Conditional Sampling for Max-Stable Processes with a Mixed Moving Maxima Representation

This paper deals with the question of conditional sampling and prediction for the class of stationary max-stable processes which allow for a mixed moving maxima representation. We develop an exact procedure for conditional sampling using the Poisson point process structure of such processes. For explicit calculations we restrict ourselves to the one-dimensional case and use a finite number of shape functions satisfying some regularity conditions. For more general shape functions approximation techniques are presented. Our algorithm is applied to the Smith process and the Brown-Resnick process. Finally, we compare our computational results to other approaches. Here, the algorithm for Gaussian processes with transformed marginals turns out to be surprisingly competitive.Comment: 35 pages; version accepted for publication in Extremes. The final publication is available at http://link.springer.co

Spatial modeling of drought events using max-stable processes

With their severe environmental and socioeconomic impact, drought events belong to the most far-reaching natural disasters. Effects are tremendous in rain-fed agricultural areas as in Africa. We analyzed and modeled the spatio-temporal statistical behavior of the Normalized Difference Vegetation Index as a risk indicator for drought, reflecting its stochastic effects on vegetation. The study used a data set for Rwanda obtained from multitemporal satellite remote sensor measurements during a 14-year period and divided into season-specific spatial random fields. Maximal deviations from average conditions were modeled with max-stable Brown–Resnick processes taking methodological and computational challenges into account. Those challenges are caused by the large spatial extent and the relatively short time span covered by the data. Extensive simulations enabled us to go beyond the observations and, thus, to estimate several important characteristics of extreme drought events, such as their expected return period

Sampling from Max-Stable Processes Conditional on a Homogeneous Functional via an MCMC Algorithm

For managing risks in climate or environmental fields, max-stable processes can be used as models for spatial and spatio-temporal extremes. When some information on the process of interest is available, conditional simulations provide probability distributions according to the information, allowing us to evaluate risks more precisely. Usually the information available is given by observed values of the process in some sites. Instead, in this work, we focus on the case that aggregated data are given. As condition, we consider a homogeneous functional like the integral or the maximum of the process. Due to the analytic intractability of the involved distributions, we propose a sampling algorithm based on MCMC techniques. The procedure consists of two steps where the second step is based on conditional sampling from a max-linear model. We illustrate the performance of the proposed algorithms in a simulation study and in an example of a real dataset of precipitation observations with a condition stemming from regional climate model outputs