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Flexible and efficient emulation of spatial extremes processes via variational autoencoders

Likun Zhang, Xiaoyu Ma, Christopher K. Wikle, Raphaël Huser

TL;DR

This work tackles the challenge of modeling and emulating high-dimensional spatial extremes with flexible tail dependence. It introduces XVAE, a variational autoencoder that embeds a novel flexible max-id spatial extremes model with nonstationary dependence into the encoding-decoding architecture, enabling fast inference and realistic joint tail simulations. A dedicated validation framework for extremes emulation, along with extensive simulations, shows that XVAE outperforms stationary Gaussian-based emulators in capturing local extremal dependence and heavy tails. The Red Sea SST application demonstrates XVAE’s utility for rapid uncertainty quantification and risk assessment of marine heatwaves under climate change, offering a scalable tool for climate emulation and beyond.

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.

Flexible and efficient emulation of spatial extremes processes via variational autoencoders

TL;DR

This work tackles the challenge of modeling and emulating high-dimensional spatial extremes with flexible tail dependence. It introduces XVAE, a variational autoencoder that embeds a novel flexible max-id spatial extremes model with nonstationary dependence into the encoding-decoding architecture, enabling fast inference and realistic joint tail simulations. A dedicated validation framework for extremes emulation, along with extensive simulations, shows that XVAE outperforms stationary Gaussian-based emulators in capturing local extremal dependence and heavy tails. The Red Sea SST application demonstrates XVAE’s utility for rapid uncertainty quantification and risk assessment of marine heatwaves under climate change, offering a scalable tool for climate emulation and beyond.

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.
Paper Structure (46 sections, 7 theorems, 75 equations, 23 figures, 2 algorithms)

This paper contains 46 sections, 7 theorems, 75 equations, 23 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Spatial extremes modeling
  4. Variational autoencoder background
  5. Methodology
  6. Flexible nonstationary max-id spatial extremes model
  7. Marginal and dependence properties
  8. XVAE: A VAE incorporating the proposed max-id model
  9. Approximate Posterior/Encoder ($q_{\boldsymbol{\phi}_e}(\boldsymbol{z}_t\mid \boldsymbol{x}_t)$):
  10. Prior on Latent Process ($p_{\boldsymbol{\phi}_d}({\boldsymbol{z}_t})$):
  11. Data Model/Decoder ($p_{\phi_d}(\boldsymbol{x}_t\mid \boldsymbol{z}_t)$):
  12. Encoder/Decoder Estimation:
  13. Uncertainty quantification:
  14. Validation framework for extremes emulation
  15. Simulation study
  16. ...and 31 more sections

Key Result

Proposition 3.1

Let $\mathcal{D} = \{k:\gamma_k=0,\; k=1,\ldots, K\}$ and $\bar{\mathcal{D}}$ be the complement of $\mathcal{D}$. For the process eqn:model, the marginal distribution function of $X_j=X(\boldsymbol{s}_j)$ can be written as As $x\rightarrow\infty$, the survival function $\bar{F}_j(x) = 1-F_j(x)\sim c_j(x/\tau)^{-{1/\alpha_0}}$ if $\mathcal{C}_j\cap \mathcal{D}= \emptyset$, and $\bar{F}_j(x) \sim c

Figures (23)

  • Figure 1: Diagram of a variational autoencoder (VAE) with the reparameterization trick.
  • Figure 2: The left panel presents knot locations used for Models \ref{['modelAI']}--\ref{['modelAD']}, and we only show the support of the one Wendland basis function centered at knot in the middle of the domain. Model \ref{['modelMaxStable']} uses the same set of knots but the basis functions are not compactly supported. The middle and right panels display the $\gamma_k$ values, $k=1,\ldots, K$, used in the expPS variables for Models \ref{['modelAI']} and \ref{['modelFlex']} respectively. The circled knots signify $\gamma_k=0$, which induces local AD.
  • Figure 3: Data replicate (left) and its corresponding emulated fields (XVAE, middle; hetGP, right) from Model \ref{['modelFlex']}. See Figure \ref{['fig:comps_across_models1']} of the Supplementary Material for comparisons for the other models. In all cases, we use data-driven knots for emulation using XVAE.
  • Figure 4: The CRPS (left) and MSPE (right) values from two emulation approaches on the datasets simulated from Models \ref{['modelGP']}--\ref{['modelMaxStable']}. For both metrics, lower values indicate better emulation results. Also, for Models \ref{['modelAD']} and \ref{['modelMaxStable']}, we plot the CRPS values on the log scale since the AD in the data generating process causes the margins to be very heavy-tailed.
  • Figure 5: From left to right, we show the empirically-estimated $\chi_h(u)$ at $h=0.5,2,5$, and $\mathrm{ARE}_\psi(u)$ with $\psi=0.05$ for Model \ref{['modelFlex']} based on data replicates (black) and XVAE emulated data (red). The $\chi_h(u)$ and $\mathrm{ARE}_\psi(u)$ estimates for the other models are shown in Figures \ref{['fig:chi_ests2']} and \ref{['fig:ARE_comps2']} of the Supplementary Material, respectively.
  • ...and 18 more figures

Theorems & Definitions (22)

  • Proposition 3.1
  • Proposition 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Remark 1
  • Remark 2
  • Lemma A.1
  • proof
  • Remark 3
  • ...and 12 more