/*
* File: MarkovInequality.java
* Authors: Kevin R. Dixon
* Company: Sandia National Laboratories
* Project: Cognitive Foundry
*
* Copyright October 1, 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.util.AbstractCloneableSerializable;
import java.util.Collection;
/**
* Implementation of the Markov Inequality hypothesis test. This is a
* distribution-free test that says "what is the probability that ANY
* random variable can take a magnitude greater than 'a', given samples
* drawn from the random variable"
* Pr{abs(X)>=a} <= mean(abs(data))/a. For example, assume we have data:
* (3 -1 2) and would like to know what range of values are possible with 95%
* confidence -> MarkovInequality.computeConfidenceInterval( (3 -1 2), 0.95 )
* The sample mean of the absolute values is (3+1+2)/3 = 2.
* 1-0.95 <= 2/a -> a=40.
* Thus Pr{ -40 <= X <= 40 } >= 0.95
*
* @author Kevin R. Dixon
* @since 2.0
*
*/
@PublicationReference(
author="Wikipedia",
title="Markov's Inequality",
type=PublicationType.WebPage,
year=2009,
url="http://en.wikipedia.org/wiki/Markov%27s_inequality"
)
public class MarkovInequality
extends AbstractCloneableSerializable
implements ConfidenceIntervalEvaluator<Collection<Double>>
{
/**
* This class has no members, so here's a static instance.
*/
public static final MarkovInequality INSTANCE =
new MarkovInequality();
/** Creates a new instance of MarkovInequality */
public MarkovInequality()
{
super();
}
/**
* Computes the Markov Inequality Bound for the given data at the
* given confidence level. Answers the question: what range of values
* can I expect at least "confidence" percent of the time?
* 1-confidence = Pr{abs(a)>=a} <= mean(abs(data))/a ->
* Pr{ -a <= X <= a } >= confidence
* @param data
* Values drawn from the underlying distribution
* @param confidence
* Confidence value to find the range of values for
* @return
* ConfidenceInterval describing the worst-case range that
* ANY random variable can take at the given confidence value
*/
public ConfidenceInterval computeConfidenceInterval(
final Collection<Double> data,
final double confidence)
{
double sampleMean = 0.0;
for( Double value : data )
{
sampleMean += Math.abs(value);
}
if( !data.isEmpty() )
{
sampleMean /= data.size();
}
return MarkovInequality.computeConfidenceInterval(
sampleMean, data.size(), confidence );
}
/**
* Computes the Markov Inequality Bound for the given data at the
* given confidence level. Answers the question: what range of values
* can I expect at least "confidence" percent of the time?
* 1-confidence = Pr{abs(a)>=a} <= mean(abs(data))/a ->
* Pr{ -a <= X <= a } >= confidence
*
* @param sampleMean
* Sample mean of the underlying data
* @param numSamples
* Number of samples used in computing the mean
* @param confidence
* Confidence value to find the range of values for
* @return
* ConfidenceInterval describing the worst-case range that
* ANY random variable can take at the given confidence value
*/
public static ConfidenceInterval computeConfidenceInterval(
final double sampleMean,
final int numSamples,
final double confidence )
{
if( (confidence <= 0.0) ||
(confidence > 1.0) )
{
throw new IllegalArgumentException(
"Confidence must be 0 < confidence <= 1" );
}
// 1-confidence = Pr{ abs(X) >= a } <= mean(abs(X)) / a
double a;
if( confidence < 1.0 )
{
a = sampleMean / (1-confidence);
}
else
{
a = Double.POSITIVE_INFINITY;
}
return new ConfidenceInterval(
sampleMean, -a, a, confidence, numSamples );
}
@Override
public ConfidenceInterval computeConfidenceInterval(
double mean,
double variance,
int numSamples,
double confidence)
{
return MarkovInequality.computeConfidenceInterval(
mean, numSamples, confidence);
}
}