Tools for Multivariate Generalized Gaussian Distributions

Description

This package provides tools for multivariate generalized Gaussian distributions (MGGD):

• Calculation of distances/divergences between multivariate generalized Gaussian distributions:

  • Kullback-Leibler divergence: kldggd

• Tools for MGGD:

  • Probability density: dmggd

  • Estimation of the parameters: estparmggd

  • Simulation from a MGGD: rmggd

  • Plot of the density of a MGGD with 2 variables: plotmggd, contourmggd

Author(s)

Pierre Santagostini pierre.santagostini@agrocampus-ouest.fr, Nizar Bouhlel nizar.bouhlel@agrocampus-ouest.fr

References

N. Bouhlel, A. Dziri, Kullback-Leibler Divergence Between Multivariate Generalized Gaussian Distributions. IEEE Signal Processing Letters, vol. 26 no. 7, July 2019. doi:10.1109/LSP.2019.2915000

E. Gomez, M. Gomez-Villegas, H. Marin. A Multivariate Generalization of the Power Exponential Family of Distribution. Commun. Statist. 1998, Theory Methods, col. 27, no. 23, p 589-600. doi:10.1080/03610929808832115

F. Pascal, L. Bombrun, J.Y. Tourneret, Y. Berthoumieu. Parameter Estimation For Multivariate Generalized Gaussian Distribution. IEEE Trans. Signal Processing, vol. 61 no. 23, p. 5960-5971, Dec. 2013. doi:10.1109/TSP.2013.2282909 #' @keywords internal

See Also

Useful links:

• https://forgemia.inra.fr/imhorphen/mggd

• Report bugs at https://forgemia.inra.fr/imhorphen/mggd/-/issues

Contour Plot of the Bivariate Generalised Gaussian Density

Description

Draws the contour plot of the probability density of the generalised Gaussian distribution with 2 variables with mean vector mu, dispersion matrix Sigma and shape parameter beta.

Usage

contourmggd(mu, Sigma, beta,
                    xlim = c(mu[1] + c(-10, 10)*Sigma[1, 1]),
                    ylim = c(mu[2] + c(-10, 10)*Sigma[2, 2]),
                    zlim = NULL, npt = 30, nx = npt, ny = npt,
                    main = "Multivariate generalised Gaussian density",
                    sub = NULL, nlevels = 10,
                    levels = pretty(zlim, nlevels), tol = 1e-6, ...)

Arguments

Value

Returns invisibly the probability density function.

Author(s)

Pierre Santagostini, Nizar Bouhlel

References

E. Gomez, M. Gomez-Villegas, H. Marin. A Multivariate Generalization of the Power Exponential Family of Distribution. Commun. Statist. 1998, Theory Methods, col. 27, no. 23, p 589-600. doi:10.1080/03610929808832115

See Also

plotmggd: plot of a bivariate generalised Gaussian density.

dmggd: Probability density of a multivariate generalised Gaussian distribution.

Examples

mu <- c(1, 4)
Sigma <- matrix(c(0.8, 0.2, 0.2, 0.2), nrow = 2)
beta <- 0.74
contourmggd(mu, Sigma, beta)

Deprecated functions

Description

Use kldggd() instead of kldiv().

Use dmggd() instead of mvdggd().

Use rmggd() instead of mvrggd().

Use estparmggd() instead of estparmvggd().

Use plotmggd() instead of plotmvggd().

Use contourmggd() instead of contourmvggd().

