BetaDist (Java Libraries for Stochastic Simulation)

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

All Implemented Interfaces:

Distribution

Direct Known Subclasses:

BetaSymmetricalDist
```
public class BetaDist
extends ContinuousDistribution
```
Extends the class ContinuousDistribution for the beta distribution with shape parameters α > 0 and β > 0, over the interval (a, b), where a < b. It has density

f (x) = (x - a)^α-1(b - x)^β-1/[B(α, β)(b - a)^α+β-1]
for a < x < b, and 0 elsewhere. It has distribution function

F(x) = I_{α, β}(x) = ∫_a^x(ξ - a)^α-1(b - ξ)^β-1/[B(α, β)(b - a)^α+β-1]dξ, for a < x < b,
where B(α, β) is the beta function defined by

B(α, β) = Γ(α)Γ(β)/Γ(α + β)
and Γ(x) is the gamma function defined in GammaDist.

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

Constructor Summary

Constructors
Constructor and Description
`BetaDist(double alpha, double beta)` Constructs a `BetaDist` object with parameters α = `alpha` and β = `beta` and default domain (0, 1).
`BetaDist(double alpha, double beta, double a, double b)` Constructs a `BetaDist` object with parameters α = `alpha` and β = `beta`, and domain (`a`, `b`).
`BetaDist(double alpha, double beta, double a, double b, int d)` Constructs a `BetaDist` object with parameters α = `alpha` and β = `beta`, and approximations of roughly `d` decimal digits of precision when computing distribution, complementary distribution, and inverse functions.
`BetaDist(double alpha, double beta, int d)` Constructs a `BetaDist` object with parameters α = `alpha` and β = `beta`, and approximations of roughly `d` decimal digits of precision when computing the distribution, complementary distribution, and inverse functions.

Method Summary

Methods
Modifier and Type	Method and Description
`static double`	`barF(double alpha, double beta, double a, double b, int d, double x)` Computes the complementary distribution function.
`static double`	`barF(double alpha, double beta, int d, double x)` Same as `barF` `(alpha, beta, 0, 1, d, x)`.
`double`	`cdf(double x)` Returns the distribution function F(x).
`static double`	`cdf(double alpha, double beta, double a, double b, int d, double x)` Computes an approximation of the distribution function, with roughly `d` decimal digits of precision.
`static double`	`cdf(double alpha, double beta, int d, double x)` Same as `cdf` `(alpha, beta, 0, 1, d, x)`.
`double`	`density(double x)` Returns f (x), the density evaluated at x.
`static double`	`density(double alpha, double beta, double x)` Same as `density` `(alpha, beta, 0, 1, x)`.
`static double`	`density(double alpha, double beta, double a, double b, double x)` Computes the density function of the beta distribution.
`double`	`getA()` Returns the parameter a of this object.
`double`	`getAlpha()` Returns the parameter α of this object.
`double`	`getB()` Returns the parameter b of this object.
`double`	`getBeta()` Returns the parameter β of this object.
`static BetaDist`	`getInstanceFromMLE(double[] x, int n)` Creates a new instance of a beta distribution with parameters α and β over the interval [0, 1] 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 alpha, double beta)` Computes and returns the mean E[X] = α/(α + β) of the beta distribution with parameters α and β, over the interval [0, 1].
`static double`	`getMean(double alpha, double beta, double a, double b)` Computes and returns the mean E[X] = (bα + aβ)/(α + β) of the beta distribution with parameters α and β over the interval [a, b].
`static double[]`	`getMLE(double[] x, int n)` Estimates the parameters (α, β) of the beta distribution over the interval [0, 1] using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
`double[]`	`getParams()` Return a table containing parameters of the current distribution.
`double`	`getStandardDeviation()` Returns the standard deviation.
`static double`	`getStandardDeviation(double alpha, double beta)` Computes the standard deviation of the beta distribution with parameters α and β, over the interval [0, 1].
`static double`	`getStandardDeviation(double alpha, double beta, double a, double b)` Computes the standard deviation of the beta distribution with parameters α and β, over the interval [a, b].
`double`	`getVariance()` Returns the variance.
`static double`	`getVariance(double alpha, double beta)` .
`static double`	`getVariance(double alpha, double beta, double a, double b)` .
`double`	`inverseF(double u)` Returns the inverse distribution function x = F^-1(u).
`static double`	`inverseF(double alpha, double beta, double a, double b, int d, double u)` Returns the inverse beta distribution function using the algorithm implemented in the Cephes math library.
`static double`	`inverseF(double alpha, double beta, int d, double u)` Same as `inverseF` `(alpha, beta, 0, 1, d, u)`.
`void`	`setParams(double alpha, double beta, double a, double b, int d)`
`java.lang.String`	`toString()`

Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
barF, getXinf, getXsup, inverseBisection, inverseBrent, setXinf, setXsup

Methods inherited from class java.lang.Object
equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

