Summary In multivariate extreme value analysis, the nature of the extremal dependence between variables should be considered when selecting appropriate statistical models. Interest often lies in determining which subsets of variables can take their largest values simultaneously while the others are of smaller order. Our approach to this problem exploits hidden regular variation properties on a collection of nonstandard cones, and provides a new set of indices that reveal aspects of the extremal dependence structure not available through existing measures of dependence. We derive theoretical properties of these indices, demonstrate their utility through a series of examples, and develop methods of inference that also estimate the proportion of extremal mass associated with each cone. We apply the methods to river flows in the U.K., estimating the probabilities of different subsets of sites being large simultaneously.

No 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 adopt a Bayesian perspective by implementing estimation through the integrated nested Laplace approximation (INLA), allowing for novel and flexible semi-parametric specifications of the Gaussian mean function. By using Gauss-Markov approximations of the Matérn covariance function (known as the Stochastic Partial Differential Equation approach) at a latent stage of the model, likelihood-based inference becomes feasible even with several thousands of observed locations. 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 posterior models, 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.

No abstract

This paper details the approach of team Lancaster to the 2019 EVA data challenge, dealing with spatio-temporal modelling of Red Sea surface temperature anomalies. We model the marginal distributions and dependence features separately; for the former, we use a combination of Gaussian and generalised Pareto distributions, while the dependence is captured using a localised Gaussian process approach. We also propose a space-time moving estimate of the cumulative distribution function that takes into account spatial variation and temporal trend in the anomalies, to be used in those regions with limited available data. The team's predictions are compared to results obtained via an empirical benchmark. Our approach performs well in terms of the threshold-weighted continuous ranked probability score criterion, chosen by the challenge organiser.