pith. sign in

arxiv: 2407.08668 · v3 · submitted 2024-07-11 · 📊 stat.ML · cs.LG

Modeling Spatial Extremal Dependence of Precipitation Using Distributional Neural Networks

Christopher B\"ulte , Lisa Leimenstoll , Melanie Schienle This is my paper

Pith reviewed 2026-05-23 22:41 UTC · model grok-4.3

classification 📊 stat.ML cs.LG
keywords max-stable processesprecipitation extremesgenerative neural networksspatial dependenceextremal coefficient functionsimulation-based estimationuncertainty quantificationextreme value theory
0
0 comments X

The pith

Generative neural networks estimate both parameters and a nonparametric spatial dependence measure for precipitation extremes under max-stable models.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a simulation-based method that trains generative neural networks to model the joint distribution of extreme precipitation values across locations. This allows recovery of the parameters of a max-stable process along with measures of uncertainty, plus a nonparametric estimate of how dependence varies with distance through the pairwise extremal coefficient function. The technique works in settings where direct likelihood calculation is impossible, as shown in simulations and applied to recent rainfall data from Western Germany that includes a major flooding event.

Core claim

Within the framework of max-stable processes, generative neural networks trained on simulations can estimate both the process parameters with uncertainty and deliver an explicit nonparametric estimate of the spatial dependence via the pairwise extremal coefficient function, performing well even when closed-form likelihood estimation is intractable.

What carries the argument

The generative neural network that approximates the finite-dimensional distributions of the max-stable process, enabling simulation-based parameter estimation and nonparametric recovery of the extremal coefficient function.

If this is right

  • The method achieves good performance in finite sample studies for complex dependence structures where closed-form estimation fails.
  • It provides uncertainty estimates for the recovered process parameters.
  • It yields an explicit nonparametric estimate of the pairwise extremal coefficient function.
  • The approach is demonstrated on monthly rainfall maxima in Western Germany for 2021-2023, including the July 2021 extreme event.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same simulation-based generative approach could be tested on other environmental extremes such as temperature or wind speed maxima.
  • The nonparametric dependence estimate might be used to diagnose whether a chosen parametric max-stable model is adequate for a given dataset.
  • Extending the network to incorporate covariates that vary in space or time could allow more flexible modeling without altering the core estimation procedure.

Load-bearing premise

The precipitation maxima are well-described by a max-stable process whose finite-dimensional distributions can be adequately approximated by the generative network in regimes where closed-form likelihoods are intractable.

What would settle it

Generate synthetic data from a known max-stable process whose parameters and pairwise extremal coefficient function are known exactly, then check whether the neural network recovers those quantities within the reported uncertainty bounds.

Figures

Figures reproduced from arXiv: 2407.08668 by Christopher B\"ulte, Lisa Leimenstoll, Melanie Schienle.

Figure 1
Figure 1. Figure 1: The figure shows the parameter estimation setup, for a two-dimensional max-stable process. In the training [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The figure shows the proposed model architecture. The spatial field is fed through three blocks convolutional [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The figure visualizes the different estimation methods for the max-stable models using a selected test sample [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The figure visualizes the different estimation methods for the robustness scenarios using a selected test sample [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The figure visualizes the energy score across the parameters [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The figure shows the maximum precipitation in mm, aggregated over the three summer months of the years [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The figure shows the parameter estimates for a powered exponential model for all three months and years [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The figure shows the different estimates for the extremal coefficient function. The black dots are the binned [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The figure shows estimations of the extremal coefficient function for July 2021. The black dots display the [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The figure visualizes the different estimation methods for the max-stable models using four randomly drawn [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The figure visualizes the different estimation methods for the two robustness scenarios using four randomly [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The figure shows the parameter estimates for different models for all three months and years using the [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The figure shows the different estimates for the extremal coefficient function. The black dots are the binned [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Estimates of pairwise extremal coefficient function of EN [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Energy score for parameters (λ, ν) and regular test data from the Brown-Resnick model. In the case of robustness scenario #1, where the range of trained and tested parameters differ, the Energy score is larger for test parameters further away from the training parameters. For the ENθ parameters especially in the upper right corner correspond to higher energy scores. A possible explanation can be obtained … view at source ↗
Figure 16
Figure 16. Figure 16: Energy scores for parameters (λ, ν) and robustness scenario #1 with test data from the Brown-Resnick model. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Visualization of the pairwise extremal coefficient function in dependence of [PITH_FULL_IMAGE:figures/full_fig_p029_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The figure visualizes the different estimation methods for the additional robustness scenario using a selected [PITH_FULL_IMAGE:figures/full_fig_p031_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The figure visualizes the different estimation methods for the additional robustness scenario using four [PITH_FULL_IMAGE:figures/full_fig_p031_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: The figure shows the observed precipitation maxima in 2022 (top row) and corresponding simulations from [PITH_FULL_IMAGE:figures/full_fig_p033_20.png] view at source ↗
read the original abstract

