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/*
* File: StudentTConfidence.java
* Authors: Kevin R. Dixon
* Company: Sandia National Laboratories
* Project: Cognitive Foundry
*
* Copyright August 16, 2007, 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.method;
import gov.sandia.cognition.annotation.PublicationReference;
import gov.sandia.cognition.annotation.PublicationType;
import gov.sandia.cognition.math.ProbabilityUtil;
import gov.sandia.cognition.math.UnivariateStatisticsUtil;
import gov.sandia.cognition.util.Summarizer;
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.Pair;
import java.util.Collection;
import java.util.Iterator;
/**
* This class implements Student's t-tests for different uses. The confidence
* test is the Paired Student's t-test to determine if the difference between
* the pairs have zero mean. The confidence interval calculation computes
* the Student-t confidence interval for the mean of the given data.
* @author Kevin R. Dixon
* @since 2.0
*/
@ConfidenceTestAssumptions(
name="Paired Student's t-test",
description="Computes the value of the null hypothesis that the differences between paired samples have zero mean and that the data are sampled from a Gaussian distributions with equal variances.",
alsoKnownAs="Dependent t-test for paired samples",
nullHypothesis="The means of the groups are equal.",
assumptions={
"The data for the pairs are iid samples from a Gaussian distribution with equal variances.",
"The common variances times the degrees of freedom is a chi-square distribution.",
"The data pairs should be sampled independently from each other."
},
distribution=StudentTDistribution.CDF.class,
dataPaired=true,
dataSameSize=true,
reference=@PublicationReference(
author="Wikipedia",
title="Student's t-test, Dependent t-test for paired samples",
type=PublicationType.WebPage,
year=2009,
url="http://en.wikipedia.org/wiki/Student_t_test#Dependent_t-test_for_paired_samples"
)
)
public class StudentTConfidence
extends AbstractCloneableSerializable
implements NullHypothesisEvaluator>,
ConfidenceIntervalEvaluator>
{
/**
* This class has no members, so here's a static instance.
*/
public static final StudentTConfidence INSTANCE =
new StudentTConfidence();
/**
* Default tolerance for the standard deviation, {@value}.
*/
public static final double DEFAULT_TOLERANCE = 1e-10;
/** Creates a new instance of StudentTConfidence */
public StudentTConfidence()
{
}
/**
* Computes a paired Student-t test for the given data. The datasets must
* be the same size.
* @param data1 First dataset to consider
* @param data2 Second dataset to consider
* @return
* ConfidenceStatistic for a Student-t test
*/
@PublicationReference(
author={
"William H. Press",
"Saul A. Teukolsky",
"William T. Vetterling",
"Brian P. Flannery"
},
title="Numerical Recipes in C, Second Edition",
type=PublicationType.Book,
year=1992,
pages=618,
notes="Function tptest()",
url="http://www.nrbook.com/a/bookcpdf.php"
)
@Override
public StudentTConfidence.Statistic evaluateNullHypothesis(
Collection data1,
Collection data2 )
{
if (data1.size() != data2.size())
{
throw new IllegalArgumentException(
"Data collections must have same number of elements" );
}
int N = data1.size();
Pair g1 = UnivariateStatisticsUtil.computeMeanAndVariance(data1);
Pair g2 = UnivariateStatisticsUtil.computeMeanAndVariance(data2);
double mean1 = g1.getFirst();
double var1 = g1.getSecond();
double mean2 = g2.getFirst();
double var2 = g2.getSecond();
double dof = N - 1;
Iterator i1 = data1.iterator();
Iterator i2 = data2.iterator();
double cov = 0.0;
for (int n = 0; n < N; n++)
{
double v1 = i1.next().doubleValue();
double v2 = i2.next().doubleValue();
cov += (v1 - mean1) * (v2 - mean2);
}
cov /= dof;
double sd = Math.sqrt( (var1 + var2 - 2 * cov) / N );
if( sd < DEFAULT_TOLERANCE )
{
sd = DEFAULT_TOLERANCE;
}
double t = Math.abs( (mean1 - mean2) / sd );
return new StudentTConfidence.Statistic( t, dof );
}
@Override
public ConfidenceInterval computeConfidenceInterval(
Collection data,
double confidence )
{
UnivariateGaussian g =
UnivariateGaussian.MaximumLikelihoodEstimator.learn(
data, UnivariateGaussian.MaximumLikelihoodEstimator.DEFAULT_VARIANCE );
return computeConfidenceInterval(
g.getMean(), g.getVariance(), data.size(), confidence );
}
@Override
public ConfidenceInterval computeConfidenceInterval(
double mean,
double variance,
int numSamples,
double confidence)
{
if ((confidence <= 0.0) ||
(confidence > 1.0))
{
throw new IllegalArgumentException(
"Confidence must be on the interval (0,1]" );
}
double alpha = 1.0 - confidence;
int dof = numSamples - 1;
StudentTDistribution.CDF cdf = new StudentTDistribution.CDF( dof );
double z = -cdf.inverse( 0.5 * alpha );
double delta = z * Math.sqrt( variance / numSamples );
if (delta < 0.0)
{
delta = 0.0;
}
return new ConfidenceInterval(
mean, mean - delta, mean + delta, confidence, numSamples );
}
/**
* Confidence statistics for a Student-t test
*/
public static class Statistic
extends AbstractConfidenceStatistic
{
/**
* Value that is used in the Student-t CDF to compute the probability.
