de.lmu.ifi.dbs.elki.math.statistics.distribution

Class PoissonDistribution

java.lang.Object

de.lmu.ifi.dbs.elki.math.statistics.distribution.AbstractDistribution

de.lmu.ifi.dbs.elki.math.statistics.distribution.PoissonDistribution

All Implemented Interfaces:

Distribution

public class PoissonDistribution
extends AbstractDistribution

INCOMPLETE implementation of the poisson distribution.

TODO: continue implementing, CDF, invcdf and nextRandom are missing

References:

Catherine Loader
Fast and Accurate Computation of Binomial Probabilities.

Since:

0.5.0

Author:

Erich Schubert

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static class	PoissonDistribution.Parameterizer Parameterization class

Field Summary

Fields

Modifier and Type	Field and Description
private int	n Number of tries
private double	p Success probability
private static double	S0 Stirling error constants: 1./12
private static double	S1 Stirling error constants: 1./360
private static double	S2 Stirling error constants: 1./1260
private static double	S3 Stirling error constants: 1./1680
private static double	S4 Stirling error constants: 1./1188
private static double[]	STIRLING_EXACT_ERROR Exact table values for n <= 15 in steps of 0.5 sfe[n] = ln( (n!

Fields inherited from class de.lmu.ifi.dbs.elki.math.statistics.distribution.AbstractDistribution

random

Constructor Summary

Constructors

Constructor and Description
PoissonDistribution(int n, double p) Constructor.
PoissonDistribution(int n, double p, java.util.Random random) Constructor.
PoissonDistribution(int n, double p, RandomFactory random) Constructor.

Method Summary

Modifier and Type	Method and Description
double	cdf(double val) Return the cumulative density function at the given value.
private static double	devianceTerm(double x, double np) Evaluate the deviance term of the saddle point approximation.
double	logpdf(double x) Return the log density of an existing value
static double	logpmf(double x, int n, double p) Poisson probability mass function (PMF) for integer values.
static double	logpoissonPDFm1(double x_plus_1, double lambda) Compute the poisson distribution PDF with an offset of + 1 log pdf(x_plus_1 - 1, lambda)
double	nextRandom() Generate a new random value
double	pdf(double x) Return the density of an existing value
static double	pmf(double x, int n, double p) Poisson probability mass function (PMF) for integer values.
double	pmf(int x) Poisson probability mass function (PMF) for integer values.
static double	poissonPDFm1(double x_plus_1, double lambda) Compute the poisson distribution PDF with an offset of + 1 pdf(x_plus_1 - 1, lambda)
double	quantile(double val) Quantile aka probit (for normal) aka inverse CDF (invcdf, cdf^-1) function.
static double	rawLogProbability(double x, double lambda) Poisson distribution probability, but also for non-integer arguments.
static double	rawProbability(double x, double lambda) Poisson distribution probability, but also for non-integer arguments.
private static double	stirlingError(double n) Calculates the Stirling Error stirlerr(n) = ln(n!)
private static double	stirlingError(int n) Calculates the Stirling Error stirlerr(n) = ln(n!)
java.lang.String	toString() Describe the distribution

Methods inherited from class java.lang.Object

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

Field Detail

n

private int n

Number of tries

p

private double p

Success probability

S0

private static final double S0

Stirling error constants: 1./12

See Also:

Constant Field Values

S1

private static final double S1

Stirling error constants: 1./360

See Also:

Constant Field Values

S2

private static final double S2

Stirling error constants: 1./1260

See Also:

Constant Field Values

S3

private static final double S3

Stirling error constants: 1./1680

See Also:

Constant Field Values

S4

private static final double S4

Stirling error constants: 1./1188

See Also:

Constant Field Values

STIRLING_EXACT_ERROR

private static final double[] STIRLING_EXACT_ERROR

Exact table values for n <= 15 in steps of 0.5

sfe[n] = ln( (n!*e^n)/((n^n)*sqrt(2*pi*n)) )

Constructor Detail

PoissonDistribution

public PoissonDistribution(int n,
                           double p)

Constructor.

Parameters:

n - Number of tries

p - Success probability

PoissonDistribution

public PoissonDistribution(int n,
                           double p,
                           java.util.Random random)

Constructor.

Parameters:

n - Number of tries

p - Success probability

random - Random generator

PoissonDistribution

public PoissonDistribution(int n,
                           double p,
                           RandomFactory random)

Constructor.

Parameters:

n - Number of tries

p - Success probability

random - Random generator

Method Detail

pmf

public double pmf(int x)

Poisson probability mass function (PMF) for integer values.

