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. By numerical examples in hundreds of dimensions, we demonstrate that our approach outperforms the standard method for vine structure selection. Supplementary material for this article is available online.

KEYWORDS:

• Dependence modeling
• Directed acyclic graph
• Graphical model
• Vine copula

Acknowledgments

The authors thank the associate editor and two anonymous referees for their helpful comments and suggestions. The VineCopula R-package (Schepsmeier et al. 2016), on which authors’ code is based, is greatly acknowledged.The first author is thankful for a research stipend of the Technische Universität München. The second author is supported by the German Research foundation (DFG grant CZ 86/4-1). Numerical computations were performed on a Linux cluster supported by DFG grant INST 95/919-1 FUGG.