High-dimensional modeling of spatial and spatio-temporal conditional extremes using INLA and Gaussian Markov random fields

Abstract

The conditional extremes framework allows for event-based stochastic modeling of dependent extremes, and has recently been extended to spatial and spatio-temporal settings. After standardizing the marginal distributions and applying an appropriate linear normalization, certain non-stationary Gaussian processes can be used as asymptotically-motivated models for the process conditioned on threshold exceedances at a fixed reference location and time. In this work, we adapt existing conditional extremes models to allow for the handling of large spatial datasets. This involves specifying the model for spatial observations at d locations in terms of a latent $$m\ll d$$ m ≪ d dimensional Gaussian model, whose structure is specified by a Gaussian Markov random field. We perform Bayesian inference for such models for datasets containing thousands of observation locations using the integrated nested Laplace approximation, or INLA. We explain how constraints on the spatial and spatio-temporal Gaussian processes, arising from the conditioning mechanism, can be implemented through the latent variable approach without losing the computationally convenient Markov property. We discuss tools for the comparison of models via their posterior distributions, and illustrate the flexibility of the approach with gridded Red Sea surface temperature data at over 6,000 observed locations. Posterior sampling is exploited to study the probability distribution of cluster functionals of spatial and spatio-temporal extreme episodes.

A model for space-time threshold exceedances with an application to extreme rainfall

“…Four main frameworks for modelling spatio-temporal extremes can be identified in the literature. A max-stable approach which employs max-stable processes (de Haan, 1984;Schlather, 2002;Kabluchko et al, 2009) to represent space and time interactions, with time treated as a third continuous dimension added to the two dimensions of space (Davis et al, 2013;Huser and Davison, 2014); a hierarchical approach where spatio-temporal dependence is built in by including a stochastic component in the model parameters (Sang and Gelfand, 2009;Turkman et al, 2010;Economou et al, 2014;Nieto-Barajas and Huerta, 2017;Morris et al, 2017;Bacro et al, 2020); a time series approach, where spatial dependence is embedded within a time series model (Davis and Mikosch, 2008;Meinguet, 2012;Embrechts et al, 2016); and a conditional approach based on an asymptotic approximation of the conditional distribution of the space-time process given one single site and time point (Wadsworth and Tawn, 2019;Simpson et al, 2020;Simpson and Wadsworth, 2021). All models developed within the max-stable approach imply spatial and temporal asymptotic dependence or exact independence at all distances and time lags, by a fundamental property of max-stable processes (Wadsworth and Tawn, 2012;Huser and Davison, 2014).…”

“…However, in the aforementioned references, the models maintain the same limiting dependence in both space and time. Within the conditional approach, Simpson et al (2020) and Simpson and Wadsworth (2021) provide examples of formulations for threshold exceedances that enable different forms of asymptotic dependence in the two domains. In this article, we follow a time series approach, as it lends itself naturally to deal with the discreteness and ordering of time.…”

An efficient workflow for modelling high-dimensional spatial extremes

“…Thus, Simpson et al (2023), replace the delta-Laplace process by a Gaussian Markov random field (Rue and Held 2005) created using the so-called stochastic partial differential equations (SPDE) approach of Lindgren et al (2011). Then, to achieve fast Bayesian inference, Simpson et al (2023) change the spatial conditional extremes model into a latent Gaussian model, to perform inference using integrated nested Laplace approximations (INLA; Rue et al 2009), implemented in the R-INLA software (Rue et al 2017). In this paper, we build upon the work of Simpson et al (2023) and develop a more general methodology for modelling spatial conditional extremes with R-INLA.…”

“…However, inference for Gaussian processes typically requires computing the inverse of the covariance matrix, whose cost scales cubicly with the model dimension. Thus, Simpson et al (2023), replace the delta-Laplace process by a Gaussian Markov random field (Rue and Held 2005) created using the so-called stochastic partial differential equations (SPDE) approach of Lindgren et al (2011). Then, to achieve fast Bayesian inference, Simpson et al (2023) change the spatial conditional extremes model into a latent Gaussian model, to perform inference using integrated nested Laplace approximations (INLA; Rue et al 2009), implemented in the R-INLA software (Rue et al 2017).…”

“…Then, to achieve fast Bayesian inference, Simpson et al (2023) change the spatial conditional extremes model into a latent Gaussian model, to perform inference using integrated nested Laplace approximations (INLA; Rue et al 2009), implemented in the R-INLA software (Rue et al 2017). In this paper, we build upon the work of Simpson et al (2023) and develop a more general methodology for modelling spatial conditional extremes with R-INLA. We also point out a theoretical weakness in the constraining methods proposed by Wadsworth and Tawn (2022) and used by Simpson et al (2023), and we demonstrate a computationally efficient way of fixing it.…”