Usage

kldiv(sigma1, beta1, sigma2, beta2, eps = 1e-06)

mvdggd(x, mu, Sigma, beta, tol = 1e-06)

mvrggd(n = 1, mu, Sigma, beta, tol = 1e-06)

estparmvggd(x, eps = 1e-06, display = FALSE, plot = display)

plotmvggd(
  mu,
  Sigma,
  beta,
  xlim = c(mu[1] + c(-10, 10) * Sigma[1, 1]),
  ylim = c(mu[2] + c(-10, 10) * Sigma[2, 2]),
  n = 101,
  xvals = NULL,
  yvals = NULL,
  xlab = "x",
  ylab = "y",
  zlab = "f(x,y)",
  col = "gray",
  tol = 1e-06,
  ...
)

contourmvggd(
  mu,
  Sigma,
  beta,
  xlim = c(mu[1] + c(-10, 10) * Sigma[1, 1]),
  ylim = c(mu[2] + c(-10, 10) * Sigma[2, 2]),
  zlim = NULL,
  npt = 30,
  nx = npt,
  ny = npt,
  main = "Multivariate generalised Gaussian density",
  sub = NULL,
  nlevels = 10,
  levels = pretty(zlim, nlevels),
  tol = 1e-06,
  ...
)

Density of a Multivariate Generalized Gaussian Distribution

Description

Density of the multivariate (p variables) generalized Gaussian distribution (MGGD) with mean vector mu, dispersion matrix Sigma and shape parameter beta.

Usage

dmggd(x, mu, Sigma, beta, tol = 1e-6)

Arguments

Details

The density function of a multivariate generalized Gaussian distribution is given by:

\displaystyle{ f(\mathbf{x}|\boldsymbol{\mu}, \Sigma, \beta) = \frac{\Gamma\left(\frac{p}{2}\right)}{\pi^\frac{p}{2} \Gamma\left(\frac{p}{2 \beta}\right) 2^\frac{p}{2\beta}} \frac{\beta}{|\Sigma|^\frac{1}{2}} e^{-\frac{1}{2}\left((\mathbf{x}-\boldsymbol{\mu})^T \Sigma^{-1} (\mathbf{x}-\boldsymbol{\mu})\right)^\beta} }

When p=1 (univariate case) it becomes:

\displaystyle{ f(x|\mu, \sigma, \beta) = \frac{\Gamma\left(\frac{1}{2}\right)}{\pi^\frac{1}{2} \Gamma\left(\frac{1}{2 \beta}\right) 2^\frac{1}{2\beta}} \frac{\beta}{\sigma^\frac{1}{2}} \ e^{-\frac{1}{2} \left(\frac{(x - \mu)^2}{2 \sigma}\right)^\beta} = \frac{\beta}{\Gamma\left(\frac{1}{2 \beta}\right) 2^\frac{1}{2 \beta} \sqrt{\sigma}} \ e^{-\frac{1}{2} \left(\frac{(x - \mu)^2}{\sigma}\right)^\beta} }

Value

The value of the density.

Author(s)

Pierre Santagostini, Nizar Bouhlel

References

E. Gomez, M. Gomez-Villegas, H. Marin. A Multivariate Generalization of the Power Exponential Family of Distribution. Commun. Statist. 1998, Theory Methods, col. 27, no. 23, p 589-600. doi:10.1080/03610929808832115

See Also

rmggd: random generation from a MGGD.

estparmggd: estimation of the parameters of a MGGD.

plotmggd, contourmggd: plot of the probability density of a bivariate generalised Gaussian distribution.

Examples

mu <- c(0, 1, 4)
Sigma <- matrix(c(0.8, 0.3, 0.2, 0.3, 0.2, 0.1, 0.2, 0.1, 0.2), nrow = 3)
beta <- 0.74
dmggd(c(0, 1, 4), mu, Sigma, beta)
dmggd(c(1, 2, 3), mu, Sigma, beta)

Estimation of the Parameters of a Multivariate Generalized Gaussian Distribution

Description

Estimation of the mean vector, dispersion matrix and shape parameter of a multivariate generalized Gaussian distribution (MGGD).

Usage

estparmggd(x, eps = 1e-6, display = FALSE, plot = display)

Arguments

Details

The \mu parameter is the mean vector of x.

The dispersion matrix \Sigma and shape parameter: \beta are computed using the method presented in Pascal et al., using an iterative algorithm.

The precision for the estimation of beta is given by the eps parameter.

