java.lang.Object
- elki.math.statistics.distribution.PoissonDistribution

All Implemented Interfaces:

Distribution
```
public class PoissonDistribution
extends java.lang.Object
implements Distribution
```
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 Description

static class PoissonDistribution.Par
Parameterization class
- Nested classes/interfaces inherited from interface elki.math.statistics.distribution.Distribution
  Distribution.Parameterizer

Field Summary

Fields
Modifier and Type	Field	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

Constructor Summary

Constructors
Constructor Description

PoissonDistribution(int n, double p)
Constructor.

Method Summary

All Methods Static Methods Instance Methods Concrete Methods
Modifier and Type	Method	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
`double`	`nextRandom(java.util.Random random)`	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
`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
`private static double`	`stirlingError(int n)`	Calculates the Stirling Error
`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
- 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
    
    Specified by:
    
    pdf in interface Distribution
    
    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
    
    Specified by:
    
    logpdf in interface Distribution
    
    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.
    
    Specified by:
    
    cdf in interface Distribution
    
    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.
    
    Specified by:
    
    quantile in interface Distribution
    
    Parameters:
    
    val - Quantile to find
    
    Returns:
    
    Quantile position
  - nextRandom
```
public double nextRandom(java.util.Random random)
```
    Description copied from interface: Distribution
    
    Generate a new random value
    
    Specified by:
    
    nextRandom in interface Distribution
    
    Parameters:
    
    random - Random number generator
    
    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

Class PoissonDistribution

Nested Class Summary

Nested classes/interfaces inherited from interface elki.math.statistics.distribution.Distribution

Field Summary

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Field Detail

n

p

S0

S1

S2

S3

S4

STIRLING_EXACT_ERROR

Constructor Detail

PoissonDistribution

Method Detail

pmf

pdf

logpdf

pmf

logpmf

cdf

quantile

nextRandom

poissonPDFm1

logpoissonPDFm1

stirlingError

stirlingError

devianceTerm

rawProbability

rawLogProbability

toString