gov.sandia.cognition.statistics.method

Class MarkovInequality

java.lang.Object

gov.sandia.cognition.util.AbstractCloneableSerializable

gov.sandia.cognition.statistics.method.MarkovInequality

All Implemented Interfaces:

ConfidenceIntervalEvaluator<java.util.Collection<java.lang.Double>>, CloneableSerializable, java.io.Serializable, java.lang.Cloneable

@PublicationReference(author="Wikipedia",
                      title="Markov\'s Inequality",
                      type=WebPage,
                      year=2009,
                      url="http://en.wikipedia.org/wiki/Markov%27s_inequality")
public class MarkovInequality
extends AbstractCloneableSerializable
implements ConfidenceIntervalEvaluator<java.util.Collection<java.lang.Double>>

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

Since:

2.0

Author:

Kevin R. Dixon

See Also:

Serialized Form

Field Summary

Fields

Modifier and Type	Field and Description
static MarkovInequality	INSTANCE This class has no members, so here's a static instance.

Constructor Summary

Constructors

Constructor and Description
MarkovInequality() Creates a new instance of MarkovInequality

Method Summary

Modifier and Type	Method and Description
ConfidenceInterval	computeConfidenceInterval(java.util.Collection<java.lang.Double> data, double confidence) Computes the Markov Inequality Bound for the given data at the given confidence level.
ConfidenceInterval	computeConfidenceInterval(double mean, double variance, int numSamples, double confidence) Computes the confidence interval given the mean and variance of the samples, number of samples, and corresponding confidence interval
static ConfidenceInterval	computeConfidenceInterval(double sampleMean, int numSamples, double confidence) Computes the Markov Inequality Bound for the given data at the given confidence level.

Methods inherited from class gov.sandia.cognition.util.AbstractCloneableSerializable

clone

Methods inherited from class java.lang.Object

equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

INSTANCE

public static final MarkovInequality INSTANCE

This class has no members, so here's a static instance.

Constructor Detail

MarkovInequality

public MarkovInequality()

Creates a new instance of MarkovInequality

Method Detail

computeConfidenceInterval

public ConfidenceInterval computeConfidenceInterval(java.util.Collection<java.lang.Double> data,
                                                    double 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

Specified by:

computeConfidenceInterval in interface ConfidenceIntervalEvaluator<java.util.Collection<java.lang.Double>>

Parameters:

data - Values drawn from the underlying distribution

confidence - Confidence value to find the range of values for

Returns:

ConfidenceInterval describing the worst-case range that ANY random variable can take at the given confidence value

computeConfidenceInterval

public static ConfidenceInterval computeConfidenceInterval(double sampleMean,
                                                           int numSamples,
                                                           double 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

Parameters:

sampleMean - Sample mean of the underlying data

numSamples - Number of samples used in computing the mean

confidence - Confidence value to find the range of values for

Returns:

ConfidenceInterval describing the worst-case range that ANY random variable can take at the given confidence value

computeConfidenceInterval

public ConfidenceInterval computeConfidenceInterval(double mean,
                                                    double variance,
                                                    int numSamples,
                                                    double confidence)

Description copied from interface: ConfidenceIntervalEvaluator

Computes the confidence interval given the mean and variance of the samples, number of samples, and corresponding confidence interval

Specified by:

computeConfidenceInterval in interface ConfidenceIntervalEvaluator<java.util.Collection<java.lang.Double>>

Parameters:

mean - Mean of the distribution.

variance - Variance of the distribution.

numSamples - Number of samples in the underlying data

confidence - Confidence value to assume for the ConfidenceInterval

Returns:

ConfidenceInterval capturing the range of the mean of the data at the desired level of confidence