SSJ
V. 2.6.

umontreal.iro.lecuyer.probdist
Class RayleighDist

java.lang.Object
  extended by umontreal.iro.lecuyer.probdist.ContinuousDistribution
      extended by 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
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
 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.
 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 String toString()
Overrides:
toString in class Object

SSJ
V. 2.6.

To submit a bug or ask questions, send an e-mail to Pierre L'Ecuyer.