In this work, we propose a simulation-based estimation approach using generative neural networks to determine dependencies of precipitation maxima and their underlying uncertainty in time and space. Within the common framework of max-stable processes for extremes under temporal and spatial dependence, our methodology allows estimating the process parameters and their respective uncertainty, but also delivers an explicit nonparametric estimate of the spatial dependence through the pairwise extremal coefficient function. We illustrate the effectiveness and robustness of our approach in a thorough finite sample study where we obtain good performance in complex settings for which closed-form likelihood estimation becomes intractable. We use the technique for studying monthly rainfall maxima in Western Germany for the period 2021-2023, which is of particular interest since it contains an extreme precipitation and consecutive flooding event in July 2021 that had a massive deadly impact. Beyond the considered setting, the presented methodology and its main generative ideas also have great potential for other applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a simulation-based estimation method using generative neural networks for parameters and uncertainty in max-stable processes modeling spatial and temporal dependence of precipitation maxima. It claims to deliver both parametric estimates and an explicit nonparametric estimate of spatial dependence via the pairwise extremal coefficient function. Effectiveness is illustrated via a finite-sample study in settings where closed-form likelihoods are intractable, followed by an application to monthly rainfall maxima in Western Germany (2021-2023), including the July 2021 flood event.

Significance. If the generative network's approximation to the finite-dimensional distributions of the max-stable process holds with accurate tail dependence, the approach would enable flexible inference and nonparametric dependence estimation in complex spatial extremes where traditional likelihood methods fail, with potential extension to other applications.

major comments (2)
  1. [Abstract] Abstract: the claim of 'good performance' and 'robustness' in finite-sample studies for intractable settings is asserted without any quantitative metrics, error bars, network architecture details, or training specifications, leaving the central claim of reliable estimation unverified.
  2. [Methodology] Methodology section (as described in abstract): both the parametric estimation with uncertainty and the nonparametric pairwise extremal coefficient estimate rest on the generative NN adequately approximating the FDDs of the underlying max-stable process; however, validation occurs only against parametric models where ground truth is available, with no diagnostics isolating tail dependence error (e.g., convergence of simulated bivariate exceedance probabilities) for truly intractable regimes.
minor comments (2)
  1. Provide explicit details on the generative network architecture, training procedure, and any simulation matching diagnostics to allow reproducibility.
  2. Clarify how the extremal coefficient function is extracted as a nonparametric estimate distinct from the fitted process parameters.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and constructive suggestions. We provide point-by-point responses to the major comments and outline the revisions we intend to make to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim of 'good performance' and 'robustness' in finite-sample studies for intractable settings is asserted without any quantitative metrics, error bars, network architecture details, or training specifications, leaving the central claim of reliable estimation unverified.

