minormal: minormal

In VecDep: Measuring Copula-Based Dependence Between Random Vectors

minormal

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 correlation-based mutual information between \mathbf{X}_{1},...,\mathbf{X}_{k} given the entire correlation matrix \mathbf{R}.

Usage

minormal(R, dim)

Arguments

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

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},

the mutual information equals

\mathcal{D}_{t \ln(t)}^{\mathcal{N}}(\mathbf{R}) = - \frac{1}{2} \ln \left (\frac{|\mathbf{R}|}{\prod_{i = 1}^{k} \left |\mathbf{R}_{ii} \right |} \right ).

The underlying assumption is that the copula of \mathbf{X} is Gaussian.

Value

The correlation-based mutual information between \mathbf{X}_{1},...,\mathbf{X}_{k}.

References

De Keyser, S. & Gijbels, I. (2024). Parametric dependence between random vectors via copula-based divergence measures. Journal of Multivariate Analysis 203:105336.
doi: https://doi.org/10.1016/j.jmva.2024.105336.

See Also

Helnormal for the computation of the Gaussian copula Hellinger distance, minormalavar for the computation of the asymptotic variance of the plug-in estimator for the Gaussian copula mutual information, miStudent for the computation of the Student-t 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)))

minormal(R,dim)

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