RayleighDist (Java Libraries for Stochastic Simulation)

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

Methods
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

Class RayleighDist

Field Summary

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

Constructor Summary

Method Summary

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

Methods inherited from class java.lang.Object

Constructor Detail

RayleighDist

RayleighDist

Method Detail

density

cdf

barF

inverseF

getMean

getVariance

getStandardDeviation

density

density

cdf

cdf

barF

barF

inverseF

inverseF

getMLE

getInstanceFromMLE

getMean

getVariance

getStandardDeviation

getA

getSigma

setParams

getParams

toString