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man/kldstudent.Rd

+% Generated by roxygen2: do not edit by hand
+% Please edit documentation in R/kldstudent.R
+\name{kldstudent}
+\alias{kldstudent}
+\title{Kullback-Leibler Divergence between Centered Multivariate \eqn{t} Distributions}
+\usage{
+kldstudent(nu1, Sigma1, nu2, Sigma2, eps = 1e-06)
+}
+\arguments{
+\item{nu1}{numeric. The degrees of freedom of the first distribution.}
+
+\item{Sigma1}{symmetric, positive-definite matrix. The scatter matrix of the first distribution.}
+
+\item{nu2}{numeric. The degrees of freedom of the second distribution.}
+
+\item{Sigma2}{symmetric, positive-definite matrix. The scatter matrix of the second distribution.}
+
+\item{eps}{numeric. Precision for the computation of the partial derivative of the Lauricella \eqn{D}-hypergeometric function (see Details). Default: 1e-06.}
+}
+\value{
+A numeric value: the Kullback-Leibler divergence between the two distributions,
+with two attributes \code{attr(, "epsilon")} (precision of the partial derivative of the Lauricella \eqn{D}-hypergeometric function,see Details)
+and \code{attr(, "k")} (number of iterations).
+}
+\description{
+Computes the Kullback-Leibler divergence between two random vectors distributed
+according to multivariate \eqn{t} distributions (MTD) with zero location vector.
+}
+\details{
+Given \eqn{X_1}, a random vector of \eqn{R^p} distributed according to the MTD
+with parameters \eqn{(0, \Sigma_1, \nu_1)}
+and \eqn{X_2}, a random vector of \eqn{R^p} distributed according to the MCD
+with parameters \eqn{(0, \Sigma_2, \nu_2)}.
+
+Let \eqn{\lambda_1, \dots, \lambda_p} the eigenvalues of the square matrix \eqn{\Sigma_1 \Sigma_2^{-1}}
+sorted in increasing order: \deqn{\lambda_1 < \dots < \lambda_{p-1} < \lambda_p}
+Depending on the values of these eigenvalues,
+the computation of the Kullback-Leibler divergence of \eqn{X_1} from \eqn{X_2}
+is given by:
+\deqn{ \displaystyle{ f(\mathbf{x}|\nu_1, \boldsymbol{\mu}_1, \Sigma_1) = \frac{\Gamma\left( \frac{\nu_1+p}{2} \right) |\Sigma_1|^{-1/2}}{\Gamma\left( \frac{\nu_1}{2} \right) (\nu_1 \pi)^{p/2}} \left( 1 + \frac{1}{\nu_1} (\mathbf{x}-\boldsymbol{\mu}_1)^T \Sigma_1^{-1} (\mathbf{x}-\boldsymbol{\mu}_1) \right)^{-\frac{\nu_1+p}{2}} \frac{\Gamma\left( \frac{\nu_1+p}{2} \right) |\Sigma_1|^{-1/2}}{\Gamma\left( \frac{\nu_1}{2} \right) (\nu_1 \pi)^{p/2}} \left( 1 + \frac{1}{\nu_1} \mathbf{x}^T \Sigma_1^{-1} \mathbf{x} \right)^{-\frac{\nu_1+p}{2}} } }
+
+where \eqn{F_D^{(p)}} is the Lauricella \eqn{D}-hypergeometric function defined for \eqn{p} variables:
+\deqn{ \displaystyle{ F_D^{(p)}\left(a; b_1, ..., b_p; g; x_1, ..., x_p\right) = \sum\limits_{m_1 \geq 0} ... \sum\limits_{m_p \geq 0}{ \frac{ (a)_{m_1+...+m_p}(b_1)_{m_1} ... (b_p)_{m_p} }{ (g)_{m_1+...+m_p} } \frac{x_1^{m_1}}{m_1!} ... \frac{x_p^{m_p}}{m_p!} } } }
+
+The computation of the partial derivative uses the \code{\link{pochhammer}} function.
+}
+\examples{
+nu1 <- 2
+Sigma1 <- matrix(c(2, 1.2, 0.4, 1.2, 2, 0.6, 0.4, 0.6, 2), nrow = 3)
+nu2 <- 4
+Sigma2 <- matrix(c(1, 0.3, 0.1, 0.3, 1, 0.4, 0.1, 0.4, 1), nrow = 3)
+
+kldstudent(nu1, Sigma1, nu2, Sigma2)
+kldstudent(nu2, Sigma2, nu1, Sigma1)
+
+}
+\references{
+N. Bouhlel and D. Rousseau (2023), Exact Rényi and Kullback-Leibler Divergences Between Multivariate t-Distributions, IEEE Signal Processing Letters.
+\doi{10.1109/LSP.2023.3324594}
+}
+\author{
+Pierre Santagostini, Nizar Bouhlel
+}