This class extends the class ContinuousDistribution for the Rayleigh distribution [56] with location parameter. More...

Inheritance diagram for umontreal.ssj.probdist.RayleighDist:

Public Member Functions
	RayleighDist (double beta)
	Constructs a RayleighDist object with parameters \(a = 0\) and.
	RayleighDist (double a, double beta)
	Constructs a RayleighDist object with parameters \(a =\) a, and \(\beta\) = beta.
double	density (double x)
	Returns \(f(x)\), the density evaluated at \(x\).
double	cdf (double x)
	Returns the distribution function \(F(x)\).
double	barF (double x)
	Returns the complementary distribution function.
double	inverseF (double u)
	Returns the inverse distribution function \(x = F^{-1}(u)\).
double	getMean ()
	Returns the mean.
double	getVariance ()
	Returns the variance.
double	getStandardDeviation ()
	Returns the standard deviation.
double	getA ()
	Returns the parameter \(a\).
double	getSigma ()
	Returns the parameter \(\beta\).
void	setParams (double a, double beta)
	Sets the parameters \(a\) and \(\beta\) for this object.
double[]	getParams ()
	Return an array containing the parameters of the current distribution in the order: [ \(a\), \(\beta\)].
String	toString ()
	Returns a String containing information about the current distribution.
Public Member Functions inherited from umontreal.ssj.probdist.ContinuousDistribution
double	inverseBrent (double a, double b, double u, double tol)
	Computes the inverse distribution function \(x = F^{-1}(u)\), using the Brent-Dekker method.
double	inverseBisection (double u)
	Computes and returns the inverse distribution function \(x = F^{-1}(u)\), using bisection.
double	getXinf ()
	Returns \(x_a\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
double	getXsup ()
	Returns \(x_b\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
void	setXinf (double xa)
	Sets the value \(x_a=\) xa, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
void	setXsup (double xb)
	Sets the value \(x_b=\) xb, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).

Static Public Member Functions
static double	density (double a, double beta, double x)
	Computes the density function ( `frayleigh` ).
static double	density (double beta, double x)
	Same as density (0, beta, x).
static double	cdf (double a, double beta, double x)
	Computes the distribution function ( `Frayleigh` ).
static double	cdf (double beta, double x)
	Same as cdf (0, beta, x).
static double	barF (double a, double beta, double x)
	Computes the complementary distribution function.
static double	barF (double beta, double x)
	Same as barF (0, beta, x).
static double	inverseF (double a, double beta, double u)
	Computes the inverse of the distribution function ( `Invrayleigh` ).
static double	inverseF (double beta, double u)
	Same as inverseF (0, beta, u).
static double[]	getMLE (double[] x, int n, double a)
	Estimates the parameter \(\beta\) of the Rayleigh distribution using the maximum likelihood method, assuming that \(a\) is known, from the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\).
static RayleighDist	getInstanceFromMLE (double[] x, int n, double a)
	Creates a new instance of a Rayleigh distribution with parameters.
static double	getMean (double a, double beta)
	Returns the mean \(a + \beta\sqrt{\pi/2}\) of the Rayleigh distribution with parameters \(a\) and \(\beta\).
static double	getVariance (double beta)
	Returns the variance of the Rayleigh distribution with parameter.
static double	getStandardDeviation (double beta)
	Returns the standard deviation \(\beta\sqrt{2 - \pi/2}\) of the Rayleigh distribution with parameter \(\beta\).

Detailed Description

This class extends the class ContinuousDistribution for the Rayleigh distribution [56] with location parameter.

\(a\), and scale parameter \(\beta> 0\). The density function is

\[ f(x) = \frac{(x-a)}{\beta^2} e^{-(x-a)^2/(2\beta^2)} \qquad\mbox{for } x \ge a, \tag{frayleigh} \]

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

\[ F(x) = 1 - e^{-(x - a)^2/(2\beta^2)} \qquad\mbox{for } x \ge a, \tag{Frayleigh} \]

and the inverse distribution function is

\[ F^{-1}(u) = x = a + \beta\sqrt{-2\ln(1-u)} \qquad\mbox{for } 0 \le u \le1. \tag{Invrayleigh} \]

Definition at line 47 of file RayleighDist.java.

