaussian() - 3, 1);
      e.addValue(r.nextGaussian() * 0.25, 3);
    }

    // Monte Carlo integration
    sum = 0;
    int points = 10000;
    for (int i = 0; i < points; i++) {
      double value = r.nextDouble() * 10.0 - 5.0;
      sum += Math.exp(e.logDensity(value));
    }
    System.out.println("Approximate integral: " + 10.0 * sum / points);

    // Check interval estimates
    double[][] Intervals = e.predictIntervals(0.9);
    
    System.out.println("Printing kernel intervals ---------------------");
    
    for (int k = 0; k < Intervals.length; k++) {
      System.out.println("Left: " + Intervals[k][0] + "\t Right: " + Intervals[k][1]);
    }
    
    System.out.println("Finished kernel printing intervals ---------------------");

    double Covered = 0;
    for (int i = 0; i < 1000; i++) {
      double val = -1;
      if (r.nextDouble() < 0.25) {
        val = 0.1 * r.nextGaussian() - 3.0;
      } else {
        val = r.nextGaussian() * 0.25;
      }
      for (int k = 0; k < Intervals.length; k++) {
        if (val >= Intervals[k][0] && val <= Intervals[k][1]) {
          Covered++;
          break;
        }
      }
    }
    System.out.println("Coverage at 0.9 level for kernel intervals: " + Covered / 1000);

    // Compare performance to normal estimator on normally distributed data
    UnivariateKernelEstimator eKernel = new UnivariateKernelEstimator();
    UnivariateNormalEstimator eNormal = new UnivariateNormalEstimator();

    for (int j = 1; j < 5; j++) {
      double numTrain = Math.pow(10, j);
      System.out.println("Number of training cases: " +
                         numTrain); 

      // Add training cases
      for (int i = 0; i < numTrain; i++) {
        double val = r.nextGaussian() * 1.5 + 0.5;
        eKernel.addValue(val, 1);
        eNormal.addValue(val, 1);
      }

      // Monte Carlo integration
      sum = 0;
      points = 10000;
      for (int i = 0; i < points; i++) {
        double value = r.nextDouble() * 20.0 - 10.0;
        sum += Math.exp(eKernel.logDensity(value));
      }
      System.out.println("Approximate integral for kernel estimator: " + 20.0 * sum / points);

      // Evaluate estimators
      double loglikelihoodKernel = 0, loglikelihoodNormal = 0;
      for (int i = 0; i < 1000; i++) {
        double val = r.nextGaussian() * 1.5 + 0.5;
        loglikelihoodKernel += eKernel.logDensity(val);
        loglikelihoodNormal += eNormal.logDensity(val);
      }
      System.out.println("Loglikelihood for kernel estimator: " +
                         loglikelihoodKernel / 1000);
      System.out.println("Loglikelihood for normal estimator: " +
                         loglikelihoodNormal / 1000);

      // Check interval estimates
      double[][] kernelIntervals = eKernel.predictIntervals(0.95);
      double[][] normalIntervals = eNormal.predictIntervals(0.95);

      System.out.println("Printing kernel intervals ---------------------");
      
      for (int k = 0; k < kernelIntervals.length; k++) {
        System.out.println("Left: " + kernelIntervals[k][0] + "\t Right: " + kernelIntervals[k][1]);
      }

      System.out.println("Finished kernel printing intervals ---------------------");

      System.out.println("Printing normal intervals ---------------------");
      
      for (int k = 0; k < normalIntervals.length; k++) {
        System.out.println("Left: " + normalIntervals[k][0] + "\t Right: " + normalIntervals[k][1]);
      }

      System.out.println("Finished normal printing intervals ---------------------");
 
      double kernelCovered = 0;
      double normalCovered = 0;
      for (int i = 0; i < 1000; i++) {
        double val = r.nextGaussian() * 1.5 + 0.5;
        for (int k = 0; k < kernelIntervals.length; k++) {
          if (val >= kernelIntervals[k][0] && val <= kernelIntervals[k][1]) {
            kernelCovered++;
            break;
          }
        }
        for (int k = 0; k < normalIntervals.length; k++) {
          if (val >= normalIntervals[k][0] && val <= normalIntervals[k][1]) {
            normalCovered++;
            break;
          }
        }
      }
      System.out.println("Coverage at 0.95 level for kernel intervals: " + kernelCovered / 1000);
      System.out.println("Coverage at 0.95 level for normal intervals: " + normalCovered / 1000);
      
      kernelIntervals = eKernel.predictIntervals(0.8);
      normalIntervals = eNormal.predictIntervals(0.8);
      kernelCovered = 0;
      normalCovered = 0;
      for (int i = 0; i < 1000; i++) {
        double val = r.nextGaussian() * 1.5 + 0.5;
        for (int k = 0; k < kernelIntervals.length; k++) {
          if (val >= kernelIntervals[k][0] && val <= kernelIntervals[k][1]) {
            kernelCovered++;
            break;
          }
        }
        for (int k = 0; k < normalIntervals.length; k++) {
          if (val >= normalIntervals[k][0] && val <= normalIntervals[k][1]) {
            normalCovered++;
            break;
          }
        }
      }
      System.out.println("Coverage at 0.8 level for kernel intervals: " + kernelCovered / 1000);
      System.out.println("Coverage at 0.8 level for normal intervals: " + normalCovered / 1000);
    }
  }
}