Modeling Spatial Extremes Using Non‐Gaussian Spatial Autoregressive Models via Convolutional Neural Networks

Abstract

Data derived from remote sensing or numerical simulations often have a regular gridded structure and are large in volume, making it challenging to find accurate spatial models that can fill in missing grid cells or simulate the process effectively, especially in the presence of spatial heterogeneity and heavy‐tailed marginal distributions. To overcome this issue, we present a spatial autoregressive modeling framework, which maps observations at a location and its neighbors to independent random variables. This is a highly flexible modeling approach and is well‐suited for non‐Gaussian fields, providing simpler interpretability than hierarchical spatial models. In particular, we consider the spatial autoregressive model with Generalized Extreme Value distribution innovations to combine the observation at a central grid location with its neighbors, capturing extreme spatial behavior based on the heavy‐tailed innovations. While these models are fast to simulate by exploiting the sparsity of the key matrices in the computations, the maximum likelihood estimation of the parameters is prohibitive due to the intractability of the likelihood, making optimization challenging. To overcome this, we train a convolutional neural network on a large training set of spatial fields that are generated using parameters covering a useful parameter space, and then use the trained network for fast parameter estimation. We further employ quantile regression for uncertainty quantification, incorporating a novel construction of the training dataset to obtain confidence bounds. Finally, we apply this model to analyze annual maximum precipitation data from ERA‐Interim‐driven Weather Research and Forecasting (WRF) simulations, allowing us to explore its spatial extreme behavior across North America.