Matrix and graph representations of vine copula structures

TL;DR

This paper explores the structures of vine copulas, demonstrating their equivalence in graph representations, introducing a unique matrix representation based on perfect elimination orderings, and providing algorithms with runtime analysis.

Contribution

It introduces a new method for constructing matrix representations of vine copulas using cherry tree sequences and proves their equivalence with existing methods.

Findings

Graph representations of vine copulas are equivalent.

A new matrix construction method via cherry tree sequences is proposed.

The algorithms for matrix construction are shown to be equivalent under the same ordering.

Abstract

Vine copulas can efficiently model multivariate probability distributions. This paper focuses on a more thorough understanding of their structures, since in the literature, vine copula representations are often ambiguous. The graph representations include the original, cherry and chordal graph sequence structures, which we show equivalence between. Importantly we also show a new result, namely that when a perfect elimination ordering of a vine structure is given, then it can always be uniquely represented with a matrix. O. M. N\'apoles has shown a way to represent vines in a matrix, and we algorithmify this previous approach, while also showing a new method for constructing such a matrix, through cherry tree sequences. We also calculate the runtime of these algorithms. Lastly, we prove that these two matrix-building algorithms are equivalent if the same perfect elimination ordering is being used.