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 data 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 data 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 speed-up 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 south-western part of Saudi Arabia, along the Red Sea coastline.