Continuous mixtures of Gaussian processes as models for spatial extremes

TL;DR

This work develops Gaussian location-scale mixtures as a robust framework to model spatial extremes by combining a Gaussian baseline with latent location and/or scale factors: $X(oldsymbol s)=S+R W(oldsymbol s)$, where $W$ is Gaussian and $(S,R)$ are independent of $W$. It introduces a general conditional simulation algorithm and a scalable two-step inference method that avoids heavy numerical integration, enabling efficient estimation of both tail and bulk behavior. A wide range of models is proposed (LM, SM, and LMS classes, plus new instances like LSM1/LSM2) with explicit extremal-dependence properties, and marginal modeling via EGPD to capture full distributional features. The methods are demonstrated on simulated data and a wildfire-weather application in Portugal, showing practical utility for predicting and simulating complete datasets for impact modelling while addressing asymptotic dependence regimes. The approach offers flexible, computationally feasible tools for multivariate and spatial extremes, with potential extensions to space-time settings.

Abstract

Spatial modelling of extreme values allows studying the risk of joint occurrence of extreme events at different locations and is of significant interest in climatic and other environmental sciences. A popular class of dependence models for spatial extremes is that of random location-scale mixtures, in which a spatial "baseline" process is multiplied or shifted by a random variable, potentially altering its extremal dependence behaviour. Gaussian location-scale mixtures retain benefits of their Gaussian baseline processes while overcoming some of their limitations, such as symmetry, light tails and weak tail dependence. We review properties of Gaussian location-scale mixtures and develop novel constructions with interesting features, together with a general algorithm for conditional simulation from these models. We leverage their flexibility to propose extended extreme-value models, that allow for appropriately modelling not only the tails but also the bulk of the data. This is important in many applications and avoids the need to explicitly select the events considered as extreme. We propose new solutions for likelihood inference in parametric models of Gaussian location-scale mixtures, in order to avoid the numerical bottleneck given by the latent location and scale variables that can lead to high computational cost of standard likelihood evaluations. The effectiveness of the models and of the inference methods is confirmed with simulated data examples, and we present an application to wildfire-related weather variables in Portugal. Although not detailed here, the approaches would also be straightforward to use for modelling multivariate (non spatial) data.

Figures (20)

• Figure 1: Simulation of 800 replications of three bivariate Gaussian random variables, with $\rho=\text{cor}(W_1,W_2)=0.25,\, 0.5,\, 0.75$, respectively (top row), and of the corresponding models LM2, SM3 and LMS2 (second to fourth row). All variables were marginally transformed to the uniform scale.
• Figure 2: Estimation of the parameters of a Laplace process on 100 simulated datasets, for each configuration, with true values $\varphi=50$ and $\eta=0.5$ (red lines). Orange boxplots report full maximum likelihood estimates (not computed for Configuration D for computational reasons), while blue boxplots show the estimates based on the likelihood of ratios explained in Section \ref{['section:new_method_step1']}.
• Figure 3: Estimation of the parameters of a Student's t process on 100 simulated datasets for each configuration, with true values (red lines) $\varphi=50$, $\eta=0.5$ and $\nu=2$. Orange boxplots report full maximum likelihood estimates (not computed for Configuration D for computational reasons), while blue boxplots show the two-steps solution for inference explained in Section \ref{['section:estimation_methods']}.
• Figure 4: Three consecutive days of observed FWI over Portugal.
• Figure 5: Three locations in the north of Portugal and the corresponding FWI histograms and fitted EGPD densities, with spatially-varying parameters $\sigma$, $\xi$ and $p$.