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Statistical deep learning for spatial extremes: partially-interpretable neural networks and neural Bayes estimation

Deep learning models remain at the forefront of machine learning, with applications in nearly all fields of science and engineering, due to their computational scalability and the capacity of neural networks to extract complex information from data. However, environmental statisticians often forego their use as a result of their “black box” nature; neural networks are indeed often heavily parameterised and lack transparency, thus making statistical inference more difficult. In an on-going effort to mitigate these well-placed concerns, we propose two methods for performing inference on spatial processes using statistical deep-learning, i.e., with a neural network embedded within a statistical model or with a neural network facilitating inference. Whilst the methodology presented is generalisable to any spatial process, we constrain our focus to their extremal behaviour and consider applications to environmental extremes and natural hazards.

We begin by proposing a framework for performing extreme quantile regression for spatio-temporal processes using partially-interpretable neural networks, which combine semi-parametric methods, e.g., generalised linear or additive models, with deep learning, and which facilitate both high predictive accuracy and statistical inference. We then use our approach to estimate extreme quantiles for a relatively high-dimensional dataset and gain insights into the drivers of extreme wildfires occurring within, and around, Europe and the Mediterranean Basin.

We then discuss neural Bayes estimation for spatial peaks-over-threshold dependence models. Traditionally, likelihood-based inference for such extremal models is computationally problematic in moderate or high dimensions, as the related likelihood functions are often intractable and/or censoring must be implemented to reduce estimation bias. We propose an alternative means of inference using neural Bayes estimators, which are likelihood-free, amortised, and drastically more computationally efficient than classical approaches. As previous neural Bayes estimators are only applicable to spatial processes sampled over a regular grid, we exploit graphical neural networks to extend the framework to allow for irregularly-spaced data.