Event-controlled constructions of random fields of maxima with non-max-stable dependence

Max-stable random fields can be constructed according to Schlather (2002) with a random function or a stationary process and a kind of random event magnitude. These are applied for the modelling of natural hazards. We simply extend these event-controlled constructions to random fields of maxima with non-max-stable dependence structure (copula). The theory for the variant with a stationary process is obvious; the parameter(s) of its correlation function is/are determined by the event magnitude. The introduced variant with random functions can only be researched numerically. The scaling of the random function is exponentially determined by the event magnitude. The location parameter of the Gumbel margins depends only on this exponential function in the researched examples; the scale parameter of the margins is normalized. In addition, we propose a method for the parameter estimation for such constructions by using Kendall’s tau. The spatial dependence in relation to the block size is considered therein. Finally, we briefly discuss some issues like the sampling.

💡 Research Summary

The paper revisits the well‑known event‑controlled construction of max‑stable random fields introduced by Schlather (2002) and proposes a systematic extension that yields random fields of maxima whose dependence structure is not max‑stable. The motivation stems from the observation that many natural‑hazard phenomena (e.g., floods, earthquakes, landslides) display spatial dependence that changes with the magnitude of the underlying event, a feature that cannot be captured by traditional max‑stable models whose copula remains invariant to event size.

Two construction pathways are examined. The first uses a stationary stochastic process (typically Gaussian or Matérn) as the underlying field. In this variant the correlation length ℓ is made a deterministic function of the event magnitude m, ℓ(m)=ℓ₀·exp(α·m). The parameter α governs whether larger events broaden (α>0) or narrow (α<0) the spatial influence. By scaling the correlation function in this way, each simulated event produces a different dependence pattern, and the resulting block‑maximum field exhibits a non‑max‑stable copula.

The second pathway replaces the stationary process with a random function fₘ(x)=exp(β·m)·g(x), where g(x) is a mean‑zero, unit‑variance random field independent of m, and β controls the exponential scaling of the function’s amplitude with event magnitude. Marginal distributions are forced to be Gumbel, G(μₘ,σ), with the location parameter μₘ directly linked to the exponential factor exp(β·m) while the scale σ is kept constant. This construction also yields a non‑max‑stable dependence structure, but its analytical tractability is limited; the authors therefore rely on extensive Monte‑Carlo experiments to explore its properties.

A central contribution of the work is a practical parameter‑estimation scheme based on Kendall’s τ, the rank‑based measure of concordance. For a given data set of block maxima, τ is computed for pairs of spatial locations. The authors pre‑compute τ‑curves as functions of α (or β) through simulation, then fit the observed τ by least‑squares matching. This approach bypasses the computationally intensive likelihood calculations required for max‑stable inference and is naturally adapted to the non‑max‑stable setting where the copula varies with block size.

The paper also discusses sampling issues. Because the scaling functions exp(α·m) or exp(β·m) can change dramatically for large m, importance sampling of the event‑size distribution (often modeled by a Pareto tail) is recommended to obtain stable estimates. Moreover, the choice of block size relative to the spatial grid resolution is critical: overly large blocks average out the magnitude‑dependent dependence, while too small blocks lead to high variance in τ estimates. The authors provide guidelines for selecting block sizes that balance bias and variance.

Through a series of numerical experiments, the authors demonstrate that (i) the τ‑based estimator reliably recovers the underlying α or β across a range of scenarios, (ii) the spatial dependence of block maxima indeed varies with block size in the proposed models, and (iii) the non‑max‑stable constructions can reproduce empirical features observed in real hazard data that standard max‑stable models cannot.

In summary, the study offers a coherent theoretical extension of event‑controlled max‑stable constructions to a broader class of random fields with magnitude‑dependent, non‑max‑stable dependence. It supplies both a modeling framework (stationary‑process and random‑function variants) and a feasible inference method (Kendall’s τ matching), thereby opening new avenues for realistic spatial extreme‑value modeling in environmental and engineering applications. Future work may incorporate multivariate event magnitudes, temporal non‑stationarity, and more complex scaling functions to further enhance the applicability of the proposed methodology.