    Authors: We agree that the abstract would benefit from more specific quantitative information to support the claims of good performance and robustness. In the revised manuscript, we will update the abstract to include key quantitative metrics from the finite-sample study, such as average estimation errors for parameters and extremal coefficients, and provide brief details on the neural network architecture and training specifications used. revision: yes

  2. Referee: [Methodology] Methodology section (as described in abstract): both the parametric estimation with uncertainty and the nonparametric pairwise extremal coefficient estimate rest on the generative NN adequately approximating the FDDs of the underlying max-stable process; however, validation occurs only against parametric models where ground truth is available, with no diagnostics isolating tail dependence error (e.g., convergence of simulated bivariate exceedance probabilities) for truly intractable regimes.

    Authors: The validation in our finite-sample study is performed in settings where the true parameters and dependence structure are known, even if the likelihood is intractable, allowing direct comparison of our estimates to the ground truth. This includes both parametric and nonparametric estimates. To further address concerns about tail dependence approximation, we will add diagnostics in the revised paper, such as evaluations of simulated bivariate exceedance probabilities and their convergence to theoretical values in the simulation studies. This will help isolate any errors in the generative network's approximation of the finite-dimensional distributions. revision: yes

Circularity Check

0 steps flagged

Simulation-based estimation via generative NN shows no reduction of extremal coefficient or parameters to fitted inputs by construction

full rationale

The paper's core methodology is a simulation-based approach using generative neural networks to approximate finite-dimensional distributions of max-stable processes for parameter estimation and nonparametric extremal coefficient computation. No load-bearing step reduces by definition or self-citation to the target quantities themselves; the extremal coefficient is obtained as an explicit output from the trained generative model rather than being fitted directly or renamed from inputs. The finite-sample validation occurs against known parametric cases but does not create a circular derivation for the reported estimates. This aligns with a minor self-citation tolerance without load-bearing circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that precipitation maxima obey a max-stable process whose dependence structure can be recovered via generative simulation; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption Precipitation maxima follow a max-stable process under temporal and spatial dependence
    Invoked as the common framework within which the neural estimation operates (abstract, first sentence of methodology paragraph).

pith-pipeline@v0.9.0 · 5687 in / 1140 out tokens · 18089 ms · 2026-05-23T22:41:47.535568+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

  1. [1]

    A., Zhang, W., and Balding, D

    Beaumont, M. A., Zhang, W., and Balding, D. J. Approximate Bayesian Computation in Population Genetics. Genetics, 162(4):2025–2035, 12

    work page 2025

  2. [2]

    Living with urban flooding: A continuous learning process for local municipalities and lessons learnt from the 2021 events in germany

    Bosseler, B., Salomon, M., Schlüter, M., and Rubinato, M. Living with urban flooding: A continuous learning process for local municipalities and lessons learnt from the 2021 events in germany. Water, 13(19):2769,

    work page 2021

  3. [3]

    Cooley, D., Naveau, P., and Poncet, P

    URL https://arxiv.org/pdf/2211.01345.pdf. Cooley, D., Naveau, P., and Poncet, P. Variograms for spatial max-stable random fields. InDependence in Probability and Statistics, volume 187, pages 373–390

    work page arXiv

  4. [4]

    Franks, J

    URL https://arxiv.org/pdf/2212.11598.pdf. Franks, J. J. Handbook of approximate bayesian computation. Journal of the American Statistical Association, 115 (532):2100–2101,

    work page arXiv

  5. [5]

    Kabluchko, Z., Schlather, M., and de Haan, L

    URL https://arxiv.org/pdf/2203.05626.pdf. Kabluchko, Z., Schlather, M., and de Haan, L. Stationary max-stable fields associated to negative definite functions. The Annals of Probability, 37(5),

    work page arXiv

  6. [6]

    Lenzi, A., Bessac, J., Rudi, J., and Stein, M

    URL https://arxiv.org/ pdf/2303.15041. Lenzi, A., Bessac, J., Rudi, J., and Stein, M. L. Neural networks for parameter estimation in intractable models. Computational Statistics & Data Analysis, 185:107762,

    work page arXiv

  7. [7]

