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/*
* File: GaussianProcessRegression.java
* Authors: Kevin R. Dixon
* Company: Sandia National Laboratories
* Project: Cognitive Foundry
*
* Copyright Mar 16, 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.PublicationReferences;
import gov.sandia.cognition.annotation.PublicationType;
import gov.sandia.cognition.collection.CollectionUtil;
import gov.sandia.cognition.evaluator.Evaluator;
import gov.sandia.cognition.learning.data.InputOutputPair;
import gov.sandia.cognition.learning.function.kernel.DefaultKernelContainer;
import gov.sandia.cognition.learning.function.kernel.Kernel;
import gov.sandia.cognition.math.matrix.Matrix;
import gov.sandia.cognition.math.matrix.MatrixFactory;
import gov.sandia.cognition.math.matrix.Vector;
import gov.sandia.cognition.math.matrix.VectorFactory;
import gov.sandia.cognition.statistics.distribution.MultivariateGaussian;
import gov.sandia.cognition.statistics.distribution.UnivariateGaussian;
import gov.sandia.cognition.util.AbstractCloneableSerializable;
import java.util.ArrayList;
import java.util.Collection;
/**
* Gaussian Process Regression, is also known as Kriging, is a nonparametric
* method to interpolate and extrapolate using Bayesian regression, where
* the expressiveness of the estimator can grow with the data. However,
* GPR does not scale well as it requires a N-cubed inversion, where N is the
* number of data points, to compute the predictive distribution.
* @author Kevin R. Dixon
* @since 3.0
* @param
* Type of inputs to map through the kernel
*/
@PublicationReferences(
references={
@PublicationReference(
author="Christopher M. Bishop",
title="Pattern Recognition and Machine Learning",
type=PublicationType.Book,
year=2006,
pages={303,312}
)
,
@PublicationReference(
author="Hanna M. Wallach",
title="Introduction to Gaussian Process Regression",
type=PublicationType.Misc,
year=2005,
url="http://www.cs.umass.edu/~wallach/talks/gp_intro.pdf"
)
,
@PublicationReference(
author="Wikipedia",
title="Kriging",
type=PublicationType.WebPage,
year=2010,
url="http://en.wikipedia.org/wiki/Kriging"
)
}
)
public class GaussianProcessRegression
extends DefaultKernelContainer
implements BayesianEstimator, Vector, MultivariateGaussian>
{
/**
* Default assumed variance of the measurements, {@value}.
*/
public static final double DEFAULT_MEASUREMENT_VARIANCE = 1.0;
/**
* Assumed known variance of the outputs (measurements),
* must be greater than or equal to zero.
*/
private double outputVariance;
/**
* Creates a new instance of GaussianProcessRegression
*/
public GaussianProcessRegression()
{
this( null, DEFAULT_MEASUREMENT_VARIANCE );
}
/**
* Creates a new instance of GaussianProcessRegression
* @param kernel
* Kernel to map the InputType to the Vector space
* @param outputVariance
* Assumed known variance of the outputs (measurements),
* must be greater than or equal to zero.
*/
public GaussianProcessRegression(
Kernel kernel,
double outputVariance )
{
super( kernel );
this.setOutputVariance(outputVariance);
}
@Override
public GaussianProcessRegression clone()
{
return (GaussianProcessRegression) super.clone();
}
public MultivariateGaussian learn(
Collection> data)
{
ArrayList> dataArray =
CollectionUtil.asArrayList(data);
final int N = dataArray.size();
Matrix C = MatrixFactory.getDefault().createMatrix(N, N);
Vector mean = VectorFactory.getDefault().createVector(N);
for( int i = 0; i < N; i++ )
{
InputOutputPair pair = dataArray.get(i);
final InputType xi = pair.getInput();
mean.setElement(i, pair.getOutput());
for( int j = 0; j < N; j++ )
{
final InputType xj = dataArray.get(j).getInput();
double kv = this.kernel.evaluate(xi, xj);
if( i == j )
{
kv += this.getOutputVariance();
}
C.setElement(i, j, kv);
}
}
return new MultivariateGaussian( mean, C );
}
/**
* Getter for outputVariance
* @return
* Assumed known variance of the outputs (measurements),
* must be greater than or equal to zero.
*/
public double getOutputVariance()
{
return this.outputVariance;
}
/**
* Getter for outputVariance
* @param outputVariance
* Assumed known variance of the outputs (measurements),
* must be greater than or equal to zero.
*/
public void setOutputVariance(
double outputVariance)
{
if( outputVariance < 0.0 )
{
throw new IllegalArgumentException(
"Output variance must be >= 0.0" );
}
this.outputVariance = outputVariance;
}
/**
* Creates the predictive distribution for future points.
* @param posterior
* Posterior from the fitting of the training data.
* @param inputs
* Training data inputs
* @return
* Predictive distribution of the Gaussian Process Regression
*/
public GaussianProcessRegression.PredictiveDistribution createPredictiveDistribution(
MultivariateGaussian posterior,
ArrayList inputs )
{
return new PredictiveDistribution(posterior, inputs);
}
/**
* Predictive distribution for Gaussian Process Regression.
*/
@PublicationReference(
author="Christopher M. Bishop",
title="Pattern Recognition and Machine Learning",
type=PublicationType.Book,
year=2006,
pages=308,
notes="Equations 6.66 and 6.67"
)
public class PredictiveDistribution
extends AbstractCloneableSerializable
implements Evaluator
{
/**
* Inputs that we've condition on.
*/
private ArrayList inputs;
/**
* Posterior distribution of the Gaussian process given the data.
*/
private MultivariateGaussian posterior;
/**
* Creates a new instance of PredictiveDistribution
* @param posterior
* Posterior distribution of the Gaussian process given the data.
* @param inputs
* Inputs that we've condition on.
*/
public PredictiveDistribution(
MultivariateGaussian posterior,
ArrayList inputs )
{
this.posterior = posterior;
this.inputs = inputs;
}
public UnivariateGaussian evaluate(
InputType input)
{
final int N = this.posterior.getInputDimensionality();
Vector k = VectorFactory.getDefault().createVector(N);
final double c = kernel.evaluate(input, input) + getOutputVariance();
for( int i = 0; i < N; i++ )
{
InputType xi = this.inputs.get(i);
k.setElement(i, kernel.evaluate(xi,input) );
}
Matrix Ci = this.posterior.getCovarianceInverse();
Vector ktCi = k.times( Ci );
double mean = ktCi.dotProduct( this.posterior.getMean() );
double variance = c - ktCi.dotProduct( k );
return new UnivariateGaussian( mean, variance );
}
}
}