Constructor & Destructor Documentation

◆ RayleighDist() [1/2]

umontreal.ssj.probdist.RayleighDist.RayleighDist ( double beta )

Constructs a RayleighDist object with parameters \(a = 0\) and.

\(\beta\) = beta.

Definition at line 56 of file RayleighDist.java.

◆ RayleighDist() [2/2]

umontreal.ssj.probdist.RayleighDist.RayleighDist	(	double	a,
		double	beta )

Constructs a RayleighDist object with parameters \(a =\) a, and \(\beta\) = beta.

Definition at line 64 of file RayleighDist.java.

Member Function Documentation

◆ barF() [1/3]

double umontreal.ssj.probdist.RayleighDist.barF	(	double	a,
		double	beta,
		double	x )

static

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

Definition at line 164 of file RayleighDist.java.

◆ barF() [2/3]

double umontreal.ssj.probdist.RayleighDist.barF	(	double	beta,
		double	x )

static

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

Definition at line 182 of file RayleighDist.java.

◆ barF() [3/3]

double umontreal.ssj.probdist.RayleighDist.barF ( double x )

Returns the complementary distribution function.

The default implementation computes \(\bar{F}(x) = 1 - F(x)\).

Parameters

x	value at which the complementary distribution function is evaluated

Returns: complementary distribution function evaluated at x

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 76 of file RayleighDist.java.

◆ cdf() [1/3]

double umontreal.ssj.probdist.RayleighDist.cdf	(	double	a,
		double	beta,
		double	x )

static

Computes the distribution function ( Frayleigh ).

Parameters

a	the location parameter
beta	the scale parameter
x	the value at which the distribution is evaluated

Returns: returns the distribution function

Definition at line 134 of file RayleighDist.java.

◆ cdf() [2/3]

double umontreal.ssj.probdist.RayleighDist.cdf	(	double	beta,
		double	x )

static

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

Definition at line 152 of file RayleighDist.java.

◆ cdf() [3/3]

double umontreal.ssj.probdist.RayleighDist.cdf ( double x )

Returns the distribution function \(F(x)\).

Parameters

x	value at which the distribution function is evaluated

Returns: distribution function evaluated at x

Implements umontreal.ssj.probdist.Distribution.

Definition at line 72 of file RayleighDist.java.

◆ density() [1/3]

double umontreal.ssj.probdist.RayleighDist.density	(	double	a,
		double	beta,
		double	x )

static

Computes the density function ( frayleigh ).

Parameters

a	the location parameter
beta	the scale parameter
x	the value at which the density is evaluated

Returns: the density function

Definition at line 105 of file RayleighDist.java.

◆ density() [2/3]

double umontreal.ssj.probdist.RayleighDist.density	(	double	beta,
		double	x )

static

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

Definition at line 121 of file RayleighDist.java.

◆ density() [3/3]

double umontreal.ssj.probdist.RayleighDist.density ( double x )

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

Parameters

x	value at which the density is evaluated

Returns: density function evaluated at x

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 68 of file RayleighDist.java.

◆ getA()

double umontreal.ssj.probdist.RayleighDist.getA ( )

Returns the parameter \(a\).

Returns: the location parameter \(a\)

Definition at line 303 of file RayleighDist.java.

◆ getInstanceFromMLE()

RayleighDist umontreal.ssj.probdist.RayleighDist.getInstanceFromMLE	(	double[]	x,
		int	n,
		double	a )

static

Creates a new instance of a Rayleigh distribution with parameters.

\(a\) and \(\hat{\beta}\). 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

Definition at line 256 of file RayleighDist.java.

◆ getMean() [1/2]

double umontreal.ssj.probdist.RayleighDist.getMean ( )

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 84 of file RayleighDist.java.

◆ getMean() [2/2]

double umontreal.ssj.probdist.RayleighDist.getMean	(	double	a,
		double	beta )

static

Returns the mean \(a + \beta\sqrt{\pi/2}\) of the Rayleigh distribution with parameters \(a\) and \(\beta\).

Parameters

a	the location parameter
beta	the scale parameter

Returns: the mean of the Rayleigh distribution

Definition at line 269 of file RayleighDist.java.

◆ getMLE()

double[] umontreal.ssj.probdist.RayleighDist.getMLE	(	double[]	x,
		int	n,
		double	a )

static

Estimates the parameter \(\beta\) 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: [ \(\hat{\beta}\)]. The maximum likelihood estimator is the value

\(\hat{\beta}\) that satisfies the equation

\[ \hat{\beta} = \sqrt{\frac{1}{2n}\sum_{i=1}^n x_i^2} \]

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 [ \(\hat{\beta}\)]

Definition at line 233 of file RayleighDist.java.

