Directional multivariate extremes in environmental phenomena

Abstract

Several environmental phenomena can be described by different correlated variables that must be considered jointly in order to be more representative of the nature of these phenomena. For such events, identification of extremes is inappropriate if it is based on marginal analysis. Extremes have usually been linked to the notion of quantile, which is an important tool to analyze risk in the univariate setting. We propose to identify multivariate extremes and analyze environmental phenomena in terms of the directional multivariate quantile, which allows us to analyze the data considering all the variables implied in the phenomena, as well as look at the data in interesting directions that can better describe an environmental catastrophe. Since there are many references in the literature that propose extremes detection based on copula models, we also generalize the copula method by introducing the directional approach. Advantages and disadvantages of the non-parametric proposal that we introduce and the copula methods are provided in the paper. We show with simulated and real data sets how by considering the first principal component direction we can improve the visualization of extremes. Finally, two cases of study are analyzed: a synthetic case of flood risk at a dam (a 3-variable case), and a real case study of sea storms (a 5-variable case). Keywords Directional Multivariate Extremes in Environmental PhenomenaRaúl Torres (ratorres@est-econ.uc3m.es) Carlo De Michele (carlo.demichele@polimi.it) Henry Laniado (hlaniado@est-econ.uc3m.es) Rosa E. Lillo (lillo@est-econ.uc3m.es) AbstractSeveral environmental phenomena can be described by different correlated variables that must be considered jointly in order to be more representative of the nature of these phenomena. For such events, identification of extremes is inappropriate if it is based on marginal analysis. Extremes have usually been linked to the notion of quantile, which is an important tool to analyze risk in the univariate setting. We propose to identify multivariate extremes and analyze environmental phenomena in terms of the directional multivariate quantile, which allows us to analyze the data considering all the variables implied in the phenomena, as well as look at the data in interesting directions that can better describe an environmental catastrophe. Since there are many references in the literature that propose extremes detection based on copula models, we also generalize the copula method by introducing the directional approach. Advantages and disadvantages of the non-parametric proposal that we introduce and the copula methods are provided in the paper. We show with simulated and real data sets how by considering the first principal component direction we can improve the visualization of extremes. Finally, two cases of study are analyzed: a synthetic case of flood risk at a dam (a 3−variable case), and a real case study of sea storms (a 5−variable case).

Estimation of the multivariate conditional tail expectation for extreme risk levels: Illustration on environmental data sets

“…In Salvadori, Durante, and Perrone (2013), multivariate RPs were estimated by using a semiparametric approximation of Kendall's distribution function. A directional multivariate quantile was proposed by Torres, De Michele, Laniado, and Lillo (2017) to detect multivariate extremes in environmental phenomena.…”