Intrinsic Whittle--Matérn fields and sparse spatial extremes

TL;DR

The paper introduces intrinsic Whittle--Matérn fields as SPDE-driven intrinsic Gaussian fields, providing a flexible mechanism to decouple short- and long-range dependence and to perform scalable inference via sparse FEM approximations. It derives variograms, establishes convergence of FEM/rational approximations, and develops likelihood-based inference alongside kriging that remains effective under extrapolation. A sparse spatial extremes framework is then built by embedding intrinsic Whittle--Matérn fields into Brown--Resnick processes, yielding sparse extremal graphical models and efficient estimation and conditional simulation in high dimensions. The authors demonstrate the methodology through kidney function longitudinal data and high-resolution marine heat waves, showing improved predictive performance and enabling extreme-value modeling on unprecedented scales. The framework opens avenues for non-Euclidean, non-stationary, and areal extensions, with broad implications for inference in statistics, spatial extremes, and climate science.

Abstract

Intrinsic Gaussian fields are used in many areas of statistics as models for spatial or spatio-temporal dependence, or as priors for latent variables. However, there are two major gaps in the literature: first, the number and flexibility of existing intrinsic models are very limited; second, theory, fast inference, and software are currently underdeveloped for intrinsic fields. We tackle these challenges by introducing the new flexible class of intrinsic Whittle--Matérn Gaussian random fields obtained as the solution to a stochastic partial differential equation (SPDE). Exploiting sparsity resulting from finite-element approximations, we develop fast estimation and simulation methods for these models. We demonstrate the benefits of this intrinsic SPDE approach for the important task of kriging under extrapolation settings. Leveraging the connection of intrinsic fields to spatial extreme value processes, we translate our theory to an SPDE approach for Brown--Resnick processes for sparse modeling of spatial extreme events. This new paradigm paves the way for efficient inference in unprecedented dimensions. To demonstrate the wide applicability of our new methodology, we apply it in two very different areas: a longitudinal study of renal function data, and the modeling of marine heat waves using high-resolution sea surface temperature data.

Figures (12)

• Figure 1: Simulations of the intrinsic Whittle--Matérn field with $\alpha=3$ and $\beta=1$ (left), and $\alpha=0.3$ and $\beta=1.5$ (right).
• Figure 2: The variogram $\gamma(h)$ in Proposition \ref{['lem:Vard2']} when $d=2$. We choose $\tau$ such that $\gamma(2)=1$ (left), $\gamma(5)-\gamma(4)=1$ (center), and $\gamma(0.7)=1$ (right).
• Figure 3: FEM approximation (points) of an intrinsic Whittle--Matérn variogram with $\alpha=3$ and $\beta=1$ (left) and $\alpha=0.3$ and $\beta=1.5$ (right) when $d=2$ and $\kappa=15$. The true variogram is in solid red. The same mesh was used for the FEM approximation in both panels. The right panel displays a fractional approximation with $m=\tilde{m}=1$ (crosses) and $m=\tilde{m}=2$ (circles).
• Figure 4: Left: Variogram with $\kappa=0.5$ and $\tau$ such that $\gamma(3)=5$. Right: Kriging estimate $\hat{u}(s)$ with observations $Y_i=u(s)+\varepsilon_i$, where $\varepsilon_i \sim N(0,1)$.
• Figure 5: Data (black dots) for two selected patients (top and bottom), and fitted/predicted values obtained by non-intrinsic (left) and intrinsic (right) models when holding out the patient's last $30\%$ of the data during training. Solid lines are posterior means and shaded areas are $90\%$ credible intervals. The vertical red lines separates data used for model fitting (left-hand side) and for validating predictions (right-hand side).

Theorems & Definitions (43)

• Definition 2.1
• Definition 2.2
• Example 2.3
• Definition 2.4
• Proposition 2.5
• Proposition 2.6
• Example 2.7
• Example 2.8
• Proposition 2.9
• Theorem 3.1