Multivariate Extreme Value Methods

TL;DR: In this paper, the authors show how the use of Copulas may help in dealing with (and, possibly, solving) these problems, and how to construct multivariate Extreme Value models suitable for applications.

Abstract: Multivariate extremes occur in several hydrologic and water resources problems. Despite their practical relevance, the real-life decision making as well as the number of designs based on an explicit treatment of multivariate variables is yet limited as compared to univariate analysis. A first problem arising when working in a multidimensional context is the lack of a “natural” definition of extreme values: essentially, this is due to the fact that different concepts of multivariate order and failure regions are possible. Also, in modeling multivariate extremes, central is the issue of dependence between the variables involved: again, several approaches are possible. A further practical problem is represented by the construction of multivariate Extreme Value models suitable for applications: the task is indeed difficult from a mathematical point of view. In addition, the calculation of multivariate Return Periods, quantiles, and design events, which represent quantities of utmost interest in applications, is rather tricky. In this Chapter we show how the use of Copulas may help in dealing with (and, possibly, solving) these problems.