Sparse regular variation
This work addresses the challenge of modeling sparse extreme events in multivariate data, which is incremental as it builds upon existing regular variation theory to better handle sparsity.
The paper tackles the problem of learning the dependence structure of extreme events in multivariate settings by introducing sparse regular variation, which captures sparsity in the spectral measure. The result is a new theoretical framework that enables detection of extremal directions, as demonstrated through numerical examples.
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 has often a sparse support 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 prove that under mild assumptions sparse regular variation and regular variation are two equivalent notions and we establish several results for sparsely regularly varying random vectors. Finally, we illustrate on numerical examples how this new concept allows one to detect extremal directions.