Representing Sparse Gaussian DAGs as Sparse R-Vines Allowing for Non-Gaussian Dependence

Abstract

Modeling dependence in high dimensional systems has become an increasingly important topic. Most approaches rely on the assumption of a multivariate Gaussian distribution such as statistical models on directed acyclic graphs (DAGs). They are based on modeling conditional independencies and are scalable to high dimensions. In contrast, vine copula models accommodate more elaborate features like tail dependence and asymmetry, as well as independent modeling of the marginals. This flexibility comes however at the cost of exponentially increasing complexity for model selection and estimation. We show a novel connection between DAGs with limited number of parents and truncated vine copulas under sufficient conditions. This motivates a more general procedure exploiting the fast model selection and estimation of sparse DAGs while allowing for non-Gaussian dependence using vine copulas. We demonstrate in a simulation study and using a high dimensional data application that our approach outperforms standard methods for vine structure estimation.

Parsimonious graphical dependence models constructed from vines

“…This includes nongreedy methods for high‐dimensional truncated vines, with and without latent variables. In the case of no latent variables, methods have been developed by Müller & Czado () and Müller ().…”

Vine Copula Based Modeling

“…This method is implemented in the vinecopulib library. A separate line of research (Müller & Czado 2018, 2019a exploits connections between vine copulas and Gaussian directed acyclic graphs to find sparsity patterns. A completely different approach is to use dimension reduction techniques before employing a copula model (as in Tagasovska et al 2019).…”

Dependence modelling in ultra high dimensions with vine copulas and the Graphical Lasso

“…We will however come back to it later for an improved version. The algorithm proposed by Müller and Czado (2017a) uses graphical models, more precisely directed acyclic graphs (DAGs), to find parsimonious structures and set the majority of pair copulas to the independence copula in larger datasets, which eases the computational effort. However, for both their and Dißmann's algorithm, more than 500 − 1000 dimensions are not solvable because of the quadratically increasing effort in terms of computation time and memory.…”