Detection of extremal directions via Euclidean projections

Regular variation provides a convenient theoretical framework to study large events. In the multivariate setting, the dependence structure of the positive extremes is characterized by a measure - the spectral measure - defined on the positive orthant of the unit sphere. This measure gathers information on the localization of extreme events and is often sparse since severe events do not simultaneously occur in all directions. However, it is defined through weak convergence which does not provide a natural way to capture this sparsity structure. In this paper, we introduce the notion of sparse regular variation which allows to better learn the dependence structure of extreme events. This concept is based on the Euclidean projection onto the simplex for which efficient algorithms are known. We show several results for sparsely regularly varying random vectors and prove that under mild assumptions sparse regular variation and regular variation are two equivalent notions. Finally, we provide numerical evidence of our theoretical findings and compare our method with a recent one developed by Goix et al. (2017).