LaplaceDist (Java Libraries for Stochastic Simulation)

java.lang.Object
- umontreal.iro.lecuyer.probdist.ContinuousDistribution
- - umontreal.iro.lecuyer.probdist.LaplaceDist

All Implemented Interfaces:

Distribution
```
public class LaplaceDist
extends ContinuousDistribution
```
Extends the class ContinuousDistribution for the Laplace distribution. It has location parameter μ and scale parameter β > 0. The density function is given by

f (x) = e^{-| x-μ|/β}/(2β)         for - ∞ < x < ∞.
The distribution function is

F(x) = (1/2)e^(x-μ)/β          if x <= μ,

F(x) = 1 - (1/2)e^(μ-x)/β          otherwise,

and its inverse is

F^-1(u) = βlog(2u) + μ          if 0 <= u <= 1/2,

F^-1(u) = μ - βlog(2(1 - u))          otherwise.

Field Summary
- Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
  decPrec

Constructor Summary

Constructors
Constructor and Description
`LaplaceDist()` Constructs a `LaplaceDist` object with default parameters μ = 0 and β = 1.
`LaplaceDist(double mu, double beta)` Constructs a `LaplaceDist` object with parameters μ = `mu` 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 mu, double beta, double x)` Computes the Laplace complementary distribution function.
`double`	`cdf(double x)` Returns the distribution function F(x).
`static double`	`cdf(double mu, double beta, double x)` Computes the Laplace distribution function.
`double`	`density(double x)` Returns f (x), the density evaluated at x.
`static double`	`density(double mu, double beta, double x)` Computes the Laplace density function.
`double`	`getBeta()` Returns the parameter β.
`static LaplaceDist`	`getInstanceFromMLE(double[] x, int n)` Creates a new instance of a Laplace distribution with parameters μ and β estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
`static double[]`	`getMaximumLikelihoodEstimate(double[] x, int n)` Deprecated.
`double`	`getMean()` Returns the mean.
`static double`	`getMean(double mu, double beta)` Computes and returns the mean E[X] = μ of the Laplace distribution with parameters μ and β.
`static double[]`	`getMLE(double[] x, int n)` Estimates the parameters (μ, β) of the Laplace distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
`double`	`getMu()` Returns the parameter μ.
`double[]`	`getParams()` Return a table containing the parameters of the current distribution.
`double`	`getPhi()` Deprecated.
`double`	`getStandardDeviation()` Returns the standard deviation.
`static double`	`getStandardDeviation(double mu, double beta)` Computes and returns the standard deviation of the Laplace distribution with parameters μ and β.
`double`	`getTheta()` Deprecated.
`double`	`getVariance()` Returns the variance.
`static double`	`getVariance(double mu, double beta)` Computes and returns the variance Var[X] = 2β² of the Laplace distribution with parameters μ and β.
`double`	`inverseF(double u)` Returns the inverse distribution function x = F^-1(u).
`static double`	`inverseF(double mu, double beta, double u)` Computes the inverse Laplace distribution function.
`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
  - LaplaceDist
```
public LaplaceDist()
```
    Constructs a LaplaceDist object with default parameters μ = 0 and β = 1.
  - LaplaceDist
```
public LaplaceDist(double mu,
           double beta)
```
    Constructs a LaplaceDist object with parameters μ = mu 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 mu,
             double beta,
             double x)
```
    Computes the Laplace density function.
  - cdf
```
public static double cdf(double mu,
         double beta,
         double x)
```
    Computes the Laplace distribution function.
  - barF
```
public static double barF(double mu,
          double beta,
          double x)
```
    Computes the Laplace complementary distribution function.
  - inverseF
```
public static double inverseF(double mu,
              double beta,
              double u)
```
    Computes the inverse Laplace distribution function.
  - getMLE
```
public static double[] getMLE(double[] x,
              int n)
```
    Estimates the parameters (μ, β) of the Laplace distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1. The estimates are returned in a two-element array, in regular order: [μ, β].
    
    Parameters:
    x - the list of observations used to evaluate parameters
    n - the number of observations used to evaluate parameters
    
    Returns:
    returns the parameters [hat(μ), hat(β)]
  - getMaximumLikelihoodEstimate
```
@Deprecated
public static double[] getMaximumLikelihoodEstimate(double[] x,
                                               int n)
```
    Deprecated.
    
    Same as getMLE.
  - getInstanceFromMLE
```
public static LaplaceDist getInstanceFromMLE(double[] x,
                             int n)
```
    Creates a new instance of a Laplace distribution with parameters μ and β estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
    
    Parameters:
    x - the list of observations to use to evaluate parameters
    n - the number of observations to use to evaluate parameters
  - getMean
```
public static double getMean(double mu,
             double beta)
```
    Computes and returns the mean E[X] = μ of the Laplace distribution with parameters μ and β.
    
    Returns:
    the mean of the Laplace distribution E[X] = μ
  - getVariance
```
public static double getVariance(double mu,
                 double beta)
```
    Computes and returns the variance Var[X] = 2β² of the Laplace distribution with parameters μ and β.
    
    Returns:
    the variance of the Laplace distribution Var[X] = 2β²
  - getStandardDeviation
```
public static double getStandardDeviation(double mu,
                          double beta)
```
    Computes and returns the standard deviation of the Laplace distribution with parameters μ and β.
    
    Returns:
    the standard deviation of the Laplace distribution
  - getMu
```
public double getMu()
```
    Returns the parameter μ.
  - getBeta
```
public double getBeta()
```
    Returns the parameter β.
  - getTheta
```
@Deprecated
public double getTheta()
```
    Deprecated.
    
    Use getMu.
  - getPhi
```
@Deprecated
public double getPhi()
```
    Deprecated.
  - getParams
```
public double[] getParams()
```
    Return a table containing the parameters of the current distribution. This table is put in regular order: [μ, β].
  - toString
```
public java.lang.String toString()
```
    Overrides:
    
    toString in class java.lang.Object

F(x) =	(1/2)e^(x-μ)/β	if x <= μ,
F(x) =	1 - (1/2)e^(μ-x)/β	otherwise,

F^-1(u) =	βlog(2u) + μ	if 0 <= u <= 1/2,
F^-1(u) =	μ - βlog(2(1 - u))	otherwise.

Class LaplaceDist

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

LaplaceDist

LaplaceDist

Method Detail

density

cdf

barF

inverseF

getMean

getVariance

getStandardDeviation

density

cdf

barF

inverseF

getMLE

getMaximumLikelihoodEstimate

getInstanceFromMLE

getMean

getVariance

getStandardDeviation

getMu

getBeta

getTheta

getPhi

getParams

toString