umontreal.iro.lecuyer.probdist

Class RayleighDist

java.lang.Object

umontreal.iro.lecuyer.probdist.ContinuousDistribution

umontreal.iro.lecuyer.probdist.RayleighDist

All Implemented Interfaces:

Distribution

public class RayleighDist
extends ContinuousDistribution

This class extends the class ContinuousDistribution for the Rayleigh distribution with location parameter a, and scale parameter β > 0. The density function is

f (x) = (x-a)/β² e^{-(x-a)2/(2β2)} for x >= a,

and f (x) = 0 for x < a. The distribution function is

F(x) = 1 - e^{-(x-a)2/(2β2)} for x >= a,

and the inverse distribution function is

F^-1(u) = x = a + β(-2ln(1-u))^1/2 for 0 <= u <= 1.

Field Summary

Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

decPrec

Constructor Summary

Constructors

Constructor and Description
RayleighDist(double beta) Constructs a RayleighDist object with parameters a = 0 and β = beta.
RayleighDist(double a, double beta) Constructs a RayleighDist object with parameters a = a, and β = beta.

Method Summary

Modifier and Type	Method and Description
double	barF(double x) Returns the complementary distribution function.
static double	barF(double beta, double x) Same as barF (0, beta, x).
static double	barF(double a, double beta, double x) Computes the complementary distribution function.
double	cdf(double x) Returns the distribution function F(x).
static double	cdf(double beta, double x) Same as cdf (0, beta, x).
static double	cdf(double a, double beta, double x) Computes the distribution function.
double	density(double x) Returns f (x), the density evaluated at x.
static double	density(double beta, double x) Same as density (0, beta, x).
static double	density(double a, double beta, double x) Computes the density function.
double	getA() Returns the parameter a.
static RayleighDist	getInstanceFromMLE(double[] x, int n, double a) Creates a new instance of a Rayleigh distribution with parameters a and hat(β).
double	getMean() Returns the mean.
static double	getMean(double a, double beta) Returns the mean a + β(π/2)1/2 of the Rayleigh distribution with parameters a and β.
static double[]	getMLE(double[] x, int n, double a) Estimates the parameter β of the Rayleigh distribution using the maximum likelihood method, assuming that a is known, from the n observations x[i], i = 0, 1,…, n - 1.
double[]	getParams() Return an array containing the parameters of the current distribution in the order: [a, β].
double	getSigma() Returns the parameter β.
double	getStandardDeviation() Returns the standard deviation.
static double	getStandardDeviation(double beta) Returns the standard deviation β(2 - π/2)1/2 of the Rayleigh distribution with parameter β.
double	getVariance() Returns the variance.
static double	getVariance(double beta) Returns the variance of the Rayleigh distribution with parameter β.
double	inverseF(double u) Returns the inverse distribution function x = F-1(u).
static double	inverseF(double beta, double u) Same as inverseF (0, beta, u).
static double	inverseF(double a, double beta, double u) Computes the inverse of the distribution function.
void	setParams(double a, double beta) Sets the parameters a and β for this object.
java.lang.String	toString()

Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

getXinf, getXsup, inverseBisection, inverseBrent, setXinf, setXsup

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

Constructor Detail

RayleighDist

public RayleighDist(double beta)

Constructs a RayleighDist object with parameters a = 0 and β = beta.

RayleighDist

public RayleighDist(double a,
                    double beta)

Constructs a RayleighDist object with parameters a = a, and β = beta.

Method Detail

density

public double density(double x)

Description copied from class: ContinuousDistribution

Returns f (x), the density evaluated at x.

Specified by:

density in class ContinuousDistribution

Parameters:

x - value at which the density is evaluated

Returns:

density function evaluated at x

cdf

public double cdf(double x)

Description copied from interface: Distribution

Returns the distribution function F(x).

Parameters:

x - value at which the distribution function is evaluated

Returns:

distribution function evaluated at x

barF

public double barF(double x)

Description copied from class: ContinuousDistribution

Returns the complementary distribution function. The default implementation computes bar(F)(x) = 1 - F(x).

Specified by:

barF in interface Distribution

Overrides:

barF in class ContinuousDistribution

Parameters:

x - value at which the complementary distribution function is evaluated

Returns:

complementary distribution function evaluated at x

inverseF

public double inverseF(double u)

Description copied from class: ContinuousDistribution

Returns the inverse distribution function x = F^-1(u). Restrictions: u∈[0, 1].

Specified by:

inverseF in interface Distribution

Overrides:

inverseF in class ContinuousDistribution

Parameters:

u - value at which the inverse distribution function is evaluated

Returns:

the inverse distribution function evaluated at u

getMean

public double getMean()

Description copied from class: ContinuousDistribution

Returns the mean.

