umontreal.iro.lecuyer.probdistmulti

Class BiNormalDist

java.lang.Object

umontreal.iro.lecuyer.probdistmulti.ContinuousDistributionMulti

umontreal.iro.lecuyer.probdistmulti.ContinuousDistribution2Dim

umontreal.iro.lecuyer.probdistmulti.BiNormalDist

Direct Known Subclasses:

BiNormalDonnellyDist, BiNormalGenzDist

public class BiNormalDist
extends ContinuousDistribution2Dim

Extends the class ContinuousDistribution2Dim for the bivariate normal distribution. It has means E[X] = μ₁, E[Y] = μ₂, and variances var [X] = σ₁², var [Y] = σ₂² such that σ₁ > 0 and σ₂ > 0. The correlation between X and Y is r. Its density function is

f (x, y) = e^-T/(2πσ₁σ₂(1-r^2)^1/2)

T = [((x - μ₁)/σ₁)² -2r((x - μ₁)/σ₁)((y - μ₂)/σ₂) + ((y - μ₂)/σ₂)²]/(2(1 - r²))

and the corresponding distribution function is (the cdf method)

Φ(μ₁, σ₁, x, μ₂, σ₂, y, r) = ∫_-∞^xdx∫_-∞^ydy e^-T/(2πσ₁σ₂(1 - r^2)^1/2).

We also define the upper distribution function (the barF method) as

bar(Φ)(μ₁, σ₁, x, μ₂, σ₂, y, r) = ∫^∞_xdx∫^∞_ydy e^-T/(2πσ₁σ₂(1 - r^2)^1/2).

When μ₁ = μ₂ = 0 and σ₁ = σ₂ = 1, we have the standard binormal distribution, with corresponding distribution function

Φ(x, y, r) = ∫^∞_xdx∫^∞_ydy e^-S/(2π(1 - r^2)^1/2)

S = (x² -2rxy + y²)/(2(1 - r²)).

Field Summary

Fields inherited from class umontreal.iro.lecuyer.probdistmulti.ContinuousDistribution2Dim

decPrec

Constructor Summary

Constructors

Constructor and Description
BiNormalDist(double rho) Constructs a BiNormalDist object with default parameters μ1 = μ2 = 0, σ1 = σ2 = 1 and correlation ρ = rho.
BiNormalDist(double mu1, double sigma1, double mu2, double sigma2, double rho) Constructs a BiNormalDist object with parameters μ1 = mu1, μ2 = mu2, σ1 = sigma1, σ2 = sigma2 and ρ = rho.

Method Summary

Modifier and Type	Method and Description
double	barF(double x, double y) .
static double	barF(double x, double y, double rho) Computes the standard upper binormal distribution with μ1 = μ2 = 0 and σ1 = σ2 = 1.
static double	barF(double mu1, double sigma1, double x, double mu2, double sigma2, double y, double rho) Computes the upper binormal distribution function with parameters μ1 = mu1, μ2 = mu2, σ1 = sigma1, σ2 = sigma2 and ρ = rho.
double	cdf(double x, double y) .
static double	cdf(double x, double y, double rho) Computes the standard binormal distribution using the fast Drezner-Wesolowsky method described in.
static double	cdf(double mu1, double sigma1, double x, double mu2, double sigma2, double y, double rho) Computes the binormal distribution function with parameters μ1 = mu1, μ2 = mu2, σ1 = sigma1, σ2 = sigma2 and ρ = rho.
double	density(double x, double y) Returns f (x, y), the density of (X, Y) evaluated at (x, y).
static double	density(double x, double y, double rho) Computes the standard binormal density function with μ1 = μ2 = 0 and σ1 = σ2 = 1.
static double	density(double mu1, double sigma1, double x, double mu2, double sigma2, double y, double rho) Computes the binormal density function with parameters μ1 = mu1, μ2 = mu2, σ1 = sigma1, σ2 = sigma2 and ρ = rho.
double[][]	getCorrelation() Returns the correlation matrix of the distribution, defined as ρij = σij/(σ_iiσ_jj)1/2.
static double[][]	getCorrelation(double mu1, double sigma1, double mu2, double sigma2, double rho) Return the correlation matrix of the binormal distribution.
double[][]	getCovariance() Returns the variance-covariance matrix of the distribution, defined as σij = E[(Xi - μi)(Xj - μj)].
static double[][]	getCovariance(double mu1, double sigma1, double mu2, double sigma2, double rho) Return the covariance matrix of the binormal distribution.
double[]	getMean() Returns the mean vector of the distribution, defined as μi = E[Xi].
static double[]	getMean(double mu1, double sigma1, double mu2, double sigma2, double rho) Return the mean vector E[X] = (μ1, μ2) of the binormal distribution.
double	getMu1() Returns the parameter μ1.
double	getMu2() Returns the parameter μ2.
double	getSigma1() Returns the parameter σ1.
double	getSigma2() Returns the parameter σ2.

Methods inherited from class umontreal.iro.lecuyer.probdistmulti.ContinuousDistribution2Dim

cdf, density

Methods inherited from class umontreal.iro.lecuyer.probdistmulti.ContinuousDistributionMulti

getDimension

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

BiNormalDist

public BiNormalDist(double rho)

Constructs a BiNormalDist object with default parameters μ₁ = μ₂ = 0, σ₁ = σ₂ = 1 and correlation ρ = rho.

