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Algorithms and components for machine learning and statistics.
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/*
* File: AdaptiveRejectionSampling.java
* Authors: Kevin R. Dixon
* Company: Sandia National Laboratories
* Project: Cognitive Foundry
*
* Copyright May 5, 2010, Sandia Corporation.
* Under the terms of Contract DE-AC04-94AL85000, there is a non-exclusive
* license for use of this work by or on behalf of the U.S. Government.
* Export of this program may require a license from the United States
* Government. See CopyrightHistory.txt for complete details.
*
*/
package gov.sandia.cognition.statistics.bayesian;
import gov.sandia.cognition.annotation.PublicationReference;
import gov.sandia.cognition.annotation.PublicationType;
import gov.sandia.cognition.evaluator.Evaluator;
import gov.sandia.cognition.learning.data.DefaultInputOutputPair;
import gov.sandia.cognition.learning.function.scalar.PolynomialFunction;
import gov.sandia.cognition.math.AbstractUnivariateScalarFunction;
import gov.sandia.cognition.math.OperationNotConvergedException;
import gov.sandia.cognition.math.ProbabilityUtil;
import gov.sandia.cognition.statistics.ProbabilityFunction;
import gov.sandia.cognition.util.AbstractCloneableSerializable;
import gov.sandia.cognition.util.ObjectUtil;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collection;
import java.util.Collections;
import java.util.Iterator;
import java.util.Random;
/**
* Samples form a univariate distribution using the method of adaptive
* rejection sampling, which is a very efficient method that iteratively
* improves the rejection and acceptance envelopes in response to additional
* points.
* @author Kevin R. Dixon
* @since 3.0
*/
@PublicationReference(
author={
"Christian P. Robert",
"George Casella"
},
title="Monte Carlo Statistical Methods, Seconds Edition",
type=PublicationType.Book,
pages={56,58, 70,71},
notes={
"Algorithm A.7",
"Algorithm A.17"
},
year=2004
)
public class AdaptiveRejectionSampling
extends AbstractCloneableSerializable
{
/**
* Default number of points, {@value}.
*/
public static final int DEFAULT_MAX_NUM_POINTS = 50;
/**
* Logarithm of the function that we want to evaluate
*/
LogEvaluator logFunction;
/**
* Input-output point pairs, sorted in ascending order by their x-axis value
*/
private ArrayList points;
/**
* Maximum number of points that will be stored
*/
private int maxNumPoints;
/**
* Minimum support (x-value) of the logFunction
*/
private double minSupport;
/**
* Maximum support (x-value) of the logFunction
*/
private double maxSupport;
/**
* Upper envelope of the logFunction
*/
UpperEnvelope upperEnvelope;
/**
* Lower envelope of the logFunction
*/
LowerEnvelope lowerEnvelope;
/**
* Creates a new instance of AdaptiveRejectionSampling
*/
public AdaptiveRejectionSampling()
{
this.maxNumPoints = DEFAULT_MAX_NUM_POINTS;
this.points = new ArrayList( DEFAULT_MAX_NUM_POINTS );
this.upperEnvelope = new UpperEnvelope();
this.lowerEnvelope = new LowerEnvelope();
}
@Override
public AdaptiveRejectionSampling clone()
{
AdaptiveRejectionSampling clone =
(AdaptiveRejectionSampling) super.clone();
clone.points = ObjectUtil.cloneSmartElementsAsArrayList(this.getPoints());
clone.upperEnvelope = clone.new UpperEnvelope();
clone.upperEnvelope.resetLines();
clone.lowerEnvelope = clone.new LowerEnvelope();
clone.lowerEnvelope.resetLines();
clone.setLogFunction( ObjectUtil.cloneSafe( this.getLogFunction() ) );
return clone;
}
/**
* Initializes the Adaptive Rejection Sampling method
* @param logFunction
* Logarithm of the evaluator to consider
* @param minSupport
* Minimum support (x-axis) of the evaluator
* @param maxSupport
* Maximum support (x-axis) of the evaluator
* @param leftPoint
* Left point to initialize
* @param midPoint
* Mid point to initialize with
* @param rightPoint
* Right point to initialize with
*/
public void initialize(
AdaptiveRejectionSampling.LogEvaluator logFunction,
double minSupport,
double maxSupport,
double leftPoint,
double midPoint,
double rightPoint )
{
this.setLogFunction(logFunction);
this.setMinSupport(minSupport);
this.setMaxSupport(maxSupport);
this.points = new ArrayList( DEFAULT_MAX_NUM_POINTS );
this.upperEnvelope = new UpperEnvelope();
this.lowerEnvelope = new LowerEnvelope();
double y = this.logFunction.evaluate(leftPoint);
this.addPoint(leftPoint, y);
y = this.logFunction.evaluate(midPoint);
this.addPoint(midPoint, y);
y = this.logFunction.evaluate(rightPoint);
this.addPoint(rightPoint, y);
}
/**
* Adds a point to the set, which will adject the upper and lower envelopes
* @param x
* X-axis value
* @param y
* Y-axis value from the logFunction
*/
public void addPoint(
double x,
double y )
{
// Only add points if we have enough space left.
