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Algorithms and components for machine learning and statistics.
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/*
* File: BayesianRobustLinearRegression.java
* Authors: Kevin R. Dixon
* Company: Sandia National Laboratories
* Project: Cognitive Foundry
*
* Copyright Mar 18, 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.evaluator.Evaluator;
import gov.sandia.cognition.learning.algorithm.IncrementalLearner;
import gov.sandia.cognition.learning.data.DatasetUtil;
import gov.sandia.cognition.learning.data.InputOutputPair;
import gov.sandia.cognition.math.RingAccumulator;
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.math.matrix.Vectorizable;
import gov.sandia.cognition.statistics.AbstractSufficientStatistic;
import gov.sandia.cognition.statistics.distribution.InverseGammaDistribution;
import gov.sandia.cognition.statistics.distribution.MultivariateGaussian;
import gov.sandia.cognition.statistics.distribution.MultivariateGaussianInverseGammaDistribution;
import gov.sandia.cognition.statistics.distribution.StudentTDistribution;
import gov.sandia.cognition.statistics.distribution.UnivariateGaussian;
import gov.sandia.cognition.util.AbstractCloneableSerializable;
import gov.sandia.cognition.util.ObjectUtil;
import java.util.Collection;
/**
* Computes a Bayesian linear estimator for a given feature function given
* a set of InputOutputPair observed values. This regression algorithm
* assumes that the conditional distribution of observations is a zero-mean
* Gaussian with unknown variance, distributed according to the standard
* (conjugate prior) inverse-Gamma distribution. The prior distribution of
* weights is a multivariate Gaussian. Given a training set, the algorithm
* computes the posterior of the weights given the training set. The system
* perform robust regression because the predictive distribution of future
* data is a Student-t distribution, resulting from the assumption that
* the variance of the outputs is unknown. Consequently, the algorithm is
* more robust against outliers than fix-variance regression.
* @author Kevin R. Dixon
* @since 3.0
*/
@PublicationReferences(
references={
@PublicationReference(
author="Christopher M. Bishop",
title="Pattern Recognition and Machine Learning",
type=PublicationType.Book,
year=2006,
pages={152,159}
)
,
@PublicationReference(
author="Jan Drugowitsch",
title="Bayesian Linear Regression",
type=PublicationType.Misc,
year=2009,
url="http://www.bcs.rochester.edu/people/jdrugowitsch/code/bayes_linear_notes_0.1.1.pdf"
)
}
)
public class BayesianRobustLinearRegression
extends AbstractCloneableSerializable
implements BayesianRegression
{
/**
* Default weight variance, {@value}.
*/
public static final double DEFAULT_WEIGHT_VARIANCE = 1.0;
/**
* Prior distribution of the weights, typically a zero-mean,
* diagonal-variance distribution.
*/
private MultivariateGaussian weightPrior;
/**
* Distribution of the output (measurement) variance
*/
private InverseGammaDistribution outputVariance;
/**
* Creates a new instance of BayesianLinearRegression
* @param dimensionality
* Sets up the parameters (except featureMap) for the given dimensionality
* of objects in feature space.
*/
public BayesianRobustLinearRegression(
int dimensionality )
{
this( new InverseGammaDistribution(),
new MultivariateGaussian( VectorFactory.getDefault().createVector(dimensionality),
MatrixFactory.getDefault().createIdentity(dimensionality,dimensionality).scale( DEFAULT_WEIGHT_VARIANCE ) ) );
}
/**
* Creates a new instance of BayesianRobustLinearRegression
* @param outputVariance
* Distribution of the output (measurement) variance
* @param weightPrior
* Prior distribution of the weights, typically a zero-mean,
* diagonal-variance distribution.
*/
public BayesianRobustLinearRegression(
InverseGammaDistribution outputVariance,
MultivariateGaussian weightPrior )
{
this.setWeightPrior(weightPrior);
this.setOutputVariance(outputVariance);
}
@Override
public BayesianRobustLinearRegression clone()
{
@SuppressWarnings("unchecked")
BayesianRobustLinearRegression clone =
(BayesianRobustLinearRegression) super.clone();
clone.setWeightPrior( ObjectUtil.cloneSafe( this.getWeightPrior() ) );
clone.setOutputVariance( ObjectUtil.cloneSafe( this.getOutputVariance() ) );
return clone;
}
/**
* Getter for weightPrior
* @return
* Prior distribution of the weights, typically a zero-mean,
* diagonal-variance distribution.
*/
public MultivariateGaussian getWeightPrior()
{
return this.weightPrior;
}
/**
* Setter for weightPrior
* @param weightPrior
* Prior distribution of the weights, typically a zero-mean,
* diagonal-variance distribution.
