Matrix and graph representations of vine copula structures
This work provides clearer structural foundations for vine copulas, which are incremental improvements for researchers in multivariate probability modeling.
The paper tackles the problem of ambiguous representations of vine copula structures by establishing equivalences between different graph representations and developing matrix construction algorithms. It shows that given a perfect elimination ordering, a vine structure can always be uniquely represented with a matrix, and proves the equivalence of two matrix-building algorithms.
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. Nápoles 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.