Vine copulas for uncertainty quantification: why and how

Abstract

Systems subject to uncertain inputs produce uncertain responses. Uncertainty quantification (UQ) deals with the estimation of the response statistics of systems for which a runnable computational model is available. Problems of interest are those where the computational model is expensive, making Monte Carlo approaches unfeasible and thus calling for cheaper solutions that require fewer runs. In these settings, an accurate representation of the input statistics, including their mutual dependencies, is critical to obtain accurate output estimates. For instance, tail dependencies among the inputs may strongly affect failure probabilities in reliability analysis. Failing to capture such forms of correlations may render accurate estimation of the output statistics hopeless, regardless of the UQ method used to carry out the analysis. The last decade saw a remarkable extension of copula models that can be effectively used to describe multivariate dependence. Among these models, copulas built by tensor product of simple pair copulas (so-called vine copulas) enable a very flexible representation of high-order dependencies [1,2]. In parallel, novel methods have been proposed to perform inference on these copula models. Here we illustrate how these relatively recent advances in copula modeling can be easily combined with virtually any UQ analysis [3], including those methods that assume the input to have a specific multivariate distribution (such as independent inputs). We showcase the approach on a variety of examples, spanning different simulated problems as well as different UQ techniques used to solve them. The analyses are fully carried out with the UQLab toolbox [4], whose simple syntax is also illustrated. [1] T. Bedford and R.M. Cooke (2002) Vines - A new graphical model for dependent random variables. The Annals of Statistics 30(4): 1031-1068. [2] K. Aas, C. Czado, A. Frigessi and H. Bakken (2009) Pair-Copula constructions of multiple dependence. Insurance, Mathematics and Economics 44:182-198. [3] E. Torre, S. Marelli, P. Embrechts and B. Sudret (2019). A general framework for data-driven uncertainty quantification under complex input dependencies using vine copulas. Probabilistic Engineering Mechanics (55): 1-16. [4] S. Marelli and B. Sudret (2014) UQLab: A framework for uncertainty quantification in Matlab. In: Vulnerability, Uncertainty, and Risk (Proc. 2nd Int. Conf. on Vulnerability, Risk Analysis and Management {(ICVRAM2014), Liverpool, United Kingdom)}, chapter 257: 2554-2563