*/
public void setWeightPrior(
MultivariateGaussian weightPrior)
{
this.weightPrior = weightPrior;
}
@Override
public MultivariateGaussianInverseGammaDistribution learn(
Collection> data)
{
MultivariateGaussian g = this.weightPrior;
RingAccumulator Cin = new RingAccumulator();
Matrix Ci = g.getCovarianceInverse();
Cin.accumulate( Ci );
RingAccumulator zn = new RingAccumulator();
Vector z = Ci.times( g.getMean() );
zn.accumulate( z );
InverseGammaDistribution ig = this.outputVariance;
double an = ig.getShape();
double bn = ig.getScale();
double sy2 = 0.0;
for (InputOutputPair pair : data)
{
Vector x1 = pair.getInput().convertToVector();
Vector x2 = x1.clone();
final double beta = DatasetUtil.getWeight(pair);
if( beta != 1.0 )
{
x2.scaleEquals(beta);
}
Cin.accumulate( x1.outerProduct(x2) );
final double y = pair.getOutput();
if( y != 1.0 )
{
x2.scaleEquals(y);
}
zn.accumulate( x2 );
sy2 += y*y;
an += 0.5;
}
Ci = Cin.getSum();
Matrix C = Ci.inverse();
z = zn.getSum();
Vector mean = C.times( z );
bn += 0.5*(sy2 - mean.times( Ci ).dotProduct(mean));
return new MultivariateGaussianInverseGammaDistribution(
new MultivariateGaussian( mean, C ),
new InverseGammaDistribution( an, bn ) );
}
/**
* Getter for outputVariance
* @return
* Distribution of the output (measurement) variance
*/
public InverseGammaDistribution getOutputVariance()
{
return this.outputVariance;
}
/**
* Setter for outputVariance
* @param outputVariance
* Distribution of the output (measurement) variance
*/
public void setOutputVariance(
InverseGammaDistribution outputVariance)
{
this.outputVariance = outputVariance;
}
/**
* Creates the distribution from which the outputs are generated, given
* the weights and the input to consider.
* @param input
* Input to condition on
* @param weights
* Weights that determine the mean
* @return
* Conditional distribution from which outputs are generated.
*/
@Override
public UnivariateGaussian createConditionalDistribution(
Vectorizable input,
Vector weights )
{
double mean = input.convertToVector().dotProduct(weights);
double variance = this.getOutputVariance().getMean();
return new UnivariateGaussian( mean, variance );
}
@Override
public BayesianRobustLinearRegression.PredictiveDistribution createPredictiveDistribution(
MultivariateGaussianInverseGammaDistribution posterior)
{
return new PredictiveDistribution(posterior);
}
/**
* Predictive distribution of future data given the posterior of
* the weights given the data.
*/
public class PredictiveDistribution
extends AbstractCloneableSerializable
implements Evaluator
{
/**
* Posterior distribution of the weights given the data.
*/
private MultivariateGaussianInverseGammaDistribution posterior;
/**
* Creates a new instance of PredictiveDistribution
* @param posterior
* Posterior distribution of the weights given the data.
*/
public PredictiveDistribution(
MultivariateGaussianInverseGammaDistribution posterior)
{
this.posterior = posterior;
}
@Override
public StudentTDistribution evaluate(
Vectorizable input)
{
Vector x = input.convertToVector();
double mean = x.dotProduct( this.posterior.getMean() );
double dofs = this.posterior.getInverseGamma().getShape() * 2.0;
double v = x.times( this.posterior.getGaussian().getCovariance() ).dotProduct(x);
double anbn = this.posterior.getInverseGamma().getShape() / this.posterior.getInverseGamma().getScale();
double precision = anbn / (1.0 + v);
return new StudentTDistribution( dofs, mean, precision );
}
}
/**
* Incremental estimator for BayesianRobustLinearRegression
*/
public static class IncrementalEstimator
extends BayesianRobustLinearRegression
implements IncrementalLearner, IncrementalEstimator.SufficientStatistic>
{
/**
* Creates a new instance of IncrementalEstimator
* @param dimensionality
* Sets up the parameters (except featureMap) for the given dimensionality
* of objects in feature space.
*/
public IncrementalEstimator(
int dimensionality )
{
super( dimensionality );
}
/**
* Creates a new instance of IncrementalEstimator
* @param outputVariance
* Distribution of the output (measurement) variance
* @param weightPrior
* Prior distribution of the weights, typically a zero-mean,
* diagonal-variance distribution.