if( this.getNumPoints() < this.getMaxNumPoints() )
{
// Note... I've tried using SortedSet here to store points.
// However, it appears to be MUCH more efficient to use ArrayList
// and re-sort every time a point is added because the ability to
// perform random-access into the ArrayList outweighs the nastiness
// of having to re-sort each time we add a point.
// In my unit-test batter, it appears about 50% faster to use
// ArrayList than TreeSet -- krdixon, 2010-05-14.
this.points.add( new Point( x, y ) );
Collections.sort( this.points );
this.upperEnvelope.resetLines();
this.lowerEnvelope.resetLines();
}
}
/**
* Gets the number of points stored
* @return
* Number of points stored
*/
public int getNumPoints()
{
return this.getPoints().size();
}
/**
* Getter for points
* @return
* Input-output point pairs, sorted in ascending order by their x-axis value
*/
protected Collection getPoints()
{
return this.points;
}
/**
* Draws a single sample by the method of adaptive rejection sampling.
* If a sample is rejected, the method will continue until a successful
* sample is selected.
* @param random
* Random number generator
* @return
* Sample drawn according to the logFunction.
*/
public double sample(
Random random )
{
final int maxRejections = 100;
for( int rejections = 0; rejections < maxRejections; rejections++ )
{
final double x = this.upperEnvelope.sampleAsDouble(random);
final double u = random.nextDouble();
final double logLower = this.lowerEnvelope.logEvaluate(x);
final double logUpper = this.upperEnvelope.logEvaluate(x);
final double envelopeRatio = Math.exp( logLower - logUpper );
// If the probability of acceptance is between the upper and lower
// envelopes, then we know we've got a winner, so just
// accept this without evaluating the function itself,
// which can be potentially costly.
if( u <= envelopeRatio )
{
return x;
}
// The "squeeze" ratio wasn't conclusive, so compute the
// acceptance ratio against the actual function
else
{
// Update the envelopes so that we have a better estimate
// of the function itself...
final double logFx = this.logFunction.evaluate(x);
if( this.getNumPoints() < this.getMaxNumPoints() )
{
this.addPoint(x, logFx);
}
// This is the according-to-Hoyle rejection ratio...
final double rejectionRatio = Math.exp( logFx - logUpper );
if( u <= rejectionRatio )
{
return x;
}
}
}
throw new OperationNotConvergedException(
"Maximum number of rejections exceeded for a single sample: " + maxRejections );
}
/**
* Draws samples by the adaptive rejection sampling method, which will
* have the distribution of the logFunction
* @param random
* Random number generator
* @param numSamples
* Number of samples to draw
* @return
* Samples from the adaptive rejection sampling method, which will
* have the distribution of the logFunction
*/
public ArrayList sample(
Random random,
int numSamples)
{
ArrayList samples = new ArrayList( numSamples );
for( int n = 0; n < numSamples; n++ )
{
samples.add( this.sample(random) );
}
return samples;
}
/**
* Getter for logFunction
* @return
* Logarithm of the function that we want to evaluate
*/
public AdaptiveRejectionSampling.LogEvaluator getLogFunction()
{
return this.logFunction;
}
/**
* Setter for logFunction
* @param logFunction
* Logarithm of the function that we want to evaluate
*/
public void setLogFunction(
AdaptiveRejectionSampling.LogEvaluator logFunction)
{
this.logFunction = logFunction;
}
/**
* Getter for maxNumPoints
* @return
* Maximum number of points that will be stored
