Principal component analysis for multivariate extremes

The aim of this project is to explore extensions of the classical dimension reduction technique of principal component analysis (PCA) to the setting of multivariate extreme value theory. In this setting, a challenging aspects is that in the natural modelling framework of non-negative max-stable vectors the orthogonal decomposition in the Euclidean space standing behind the PCA for normally distributed data is no longer applicable. Instead, the max-times-algebra lends itself to a more suitable framework for a decomposition of the dependence structure. This project explores how an optimal projection of a max-stable vector into a lower dimensional space can be implemented efficiently, justified theoretically and how we can interpret the result for specific classes of models