June 2024 Modeling extremal streamflow using deep learning approximations and a flexible spatial process
Reetam Majumder, Brian J. Reich, Benjamin A. Shaby
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Ann. Appl. Stat. 18(2): 1519-1542 (June 2024). DOI: 10.1214/23-AOAS1847


Quantifying changes in the probability and magnitude of extreme flooding events is key to mitigating their impacts. While hydrodynamic data are inherently spatially dependent, traditional spatial models, such as Gaussian processes, are poorly suited for modeling extreme events. Spatial extreme value models with more realistic tail dependence characteristics are under active development. They are theoretically justified but give intractable likelihoods, making computation challenging for small datasets and prohibitive for continental-scale studies. We propose a process mixture model (PMM) which specifies spatial dependence in extreme values as a convex combination of a Gaussian process and a max-stable process, yielding desirable tail dependence properties but intractable likelihoods. To address this, we employ a unique computational strategy where a feed-forward neural network is embedded in a density regression model to approximate the conditional distribution at one spatial location, given a set of neighbors. We then use this univariate density function to approximate the joint likelihood for all locations by way of a Vecchia approximation. The PMM is used to analyze changes in annual maximum streamflow within the U.S. over the last 50 years and is able to detect areas which show increases in extreme streamflow over time.

Funding Statement

This work was supported by grants from the Southeast National Synthesis Wildfire and the United States Geological Survey’s National Climate Adaptation Science Center (G21AC10045), the National Science Foundation (CBET2151651, DMS2152887, and DMS2001433), and the National Institutes of Health (R01ES031651-01).


The authors thank Prof. Sankarasubramanian Arumugam of North Carolina State University for discussion of the data and scope of the project.


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Reetam Majumder. Brian J. Reich. Benjamin A. Shaby. "Modeling extremal streamflow using deep learning approximations and a flexible spatial process." Ann. Appl. Stat. 18 (2) 1519 - 1542, June 2024. https://doi.org/10.1214/23-AOAS1847


Received: 1 August 2022; Revised: 1 October 2023; Published: June 2024
First available in Project Euclid: 5 April 2024

Digital Object Identifier: 10.1214/23-AOAS1847

Keywords: Gaussian process , Max-stable process , neural networks , spatial extremes , Vecchia approximation

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.18 • No. 2 • June 2024
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