Measuring dependence between random vectors via optimal transport

Gilles Mordant and Johan Segers
Additional contact information
Johan Segers: Université catholique de Louvain, LIDAM/ISBA, Belgium

No 2021053, LIDAM Reprints ISBA from Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA)

Abstract: To quantify the dependence between two random vectors of possibly different dimensions, we propose to rely on the properties of the 2-Wasserstein distance. We first propose two coefficients that are based on the Wasserstein distance between the actual distribution and a reference distribution with independent components. The coefficients are normalized to take values between 0 and 1, where 1 represents the maximal amount of dependence possible given the two multivariate margins. We then make a quasi-Gaussian assumption that yields two additional coefficients rooted in the same ideas as the first two. These different coefficients are more amenable for distributional results and admit attractive formulas in terms of the joint covariance or correlation matrix. Furthermore, maximal dependence is proved to occur at the covariance matrix with minimal von Neumann entropy given the covariance matrices of the two multivariate margins. This result also helps us revisit the RV coefficient by proposing a sharper normalisation. The two coefficients based on the quasi-Gaussian approach can be estimated easily via the empirical covariance matrix. The estimators are asymptotically normal and their asymptotic variances are explicit functions of the covariance matrix, which can thus be estimated consistently too. The results extend to the Gaussian copula case, in which case the estimators are rank-based. The results are illustrated through theoretical examples. Monte Carlo simulations and a case study involving electroencephalography data are proposed in the supplementary material.

Keywords: Bures-Wasserstein distance; Copula; Delta method; Normal scores rank correlation; RV coefficient (search for similar items in EconPapers)
Pages: 36
Date: 2021-12-01
Note: In: Journal of Multivariate Analysis, 2022, vol. 189, 104912
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:aiz:louvar:2021053

DOI: 10.1016/j.jmva.2021.104912

Access Statistics for this paper

More papers in LIDAM Reprints ISBA from Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA) Voie du Roman Pays 20, 1348 Louvain-la-Neuve (Belgium). Contact information at EDIRC.
Bibliographic data for series maintained by Nadja Peiffer ().