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Algorithms and components for machine learning and statistics.
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/*
* File: GaussianConfidence.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.UnivariateStatisticsUtil;
import gov.sandia.cognition.statistics.UnivariateDistribution;
import gov.sandia.cognition.statistics.distribution.UnivariateGaussian;
import gov.sandia.cognition.util.AbstractCloneableSerializable;
import gov.sandia.cognition.util.Pair;
import java.util.Collection;
/**
* This test is sometimes called the "Z test"
* Defines a range of values that the statistic can take, as well as the
* confidence that the statistic is between the lower and upper bounds. This
* test is useful in those situations where the tested data were generated by
* a (univariate) Gaussian distribution.
* @author Kevin R. Dixon
* @since 2.0
*
*/
@ConfidenceTestAssumptions(
name="Gaussian Z-test",
alsoKnownAs="Z-test",
description="Determines if two populations have the same mean, if the populations are Gaussian and relatively large, at least 30 or so.",
assumptions={
"The two groups are sampled independently of each other.",
"The two groups are sampled from a Gaussian distribution, or the underlying distributions are non-Gaussian but obey the weak law of large numbers.",
"The variances of the two groups are equal."
},
nullHypothesis="The means of the groups are equal.",
dataPaired=false,
dataSameSize=false,
distribution=UnivariateGaussian.CDF.class,
reference=@PublicationReference(
author="Wikipedia",
title="Z-test",
type=PublicationType.WebPage,
year=2009,
url="http://en.wikipedia.org/wiki/Z-test"
)
)
public class GaussianConfidence
extends AbstractCloneableSerializable
implements NullHypothesisEvaluator>,
ConfidenceIntervalEvaluator>
{
/**
* This class has no members, so here's a static instance.
*/
public static final GaussianConfidence INSTANCE =
new GaussianConfidence();
/** Creates a new instance of GaussianConfidence */
public GaussianConfidence()
{
}
@Override
public GaussianConfidence.Statistic evaluateNullHypothesis(
Collection data1,
Collection data2 )
{
int N1 = data1.size();
UnivariateGaussian g1 =
UnivariateGaussian.MaximumLikelihoodEstimator.learn( data1, 0.0 );
double std1 = Math.sqrt( g1.getVariance() );
int N2 = data2.size();
UnivariateGaussian g2 =
UnivariateGaussian.MaximumLikelihoodEstimator.learn( data2, 0.0 );
double std2 = Math.sqrt( g2.getVariance() );
double numerator = Math.abs( g1.getMean() - g2.getMean() );
double denominator =
Math.sqrt( ((std1 * std1) / N1) + ((std2 * std2) / N2) );
double z = numerator / denominator;
return new GaussianConfidence.Statistic( z );
}
/**
* Computes the probability that the input was drawn from the estimated
* UnivariateGaussian distribution. That is, what is the probability
* that the UnivariateGaussian could produce a MORE UNLIKELY sample than
* the given "input". For example, the probability of drawing a more
* unlikely sample that the mean is 1.0 and infinity is 0.0
*
* @param data1 Dataset to consider
* @param data2 Sample to compute the probability that a
* UnivariateGaussian would produce a more unlikely sample than "data2"
* @return probability that the input was drawn from this estimated
* UnivariateGaussian distribution. That is, what is the probability
* that the UnivariateGaussian could produce a MORE UNLIKELY sample than
* the given input
*/
public static GaussianConfidence.Statistic evaluateNullHypothesis(
Collection data1,
double data2 )
{
Pair result =
UnivariateStatisticsUtil.computeMeanAndVariance(data1);
double mean = result.getFirst();
double variance = result.getSecond();
// This will tell us the probability of the left tail of the Gaussian
double delta = Math.abs( mean - data2 );
double z;
if( variance != 0.0 )
{
z = delta / Math.sqrt( variance );
}
else if( delta != 0.0 )
{
z = Double.POSITIVE_INFINITY;
}
else
{
z = 0.0;
}
// This should actually be the student-t distribution with
// this.getNumSamples() number of degrees of freedom. However, since
// we expect the number of samples to be >30 or so, then the
// Gaussian and Student-t distribution (N>=30) are approximately
// equal.
return new GaussianConfidence.Statistic( z );
}
@Override
public ConfidenceInterval computeConfidenceInterval(
Collection data,
double confidence )
{
UnivariateGaussian g = UnivariateGaussian.MaximumLikelihoodEstimator.learn(
data, UnivariateGaussian.MaximumLikelihoodEstimator.DEFAULT_VARIANCE );
return GaussianConfidence.computeConfidenceInterval(
g, data.size(), confidence );
}
/**
* Computes the Gaussian confidence interval given a distribution of
* data, number of samples, and corresponding confidence interval
* @param dataDistribution
* UnivariateGaussian describing the distribution of the underlying data
* @param numSamples
* Number of samples in the underlying data
* @param confidence
* Confidence value to assume for the ConfidenceInterval
* @return
* ConfidenceInterval capturing the range of the mean of the data
* at the desired level of confidence
*/
public static ConfidenceInterval computeConfidenceInterval(
UnivariateDistribution dataDistribution,
int numSamples,
double confidence )
{
return INSTANCE.computeConfidenceInterval(
dataDistribution.getMean().doubleValue(),
dataDistribution.getVariance(),
numSamples, confidence );
}
@PublicationReference(
author="Wikipedia",
title="Standard error (statistics)",
type=PublicationType.WebPage,
year=2009,
url="http://en.wikipedia.org/wiki/Standard_error_(statistics)"
)
@Override
public ConfidenceInterval computeConfidenceInterval(
double mean,
double variance,
int numSamples,
double confidence )
{
double alpha = 1.0 - confidence;
double z = -UnivariateGaussian.CDF.Inverse.evaluate(
0.5 * alpha, 0.0, 1.0 );
double delta = z * Math.sqrt( variance / numSamples );
return new ConfidenceInterval(
mean, mean - delta, mean + delta, confidence, numSamples );
}
/**
* Confidence statistics for a Gaussian distribution
*/
public static class Statistic
extends AbstractConfidenceStatistic
{
/**
* Value that is used in the Gaussian CDF to compute
* probability. Usually just called "z-statistic"
*/
private double z;
/**
* Creates a new instance of Statistic
* @param z
* Value that is used in the Inverse Gaussian CDF to compute
* probability. Usually just called "z-statistic"
*/
public Statistic(
double z )
{
super( 2.0 * UnivariateGaussian.CDF.evaluate( -z, 0.0, 1.0 ) );
this.setZ( z );
}
/**
* Setter for z
* @return
* Value that is used in the Inverse Gaussian CDF to compute
* probability. Usually just called "z-statistic"
*/
public double getZ()
{
return this.z;
}
/**
* Getter for z
* @param z
* Value that is used in the Inverse Gaussian CDF to compute
* probability. Usually just called "z-statistic"
*/
protected void setZ(
double z )
{
this.z = z;
}
@Override
public double getTestStatistic()
{
return this.getZ();
}
}
}