Exact simulation of max-stable processes

Abstract

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 simulation highly nontrivial. Algorithms based on finite approximations that are used in practice are often not exact and computationally inefficient. We will present two algorithms for exact simulation of a max-stable process at a finite number of locations. The first algorithm generalizes the approach by Dieker and Mikosch [2015] for Brown-Resnick processes and it is based on simulation from the spectral measure. The second algorithm relies on the idea to simulate only the extremal functions, that is, those functions in the construction of a max-stable process that effectively contribute to the pointwise maximum. We study the complexity of both algorithms and prove that the second procedure is always more efficient. Moreover, we provide closed expressions for their implementation that cover the most popular models for max-stable processes and extreme value copulas. For simulation on dense grids, an adaptive design of the second algorithm is proposed.

Generalized Pickands constants and stationary max-stable processes

“…The above Corollary states that the simulation problem of Pickands constant H δ W is equivalent to the problem of simulating the constants C δ W in spatial extreme value theory, provided that ξ W admits an M3 representation and the DiekerYakir representation for H δ W holds. This is a fruitful observation since there is active research on the simulation of max-stable processes (Dieker and Mikosch, 2015;Dombry et al, 2016) and even of the constant C δ W (Oesting et al, 2012). We conclude this section with several examples.…”

When Gumbel met Galambos

“…Beyond highlighting the kinship between the Gumbel and Galambos families of copulas, formulas (6.12)-(6.13) lead to a unified simulation algorithm for these two dependence structures. This procedure, adapted from [7,35], is presented in a broader context in [2]. Gumbel and Galambos are thereby united, at last.…”

Full likelihood inference for max‐stable data

“…In the remainder of the paper, we shall always consider max‐stable processes Z ( s ) with unit Fréchet marginal distributions. Representation is useful to construct a wide variety of max‐stable processes, such as the Smith model (Smith, ), the Schlather model (Schlather, ), the Brown–Resnick model (Kabluchko, Schlather, & de Haan, ), the extremal‐ t model (Opitz, ), and the Tukey g ‐and‐ h model (Xu & Genton, ), and to simulate from them (Dombry, Engelke, & Oesting, ; Schlather, ). Multivariate max‐stable models can be constructed similarly by substituting the processes W j ( s ) in Equation by analogous random vectors W j = ( W j 1 ,…, W jD ) ⊤ .…”

“…Realizations from the Brown–Resnick model on [0,1] 2 , with semivariogram γ ( h ) = (‖ h ‖/ λ ) ν , and parameter values taken according to Table , covering short‐ to long‐range‐dependent processes (top to bottom) and rough to smooth processes (left to right). Realizations, simulated exactly using the algorithm of Dombry et al (), are displayed on standard Gumbel margins…”