Realistic and Fast Modeling of Spatial Extremes over Large Geographical Domains

Abstract

Various natural phenomena, such as precipitation, generally exhibit spatial extremal dependence at short distances only, while the dependence usually fades away as the distance between sites increases arbitrarily. However, the available models proposed in the literature for spatial extremes, which are based on max-stable or Pareto processes or comparatively less computationally demanding "sub-asymptotic" models based on Gaussian location and/or scale mixtures, generally assume that spatial extremal dependence persists across the entire spatial domain. This is a clear limitation when modeling extremes over large geographical domains, but surprisingly, it has been mostly overlooked in the literature. In this paper, we develop a more realistic Bayesian framework based on a novel Gaussian scale mixture model, where the Gaussian process component is defined by a stochastic partial differential equation that yields a sparse precision matrix, and the random scale component is modeled as a lowrank Pareto-tailed or Weibull-tailed spatial process determined by compactly supported basis functions. We show that our proposed model is approximately tail-stationary despite its nonstationary construction in terms of basis functions, and we demonstrate that it can capture a wide range of extremal dependence structures as a function of distance. Furthermore, the inherently sparse structure of our spatial model allows fast Bayesian computations, even in high spatial dimensions, based on a customized Markov chain Monte Carlo algorithm, which prioritize calibration in the tail. In our application, we fit our model to analyze heavy monsoon rainfall data in Bangladesh. Our study indicates that the proposed model outperforms some natural alternatives, and that the model fits precipitation extremes satisfactorily well. Finally, we use the fitted model to draw inferences on long-term return levels for marginal precipitation at each site, and for spatial aggregates.

Advances in statistical modeling of spatial extremes

“…These limitations also make it difficult to adapt such models to the case of spatio-temporal data. Hazra et al (2022) recently attempted to address these issues by proposing a Gaussian scale mixture model extension, which can capture short-range asymptotic dependence, mid-range asymptotic independence, and long-range exact independence, by replacing R with a suitable spatial process R(s); see also Krupskii and Huser (2022).…”

An efficient workflow for modelling high-dimensional spatial extremes

“…Additionally, these models must often rely on less efficient inference methods. Further improvements are given by the kernel convolution model of Krupskii and Huser (2022), more recent scale-mixture models such as that of Hazra et al (2021), and the spatial conditional extremes model of Wadsworth and Tawn (2022), all allowing for flexible modelling of different extremal dependence classes as a function of distance. The spatial conditional extremes model allows for a particularly simple way of modelling spatial extremes.…”