BiNormalDist

public BiNormalDist(double mu1,
                    double sigma1,
                    double mu2,
                    double sigma2,
                    double rho)

Constructs a BiNormalDist object with parameters μ₁ = mu1, μ₂ = mu2, σ₁ = sigma1, σ₂ = sigma2 and ρ = rho.

Method Detail

density

public double density(double x,
                      double y)

Description copied from class: ContinuousDistribution2Dim

Returns f (x, y), the density of (X, Y) evaluated at (x, y).

Specified by:

density in class ContinuousDistribution2Dim

Parameters:

x - value x at which the density is evaluated

y - value y at which the density is evaluated

Returns:

density function evaluated at (x, y)

density

public static double density(double x,
                             double y,
                             double rho)

Computes the standard binormal density function with μ₁ = μ₂ = 0 and σ₁ = σ₂ = 1.

density

public static double density(double mu1,
                             double sigma1,
                             double x,
                             double mu2,
                             double sigma2,
                             double y,
                             double rho)

Computes the binormal density function with parameters μ₁ = mu1, μ₂ = mu2, σ₁ = sigma1, σ₂ = sigma2 and ρ = rho.

cdf

public static double cdf(double x,
                         double y,
                         double rho)

Computes the standard binormal distribution using the fast Drezner-Wesolowsky method described in. The absolute error is expected to be smaller than 2⋅10^-7.

cdf

public double cdf(double x,
                  double y)

Description copied from class: ContinuousDistribution2Dim

. Computes the distribution function F(x, y):

F(x, y) = P[X <= x, Y <= y] = ∫_-∞^xds∫_-∞^ydt f (s, t).

Specified by:

cdf in class ContinuousDistribution2Dim

Parameters:

x - value x at which the distribution function is evaluated

y - value y at which the distribution function is evaluated

Returns:

distribution function evaluated at (x, y)

cdf

public static double cdf(double mu1,
                         double sigma1,
                         double x,
                         double mu2,
                         double sigma2,
                         double y,
                         double rho)

Computes the binormal distribution function with parameters μ₁ = mu1, μ₂ = mu2, σ₁ = sigma1, σ₂ = sigma2 and ρ = rho. Uses the fast Drezner-Wesolowsky method described in. The absolute error is expected to be smaller than 2⋅10^-7.

barF

public static double barF(double x,
                          double y,
                          double rho)

Computes the standard upper binormal distribution with μ₁ = μ₂ = 0 and σ₁ = σ₂ = 1. Uses the fast Drezner-Wesolowsky method described in. The absolute error is expected to be smaller than 2⋅10^-7.

barF

public double barF(double x,
                   double y)

Description copied from class: ContinuousDistribution2Dim

. Computes the upper cumulative distribution function bar(F)(x, y):

bar(F)(x, y) = P[X >= x, Y >= y] = ∫^∞_xds∫^∞_ydt f (s, t).

Overrides:

barF in class ContinuousDistribution2Dim

Parameters:

x - value x at which the upper distribution is evaluated

y - value y at which the upper distribution is evaluated

Returns:

upper distribution function evaluated at (x, y)

barF

public static double barF(double mu1,
                          double sigma1,
                          double x,
                          double mu2,
                          double sigma2,
                          double y,
                          double rho)

Computes the upper binormal distribution function with parameters μ₁ = mu1, μ₂ = mu2, σ₁ = sigma1, σ₂ = sigma2 and ρ = rho. Uses the fast Drezner-Wesolowsky method described in. The absolute error is expected to be smaller than 2⋅10^-7.

getMean

public double[] getMean()

Description copied from class: ContinuousDistributionMulti

Returns the mean vector of the distribution, defined as μ_i = E[X_i].

Specified by:

getMean in class ContinuousDistributionMulti

getMean

public static double[] getMean(double mu1,
                               double sigma1,
                               double mu2,
                               double sigma2,
                               double rho)

Return the mean vector E[X] = (μ₁, μ₂) of the binormal distribution.

getCovariance

public double[][] getCovariance()

Description copied from class: ContinuousDistributionMulti

Returns the variance-covariance matrix of the distribution, defined as
σ_ij = E[(X_i - μ_i)(X_j - μ_j)].

Specified by:

getCovariance in class ContinuousDistributionMulti

getCovariance

public static double[][] getCovariance(double mu1,
                                       double sigma1,
                                       double mu2,
                                       double sigma2,
                                       double rho)

Return the covariance matrix of the binormal distribution.

getCorrelation

public double[][] getCorrelation()

Description copied from class: ContinuousDistributionMulti

Returns the correlation matrix of the distribution, defined as ρ_ij = σ_ij/(σ_iiσ_jj)^1/2.

Specified by:

getCorrelation in class ContinuousDistributionMulti

getCorrelation

public static double[][] getCorrelation(double mu1,
                                        double sigma1,
                                        double mu2,
                                        double sigma2,
                                        double rho)

Return the correlation matrix of the binormal distribution.

getMu1

public double getMu1()

Returns the parameter μ₁.

getMu2

public double getMu2()

Returns the parameter μ₂.

getSigma1

public double getSigma1()

Returns the parameter σ₁.

getSigma2

public double getSigma2()

Returns the parameter σ₂.