/*
* 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/>.
*/
/*
* UnivariateKernelEstimator.java
* Copyright (C) 2009-2012 University of Waikato, Hamilton, New Zealand
*
*/
package weka.estimators;
import java.util.ArrayList;
import java.util.Iterator;
import java.util.Map;
import java.util.Random;
import java.util.Set;
import java.util.TreeMap;
import weka.core.Statistics;
import weka.core.Utils;
/**
* Simple weighted kernel density estimator.
*
* @author Eibe Frank (eibe@cs.waikato.ac.nz)
* @version $Revision: 8034 $
*/
public class UnivariateKernelEstimator implements UnivariateDensityEstimator,
UnivariateIntervalEstimator,
UnivariateQuantileEstimator {
/** 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]);
}
/**
* Returns the quantile for the given percentage.
*
* @param percentage the percentage
* @return the quantile
*/
public double predictQuantile(double percentage) {
// Update the bandwidth
updateWidth();
// Compute minimum and maximum value, and delta
double val = Statistics.normalInverse(1.0 - (1.0 - 0.95) / 2);
double min = m_TM.firstKey() - val * m_Width;
double max = m_TM.lastKey() + val * m_Width;
double delta = (max - min) / m_NumIntervals;
// Compute approximate quantile
double[] probabilities = new double[m_NumIntervals];
double sum = 0;
double leftVal = Math.exp(logDensity(min));
for (int i = 0; i < m_NumIntervals; i++) {
if (sum >= percentage) {
return min + i * delta;
}
double rightVal = Math.exp(logDensity(min + (i + 1) * delta));
sum += 0.5 * (leftVal + rightVal) * delta;
leftVal = rightVal;
}
return max;
}
/**
* 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);
}
}
}