Geometric modelling of spatial extremes

Recent developments in extreme value statistics have established the so-called geometric approach as a powerful modelling tool for multivariate extremes. We tailor these methods to the case of spatial modelling and examine their efficacy at inferring extremal dependence and performing extrapolation. The geometric approach is based around a limit set described by a gauge function, which is a key target for inference. We consider a variety of spatially-parameterised gauge functions and perform inference on them by building on the framework of Wadsworth and Campbell (2024), where extreme radii are modelled via a truncated gamma distribution. We also consider spatial modelling of the angular distribution, for which we propose two candidate models. Estimation of extreme event probabilities is possible by combining draws from the radial and angular models respectively. We compare our method with two other established frameworks for spatial extreme value analysis and show that our approach generally allows for unbiased, albeit more uncertain, inference compared to the more classical models. We illustrate the methodology on a space weather dataset of daily geomagnetic field fluctuations.

💡 Research Summary

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The paper introduces a novel spatial extreme‑value modelling framework that adapts the geometric approach—originally developed for multivariate extremes—to spatial data. The central idea is to transform observations to standard exponential margins, then decompose each d‑dimensional observation vector X into a radial component R = ‖X‖ (typically the ℓ1‑norm) and an angular component W = X / R that lies on the unit simplex. Under mild regularity conditions, the joint density of (R, W) for large R can be approximated by

 f_{R|W}(r | w) ∝ r^{d‑1} exp{‑r g(w)}

where g(·) is a homogeneous gauge function that characterises the limit set G = {x : g(x) ≤ 1}. Building on Wadsworth and Campbell (2024), the authors assume that exceedances of a high radial threshold r_τ(w) follow a left‑truncated Gamma distribution

 R | {W = w, R > r_τ(w)} ∼ truncGamma(a, g(w))

with shape a (often set to the dimension d) and rate equal to the gauge evaluated at the observed angle. Consequently, inference on extreme dependence reduces to estimating the gauge function across the spatial domain.

To make the approach practical for spatial processes, the authors propose several parametric families of spatially‑parameterised gauge functions:

1. Spatial Gaussian gauge: g_G(x) = (x^{1/2})ᵀ Σ^{-1} x^{1/2}, where Σ is a correlation matrix derived from a powered‑exponential correlation function ρ(h) = exp{‑(h/λ)^κ}. This form captures asymptotic independence (AI) and is computationally cheap.

2. Spatial Laplace gauge: g_L(x) = √{xᵀ Σ^{-1} x}, also suitable for AI but derived from a Laplace random field.

3. Spatial Generalised Gaussian gauge: g_{GG}(x) =