Modeling Spatial Extremes using Non-Gaussian Spatial Autoregressive Models via Convolutional Neural Networks

TL;DR

The paper tackles the challenge of modeling spatial extremes on large regular grids with heavy-tailed marginals, where traditional max-stable or Gaussian approaches are either inflexible or computationally prohibitive. It develops a non-Gaussian SAR framework, termed GEV-SAR, that embeds Generalized Extreme Value innovations into a sparse basis-function representation, enabling fast simulation and flexible non-Gaussian dependence. To overcome intractable likelihoods, it trains a convolutional neural network to perform amortized parameter estimation for $\theta=(ξ, κ^2, τ^2)$, with uncertainty quantified via quantile regression, and validates the approach on ERA-Interim-driven NA-CORDEX precipitation data. The results show the CNN-based estimator dramatically speeds up inference (milliseconds vs hours for MLE) while accurately recovering extremal spatial patterns, indicating a scalable tool for climate risk assessment and real-time spatial extremes analysis. Code and data are made available for replication and further study.

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 well-suited for non-Gaussian fields, providing simpler interpretability. In particular, we consider the SAR 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 that covers a useful parameter space, and then use the trained network for fast parameter estimation. 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.

Figures (14)

• Figure 1: Expanding a Gaussian SAR process to a Non-Gaussian SAR process using Gumbel$(0,1)$ and GEV$\bigl(1,\xi,\xi\bigr)$ innovations $\bigl(\xi>0\bigr)$ to generate spatial extreme fields. A fixed uniform sample is used, and then transformed for different scenarios, and the resulting spatial fields are subsequently scaled to the range $[0,1]$.
• Figure 2: Spatial fields under fixed $\xi = 0.5$ and $\kappa^2 = 0.1$ with varying nugget variance $\tau^2$, ranging from $0.01\%$ to $10\%$ of the total variability. This highlights the influence of the nugget term on the spatial field.
• Figure 3: Parameter configurations are generated for training and validating the CNN model. The spatial parameters $\kappa^2$ and $\tau^2$ are generated on the log scale.
• Figure 4: Performance of the CNN model on the test set of size 10,000, illustrating the bias in parameter estimates for different sample sizes.
• Figure 5: RMSE of the CNN estimates on the test set with repetition, illustrating the bias and variability in parameter estimates across different replication sizes.