miStudent: miStudent

In VecDep: Measuring Copula-Based Dependence Between Random Vectors

miStudent

Description

Given a q-dimensional random vector \mathbf{X} = (\mathbf{X}_{1},...,\mathbf{X}_{k}) with \mathbf{X}_{i} a d_{i}-dimensional random vector, i.e., q = d_{1} + ... + d_{k}, this function computes the Student-t mutual information between \mathbf{X}_{1},...,\mathbf{X}_{k} given the entire correlation matrix \mathbf{R} and the degrees of freedom nu.

Usage

miStudent(R, dim, nu)

Arguments

R	The correlation matrix of \mathbf{X}.
dim	The vector of dimensions (d_{1},...,d_{k}).
nu	The degrees of freedom.

Details

Given a correlation matrix

\mathbf{R} = \begin{pmatrix} \mathbf{R}_{11} & \mathbf{R}_{12} & \cdots & \mathbf{R}_{1k} \\ \mathbf{R}_{12}^{\text{T}} & \mathbf{R}_{22} & \cdots & \mathbf{R}_{2k} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{R}_{1k}^{\text{T}} & \mathbf{R}_{2k}^{\text{T}} & \cdots & \mathbf{R}_{kk} \end{pmatrix},

and a certain amount of degrees of freedom \nu > 0, the Student-t mutual information equals

\mathcal{D}_{t \ln(t)}^{\text{S}}(\mathbf{R},\nu) = - \frac{1}{2} \ln \left (\frac{|\mathbf{R}|}{\prod_{i = 1}^{k} \left |\mathbf{R}_{ii} \right |} \right ) + K(\nu),

where

\hspace{-2cm} K(\nu) = \ln \left (\frac{\Gamma((q+\nu)/2) \Gamma(\nu/2)^{k-1}}{\prod_{i = 1}^{k} \Gamma((d_{i} + \nu)/2)} \right ) + \sum_{i = 1}^{k} \left [\frac{d_{i} + \nu}{2} \psi((d_{i} + \nu)/2) \right ]

\hspace{5.4cm} - \frac{q + \nu}{2} \psi((q + \nu)/2) - \frac{\nu}{2}(k-1)\psi(\nu/2),

with \Gamma the gamma function and \psi the digamma function. The underlying assumption is that the copula of \mathbf{X} is Student-t.

Value

The Student-t mutual information between \mathbf{X}_{1},...,\mathbf{X}_{k}.

References

De Keyser, S. & Gijbels, I. (2024). Hierarchical variable clustering via copula-based divergence measures between random vectors. International Journal of Approximate Reasoning 165:109090. doi: https://doi.org/10.1016/j.ijar.2023.109090.

See Also

minormal for the computation of the Gaussian copula mutual information.

Examples

q = 10
dim = c(1,2,3,4)

# AR(1) correlation matrix with correlation 0.5
R = 0.5^(abs(matrix(1:q-1,nrow = q, ncol = q, byrow = TRUE) - (1:q-1)))

# Degrees of freedom
nu = 7

miStudent(R,dim,nu)

VecDep documentation built on April 4, 2025, 5:14 a.m.