Pseudo-polar representations in extreme value analysis

I will elaborate on the benefits of certain pseudo-polar representations useful in extreme value analysis. In the context considered here, a pseudo-polar transformation identifies a stochastic vector X with a pair (R,W) of a nonnegative scalar "radial" variable R and an "angular" vector W obtained from projecting the vector X onto a certain unit sphere. Whereas the radial variable R characterizes the severity of an event X, the angular variable W can be interpreted as a direction that indicates relative contributions of components X_j to R. 
I propose to define the radial variable R=rad(X) through a  nonnegative  continuous  mapping rad that is homogeneous, i.e. rad(tx) = t rad(x). Regarding extreme events, we can rely on the theoretical framework of multivariate regular variation  and consider radial excesses above a high radial threshold. In particular, different choices of rad correspond to different kinds of extreme events to be considered. As the radial threshold tends to infinity, this yields limiting behavior with independence between R and W, where R follows the Pareto distribution. An  extension of this approach allows  for modeling of extremes when components X_j of X are upper-bounded with  tail decay of polynomial type towards the upper bound. 
Finally, the practical utility of such modeling will be illustrated on real data.