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

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 proposed approach overcomes this burden and makes the first step into ultra high dimensional non-Gaussian dependence modelling by using a divide-and-conquer approach. First, Gaussian methods are applied to split datasets into feasibly small subsets and second, parsimonious and flexible vine copulas are applied thereon. Finally, these sub-models are reconciled into one joint model. Numerical results demonstrating the feasibility of the novel approach in moderate dimensions are provided. The ability of the approach to estimate ultra high dimensional non-Gaussian dependence models in thousands of dimensions is presented.