* Usually just called the "t-statistic"
*/
private double t;
/**
* Number of degrees of freedom in the Student-t distribution, usually
* the number of data points - 1
*/
private double degreesOfFreedom;
/**
* Creates a new instance of Statistic
* @param t
* Value that is used in the Student-t CDF to compute the probability.
* Usually just called the "t-statistic"
* @param degreesOfFreedom
* Number of degrees of freedom in the Student-t distribution, usually
* the number of data points - 1
*/
public Statistic(
double t,
double degreesOfFreedom )
{
super( Statistic.twoTailTStatistic( t, degreesOfFreedom ) );
this.setT( t );
this.setDegreesOfFreedom( degreesOfFreedom );
}
/**
* Copy Constructor
* @param other Statistic to copy
*/
public Statistic(
Statistic other )
{
this( other.getT(), other.getDegreesOfFreedom() );
}
@Override
public Statistic clone()
{
return (Statistic) super.clone();
}
/**
* Computes the likelihood that a StudentTDistribution would generate
* a LESS LIKELY sample as "t", given the degrees of freedom. This is a
* two tailed test, thus, we're computing the probability that a Student-t
* distribution would be as far away as "t" (both tails)
* @param t
* Sample to determine how likely a worse sample is than "t"
* @param degreesOfFreedom
* Number of degrees of freedom in the Student-t distribution
* @return
* Probability that a Student-t distribution would generate as bad of
* a sample as "t"
*/
public static double twoTailTStatistic(
double t,
double degreesOfFreedom )
{
StudentTDistribution.CDF cdf =
new StudentTDistribution.CDF( degreesOfFreedom );
return 2.0 * cdf.evaluate( -t );
}
/**
* Getter for t
* @return
* Value that is used in the Student-t CDF to compute the probability.
* Usually just called the "t-statistic"
*/
public double getT()
{
return this.t;
}
/**
* Setter for t
* @param t
* Value that is used in the Student-t CDF to compute the probability.
* Usually just called the "t-statistic"
*/
protected void setT(
double t )
{
this.t = t;
}
/**
* Getter for degreesOfFreedom
* @return
* Number of degrees of freedom in the Student-t distribution, usually
* the number of data points - 1
*/
public double getDegreesOfFreedom()
{
return this.degreesOfFreedom;
}
/**
* Setter for degreesOfFreedom
* @param degreesOfFreedom
* Number of degrees of freedom in the Student-t distribution, usually
* the number of data points - 1
*/
protected void setDegreesOfFreedom(
double degreesOfFreedom )
{
if (degreesOfFreedom <= 0.0)
{
throw new IllegalArgumentException( "degreesOfFreedom > 0.0" );
}
this.degreesOfFreedom = degreesOfFreedom;
}
@Override
public double getTestStatistic()
{
return this.getT();
}
}
/**
* An implementation of the {@code Summarizer} interface for creating a
* {@code ConfidenceInterval}
*/
public static class Summary
extends AbstractCloneableSerializable
implements Summarizer
{
/** The confidence for the created interval. */
private double confidence;
/**
* Creates a new Summarizer.
*
* @param confidence The confidence for the interval.
*/
public Summary(
final double confidence )
{
super();
this.setConfidence( confidence );
}
@Override
public ConfidenceInterval summarize(
final Collection data )
{
return new StudentTConfidence().computeConfidenceInterval(
data, this.getConfidence() );
}
/**
* Gets the confidence for created the interval.
*
* @return The confidence for the created interval.
*/
public double getConfidence()
{
return this.confidence;
}
/**
* Sets the confidence for created the interval.
*
* @param confidence The confidence for the created interval.
*/
public void setConfidence(
final double confidence )
{
ProbabilityUtil.assertIsProbability( confidence );
this.confidence = confidence;
}
}
}