Extrapolation of max-stable random fields with Fr\'echet marginals

TL;DR

This paper introduces a novel method for predicting stationary max-stable random fields with heavy-tailed Fréchet marginals, leveraging level set extrapolation without moment assumptions, and demonstrates its effectiveness on simulated and real data.

Contribution

It develops an extrapolation-based prediction method for max-stable fields with explicit connections to excursion metrics, applicable to heavy-tailed distributions, and proves the existence of the predictor.

Findings

Method effectively predicts max-stable fields with heavy tails.

Explicit connection established between excursion metric and Davis-Resnick distance.

Validated on simulated data and real precipitation records.

Abstract

We propose a method for the prediction of stationary max--stable random fields with $\alpha$-Fr\'echet marginal distribution $H_\alpha$. The method is suitable to cope with heavy tails for $\alpha\in(0,2)$ and is (approximately) exact in marginal distributions. It is based on a recent extrapolation approach via level sets which requires no moment assumptions. An explicit connection between the excursion metric and the Davis-Resnick distance is established. The existence of the predictor is proven. The non-uniqueness of the forecast is demonstrated on several examples. The method is tested on multiple simulated time series and random fields as well as applied to real data of annual maximum precipitation.