kldggd function

Kullback-Leibler Divergence between Centered Multivariate generalized Gaussian Distributions

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

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

Arguments

• Sigma1: symmetric, positive-definite matrix. The dispersion matrix of the first distribution.
• beta1: positive real number. The shape parameter of the first distribution.
• Sigma2: symmetric, positive-definite matrix. The dispersion matrix of the second distribution.
• beta2: positive real number. The shape parameter of the second distribution.
• eps: numeric. Precision for the computation of the Lauricella $D$-hypergeometric function (see lauricella). Default: 1e-06.

Returns

A numeric value: the Kullback-Leibler divergence between the two distributions, with two attributes attr(, "epsilon") (precision of the result of the Lauricella $D$-hypergeometric Function) and attr(, "k") (number of iterations) except when the distributions are univariate.

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:

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 defined for $p$ variables:

This computation uses the lauricella function.

When $p = 1$ (univariate case): let $X_1$, a random variable distributed according to the centered 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 $(0, \sigma_2, \beta_2)$.

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)

References

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

See Also

dmggd : probability density of a MGGD.

Author(s)

Pierre Santagostini, Nizar Bouhlel