Simulation of Brown–Resnick processes

Abstract

Brown-Resnick processes form a flexible class of stationary max-stable processes based on Gaussian random fields. With regard to applications, fast and accurate simulation of these processes is an important issue. In fact, Brown-Resnick processes that are generated by a dissipative flow do not allow for good finite approximations using the definition of the processes. On large intervals we get either huge approximation errors or very long operating times. Looking for solutions of this problem, we give different representations of the Brown-Resnick processes-including random shifting and a mixed moving maxima representation-and derive various kinds of finite approximations that can be used for simulation purposes. Furthermore, error bounds are calculated in the case of the original process by Brown and Resnick (J Appl Probab 14(4): [732][733][734][735][736][737][738][739] 1977). For a one-parametric class of BrownResnick processes based on the fractional Brownian motion we perform a simulation study and compare the results of the different methods concerning their approximation quality. The presented simulation techniques turn out to provide remarkable improvements.

Generalized Pickands constants and stationary max-stable processes

“…Furthermore, C W ensures that the margins of ξ W are standard Gumbel distributions and it appears thus naturally in the theory of max-stable processes. It plays a crucial role in the simulation of such processes but its numerical evaluation is time intensive and the exact value is, apart from special cases, unknown (Oesting et al, 2012).…”

“…Surprisingly, the constant C δ W appears in the moving maxima representation of ξ W restricted on δZ; see Theorem 8 and Remark 9 in Oesting et al (2012). In the aforementioned contribution, the constant C δ W has already been evaluated numerically for different values of δ in order to simulate samples from the max-stable process ξ W .…”

A software review for extreme value analysis

“…For example, the lack of availability of efficient simulation algorithms for certain classes of max-stable processes (e.g. Oesting et al 2012) is an important theoretical constraint for SpatialExtremes. Stephenson and Gilleland (2005) proposed the creation of a software initiative to develop a reliable and coherent set of tools in order to further increase the use of extreme value methods in various applied fields.…”

Efficient inference for spatial extreme value processes associated to log-Gaussian random functions

“…This idea does not extend easily to nonstationary underlying random fields. We simulate these max-stable processes using representation (3) and Method 2 of Oesting et al (2012), which involves random shifting of the centres of the Brownian motion. We then record which maxima occur simultaneously.…”

“…By contrast, for larger λ values, K could be taken much smaller and accurate representations attained. The best way to simulate Brown-Resnick processes is open; Oesting et al (2012) examine different methods, which perform best on different regions of the parameter space. Their Method 4 appears to perform better on larger spatial domains, analogous to smaller scale parameters in our case.…”