Multivariate design via Copulas

Abstract

Calculating return periods and design quantiles in a multivariate framework is a difficult problem: essentially, this is due to the lack of a natural total order in multi-dimensional Euclidean spaces. This paper tries to make the issue clear. First, we outline a possible way to introduce a coherent notion of multivariate total order, and discuss its conse-5 quences on the calculation of multivariate return period: in particular, the latter is based on Copulas and the Kendall's measure, which provides a consistent notion of multivari-ate quantile. Secondly, we introduce several approaches for the identification of critical design events: these latter quantities are of utmost importance in practical applications, but their calculation is yet limited, due to the lack of a suitable theoretical setting where 10 to embed the problem. Throughout the paper, a case study involving the behavior of a dam is used to illustrate the new concepts outlined in this work.