Prediction of max--stable random functions via excursion sets

Spodarev Evgeny, Ulm University
Makogin Vitalii, Ulm University
Donhauser Tim, Ulm University

We use the concept of excursion sets for the prediction of stationary random functions without any moment existence assumptions. To do so, an excursion metric on the space of random variables is defined which appears to be a kind of a madogram. Using equivalent forms of this metric and the specific choice of excursion levels, we formulate the prediction problem as a (constrained) minimization of a certain target functional which involves the tail dependence function of the observed data as well as the 2--Wasserstein distance on the marginals. The method yields a prediction which is (nearly) exact in marginal distribution. Existence of the solution and a fast stochastic gradient descent method to compute the predictor are discussed. An application to the extrapolation of stationary max--stable random functions illustrates the use of the aforementioned theory. Numerical experiments with simulated and real data exhibiting Fr\'echet--distributed marginals show the practical merits of the approach.

Area: CS60 - Extreme value theory (Evgeny Spodarev)

Keywords: Max-stable random vector, excursion, Gini metric, Wasserstein distance, D-norm, copula, tail dependence function, stochastic gradient descent, Frechet marginal distribution, precipitation data

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