Bayesian Latent Gaussian Models for High-Dimensional Spatial Extremes

In this chapter, we show how to efficiently model high-dimensional extreme peaks-over-threshold events over space in complex non-stationary settings, using extended latent Gaussian models (LGMs), and how to exploit the fitted model in practice for the computation of long-term return levels. The extended LGM framework assumes that the data follow a specific parametric distribution, whose unknown parameters are transformed using a multivariate link function and are then further modeled at the latent level in terms of fixed and random effects that have a joint Gaussian distribution. In the extremal context, we here assume that the response level distribution is described in terms of a Poisson point process likelihood, motivated by asymptotic extreme-value theory, and which conveniently exploits information from all threshold exceedances. This contrasts with the more common data-wasteful approach based on block maxima, which are typically modeled with the generalized extreme-value (GEV) distribution. When conditional independence can be assumed at the response level and latent random effects have a sparse probabilistic structure, fast approximate Bayesian inference becomes possible in very high dimensions, and we here present the recently proposed inference approach called “Max-and-Smooth,” which provides exceptional speedup compared to alternative methods. The proposed methodology is illustrated by application to satellite-derived precipitation data over Saudi Arabia, obtained from the Tropical Rainfall Measuring Mission, with 2738 grid cells and about 20 million spatio-temporal observations in total. Our fitted model captures the spatial variability of extreme precipitation satisfactorily, and our results show that the most intense precipitation events are expected near the southwestern part of Saudi Arabia, along the Red Sea coastline.