Details section, equations: some characters in bold.

See also: mvdggd.

R/kldggd.R

@@ -18,13 +18,13 @@ kldggd <- function(Sigma1, beta1, Sigma2, beta2, eps = 1e-06) {
   #' 0 if the distributions are univariate) 
   #' and \code{attr(, "k")} (number of iterations).
   #'
-  #' @details Given \eqn{X_1}, a random vector of \eqn{R^p} (\eqn{p > 1}) distributed according to the MGGD
-  #' with parameters \eqn{(0, \Sigma_1, \beta_1)}
-  #' and \eqn{X_2}, a random vector of \eqn{R^p} distributed according to the MGGD
-  #' with parameters \eqn{(0, \Sigma_2, \beta_2)}.
+  #' @details Given \eqn{\mathbf{X}_1}, a random vector of \eqn{\mathbb{R}^p} (\eqn{p > 1}) distributed according to the MGGD
+  #' with parameters \eqn{(\mathbf{0}, \Sigma_1, \beta_1)}
+  #' and \eqn{\mathbf{X}_2}, a random vector of \eqn{\mathbb{R}^p} distributed according to the MGGD
+  #' with parameters \eqn{(\mathbf{0}, \Sigma_2, \beta_2)}.
   #'
   #' The Kullback-Leibler divergence between \eqn{X_1} and \eqn{X_2} is given by:
-  #' \deqn{ \displaystyle{ KL(X_1||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} } }
+  #' \deqn{ \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} } }
   #' \deqn{ \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 \eqn{\lambda_1 < ... < \lambda_{p-1} < \lambda_p} are the eigenvalues
@@ -39,6 +39,7 @@ kldggd <- function(Sigma1, beta1, Sigma2, beta2, eps = 1e-06) {
   #' and \eqn{X_2}, a random variable distributed according to the generalized Gaussian distribution
   #' with parameters \eqn{(0, \sigma_2, \beta_2)}.
   #' \deqn{ 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} } }
+  #' @seealso [mvdggd]: probability density of a MGGD.
   #'
   #' @author Pierre Santagostini, Nizar Bouhlel
   #' @references N. Bouhlel, A. Dziri, Kullback-Leibler Divergence Between Multivariate Generalized Gaussian Distributions.

man/kldggd.Rd

@@ -29,13 +29,13 @@ Computes the Kullback- Leibler divergence between two random variables distribut
 according to multivariate generalized Gaussian distributions (MGGD) with zero means.
 }
 \details{
-Given \eqn{X_1}, a random vector of \eqn{R^p} (\eqn{p > 1}) distributed according to the MGGD
-with parameters \eqn{(0, \Sigma_1, \beta_1)}
-and \eqn{X_2}, a random vector of \eqn{R^p} distributed according to the MGGD
-with parameters \eqn{(0, \Sigma_2, \beta_2)}.
+Given \eqn{\mathbf{X}_1}, a random vector of \eqn{\mathbb{R}^p} (\eqn{p > 1}) distributed according to the MGGD
+with parameters \eqn{(\mathbf{0}, \Sigma_1, \beta_1)}
+and \eqn{\mathbf{X}_2}, a random vector of \eqn{\mathbb{R}^p} distributed according to the MGGD
+with parameters \eqn{(\mathbf{0}, \Sigma_2, \beta_2)}.
 
 The Kullback-Leibler divergence between \eqn{X_1} and \eqn{X_2} is given by:
-\deqn{ \displaystyle{ KL(X_1||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} } }
+\deqn{ \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} } }
 \deqn{ \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 \eqn{\lambda_1 < ... < \lambda_{p-1} < \lambda_p} are the eigenvalues
@@ -73,6 +73,9 @@ N. Bouhlel, A. Dziri, Kullback-Leibler Divergence Between Multivariate Generaliz
 IEEE Signal Processing Letters, vol. 26 no. 7, July 2019.
 \doi{10.1109/LSP.2019.2915000}
 }
+\seealso{
+\link{mvdggd}: probability density of a MGGD.
+}
 \author{
 Pierre Santagostini, Nizar Bouhlel
 }