Invited talk by Emiliano Torre
Emiliano Torre gives an invited talk at Università degli Studi di Torino titled "Modeling high-dimensional system inputs with copulas for uncertainty quantification problems".
Abstract
The quantification of uncertainty (UQ) in the output Y of a system subject to a multivariate stochastic input X requires to build a map from X to Y , and to study how the components/parameters of X influence Y.
Advanced UQ methods based on spectral decomposition, such as polynomial chaos expansions (PCE, [1]), allow one to accomplish this task when the components of X are mutually independent, or when they can be mapped onto independent variables by means of isoprobabilistic transformations like Rosenblatt or Nataf [2]. The Rosenblatt transform, however, is often difficult to compute, both analytically and numerically, especially in large dimensions. The Nataf and related transforms, instead, are applicable under restrictive assumptions about the joint distribution of the input only (such as normality).
In this contribution we propose an effective approach to model the dependence structure of X via vine copulas. Vine copulas consist of a factorization of the joint dependence structure (copula) of X into pair copulas of its components [3,4]. The advantage of this approach is two-fold: it grants great flexibility in modeling the pairwise dependencies of the data, while at the same time providing a natural framework to transform the input model into the independent unit hypercube via the Rosenblatt transform. The latter can be finally used to build a map of the input X onto the output Y by, e.g., PCE.
References
[1] R.G. Ghanem and P.D. Spanos (1991) Stochastic finite elements: a spectral approach. Springer-Verlag, New York.
[2] R. Lebrun and A. Dutfoy, A. (2009) An innovating analysis of the Nataf transformation from the copula viewpoint Prob. Eng. Mech., 24, 312-320.
[3] T. Bedford and R.M. Cooke (2002) Vines - A new graphical model for dependent random variables. The Annals of Statistics 30(4): 1031-1068.
[4] K. Aas, C. Czado, A. Frigessi and H. Bakken (2009) Pair-Copula constructions of multiple dependence. Insurance, Mathematics and Economics 44:182-198.