    E., Ehmele, F., Feldmann, H., Franca, M

    19 ESTIMATION OF SPATIO -TEMPORAL EXTREMES VIA GENERATIVE NEURAL NETWORKS Mohr, S., Ehret, U., Kunz, M., Ludwig, P., Caldas-Alvarez, A., Daniell, J. E., Ehmele, F., Feldmann, H., Franca, M. J., Gattke, C., Hundhausen, M., Knippertz, P., Küpfer, K., Mühr, B., Pinto, J. G., Quinting, J., Schäfer, A. M., Scheibel, M., Seidel, F., and Wisotzky, C. A multi-dis...

    work page 2021

  8. [8]

    Pacchiardi, L., Adewoyin, R

    URL https://arxiv.org/abs/2205.15784. Pacchiardi, L., Adewoyin, R. A., Dueben, P., and Dutta, R. Probabilistic forecasting with generative networks via scoring rule minimization. Journal of Machine Learning Research, 25(45):1–64,

    work page arXiv

  9. [9]

    The Effectiveness of Data Augmentation in Image Classification using Deep Learning

    URL https://arxiv.org/pdf/1712.04621.pdf. Radev, S. T., Mertens, U. K., V oss, A., Ardizzone, L., and Köthe, U. Bayesflow: Learning complex stochastic models with invertible neural networks. IEEE Transactions on Neural Networks and Learning Systems, 33(4):1452–1466,

    work page internal anchor Pith review Pith/arXiv arXiv

  10. [10]

    Ramesh, P., Lueckmann, J.-M., Boelts, J., Tejero-Cantero, Á., Greenberg, D

    URL https://arxiv.org/ pdf/2305.04341.pdf. Ramesh, P., Lueckmann, J.-M., Boelts, J., Tejero-Cantero, Á., Greenberg, D. S., Goncalves, P. J., and Macke, J. H. GATSBI: Generative adversarial training for simulation-based inference. In International Conference on Learning Representations,

    work page arXiv

  11. [11]

    URL https://arxiv.org/pdf/2208.12942. Sang, H. and Gelfand, A. E. Continuous spatial process models for spatial extreme values. Journal of Agricultural, Biological, and Environmental Statistics, 15(1):49–65,

    work page arXiv

  12. [12]

    The years are from 2021-2023 from top to bottom

    23 ESTIMATION OF SPATIO -TEMPORAL EXTREMES VIA GENERATIVE NEURAL NETWORKS (a) Brown-Resnick /uni00000013/uni00000018/uni00000014/uni00000013/uni00000014/uni00000018/uni00000015/uni00000013/uni00000015/uni00000018/uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014/uni00000011/uni00000013 /uni00000014/uni00000011/uni000000...

    work page 2021

  13. [13]

    ,900 is the index of the corresponding location, t is the year of the observation and lat, lon describe the latitude and longitude, respectively

    31 ESTIMATION OF SPATIO -TEMPORAL EXTREMES VIA GENERATIVE NEURAL NETWORKS E GEV fit The GEV parameters are modeled by the following equations µ(i, t) = β0,µ + β1,µlat(i) + β2,µ + lon(i) + β3,µt σ = β0,σ γ = β0,γ, where i = 1, . . . ,900 is the index of the corresponding location, t is the year of the observation and lat, lon describe the latitude and long...

    work page 2022

  14. [14]

    The parameters were fit on the three summer months over the years of 1931-2020

    First of, note that the shape β0,µ β1,µ β2,µ β3,µ β0,σ β0,γ Estimation 64.7996 -0.9997 0.0149 0.0011 7.0045 0.1052 Standard error 27.3724 0.5433 0.3192 0.0104 0.2126 0.0224 Table 6: The estimated GEV parameters and corresponding standard errors of the model described above. The parameters were fit on the three summer months over the years of 1931-2020. pa...

    work page 1931