SSJ
V. labo.

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

RayleighDist

This class extends the class ContinuousDistribution for the Rayleigh distribution[#!tEVA00a!#] with location parameter a, and scale parameter β > 0. The density function is

f (x) = (x-a)/β2 e-(x-a)2/(2β2)        for xa,$\displaystyle \eqlabel$eq:frayleigh

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

F(x) = 1 - e-(x-a)2/(2β2)        for xa,[tex2html_wrap_indisplay341]eq:Frayleigh

and the inverse distribution function is

F-1(u) = x = a + β$\displaystyle \sqrt{{-2ln(1-u)}}$        for 0≤u≤1.[tex2html_wrap_indisplay344]eq:Invrayleigh


Field Summary
 
Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
decPrec
 
Constructor Summary
RayleighDist(double beta)
          .
RayleighDist(double a, double beta)
          .
 
Method Summary
 double barF(double x)
          Returns the complementary distribution function.
static double barF(double beta, double x)
          .
static double barF(double a, double beta, double x)
          .
 double cdf(double x)
          .
static double cdf(double beta, double x)
          .
static double cdf(double a, double beta, double x)
          .
 double density(double x)
          Returns f (x), the density evaluated at x.
static double density(double beta, double x)
          .
static double density(double a, double beta, double x)
          .
 double getA()
          .
static RayleighDist getInstanceFromMLE(double[] x, int n, double a)
          .
 double getMean()
          Returns the mean.
static double getMean(double a, double beta)
          .
static double[] getMLE(double[] x, int n, double a)
          .
 double[] getParams()
          .
 double getSigma()
          .
 double getStandardDeviation()
          Returns the standard deviation.
static double getStandardDeviation(double beta)
          .
 double getVariance()
          Returns the variance.
static double getVariance(double beta)
          .
 double inverseF(double u)
          Returns the inverse distribution function x = F-1(u).
static double inverseF(double beta, double u)
          .
static double inverseF(double a, double beta, double u)
          .
 void setParams(double a, double beta)
          .
 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)
. \begin{tabb}Constructs a \texttt{RayleighDist} object with parameters
 $a = 0$\ and $\beta$\ = \texttt{beta}.
 \end{tabb}


RayleighDist

public RayleighDist(double a,
                    double beta)
. \begin{tabb}Constructs a \texttt{RayleighDist} object with parameters
 $a =$\ \texttt{a}, and $\beta$\ = \texttt{beta}.
 \end{tabb}

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
. \begin{tabb}Returns the distribution function $F(x)$.
 \end{tabb}
xvalue at which the distribution function is evaluated 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)
. \begin{tabb}Computes the density function.
 \end{tabb}
athe location parameter betathe scale parameter xthe value at which the density is evaluated the density function


density

public static double density(double beta,
                             double x)
. \begin{tabb}Same as \texttt{density (0, beta, x)}.
 \end{tabb}
betathe scale parameter xthe value at which the density is evaluated returns the density function


cdf

public static double cdf(double a,
                         double beta,
                         double x)
. \begin{tabb}
 Computes the distribution function.
 \end{tabb}
athe location parameter betathe scale parameter xthe value at which the distribution is evaluated returns the distribution function


cdf

public static double cdf(double beta,
                         double x)
. \begin{tabb}Same as \texttt{cdf (0, beta, x)}.
 \end{tabb}
betathe scale parameter xthe value at which the distribution is evaluated returns the distribution function


barF

public static double barF(double a,
                          double beta,
                          double x)
. \begin{tabb}
 Computes the complementary distribution function.
 \end{tabb}
athe location parameter betathe scale parameter xthe value at which the complementary distribution is evaluated returns the complementary distribution function


barF

public static double barF(double beta,
                          double x)
. \begin{tabb}Same as \texttt{barF (0, beta, x)}.
 \end{tabb}
betathe scale parameter xthe value at which the complementary distribution is evaluated returns the complementary distribution function


inverseF

public static double inverseF(double a,
                              double beta,
                              double u)
. \begin{tabb}
 Computes the inverse of the distribution function.
 \end{tabb}
athe location parameter betathe scale parameter uthe value at which the inverse distribution is evaluated returns the inverse of the distribution function


inverseF

public static double inverseF(double beta,
                              double u)
. \begin{tabb}Same as \texttt{inverseF (0, beta, u)}.
 \end{tabb}
betathe scale parameter uthe value at which the inverse distribution is evaluated returns the inverse of the distribution function


getMLE

public static double[] getMLE(double[] x,
                              int n,
                              double a)
. \begin{tabb}
 Estimates the parameter $\beta$\ of the Rayleigh distribution
 usi...
 ...rt{\frac1{2n}\htsum_{i=1}^{n} x_i^2}
 \end{displaymath} \end{detailed}\end{tabb}
xthe list of observations to use to evaluate parameters nthe number of observations to use to evaluate parameters athe location parameter returns the parameter [ $ \hat{{\beta}}$]


getInstanceFromMLE

public static RayleighDist getInstanceFromMLE(double[] x,
                                              int n,
                                              double a)
. \begin{tabb}
 Creates a new instance of a Rayleigh distribution with parameters ...
 ... method
 based on the $n$\ observations $x[i]$, $i = 0, \ldots, n-1$.
 \end{tabb}
xthe list of observations to use to evaluate parameters nthe number of observations to use to evaluate parameters athe location parameter


getMean

public static double getMean(double a,
                             double beta)
. \begin{tabb}Returns the mean $a + \beta\sqrt{\pi/2}$\ of the
 Rayleigh distribution with parameters $a$\ and $\beta$.
 \end{tabb}
athe location parameter betathe scale parameter the mean of the Rayleigh distribution


getVariance

public static double getVariance(double beta)
. \begin{tabb}Returns the variance
 of the Rayleigh distribution with parameter $\beta$.
 \end{tabb}
betathe scale parameter the variance of the Rayleigh distribution


getStandardDeviation

public static double getStandardDeviation(double beta)
. \begin{tabb}Returns the standard deviation $\beta\sqrt{2 - \pi/2}$\ of
 the Rayleigh distribution with parameter $\beta$.
 \end{tabb}
betathe scale parameter the standard deviation of the Rayleigh distribution


getA

public double getA()
. \begin{tabb}Returns the parameter $a$.
 \end{tabb}
the location parameter a


getSigma

public double getSigma()
. \begin{tabb}Returns the parameter $\beta$.
 \end{tabb}
the scale parameter beta


setParams

public void setParams(double a,
                      double beta)
. \begin{tabb}Sets the parameters $a$\ and $\beta$\ for this object.
 \end{tabb}
athe location parameter betathe scale parameter


getParams

public double[] getParams()
. \begin{tabb}
 Return an array containing the parameters of the current distribution
 in the order: [$a$, $\beta$].
 \end{tabb}
[a, β]
 


toString

public String toString()
. \begin{hide}
 \par
 \begin{tabb}
 Returns a \texttt{String} containing information...
 ...texttt{String} containing information about the current distribution}
 \end{hide}

Overrides:
toString in class Object

SSJ
V. labo.

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