Value

A list of 3 elements:

• mu the mean vector.

• Sigma: symmetric positive-definite matrix. The dispersion matrix.

• beta non-negative numeric value. The shape parameter.

with two attributes attr(, "epsilon") (precision of the result) and attr(, "k") (number of iterations).

Author(s)

Pierre Santagostini, Nizar Bouhlel

References

F. Pascal, L. Bombrun, J.Y. Tourneret, Y. Berthoumieu. Parameter Estimation For Multivariate Generalized Gaussian Distribution. IEEE Trans. Signal Processing, vol. 61 no. 23, p. 5960-5971, Dec. 2013. doi: 10.1109/TSP.2013.2282909

See Also

dmggd: probability density of a MGGD.

rmggd: random generation from a MGGD.

Examples

mu <- c(0, 1, 4)
Sigma <- matrix(c(0.8, 0.3, 0.2, 0.3, 0.2, 0.1, 0.2, 0.1, 0.2), nrow = 3)
beta <- 0.74
x <- rmggd(100, mu, Sigma, beta)

# Estimation of the parameters
estparmggd(x)

Kullback-Leibler Divergence between Centered Multivariate generalized Gaussian Distributions

Description

Computes the Kullback- Leibler divergence between two random variables distributed according to multivariate generalized Gaussian distributions (MGGD) with zero means.

Usage

kldggd(Sigma1, beta1, Sigma2, beta2, eps = 1e-06)

Arguments

Details

Given \mathbf{X}_1, a random vector of \mathbb{R}^p (p > 1) distributed according to the MGGD with parameters (\mathbf{0}, \Sigma_1, \beta_1) and \mathbf{X}_2, a random vector of \mathbb{R}^p distributed according to the MGGD with parameters (\mathbf{0}, \Sigma_2, \beta_2).

The Kullback-Leibler divergence between X_1 and X_2 is given by:

\displaystyle{ KL(\mathbf{X}_1||\mathbf{X}_2) = \ln{\left(\frac{\beta_1 |\Sigma_1|^{-1/2} \Gamma\left(\frac{p}{2\beta_2}\right)}{\beta_2 |\Sigma_2|^{-1/2} \Gamma\left(\frac{p}{2\beta_1}\right)}\right)} + \frac{p}{2} \left(\frac{1}{\beta_2} - \frac{1}{\beta_1}\right) \ln{2} - \frac{p}{2\beta_2} + 2^{\frac{\beta_2}{\beta_1}-1} \frac{\Gamma{\left(\frac{\beta_2}{\beta_1} + \frac{p}{\beta_1}\right)}}{\Gamma{\left(\frac{p}{2 \beta_1}\right)}} \lambda_p^{\beta_2} }

\displaystyle{ \times F_D^{(p-1)}\left(-\beta_1; \underbrace{\frac{1}{2},\dots,\frac{1}{2}}_{p-1}; \frac{p}{2}; 1-\frac{\lambda_{p-1}}{\lambda_p},\dots,1-\frac{\lambda_{1}}{\lambda_p}\right) }

where \lambda_1 < ... < \lambda_{p-1} < \lambda_p are the eigenvalues of the matrix \Sigma_1 \Sigma_2^{-1}
and F_D^{(p-1)} is the Lauricella D-hypergeometric Function.

This computation uses the lauricella function.

When p = 1 (univariate case): let X_1, a random variable distributed according to the generalized Gaussian distribution with parameters (0, \sigma_1, \beta_1) and X_2, a random variable distributed according to the generalized Gaussian distribution with parameters (0, \sigma_2, \beta_2).