◆ getParams()

double[] umontreal.ssj.probdist.RayleighDist.getParams ( )

Return an array containing the parameters of the current distribution in the order: [ \(a\), \(\beta\)].

Returns: [ \(a\), \(\beta\)]

Implements umontreal.ssj.probdist.Distribution.

Definition at line 336 of file RayleighDist.java.

◆ getSigma()

double umontreal.ssj.probdist.RayleighDist.getSigma ( )

Returns the parameter \(\beta\).

Returns: the scale parameter \(beta\)

Definition at line 312 of file RayleighDist.java.

◆ getStandardDeviation() [1/2]

double umontreal.ssj.probdist.RayleighDist.getStandardDeviation ( )

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 92 of file RayleighDist.java.

◆ getStandardDeviation() [2/2]

double umontreal.ssj.probdist.RayleighDist.getStandardDeviation ( double beta )

static

Returns the standard deviation \(\beta\sqrt{2 - \pi/2}\) of the Rayleigh distribution with parameter \(\beta\).

Parameters

beta	the scale parameter

Returns: the standard deviation of the Rayleigh distribution

Definition at line 294 of file RayleighDist.java.

◆ getVariance() [1/2]

double umontreal.ssj.probdist.RayleighDist.getVariance ( )

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 88 of file RayleighDist.java.

◆ getVariance() [2/2]

double umontreal.ssj.probdist.RayleighDist.getVariance ( double beta )

static

Returns the variance of the Rayleigh distribution with parameter.

\(\beta\).

Parameters

beta	the scale parameter

Returns: the variance of the Rayleigh distribution

Definition at line 281 of file RayleighDist.java.

◆ inverseF() [1/3]

double umontreal.ssj.probdist.RayleighDist.inverseF	(	double	a,
		double	beta,
		double	u )

static

Computes the inverse of the distribution function ( Invrayleigh ).

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

Definition at line 195 of file RayleighDist.java.

◆ inverseF() [2/3]

double umontreal.ssj.probdist.RayleighDist.inverseF	(	double	beta,
		double	u )

static

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

Definition at line 215 of file RayleighDist.java.

◆ inverseF() [3/3]

double umontreal.ssj.probdist.RayleighDist.inverseF ( double u )

Returns the inverse distribution function \(x = F^{-1}(u)\).

Restrictions: \(u \in[0,1]\).

Parameters

u	value at which the inverse distribution function is evaluated

Returns: the inverse distribution function evaluated at u

Exceptions

IllegalArgumentException if \(u\) is not in the interval \([0,1]\)

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 80 of file RayleighDist.java.

◆ setParams()

void umontreal.ssj.probdist.RayleighDist.setParams	(	double	a,
		double	beta )

Sets the parameters \(a\) and \(\beta\) for this object.

Parameters

a	the location parameter
beta	the scale parameter

Definition at line 322 of file RayleighDist.java.

◆ toString()

String umontreal.ssj.probdist.RayleighDist.toString ( )

Returns a String containing information about the current distribution.

Returns: a String containing information about the current distribution

Definition at line 346 of file RayleighDist.java.

The documentation for this class was generated from the following file:

src/main/java/umontreal/ssj/probdist/RayleighDist.java

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ RayleighDist() [1/2]

◆ RayleighDist() [2/2]

Member Function Documentation

◆ barF() [1/3]

◆ barF() [2/3]

◆ barF() [3/3]

◆ cdf() [1/3]

◆ cdf() [2/3]

◆ cdf() [3/3]

◆ density() [1/3]

◆ density() [2/3]

◆ density() [3/3]

◆ getA()

◆ getInstanceFromMLE()

◆ getMean() [1/2]

◆ getMean() [2/2]

◆ getMLE()

◆ getParams()

◆ getSigma()

◆ getStandardDeviation() [1/2]

◆ getStandardDeviation() [2/2]

◆ getVariance() [1/2]

◆ getVariance() [2/2]

◆ inverseF() [1/3]

◆ inverseF() [2/3]

◆ inverseF() [3/3]

◆ setParams()

◆ toString()