Helnormal: Helnormal

In VecDep: Measuring Copula-Based Dependence Between Random Vectors

Helnormal

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 Hellinger distance between \mathbf{X}_{1},...,\mathbf{X}_{k} given the entire correlation matrix \mathbf{R}.

Usage

Helnormal(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 Hellinger distance equals

\mathcal{D}_{(\sqrt{t}-1)^{2}}^{\mathcal{N}}(\mathbf{R}) = 2 - 2 \frac{2^{q/2} |\mathbf{R}|^{1/4}}{\left |\mathbf{I}_{q} + \mathbf{R}_{0}^{-1} \mathbf{R} \right |^{1/2} \prod_{i = 1}^{k} \left | \mathbf{R}_{ii} \right |^{1/4}},

where \mathbf{I}_{q} denotes the identity matrix, and \mathbf{R}_{0} = \text{diag}(\mathbf{R}_{11},\dots,\mathbf{R}_{kk}) is the correlation matrix under independence of \mathbf{X}_{1}, \dots, \mathbf{X}_{k}. The underlying assumption is that the copula of \mathbf{X} is Gaussian.

Value

The correlation-based Hellinger distance 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

minormal for the computation of the Gaussian copula mutual information, Helnormalavar for the computation of the asymptotic variance of the plug-in estimator for the Gaussian copula Hellinger distance.

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)))

Helnormal(R,dim)

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