Exact simulation of max-stable processes.
Max-stable processes play an important role as models for spatial extreme events. Their complex structure as the pointwise maximum over an infinite number of random functions makes their simulation difficult. Algorithms based on finite approximations are often inexact and computationally inefficient. We present a new algorithm for exact simulation of a max-stable process at a finite number of locations. It relies on the idea of simulating only the extremal functions, that is, those functions in the construction of a max-stable process that effectively contribute to the pointwise maximum. We further generalize the algorithm by Dieker & Mikosch (2015) for Brown-Resnick processes and use it for exact simulation via the spectral measure. We study the complexity of both algorithms, prove that our new approach via extremal functions is always more efficient, and provide closed-form expressions for their implementation that cover most popular models for max-stable processes and multivariate extreme value distributions. For simulation on dense grids, an adaptive design of the extremal function algorithm is proposed.
Université de Franche-Comté, Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, 16 Route de Gray, 25030 Besançon cedex, France.
Clément
C
Dombry
École Polytechnique Fédérale de Lausanne, EPFL-FSB-MATHAA-STAT, Station 8, 1015 Lausanne, Switzerland , sebastian.engelke@epfl.ch.
Sebastian
S
Engelke
Universität Siegen, Department Mathematik, Walter-Flex-Straße 3, 57068 Siegen, Germany , oesting@mathematik.uni-siegen.de.
Marco
M
Oesting
Exact simulation; Extremal function; Extreme value distribution; Max-stable process; Spectral measure