*/
public int getMaxNumPoints()
{
return this.maxNumPoints;
}
/**
* Setter for maxNumPoints
* @param maxNumPoints
* Maximum number of points that will be stored
*/
public void setMaxNumPoints(
int maxNumPoints)
{
this.maxNumPoints = maxNumPoints;
}
/**
* Getter for minSupport
* @return
* Minimum support (x-value) of the logFunction
*/
public double getMinSupport()
{
return this.minSupport;
}
/**
* Setter for minSupport
* @param minSupport
* Minimum support (x-value) of the logFunction
*/
public void setMinSupport(
double minSupport)
{
this.minSupport = minSupport;
}
/**
* Getter for maxSupport
* @return
* Maximum support (x-value) of the logFunction
*/
public double getMaxSupport()
{
return this.maxSupport;
}
/**
* Setter for maxSupport
* @param maxSupport
* Maximum support (x-value) of the logFunction
*/
public void setMaxSupport(
double maxSupport)
{
this.maxSupport = maxSupport;
}
/**
* Describes an enveloping function comprised of a sorted sequence of lines
*/
public abstract class AbstractEnvelope
extends AbstractUnivariateScalarFunction
{
/**
* Line segments that comprise the envelope
*/
protected ArrayList lines;
/**
* Default constructor
*/
public AbstractEnvelope()
{
this.lines = null;
}
@Override
public AbstractEnvelope clone()
{
AbstractEnvelope clone = (AbstractEnvelope) super.clone();
clone.lines = ObjectUtil.cloneSmartElementsAsArrayList(this.getLines());
return clone;
}
/**
* Getter for lines
* @return
* Line segments that comprise the envelope
*/
protected ArrayList getLines()
{
if( this.lines == null )
{
this.computeLines();
}
return this.lines;
}
/**
* Resets the line segments
*/
public void resetLines()
{
this.lines = null;
}
/**
* Computes the line segments comprising this Envelope
*/
abstract protected void computeLines();
/**
* Evaluates the logarithm of the Envelope
* @param input
* Input to consider
* @return
* Logarithm of the Envelope
*/
public double logEvaluate(
Double input)
{
return this.findLineSegment(input).evaluate(input);
}
public double evaluate(
double input)
{
return Math.exp( this.logEvaluate(input) );
}
/**
* Finds the line segment that contains the input
* @param input
* Input to consider
* @return
* Line segment that contains the input
*/
protected LineSegment findLineSegment(
Double input )
{
ArrayList ls = this.getLines();
final int index = Collections.binarySearch(ls, input);
return ls.get(index);
}
}
/**
* Constructs the upper envelope for sampling.
*/
public class UpperEnvelope
extends AbstractEnvelope
implements ProbabilityFunction
{
/**
* Cumulative sums of the normalized weights of the lines...
* This is automatically computed by computeSegments method.
*/
protected double[] segmentCDF;
/**
* Default constructor
*/
public UpperEnvelope()
{
super();
this.segmentCDF = null;
}
@Override
public UpperEnvelope clone()
{
UpperEnvelope clone = (UpperEnvelope) super.clone();
clone.segmentCDF = ObjectUtil.cloneSmart(this.segmentCDF);
return clone;
}
public UpperEnvelope getProbabilityFunction()
{
return this;
}
/**
* Gets the mean, which is not a supported operation. An exception is
* thrown.
*
* @return Nothing.
*/
public Double getMean()
{
throw new UnsupportedOperationException("Not supported yet.");
}
/**
* Samples from this distribution as a double.
*
* @param random
* The random number generator to use.
* @return
* A sample from this distribution.
*/
public double sampleAsDouble(
Random random)
{
// This is really just a trick to make sure we re-compute the lines
// AND the segmentCDF!
