Regular variation provides a convenient theoretical framework to study large events. The concept is based on the Euclidean projection onto the simplex for efficient algorithms.
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.