Dependence Modeling in Ultra High Dimensions with Vine Copulas and the Graphical Lasso

TL;DR

This paper introduces a divide-and-conquer approach combining Gaussian methods and vine copulas to efficiently model ultra high dimensional non-Gaussian dependencies, overcoming computational challenges.

Contribution

It presents a novel scalable method for ultra high dimensional dependence modeling using a combination of Gaussian splitting and vine copulas, enabling analysis in thousands of dimensions.

Findings

Feasibility demonstrated in moderate dimensions

Ability to estimate models in thousands of dimensions

Efficient handling of non-Gaussian dependencies

Abstract

To model high dimensional data, Gaussian methods are widely used since they remain tractable and yield parsimonious models by imposing strong assumptions on the data. Vine copulas are more flexible by combining arbitrary marginal distributions and (conditional) bivariate copulas. Yet, this adaptability is accompanied by sharply increasing computational effort as the dimension increases. The approach proposed in this paper overcomes this burden and makes the first step into ultra high dimensional non-Gaussian dependence modeling by using a divide-and-conquer approach. First, we apply Gaussian methods to split datasets into feasibly small subsets and second, apply parsimonious and flexible vine copulas thereon. Finally, we reconcile them into one joint model. We provide numerical results demonstrating the feasibility of our approach in moderate dimensions and showcase its ability to estimate ultra high dimensional non-Gaussian dependence models in thousands of dimensions.