ArrayList ls = this.getLines();
final double p1 = random.nextDouble();
int index = Arrays.binarySearch( this.segmentCDF, p1 );
if( index < 0 )
{
int insertionPoint = -index - 1;
index = insertionPoint;
}
LineSegment segment = ls.get(index);
final double p2 = random.nextDouble();
return segment.sampleExp(p2);
}
@Override
public Double sample(
final Random random)
{
return this.sampleAsDouble(random);
}
@Override
public ArrayList sample(
final Random random,
final int numSamples)
{
final ArrayList result = new ArrayList(numSamples);
this.sampleInto(random, numSamples, result);
return result;
}
@Override
public void sampleInto(
final Random random,
final int sampleCount,
final Collection output)
{
for (int i = 0; i < sampleCount; i++)
{
output.add(this.sampleAsDouble(random));
}
}
/**
* Recomputes the line segments that comprise the upper envelope
*/
protected void computeLines()
{
final int numLines = (points.size()-1) * 2;
this.lines = new ArrayList( numLines );
this.segmentCDF = new double[ numLines ];
double totalMass = 0.0;
Iterator iterator = points.iterator();
double left = getMinSupport();
double right = iterator.next().getInput();
PolynomialFunction.Linear leftLine = Point.line(0,points);
LineSegment leftMost = new LineSegment(
leftLine, left, right );
double weight = leftMost.integrateExp();
totalMass += weight;
this.lines.add( leftMost );
this.segmentCDF[this.lines.size()-1] = totalMass;
PolynomialFunction.Linear rightLine = Point.line(1, points);
left = right;
right = iterator.next().getInput();
LineSegment segment = new LineSegment(
rightLine, left, right );
weight = segment.integrateExp();
totalMass += weight;
this.lines.add( segment );
this.segmentCDF[this.lines.size()-1] = totalMass;
final int N = points.size();
for( int index = 1; index < N-2; index++ )
{
left = right;
leftLine = Point.line(index-1, points);
rightLine = Point.line(index+1, points);
right = Point.intercept(leftLine, rightLine);
segment = new LineSegment(leftLine, left, right);
weight = segment.integrateExp();
totalMass += weight;
this.lines.add( segment );
this.segmentCDF[this.lines.size()-1] = totalMass;
left = right;
right = iterator.next().getInput();
segment = new LineSegment(rightLine, left, right);
weight = segment.integrateExp();
totalMass += weight;
lines.add( segment );
this.segmentCDF[this.lines.size()-1] = totalMass;
}
left = right;
right = iterator.next().getInput();
segment = new LineSegment( Point.line(N-3,points), left, right);
weight = segment.integrateExp();
totalMass += weight;
this.lines.add( segment );
this.segmentCDF[this.lines.size()-1] = totalMass;
left = right;
right = getMaxSupport();
LineSegment rightMost = new LineSegment(Point.line(N-2, points), left, right);
weight = rightMost.integrateExp();
totalMass += weight;
this.lines.add( rightMost );
this.segmentCDF[this.lines.size()-1] = totalMass;
for( int i = 0; i < this.lines.size(); i++ )
{
this.segmentCDF[i] /= totalMass;
}
}
}
/**
* Define the lower envelope for Adaptive Rejection Sampling
*/
public class LowerEnvelope
extends AbstractEnvelope
{
/**
* Default constructor
*/
public LowerEnvelope()
{
super();
}
/**
* Recomputes the line segments that comprise the upper envelope
*/
protected void computeLines()
{
final int numPoints = points.size();
final int numLines = numPoints+1;
this.lines = new ArrayList( numLines );
Iterator iterator = points.iterator();
double left = minSupport;
double right = iterator.next().getInput();
PolynomialFunction.Linear line = new PolynomialFunction.Linear(
0.0,Double.NEGATIVE_INFINITY);
this.lines.add( new LineSegment(line, left, right) );
for( int i = 0; i < numPoints-1; i++ )
{
left = right;
right = iterator.next().getInput();
line = Point.line(i, points);
this.lines.add( new LineSegment(line, left, right) );
}
left = right;
right = maxSupport;
line = new PolynomialFunction.Linear( 0.0,Double.NEGATIVE_INFINITY);
this.lines.add( new LineSegment(line, left, right) );
}
}
/**
* A line that has a minimum and maximum support (x-axis) value.
*/
public static class LineSegment
extends PolynomialFunction.Linear
implements Comparable
{
/**
* Left (minimum) x-axis value
*/
double left;
/**
* Right (maximum) x-axis value
*/
double right;
/**
* Creates a new instance of LineSegment
* @param line
* The line
* @param left
* Left (minimum) x-axis value
* @param right
* Right (maximum) x-axis value
*/
public LineSegment(
PolynomialFunction.Linear line,
double left,
double right )
{
super( line.getQ0(), line.getQ1() );
this.left = left;
this.right = right;
}
/**
* Sample from the exponent of the line segment
* @param p
* Probability into the line segment
* @return
* Sample (x-axis) value into the line segment
*/
public double sampleExp(
double p )
{
ProbabilityUtil.assertIsProbability(p);
double q1 = this.getQ1();
if( Math.abs(q1) >= COLLINEAR_TOLERANCE )
{
double l = Math.exp( q1*this.left );
double r = Math.exp( q1*this.right );
double delta = p*(r-l);
double x = Math.log( l + delta ) / q1;
return x;
}
else
{
// Straight line
double l = this.left;
double r = this.right;
return l + p*(r-l);
}
}
/**
* Integrates the exponent of the line segment
* @return
* Integral of the exponent of the line segment
*/
public double integrateExp()
{
// integral( exp( m*x + b ) )
double q0 = this.getQ0();
double q1 = this.getQ1();
if( Math.abs(q1) >= COLLINEAR_TOLERANCE )
{
double l = Math.exp( q1*this.left );
double r = Math.exp( q1*this.right );
double coeff = Math.exp(q0) / q1;
return coeff * (r-l);
}
else
{
// Slope is zero... this is a line segment
double l = this.left;
double r = this.right;
double coeff = Math.exp(q0);
return coeff * (r-l);
}
}
public int compareTo(
Double o)
{
final double x = o;
if( x < this.left )
{
return 1;
}
else if( x > this.right )
{
return -1;
}
else
{
return 0;
}
}
}
/**
* An InputOutputPair that has a natural ordering according to their
* input (x-axis) values.