- Constructor Detail
  - BetaDist
```
public BetaDist(double alpha,
        double beta)
```
    Constructs a BetaDist object with parameters α = alpha and β = beta and default domain (0, 1).
  - BetaDist
```
public BetaDist(double alpha,
        double beta,
        double a,
        double b)
```
    Constructs a BetaDist object with parameters α = alpha and β = beta, and domain (a, b).
  - BetaDist
```
public BetaDist(double alpha,
        double beta,
        int d)
```
    Constructs a BetaDist object with parameters α = alpha and β = beta, and approximations of roughly d decimal digits of precision when computing the distribution, complementary distribution, and inverse functions. The default domain (0, 1) is used.
  - BetaDist
```
public BetaDist(double alpha,
        double beta,
        double a,
        double b,
        int d)
```
    Constructs a BetaDist object with parameters α = alpha and β = beta, and approximations of roughly d decimal digits of precision when computing distribution, complementary distribution, and inverse functions. The domain (a, b) is used.
- 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
  - 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 alpha,
             double beta,
             double x)
```
    Same as density (alpha, beta, 0, 1, x).
  - density
```
public static double density(double alpha,
             double beta,
             double a,
             double b,
             double x)
```
    Computes the density function of the beta distribution.
  - cdf
```
public static double cdf(double alpha,
         double beta,
         int d,
         double x)
```
    Same as cdf (alpha, beta, 0, 1, d, x).
  - cdf
```
public static double cdf(double alpha,
         double beta,
         double a,
         double b,
         int d,
         double x)
```
    Computes an approximation of the distribution function, with roughly d decimal digits of precision.
  - barF
```
public static double barF(double alpha,
          double beta,
          int d,
          double x)
```
    Same as barF (alpha, beta, 0, 1, d, x).
  - barF
```
public static double barF(double alpha,
          double beta,
          double a,
          double b,
          int d,
          double x)
```
    Computes the complementary distribution function.
  - inverseF
```
public static double inverseF(double alpha,
              double beta,
              int d,
              double u)
```
    Same as inverseF (alpha, beta, 0, 1, d, u).
  - inverseF
```
public static double inverseF(double alpha,
              double beta,
              double a,
              double b,
              int d,
              double u)
```
    Returns the inverse beta distribution function using the algorithm implemented in the Cephes math library. The method performs interval halving or Newton iterations to compute the inverse. The precision depends on the accuracy of the cdf method. The argument d gives a good idea of the precision attained.
  - getMLE
```
public static double[] getMLE(double[] x,
              int n)
```
    Estimates the parameters (α, β) of the beta distribution over the interval [0, 1] 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: [α, β].
    
    Parameters:
    x - the list of observations to use to evaluate parameters
    n - the number of observations to use to evaluate parameters
    
    Returns:
    returns the parameters [ hat(α), hat(β)]
  - getMaximumLikelihoodEstimate
```
@Deprecated
public static double[] getMaximumLikelihoodEstimate(double[] x,
                                               int n)
```
    Deprecated.
    
    Same as getMLE.
  - getInstanceFromMLE
```
public static BetaDist getInstanceFromMLE(double[] x,
                          int n)
```
    Creates a new instance of a beta distribution with parameters α and β over the interval [0, 1] 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 alpha,
             double beta)
```
    Computes and returns the mean E[X] = α/(α + β) of the beta distribution with parameters α and β, over the interval [0, 1].
    
    Returns:
    the mean of the Beta distribution
  - getMean
```
public static double getMean(double alpha,
             double beta,
             double a,
             double b)
```
    Computes and returns the mean E[X] = (bα + aβ)/(α + β) of the beta distribution with parameters α and β over the interval [a, b].
    
    Returns:
    the mean of the Beta distribution
  - getVariance
```
public static double getVariance(double alpha,
                 double beta)
```
    . Computes and returns the variance Var[X] = ${\frac{{\alpha\beta}}{{(\alpha + \beta)^2 (\alpha + \beta + 1)}}}$ of the beta distribution with parameters α and β, over the interval [0, 1].
    
    Returns:
    the variance of the beta distribution Var[X] = αβ/[(α + β)²(α + β + 1)].
  - getVariance
```
public static double getVariance(double alpha,
                 double beta,
                 double a,
                 double b)
```
    . Computes and returns the variance Var[X] = ${\frac{{\alpha\beta(b-a)^2}}{{(\alpha + \beta)^2 (\alpha + \beta + 1)}}}$ of the beta distribution with parameters α and β, over the interval [a, b].
    
    Returns:
    the variance of the beta distribution Var[X] = αβ/[(α + β)²(α + β + 1)].
  - getStandardDeviation
```
public static double getStandardDeviation(double alpha,
                          double beta)
```
    Computes the standard deviation of the beta distribution with parameters α and β, over the interval [0, 1].
    
    Returns:
    the standard deviation of the Beta distribution
  - getStandardDeviation
```
public static double getStandardDeviation(double alpha,
                          double beta,
                          double a,
                          double b)
```
    Computes the standard deviation of the beta distribution with parameters α and β, over the interval [a, b].
    
    Returns:
    the standard deviation of the Beta distribution
  - getAlpha
```
public double getAlpha()
```
    Returns the parameter α of this object.
  - getBeta
```
public double getBeta()
```
    Returns the parameter β of this object.
  - getA
```
public double getA()
```
    Returns the parameter a of this object.
  - getB
```
public double getB()
```
    Returns the parameter b of this object.
  - setParams
```
public void setParams(double alpha,
             double beta,
             double a,
             double b,
             int d)
```
  - getParams
```
public double[] getParams()
```
    Return a table containing parameters of the current distribution. This table is put in regular order: [α, β].
  - toString
```
public java.lang.String toString()
```
    Overrides:
    
    toString in class java.lang.Object

Class BetaDist

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

BetaDist

BetaDist

BetaDist

BetaDist

Method Detail

density

cdf

inverseF

getMean

getVariance

getStandardDeviation

density

density

cdf

cdf

barF

barF

inverseF

inverseF

getMLE

getMaximumLikelihoodEstimate

getInstanceFromMLE

getMean

getMean

getVariance

getVariance

getStandardDeviation

getStandardDeviation

getAlpha

getBeta

getA

getB

setParams

getParams

toString