KL(X_1||X_2) = \displaystyle{ \ln{\left(\frac{\frac{\beta_1}{\sqrt{\sigma_1}} \Gamma\left(\frac{1}{2\beta_2}\right)}{\frac{\beta_2}{\sqrt{\sigma_2}} \Gamma\left(\frac{1}{2\beta_1}\right)}\right)} + \frac{1}{2} \left(\frac{1}{\beta_2} - \frac{1}{\beta_1}\right) \ln{2} - \frac{1}{2\beta_2} + 2^{\frac{\beta_2}{\beta_1}-1} \frac{\Gamma{\left(\frac{\beta_2}{\beta_1} + \frac{1}{\beta_1}\right)}}{\Gamma{\left(\frac{1}{2 \beta_1}\right)}} \left(\frac{\sigma_1}{\sigma_2}\right)^{\beta_2} }

Value

A numeric value: the Kullback-Leibler divergence between the two distributions, with two attributes attr(, "epsilon") (precision of the result of the Lauricella function; 0 if the distributions are univariate) and attr(, "k") (number of iterations).

Author(s)

Pierre Santagostini, Nizar Bouhlel

References

N. Bouhlel, A. Dziri, Kullback-Leibler Divergence Between Multivariate Generalized Gaussian Distributions. IEEE Signal Processing Letters, vol. 26 no. 7, July 2019. doi:10.1109/LSP.2019.2915000

See Also

dmggd: probability density of a MGGD.

Examples

beta1 <- 0.74
beta2 <- 0.55
Sigma1 <- matrix(c(0.8, 0.3, 0.2, 0.3, 0.2, 0.1, 0.2, 0.1, 0.2), nrow = 3)
Sigma2 <- matrix(c(1, 0.3, 0.2, 0.3, 0.5, 0.1, 0.2, 0.1, 0.7), nrow = 3)

# Kullback-Leibler divergence
kl12 <- kldggd(Sigma1, beta1, Sigma2, beta2)
kl21 <- kldggd(Sigma2, beta2, Sigma1, beta1)
print(kl12)
print(kl21)

# Distance (symmetrized Kullback-Leibler divergence)
kldist <- as.numeric(kl12) + as.numeric(kl21)
print(kldist)

Lauricella D-Hypergeometric Function

Description

Computes the Lauricella D-hypergeometric Function function.

Usage

lauricella(a, b, g, x, eps = 1e-06)

Arguments

Details

If n is the length of the b and x vectors, the Lauricella D-hypergeometric Function function is given by:

\displaystyle{F_D^{(n)}\left(a, b_1, ..., b_n, g; x_1, ..., x_n\right) = \sum_{m_1 \geq 0} ... \sum_{m_n \geq 0}{ \frac{ (a)_{m_1+...+m_n}(b_1)_{m_1} ... (b_n)_{m_n} }{ (g)_{m_1+...+m_n} } \frac{x_1^{m_1}}{m_1!} ... \frac{x_n^{m_n}}{m_n!} } }

where (x)_p is the Pochhammer symbol (see pochhammer).

If |x_i| < 1, i = 1, \dots, n, this sum converges. Otherwise there is an error.

The eps argument gives the required precision for its computation. It is the attr(, "epsilon") attribute of the returned value.

Sometimes, the convergence is too slow and the required precision cannot be reached. If this happens, the attr(, "epsilon") attribute is the precision that was really reached.

Value

A numeric value: the value of the Lauricella function, with two attributes attr(, "epsilon") (precision of the result) and attr(, "k") (number of iterations).

Author(s)

Pierre Santagostini, Nizar Bouhlel

References

N. Bouhlel, A. Dziri, Kullback-Leibler Divergence Between Multivariate Generalized Gaussian Distributions. IEEE Signal Processing Letters, vol. 26 no. 7, July 2019. doi:10.1109/LSP.2019.2915000

N. Bouhlel and D. Rousseau (2023), Exact Rényi and Kullback-Leibler Divergences Between Multivariate t-Distributions. IEEE Signal Processing Letters, vol. 30, pp. 1672-1676, October 2023. doi:10.1109/LSP.2023.3324594

Logarithm of the Pochhammer Symbol

Description

Computes the logarithm of the Pochhammer symbol.

