[Paper Review] Flexible and efficient emulation of spatial extremes processes via variational autoencoders

Likun Zhang, Xiaoyu Ma+2 more|arXiv (Cornell University)|Jul 16, 2023

Climate variability and modelsEnvironmental Science3 citations

TL;DR

This paper proposes a novel spatial extremes emulator, XVAE, which integrates a flexible max-infinitely divisible process into a variational autoencoder framework to efficiently model non-stationary, dependent extreme events in high-dimensional spatial data. By combining extreme value theory with deep generative modeling, XVAE enables fast Bayesian inference and accurate emulation of complex climate extremes, outperforming traditional methods in computational efficiency and tail dependence modeling on a 30-year Red Sea sea surface temperature dataset.

ABSTRACT

Many real-world processes have complex tail dependence structures that cannot be characterized using classical Gaussian processes. More flexible spatial extremes models exhibit appealing extremal dependence properties but are often exceedingly prohibitive to fit and simulate from in high dimensions. In this paper, we aim to push the boundaries on computation and modeling of high-dimensional spatial extremes via integrating a new spatial extremes model that has flexible and non-stationary dependence properties in the encoding-decoding structure of a variational autoencoder called the XVAE. The XVAE can emulate spatial observations and produce outputs that have the same statistical properties as the inputs, especially in the tail. Our approach also provides a novel way of making fast inference with complex extreme-value processes. Through extensive simulation studies, we show that our XVAE is substantially more time-efficient than traditional Bayesian inference while outperforming many spatial extremes models with a stationary dependence structure. Lastly, we analyze a high-resolution satellite-derived dataset of sea surface temperature in the Red Sea, which includes 30 years of daily measurements at 16703 grid cells. We demonstrate how to use XVAE to identify regions susceptible to marine heatwaves under climate change and examine the spatial and temporal variability of the extremal dependence structure.

Motivation & Objective

• To develop a surrogate model that accurately emulates high-dimensional spatial extremes with flexible, non-stationary dependence structures.
• To overcome the computational infeasibility of fitting and simulating complex spatial extremes models in high dimensions.
• To enable fast Bayesian inference and uncertainty quantification for extreme events in climate and environmental modeling.
• To provide a validation framework tailored to assessing extremal dependence in spatial field emulation.
• To demonstrate the method’s effectiveness on a high-resolution 30-year satellite-derived sea surface temperature dataset in the Red Sea.

Proposed method

• Proposes a new max-infinitely divisible (max-id) process with flexible, non-stationary extremal dependence properties, grounded in extreme value theory.
• Embeds the max-id process within a variational autoencoder (VAE) architecture, forming the XVAE framework, to enable efficient posterior approximation.
• Uses variational Bayes and stochastic gradient descent to jointly estimate latent variables and model parameters, enabling scalable inference.
• Applies a marginal transformation using site-specific GEV distributions to standardize monthly maxima across grid cells before input to the XVAE.
• Employs a generalized likelihood-ratio test with Wilk’s theorem to assess goodness-of-fit of marginal distributions (GEV vs. t) at each location.
• Validates emulation performance using a custom framework that evaluates tail dependence structure and extremal behavior across spatial lags.

Figure 1 : A schematic diagram for the XVAE, in which the parameters $\boldsymbol{\phi}_{e}$ and $\boldsymbol{\phi}_{d}$ in the encoder $q_{\boldsymbol{\phi}_{e}}(\boldsymbol{z}_{t}\mid\boldsymbol{x}_{t})$ and the decoder $p_{\boldsymbol{\phi}_{d}}(\boldsymbol{x}_{t}\mid\boldsymbol{z}_{t})$ are both

Experimental results

Research questions

• RQ1 Can a deep generative model effectively capture and emulate complex, non-stationary extremal dependence in high-dimensional spatial processes?
• RQ2 How does the XVAE framework compare to traditional Gaussian process emulators in terms of computational efficiency and accuracy for extreme value modeling?
• RQ3 To what extent can the XVAE reproduce the true tail dependence structure of spatial extremes, especially in regions with weak or time-varying extremal dependence?
• RQ4 Does the model outperform existing GAN-based or copula-based emulators in capturing the joint tail behavior of spatial extremes?
• RQ5 What are the spatial and temporal patterns of extremal dependence in high-resolution sea surface temperature data from the Red Sea?

Key findings

• The XVAE achieves substantial computational speedups over traditional Bayesian inference, enabling efficient posterior simulation and uncertainty quantification for complex spatial extremes.
• The XVAE outperforms stationary dependence models and existing GAN-based emulators in capturing the true extremal dependence structure, especially for concurrent extreme events.
• On the Red Sea SST dataset, the XVAE reveals that extremal dependence is weaker in the interior of the sea and has decreased slightly over the 30-year period.
• Goodness-of-fit tests confirm that the GEV distribution provides a better fit to monthly maxima than the t distribution at all locations, with p-values > 0.05 across all sites under the GEV model.
• Empirical tail dependence measures (χh(u)) show that XVAE-generated data closely match observed data across spatial lags of 50 km, 200 km, and 500 km.
• The model successfully handles high-dimensional data with 16,703 grid cells and 372 monthly maxima, demonstrating scalability and robustness in real-world climate applications.

Figure 3 : Data replicate (left) and its corresponding emulated fields (XVAE, middle; hetGP , right) from Model III . See Figure E.1 of the Supplementary Material for comparisons for the other models. In all cases, we use data-driven knots for emulation using XVAE.