*/
public IncrementalEstimator(
InverseGammaDistribution outputVariance,
MultivariateGaussian weightPrior )
{
super( outputVariance, weightPrior);
}
@Override
public IncrementalEstimator.SufficientStatistic createInitialLearnedObject()
{
return new SufficientStatistic(
new MultivariateGaussianInverseGammaDistribution(
this.getWeightPrior(), this.getOutputVariance() ));
}
@Override
public MultivariateGaussianInverseGammaDistribution learn(
Collection> data)
{
IncrementalEstimator.SufficientStatistic target =
this.createInitialLearnedObject();
this.update(target, data);
return target.create();
}
@Override
public void update(
IncrementalEstimator.SufficientStatistic target,
InputOutputPair data)
{
target.update(data);
}
@Override
public void update(
SufficientStatistic target,
Iterable> data)
{
target.update(data);
}
/**
* SufficientStatistic for incremental Bayesian linear regression
*/
public class SufficientStatistic
extends AbstractSufficientStatistic, MultivariateGaussianInverseGammaDistribution>
{
/**
* Sum of the output squared
*/
private double outputSumSquared;
/**
* "z" statistic, proportional to the mean
*/
private Vector z;
/**
* Covariance inverse, sometimes called "precision"
*/
private Matrix covarianceInverse;
/**
* Creates a new instance of SufficientStatistic
* @param prior
* Prior on the weights
*/
public SufficientStatistic(
MultivariateGaussianInverseGammaDistribution prior )
{
super();
if( prior != null )
{
Vector mean = prior.getMean();
this.covarianceInverse =
prior.getGaussian().getCovarianceInverse().clone();
this.z = this.covarianceInverse.times( mean );
double a0 = prior.getInverseGamma().getShape();
double b0 = prior.getInverseGamma().getScale();
this.count = (long) Math.ceil(2.0*a0);
this.outputSumSquared = 2.0*b0 + mean.dotProduct(this.z);
}
else
{
this.covarianceInverse = null;
this.z = null;
this.count = 0;
this.outputSumSquared = 0.0;
}
}
@Override
public void update(
InputOutputPair value)
{
this.count++;
Vector v = value.getInput().convertToVector();
Vector x1 = v;
Vector x2 = v.clone();
final double y = value.getOutput();
final double beta = DatasetUtil.getWeight(value);
if( beta != 1.0 )
{
x2.scaleEquals(beta);
}
if( this.covarianceInverse == null )
{
this.covarianceInverse = x1.outerProduct(x2);
}
else
{
this.covarianceInverse.plusEquals( x1.outerProduct(x2) );
}
if( y != 1.0 )
{
x2.scaleEquals( y );
}
if( this.z == null )
{
this.z = x2;
}
else
{
this.z.plusEquals( x2 );
}
this.outputSumSquared += y*y;
}
@Override
public MultivariateGaussianInverseGammaDistribution create()
{
MultivariateGaussianInverseGammaDistribution g =
new MultivariateGaussianInverseGammaDistribution( this.getDimensionality() );
this.create(g);
return g;
}
@Override
public void create(
MultivariateGaussianInverseGammaDistribution distribution)
{
distribution.getGaussian().setMean(this.getMean());
distribution.getGaussian().setCovarianceInverse(this.getCovarianceInverse());
distribution.getInverseGamma().setShape(this.getShape());
distribution.getInverseGamma().setScale(this.getScale());
}
/**
* Getter for covarianceInverse
* @return
* Covariance inverse, sometimes called "precision"
*/
public Matrix getCovarianceInverse()
{
return this.covarianceInverse;
}
/**
* Getter for z
* @return
* "z" statistic, proportional to the mean
*/
public Vector getZ()
{
return this.z;
}
/**
* Computes the mean of the Gaussian, but involves a matrix
* inversion and multiplication, so it's expensive.
* @return
* Mean of the Gaussian.
*/
public Vector getMean()
{
return this.covarianceInverse.inverse().times( this.z );
}
/**
* Gets the dimensionality of the underlying Gaussian
* @return
* Dimensionality of the underlying Gaussian
*/
public int getDimensionality()
{
return this.getZ().getDimensionality();
}
/**
* Getter for outputSumSquared
* @return
* Sum of the output squared
*/
public double getOutputSumSquared()
{
return this.outputSumSquared;
}
/**
* Computes the shape component for the inverse-gamma distribution
* @return
* Shape component for the inverse-gamma distribution
*/
public double getShape()
{
return this.getCount()/2.0;
}
/**
* Computes the scale component for the inverse-gamma distribution
* @return
* Scale component for the inverse-gamma distribution
*/
public double getScale()
{
Vector mean = this.getMean();
Matrix Ci = this.covarianceInverse;
return 0.5 * (this.outputSumSquared - mean.times(Ci).dotProduct(mean));
}
}
}
}