diststudent() R function from [multvardiv]

Distance/Divergence between Centered Multivariate $t$ Distributions

Computes the distance or divergence (Renyi divergence, Bhattacharyya distance or Hellinger distance) between two random vectors distributed according to multivariate $t$ distributions (MTD) with zero mean vector.


diststudent(nu1, Sigma1, nu2, Sigma2,
                   dist = c("renyi", "bhattacharyya", "hellinger"),
                   bet = NULL, eps = 1e-06)

Arguments

nu1: numeric. The degrees of freedom of the first distribution.
Sigma1: symmetric, positive-definite matrix. The correlation matrix of the first distribution.
nu2: numeric. The degrees of freedom of the second distribution.
Sigma2: symmetric, positive-definite matrix. The correlation matrix of the second distribution.
dist: character. The distance or divergence used. One of "renyi" (default), "battacharyya" or "hellinger".
bet: numeric, positive and not equal to 1. Order of the Renyi divergence. Ignored if distance="bhattacharyya" or distance="hellinger".
eps: numeric. Precision for the computation of the partial derivative of the Lauricella $D$ -hypergeometric function (see Details). Default: 1e-06.

Returns

A numeric value: the divergence between the two distributions, with two attributes attr(, "epsilon") (precision of the result of the Lauricella $D$ -hypergeometric function,see Details) and attr(, "k") (number of iterations).

Details

Given $X_1$ , a random vector of $\mathbb{R}^p$ distributed according to the MTD with parameters $(\nu_1, \mathbf{0}, \Sigma_1)$

and $X_2$ , a random vector of $\mathbb{R}^p$ distributed according to the MTD with parameters $(\nu_2, \mathbf{0}, \Sigma_2)$ .

Let $\delta_1 = \frac{\nu_1 + p}{2} \beta$ , $\delta_2 = \frac{\nu_2 + p}{2} (1 - \beta)$

and $\lambda_1, \dots, \lambda_p$ the eigenvalues of the square matrix $\Sigma_1 \Sigma_2^{-1}$

sorted in increasing order:

\lambda_1 < \dots < \lambda_{p-1} < \lambda_p

The Renyi divergence between $X_1$ and $X_2$ is:

\begin{aligned}D_R^\beta(\mathbf{X}_1||\mathbf{X}_1) & = & \displaystyle{\frac{1}{\beta - 1} \bigg[ \beta \ln\left(\frac{\Gamma\left(\frac{\nu_1+p}{2}\right) \Gamma\left(\frac{\nu_2}{2}\right) \nu_2^{\frac{p}{2}}}{\Gamma\left(\frac{\nu_2+p}{2}\right) \Gamma\left(\frac{\nu_1}{2}\right) \nu_1^{\frac{p}{2}}}\right) + \ln\left(\frac{\Gamma\left(\frac{\nu_2+p}{2}\right)}{\Gamma\left(\frac{\nu_2}{2}\right)}\right) + \ln\left(\frac{\Gamma\left(\delta_1 + \delta_2 - \frac{p}{2}\right)}{\Gamma(\delta_1 + \delta_2)}\right) } \\&& \displaystyle{- \frac{\beta}{2} \sum_{i=1}^p{\ln\lambda_i} + \ln F_D \bigg]}\end{aligned}

with $F_D$ given by:

If $\displaystyle{\frac{\nu_1}{\nu_2} \lambda_1 \> 1}$ :

c(" $\n$ ", " $\\displaystyle{ F_D = F_D^{(p)}{\\bigg( \\delta_1, \\underbrace{\\frac{1}{2}, \\dots, \\frac{1}{2}}_p; \\delta_1+\\delta_2; 1-\\frac{\\nu_2}{\\nu_1 \\lambda_1}, \\dots, 1-\\frac{\\nu_2}{\\nu_1 \\lambda_p} \\bigg)} }\n$ ")
If $\displaystyle{\frac{\nu_1}{\nu_2} \lambda_p \< 1}$ :

c(" $\n$ ", " $\\displaystyle{ F_D = \\prod_{i=1}^p{\\left(\\frac{\\nu_1}{\\nu_2} \\lambda_i\\right)^{\\frac{1}{2}}} F_D^{(p)}\\bigg(\\delta_2, \\underbrace{\\frac{1}{2}, \\dots, \\frac{1}{2}}_p; \\delta_1+\\delta_2; 1-\\frac{\\nu_1}{\\nu_2}\\lambda_1, \\dots, 1-\\frac{\\nu_1}{\\nu_2}\\lambda_p\\bigg) }\n$ ")
If $\displaystyle{\frac{\nu_1}{\nu_2} \lambda_1 \< 1}$ and $\displaystyle{\frac{\nu_1}{\nu_2} \lambda_p \> 1}$ :

c(" $\n$ ", " $\\displaystyle{ F_D = \\left(\\frac{\\nu_2}{\\nu_1} \\frac{1}{\\lambda_p}\\right)^{\\delta_2} \\prod_{i=1}^p\\left(\\frac{\\nu_1}{\\nu_2}\\lambda_i\\right)^\\frac{1}{2} F_D^{(p)}\\bigg(\\delta_2, \\underbrace{\\frac{1}{2}, \\dots, \\frac{1}{2}}_p, \\delta_1+\\delta_2-\\frac{p}{2}; \\delta_1+\\delta2; 1-\\frac{\\lambda_1}{\\lambda_p}, \\dots, 1-\\frac{\\lambda_{p-1}}{\\lambda_p}, 1-\\frac{\\nu_2}{\\nu_1}\\frac{1}{\\lambda_p}\\bigg) }\n$ ")

where $F_D^{(p)}$ is the Lauricella $D$ -hypergeometric function defined for $p$ variables:

\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!} } }

Its computation uses the lauricella function.

The Bhattacharyya distance is given by:

D_B(\mathbf{X}_1||\mathbf{X}_2) = \frac{1}{2} D_R^{1/2}(\mathbf{X}_1||\mathbf{X}_2)

And the Hellinger distance is given by:

D_H(\mathbf{X}_1||\mathbf{X}_2) = 1 - \exp{\left(-\frac{1}{2} D_R^{1/2}(\mathbf{X}_1||\mathbf{X}_2)\right)}

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)

# Renyi divergence
diststudent(nu1, Sigma1, nu2, Sigma2, bet = 0.25)
diststudent(nu2, Sigma2, nu1, Sigma1, bet = 0.25)

# Bhattacharyya distance
diststudent(nu1, Sigma1, nu2, Sigma2, dist = "bhattacharyya")
diststudent(nu2, Sigma2, nu1, Sigma1, dist = "bhattacharyya")

# Hellinger distance
diststudent(nu1, Sigma1, nu2, Sigma2, dist = "hellinger")
diststudent(nu2, Sigma2, nu1, Sigma1, dist = "hellinger")

References

N. Bouhlel and D. Rousseau (2023), Exact Rényi and Kullback-Leibler Divergences Between Multivariate t-Distributions, IEEE Signal Processing Letters. tools:::Rd_expr_doi("10.1109/LSP.2023.3324594")

Author(s)

Pierre Santagostini, Nizar Bouhlel

multvardiv package Read PDF manual

Maintainer: Pierre Santagostini
License: GPL (>= 3)
Last published: 2024-12-20

Useful links

diststudent function

Distance/Divergence between Centered Multivariate ttt Distributions