*/
public static class Point
extends DefaultInputOutputPair
implements Comparable
{
/**
* Creates a new instance of Point
* @param x
* Input (x-axis) value
* @param y
* Output (y-axis) value, most likely the natural logarithm of the
* function output.
*/
public Point(
double x,
double y )
{
super( x, y );
}
public int compareTo(
Point o)
{
double x0 = this.getInput();
double x1 = o.getInput();
if( x0 < x1 )
{
return -1;
}
else if( x0 > x1 )
{
return 1;
}
else
{
return 0;
}
}
/**
* Connects the points at index and index + 1 with a straight line.
* If index is 0, then we connect
* @param index
* Lower index to connect to index + 1
* @param points
* Points sorted along the x-axis
* @return
* Linear fit between the two given points.
*/
public static PolynomialFunction.Linear line(
int index,
ArrayList points )
{
Point pi = points.get(index);
Point pip1 = points.get(index+1);
return PolynomialFunction.Linear.fit(pi, pip1);
}
/**
* Computes the x-axis value where the two lines intersect
* @param line1
* First line segment
* @param line2
* Second line segment
* @return
* X-axis value where the two lines intersect
*/
public static double intercept(
PolynomialFunction.Linear line1,
PolynomialFunction.Linear line2 )
{
double a1 = line1.getQ1();
double b1 = line1.getQ0();
double a2 = line2.getQ1();
double b2 = line2.getQ0();
// The lines are collinear
if( a1 == a2 )
{
// I suppose we could see if b1==b2, but then lines
// intersect everywhere, so I'm not sure what the answer
// would be
throw new IllegalArgumentException( "Lines are collinear" );
}
else
{
// y = a1 * x + b1;
// y = a2 * x + b2;
// a1 * x + b1 = a2 * x + b2
// (a1-a2)*x = b2-b1
// x = (b2-b1)/(a1-a2)
return (b2-b1)/(a1-a2);
}
}
}
/**
* Wraps an Evaluator and takes the natural logarithm of the evaluate method
* @param
* Type of Evaluator to wrap
*/
public static abstract class LogEvaluator>
extends AbstractUnivariateScalarFunction
{
/**
* Evaluator to wrap and compute the natural logarithm of.
*/
protected EvaluatorType function;
/**
* Creates a new instance of LogEvaluator
* @param function
* Evaluator to wrap and compute the natural logarithm of.
*/
public LogEvaluator(
EvaluatorType function)
{
this.setFunction(function);
}
@Override
public LogEvaluator clone()
{
@SuppressWarnings("unchecked")
LogEvaluator clone =
(LogEvaluator) super.clone();
clone.setFunction( ObjectUtil.cloneSmart( this.getFunction() ) );
return clone;
}
/**
* Getter for function
* @return
* Evaluator to wrap and compute the natural logarithm of.
*/
public EvaluatorType getFunction()
{
return this.function;
}
/**
* Setter for function
* @param function
* Evaluator to wrap and compute the natural logarithm of.
*/
public void setFunction(
EvaluatorType function)
{
this.function = function;
}
public double evaluate(
double input)
{
return Math.log(this.function.evaluate(input));
}
}
/**
* Wraps a PDF so that it returns the logEvaluate method.
*/
public static class PDFLogEvaluator
extends LogEvaluator>
{
/**
* Creates a new instance of PDFLogEvaluator
* @param function
* PDF to wrap
*/
public PDFLogEvaluator(
ProbabilityFunction function )
{
super( function );
}
@Override
public double evaluate(
double input)
{
return this.function.logEvaluate(input);
}
}
}