Parameters:

x - integer values

Returns:

Probability

pdf

public double pdf(double x)

Description copied from interface: Distribution

Return the density of an existing value

Parameters:

x - existing value

Returns:

distribution density

logpdf

public double logpdf(double x)

Description copied from interface: Distribution

Return the log density of an existing value

Parameters:

x - existing value

Returns:

log distribution density

pmf

@Reference(title="Fast and accurate computation of binomial probabilities",
           authors="C. Loader",
           booktitle="",
           url="http://projects.scipy.org/scipy/raw-attachment/ticket/620/loader2000Fast.pdf",
           bibkey="web/Loader00")
public static double pmf(double x,
                                                                                                                                                                                                                                                                                               int n,
                                                                                                                                                                                                                                                                                               double p)

Poisson probability mass function (PMF) for integer values.

Parameters:

x - integer values

Returns:

Probability

logpmf

public static double logpmf(double x,
                            int n,
                            double p)

Poisson probability mass function (PMF) for integer values.

Parameters:

x - integer values

Returns:

Probability

cdf

public double cdf(double val)

Description copied from interface: Distribution

Return the cumulative density function at the given value.

Parameters:

val - existing value

Returns:

cumulative density

quantile

public double quantile(double val)

Description copied from interface: Distribution

Quantile aka probit (for normal) aka inverse CDF (invcdf, cdf^-1) function.

Parameters:

val - Quantile to find

Returns:

Quantile position

nextRandom

public double nextRandom()

Description copied from interface: Distribution

Generate a new random value

Specified by:

nextRandom in interface Distribution

Overrides:

nextRandom in class AbstractDistribution

Returns:

new random value

poissonPDFm1

public static double poissonPDFm1(double x_plus_1,
                                  double lambda)

Compute the poisson distribution PDF with an offset of + 1

pdf(x_plus_1 - 1, lambda)

Parameters:

x_plus_1 - x+1

lambda - Lambda

Returns:

pdf

logpoissonPDFm1

public static double logpoissonPDFm1(double x_plus_1,
                                     double lambda)

Compute the poisson distribution PDF with an offset of + 1

log pdf(x_plus_1 - 1, lambda)

Parameters:

x_plus_1 - x+1

lambda - Lambda

Returns:

pdf

stirlingError

@Reference(title="Fast and accurate computation of binomial probabilities",
           authors="C. Loader",
           booktitle="",
           url="http://projects.scipy.org/scipy/raw-attachment/ticket/620/loader2000Fast.pdf",
           bibkey="web/Loader00")
private static double stirlingError(int n)

Calculates the Stirling Error

stirlerr(n) = ln(n!) - ln(sqrt(2*pi*n)*(n/e)^n)

Parameters:

n - Parameter n

Returns:

Stirling error

stirlingError

@Reference(title="Fast and accurate computation of binomial probabilities",
           authors="C. Loader",
           booktitle="",
           url="http://projects.scipy.org/scipy/raw-attachment/ticket/620/loader2000Fast.pdf",
           bibkey="web/Loader00")
private static double stirlingError(double n)

Calculates the Stirling Error

stirlerr(n) = ln(n!) - ln(sqrt(2*pi*n)*(n/e)^n)

Parameters:

n - Parameter n

Returns:

Stirling error

devianceTerm

@Reference(title="Fast and accurate computation of binomial probabilities",
           authors="C. Loader",
           booktitle="",
           url="http://projects.scipy.org/scipy/raw-attachment/ticket/620/loader2000Fast.pdf",
           bibkey="web/Loader00")
private static double devianceTerm(double x,
                                                                                                                                                                                                                                                                                                         double np)

Evaluate the deviance term of the saddle point approximation.

bd0(x,np) = x*ln(x/np)+np-x

Parameters:

x - probability density function position

np - product of trials and success probability: n*p

Returns:

Deviance term

rawProbability

public static double rawProbability(double x,
                                    double lambda)

Poisson distribution probability, but also for non-integer arguments.

lb^x exp(-lb) / x!

Parameters:

x - X

lambda - lambda

Returns:

Poisson distribution probability

rawLogProbability

public static double rawLogProbability(double x,
                                       double lambda)

Poisson distribution probability, but also for non-integer arguments.

lb^x exp(-lb) / x!

Parameters:

x - X

lambda - lambda

Returns:

Poisson distribution probability

toString

public java.lang.String toString()

Description copied from interface: Distribution

Describe the distribution

Specified by:

toString in interface Distribution

Overrides:

toString in class java.lang.Object

Returns:

description