bwd1: bwd1

In VecDep: Measuring Copula-Based Dependence Between Random Vectors

bwd1

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 Bures-Wasserstein coefficient \mathcal{D}_{1} between \mathbf{X}_{1},...,\mathbf{X}_{k} given the entire correlation matrix \mathbf{R}.

Usage

bwd1(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 coefficient \mathcal{D}_{1} equals

\mathcal{D}_{1}(\mathbf{R}) = \frac{d_{W}^{2}(\mathbf{R},\mathbf{I}_{q}) - \sum_{i=1}^{k}d_{W}^{2}(\mathbf{R}_{ii},\mathbf{I}_{d_{i}})}{\text{sup}_{\mathbf{A} \in \Gamma(\mathbf{R}_{11}, \dots, \mathbf{R}_{kk})}d_{W}^{2}(\mathbf{A},\mathbf{I}_{q}) - \sum_{i=1}^{k}d_{W}^{2}(\mathbf{R}_{ii},\mathbf{I}_{d_{i}})},

where d_{W} stands for the Bures-Wasserstein distance, \Gamma(\mathbf{R}_{11}, \dots, \mathbf{R}_{kk}) denotes the set of all correlation matrices with diagonal blocks \mathbf{R}_{ii} for i = 1, \dots, k, and \mathbf{I}_{q} is the identity matrix. The underlying assumption is that the copula of \mathbf{X} is Gaussian.

Value

The first Bures-Wasserstein dependence coefficient \mathcal{D}_{1} between \mathbf{X}_{1},...,\mathbf{X}_{k}.

References

De Keyser, S. & Gijbels, I. (2024). High-dimensional copula-based Wasserstein dependence. doi: https://doi.org/10.48550/arXiv.2404.07141.

See Also

bwd2 for the computation of the second Bures-Wasserstein dependence coefficient \mathcal{D}_{2}, bwd1avar for the computation of the asymptotic variance of the plug-in estimator for \mathcal{D}_{1}.

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)))

bwd1(R,dim)

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