Usage

lnpochhammer(x, n)

Arguments

Details

The Pochhammer symbol is given by:

\displaystyle{ (x)_n = \frac{\Gamma(x+n)}{\Gamma(x)} = x (x+1) ... (x+n-1) }

So, if n > 0:

\displaystyle{ log\left((x)_n\right) = log(x) + log(x+1) + ... + log(x+n-1) }

If n = 0, \displaystyle{ log\left((x)_n\right) = log(1) = 0}

Value

Numeric value. The logarithm of the Pochhammer symbol.

Author(s)

Pierre Santagostini, Nizar Bouhlel

See Also

pochhammer()

Examples

lnpochhammer(2, 0)
lnpochhammer(2, 1)
lnpochhammer(2, 3)

Plot of the Bivariate Generalised Gaussian Density

Description

Plots the probability density of the generalised Gaussian distribution with 2 variables with mean vector mu, dispersion matrix Sigma and shape parameter beta.

Usage

plotmggd(mu, Sigma, beta, xlim = c(mu[1] + c(-10, 10)*Sigma[1, 1]),
                 ylim = c(mu[2] + c(-10, 10)*Sigma[2, 2]), n = 101,
                 xvals = NULL, yvals = NULL, xlab = "x", ylab = "y",
                 zlab = "f(x,y)", col = "gray", tol = 1e-6, ...)

Arguments

Value

Returns invisibly the probability density function.

Author(s)

Pierre Santagostini, Nizar Bouhlel

References

E. Gomez, M. Gomez-Villegas, H. Marin. A Multivariate Generalization of the Power Exponential Family of Distribution. Commun. Statist. 1998, Theory Methods, col. 27, no. 23, p 589-600. doi:10.1080/03610929808832115

See Also

contourmggd: contour plot of a bivariate generalised Gaussian density.

dmggd: Probability density of a multivariate generalised Gaussian distribution.

Examples

mu <- c(1, 4)
Sigma <- matrix(c(0.8, 0.2, 0.2, 0.2), nrow = 2)
beta <- 0.74
plotmggd(mu, Sigma, beta)

Pochhammer Symbol

Description

Computes the Pochhammer symbol.

Usage

pochhammer(x, n)

Arguments

Details

The Pochhammer symbol is given by:

\displaystyle{ (x)_n = \frac{\Gamma(x+n)}{\Gamma(x)} = x (x+1) ... (x+n-1) }

Value

Numeric value. The value of the Pochhammer symbol.

Author(s)

Pierre Santagostini, Nizar Bouhlel

Examples

pochhammer(2, 0)
pochhammer(2, 1)
pochhammer(2, 3)

Simulate from a Multivariate Generalized Gaussian Distribution

Description

Produces one or more samples from a multivariate (p variables) generalized Gaussian distribution (MGGD).

Usage

rmggd(n = 1 , mu, Sigma, beta, tol = 1e-6)

Arguments

Details

A sample from a centered MGGD with dispersion matrix \Sigma and shape parameter \beta can be generated using:

\displaystyle{X = \tau \ \Sigma^{1/2} \ U}

where U is a random vector uniformly distributed on the unit sphere and \tau is such that \tau^{2\beta} is generated from a distribution Gamma with shape parameter \displaystyle{\frac{p}{2\beta}} and scale parameter 2.

This property is used to generate a sample from a MGGD.

Value

A matrix with p columns and n rows.

Author(s)

Pierre Santagostini, Nizar Bouhlel

References

E. Gomez, M. Gomez-Villegas, H. Marin. A Multivariate Generalization of the Power Exponential Family of Distribution. Commun. Statist. 1998, Theory Methods, col. 27, no. 23, p 589-600. doi:10.1080/03610929808832115

See Also

dmggd: probability density of a MGGD..

estparmggd: estimation of the parameters of a MGGD.

Examples

mu <- c(0, 0, 0)
Sigma <- matrix(c(0.8, 0.3, 0.2, 0.3, 0.2, 0.1, 0.2, 0.1, 0.2), nrow = 3)
beta <- 0.74
rmggd(100, mu, Sigma, beta)