# Kullback-Leibler Divergence between Centered Multivariate generalized Gaussian Distributions Computes the Kullback- Leibler divergence between two random variables distributed according to multivariate generalized Gaussian distributions (MGGD) with zero means. ```r 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 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 function; 0 if the distributions are univariate) and `attr(, "k")` (number of iterations). ## 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 $(0, \sigma_2, \beta_2)$. $$ KL(X_1||X_2) = \displaystyle{ \ln{\left(\frac{\frac{\beta_1}{\sqrt{\sigma_1}} \Gamma\left(\frac{1}{2\beta_2}\right)}{\frac{\beta_2}{\sqrt{\sigma_2}} \Gamma\left(\frac{1}{2\beta_1}\right)}\right)} + \frac{1}{2} \left(\frac{1}{\beta_2} - \frac{1}{\beta_1}\right) \ln{2} - \frac{1}{2\beta_2} + 2^{\frac{\beta_2}{\beta_1}-1} \frac{\Gamma{\left(\frac{\beta_2}{\beta_1} + \frac{1}{\beta_1}\right)}}{\Gamma{\left(\frac{1}{2 \beta_1}\right)}} \left(\frac{\sigma_1}{\sigma_2}\right)^{\beta_2} } $$ ## Examples ```r 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

kldggd function