BetaDist (Java Libraries for Stochastic Simulation)

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umontreal.iro.lecuyer.probdist
Class BetaDist

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 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
`BetaDist(double alpha, double beta)` Constructs a `BetaDist` object with parameters α = `alpha`, β = `beta` and default domain [0, 1].
`BetaDist(double alpha, double beta, double a, double b)` Constructs a `BetaDist` object with parameters α = `alpha`, β = `beta` and domain [`a`, `b`].
`BetaDist(double alpha, double beta, double a, double b, int d)` Deprecated.
`BetaDist(double alpha, double beta, int d)` Deprecated.

Method Summary
`double`	`barF(double x)` Returns the complementary distribution function.
`static double`	`barF(double alpha, double beta, double x)` Same as `barF` `(alpha, beta, 0, 1, x)`.
`static double`	`barF(double alpha, double beta, double a, double b, double x)` Computes the complementary distribution function.
`static double`	`barF(double alpha, double beta, double a, double b, int d, double x)` Deprecated.
`static double`	`barF(double alpha, double beta, int d, double x)` Deprecated.
`double`	`cdf(double x)` Returns the distribution function F(x).
`static double`	`cdf(double alpha, double beta, double x)` Same as `cdf` `(alpha, beta, 0, 1, x)`.
`static double`	`cdf(double alpha, double beta, double a, double b, double x)` Computes the distribution function.
`static double`	`cdf(double alpha, double beta, double a, double b, int d, double x)` Deprecated.
`static double`	`cdf(double alpha, double beta, int d, double x)` Deprecated.
`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.
`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 an array containing the parameters of the current distribution as [α, β, a, b].
`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 u)` Same as `inverseF` `(alpha, beta, 0, 1, u)`.
`static double`	`inverseF(double alpha, double beta, double a, double b, double u)` Returns the inverse beta distribution function using the algorithm implemented in the Cephes math library.
`static double`	`inverseF(double alpha, double beta, double a, double b, int d, double u)` Deprecated.
`static double`	`inverseF(double alpha, double beta, int d, double u)` Deprecated.
`void`	`setParams(double alpha, double beta, double a, double b)` Sets the parameters of the current distribution.
`void`	`setParams(double alpha, double beta, double a, double b, int d)` Deprecated.
`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

BetaDist

public BetaDist(double alpha,
                double beta)

Constructs a BetaDist object with parameters α = alpha, 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, β = beta and domain [a, b].

BetaDist

@Deprecated
public BetaDist(double alpha,
                           double beta,
                           int d)

Deprecated.

BetaDist

@Deprecated
public BetaDist(double alpha,
                           double beta,
                           double a,
                           double b,
                           int d)

Deprecated.

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 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

@Deprecated
public static double cdf(double alpha,
                                    double beta,
                                    int d,
                                    double x)

Deprecated.

cdf

@Deprecated
public static double cdf(double alpha,
                                    double beta,
                                    double a,
                                    double b,
                                    int d,
                                    double x)

Deprecated.

barF

@Deprecated
public static double barF(double alpha,
                                     double beta,
                                     int d,
                                     double x)

Deprecated.

barF

@Deprecated
public static double barF(double alpha,
                                     double beta,
                                     double a,
                                     double b,
                                     int d,
                                     double x)

Deprecated.

cdf

public static double cdf(double alpha,
                         double beta,
                         double x)

Same as cdf (alpha, beta, 0, 1, x).

cdf

public static double cdf(double alpha,
                         double beta,
                         double a,
                         double b,
                         double x)

Computes the distribution function.

barF

public static double barF(double alpha,
                          double beta,
                          double x)

Same as barF (alpha, beta, 0, 1, x).

barF

public static double barF(double alpha,
                          double beta,
                          double a,
                          double b,
                          double x)

Computes the complementary distribution function.

inverseF

@Deprecated
public static double inverseF(double alpha,
                                         double beta,
                                         int d,
                                         double u)

Deprecated.

inverseF

public static double inverseF(double alpha,
                              double beta,
                              double u)

Same as inverseF (alpha, beta, 0, 1, u).

inverseF

@Deprecated
public static double inverseF(double alpha,
                                         double beta,
                                         double a,
                                         double b,
                                         int d,
                                         double u)

Deprecated.

inverseF

public static double inverseF(double alpha,
                              double beta,
                              double a,
                              double b,
                              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.

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)]

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

@Deprecated
public void setParams(double alpha,
                                 double beta,
                                 double a,
                                 double b,
                                 int d)

Deprecated.

setParams

public void setParams(double alpha,
                      double beta,
                      double a,
                      double b)

Sets the parameters of the current distribution. See the constructor.

getParams

public double[] getParams()

Return an array containing the parameters of the current distribution as [α, β, a, b].

toString

public String toString()

Overrides:: toString in class Object

Overview

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Deprecated

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V. 2.6.2.

PREV CLASS NEXT CLASS

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SUMMARY: NESTED | FIELD | CONSTR | METHOD

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

umontreal.iro.lecuyer.probdist Class BetaDist

BetaDist

BetaDist

BetaDist

BetaDist

density

cdf

barF

inverseF

getMean

getVariance

getStandardDeviation

density

density

cdf

cdf

barF

barF

cdf

cdf

barF

barF

inverseF

inverseF

inverseF

inverseF

getMLE

getInstanceFromMLE

getMean

getMean

getVariance

getVariance

getStandardDeviation

getStandardDeviation

getAlpha

getBeta

getA

getB

setParams

setParams

getParams

toString

umontreal.iro.lecuyer.probdist
Class BetaDist