com.aliasi.stats

Class PoissonDistribution

java.lang.Object

com.aliasi.stats.AbstractDiscreteDistribution

com.aliasi.stats.PoissonDistribution

All Implemented Interfaces:

DiscreteDistribution

Direct Known Subclasses:

PoissonConstant, PoissonEstimator

public abstract class PoissonDistribution
extends AbstractDiscreteDistribution

The PoissonDistribution abstract class is used for calculating Poisson distributions. Poisson distributions are limits of Poisson processes, and are used to model rates of occurrences of events within a fixed period (of time, space, etc.). Poisson distributions are good models of lengths of texts or the rate of occurrence of words in text, as well as many other natural phenomena.

The Poisson distribution is a parametric discrete distribution with a single parameter λ > 0 which is the average rate of occurrence of events in a period. The resulting distribution provides a likelihood for each non-negative number of outcomes. Specifically, the Poisson distribution with rate parameter λ is defined for k > 0 by:

Poissonλ(k) = e-λ λk / k!

Note that this definition produces a properly normalized probability distribution over natural numbers; if λ > 0, then:

Σk >= 0 Poissonλ(k) = 1.0

The expected value of a Poisson distribution is equal to the rate parameter:

E(Poissonλ) = λ

The variance is also equal to the rate parameter:

Var(Poissonλ) =def E([Poissonλ - E(Poissonλ)]2) = λ

Concrete subclasses need only implement the abstract mean() method; the method log2Probability(long) computes the log (base 2) of the Poisson probability estimate for a given number of outcomes in terms of the value of the rate parameter lambda(). Logarithms are used to prevent over- and underflow in calculations.

For more information, see:

• Eric W. Weisstein. Poisson Distribution. From MathWorld--A Wolfram Web Resource.

Since:

LingPipe2.0

Version:

2.0

Author:

Bob Carpenter

Constructor Summary

Constructors

Modifier	Constructor and Description
protected	PoissonDistribution() Construct an abstract Poisson distribution.

Method Summary

Methods

Modifier and Type	Method and Description
double	log2Probability(long outcome) Returns the log (base 2) probability estimate in this Poisson distribution for the specified outcome.
abstract double	mean() Returns the mean of this Poisson distribution, which is equal to the rate parameter λ.
long	minOutcome() Returns the minimum outcome with non-zero probability, 0.
double	probability(long outcome) Returns the probability estimate in this Poisson distribution for the specified outcome.
double	variance() Returns the variance of this Poisson distribution, which is equal to the mean.

Methods inherited from class com.aliasi.stats.AbstractDiscreteDistribution

cumulativeProbability, cumulativeProbabilityGreater, cumulativeProbabilityLess, entropy, maxOutcome

Methods inherited from class java.lang.Object

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

Constructor Detail

PoissonDistribution

protected PoissonDistribution()

Construct an abstract Poisson distribution.

Method Detail

mean

public abstract double mean()

Returns the mean of this Poisson distribution, which is equal to the rate parameter λ. Concrete implementations are responsible for ensuring that the mean is positive and finite.

Specified by:

mean in interface DiscreteDistribution

Overrides:

mean in class AbstractDiscreteDistribution

Returns:

The mean of this distribution.

variance

public double variance()

Returns the variance of this Poisson distribution, which is equal to the mean.

Specified by:

variance in interface DiscreteDistribution

Overrides:

variance in class AbstractDiscreteDistribution

Returns:

The variance of this distribution.

minOutcome

public long minOutcome()

Returns the minimum outcome with non-zero probability, 0.

Specified by:

minOutcome in interface DiscreteDistribution

Overrides:

minOutcome in class AbstractDiscreteDistribution

Returns:

Zero.

log2Probability

public final double log2Probability(long outcome)

Returns the log (base 2) probability estimate in this Poisson distribution for the specified outcome. This method will throw an illegal state exception if the mean implementation returns a non-positive number. If the outcome is negative, the result will be negative-infinity.

Specified by:

log2Probability in interface DiscreteDistribution

Overrides:

log2Probability in class AbstractDiscreteDistribution

Parameters:

outcome - The outcome being estimated.

Returns:

The log (base 2) probability of finding the specified number of outcomes given this distribution's rate parameter.

Throws:

IllegalStateException - if the mean is not a positive finite value.

probability

public final double probability(long outcome)

Returns the probability estimate in this Poisson distribution for the specified outcome. Note that if the outcome is negative, the result will be zero.

Specified by:

probability in interface DiscreteDistribution

Specified by:

probability in class AbstractDiscreteDistribution

Parameters:

outcome - The outcome whose probability is returned.

Returns:

The log (base 2) probability of finding the specified number of outcomes given this distribution's rate parameter.

Throws:

IllegalStateException - If the mean is not a positive finite value.