Specified by:

getMean in interface Distribution

Overrides:

getMean in class ContinuousDistribution

Returns:

the mean

getVariance

public double getVariance()

Description copied from class: ContinuousDistribution

Returns the variance.

Specified by:

getVariance in interface Distribution

Overrides:

getVariance in class ContinuousDistribution

Returns:

the variance

getStandardDeviation

public double getStandardDeviation()

Description copied from class: ContinuousDistribution

Returns the standard deviation.

Specified by:

getStandardDeviation in interface Distribution

Overrides:

getStandardDeviation in class ContinuousDistribution

Returns:

the standard deviation

density

public static double density(double a,
                             double beta,
                             double x)

Computes the density function.

Parameters:

a - the location parameter

beta - the scale parameter

x - the value at which the density is evaluated

Returns:

the density function

density

public static double density(double beta,
                             double x)

Same as density (0, beta, x).

Parameters:

beta - the scale parameter

x - the value at which the density is evaluated

Returns:

returns the density function

cdf

public static double cdf(double a,
                         double beta,
                         double x)

Computes the distribution function.

Parameters:

a - the location parameter

beta - the scale parameter

x - the value at which the distribution is evaluated

Returns:

returns the distribution function

cdf

public static double cdf(double beta,
                         double x)

Same as cdf (0, beta, x).

Parameters:

beta - the scale parameter

x - the value at which the distribution is evaluated

Returns:

returns the distribution function

barF

public static double barF(double a,
                          double beta,
                          double x)

Computes the complementary distribution function.

Parameters:

a - the location parameter

beta - the scale parameter

x - the value at which the complementary distribution is evaluated

Returns:

returns the complementary distribution function

barF

public static double barF(double beta,
                          double x)

Same as barF (0, beta, x).

Parameters:

beta - the scale parameter

x - the value at which the complementary distribution is evaluated

Returns:

returns the complementary distribution function

inverseF

public static double inverseF(double a,
                              double beta,
                              double u)

Computes the inverse of the distribution function.

Parameters:

a - the location parameter

beta - the scale parameter

u - the value at which the inverse distribution is evaluated

Returns:

returns the inverse of the distribution function

inverseF

public static double inverseF(double beta,
                              double u)

Same as inverseF (0, beta, u).

Parameters:

beta - the scale parameter

u - the value at which the inverse distribution is evaluated

Returns:

returns the inverse of the distribution function

getMLE

public static double[] getMLE(double[] x,
                              int n,
                              double a)

Estimates the parameter β of the Rayleigh distribution using the maximum likelihood method, assuming that a is known, from the n observations x[i], i = 0, 1,…, n - 1. The estimate is returned in a one-element array: [hat(β)].

Parameters:

x - the list of observations to use to evaluate parameters

n - the number of observations to use to evaluate parameters

a - the location parameter

Returns:

returns the parameter [ hat(β)]

getInstanceFromMLE

public static RayleighDist getInstanceFromMLE(double[] x,
                                              int n,
                                              double a)

Creates a new instance of a Rayleigh distribution with parameters a and hat(β). This last is estimated using the maximum likelihood method based on the n observations x[i], i = 0,…, n - 1.

Parameters:

x - the list of observations to use to evaluate parameters

n - the number of observations to use to evaluate parameters

a - the location parameter

getMean

public static double getMean(double a,
                             double beta)

Returns the mean a + β(π/2)^1/2 of the Rayleigh distribution with parameters a and β.

Parameters:

a - the location parameter

beta - the scale parameter

Returns:

the mean of the Rayleigh distribution

getVariance

public static double getVariance(double beta)

Returns the variance of the Rayleigh distribution with parameter β.

Parameters:

beta - the scale parameter

Returns:

the variance of the Rayleigh distribution

getStandardDeviation

public static double getStandardDeviation(double beta)

Returns the standard deviation β(2 - π/2)^1/2 of the Rayleigh distribution with parameter β.

Parameters:

beta - the scale parameter

Returns:

the standard deviation of the Rayleigh distribution

getA

public double getA()

Returns the parameter a.

Returns:

the location parameter a

getSigma

public double getSigma()

Returns the parameter β.

Returns:

the scale parameter beta

setParams

public void setParams(double a,
                      double beta)

Sets the parameters a and β for this object.

Parameters:

a - the location parameter

beta - the scale parameter

getParams

public double[] getParams()

Return an array containing the parameters of the current distribution in the order: [a, β].

Returns:

[a, β]

toString

public java.lang.String toString()

Overrides:

toString in class java.lang.Object