Pith. sign in

REVIEW 4 major objections 5 minor 41 references

Forecasting with Markovian max-stable fields in space and time: An application to wind gust speeds

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proposes a Markovian max-stable field defined by a max-autoregressive recurrence with advection, proves that its pairwise likelihood estimator is consistent and asymptotically normal, and shows that the corresponding forecasting…

desk verdict Real asymptotic theory for a Markovian max-stable model, but the forecasting comparison is not yet convincing because the two models receive different conditioning information. read the letter →

arxiv 2411.15511 v1 pith:HPKXDFHY submitted 2024-11-23 stat.ME stat.AP

Ryan Cotsakis , Erwan Koch , Christian-Yann Robert This is my paper · ORCID
classification stat.MEstat.AP MSC 62G3262M3062M1060G7062F12
keywords space-timemax-stablemodelmax-autoregressiveadvectionMarkovpropertypairwiselikelihoodBrown-Resnickwindgustforecastingensemble
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that a Markovian space-time max-stable model built from a max-autoregressive recurrence with advection — $Z(s,t)=\max\{aZ(s-\tau,t-1),(1-a)W_t(s)\}$ — is both theoretically tractable and practically suited to forecasting extreme wind gusts. It proves that the space-time pairwise likelihood estimator is strongly consistent and asymptotically normal as the grid and observation window grow, and it develops a forecasting strategy that exploits the Markov property to sample exactly from the predictive distribution. Applied to hourly maxima of 3-second wind gust speeds from reanalysis data over northwestern France in December 1999, the model captures the observed east/south-east propagation of the storm and outperforms the space-time Brown-Resnick (DKS) competitor on CRPS and RMSE at lead times of 1-7 hours. If correct, this gives forecasters a parsimonious, interpretable statistical complement to numerical weather prediction and AI models for storm-scale extremes.

What carries the argument

The central object is the max-autoregressive recurrence with advection, $Z(s,t)=\max\{aZ(s-\tau,t-1),(1-a)W_t(s)\}$, which makes the field time-Markovian: conditioning on the entire past reduces to conditioning on the single value $Z(s-u\tau,t)$ at the advected upstream location. This yields the closed-form conditional distribution $P(Z(s,t+u)\le z_2\mid Z(s-u\tau,t)=z_1)=1_{\{z_2\ge a^u z_1\}}\exp(-(1-a^u)/z_2)$, which is what makes exact ensemble forecasting possible. The companion object is the bivariate exponent measure, which gives the pairwise densities used for estimation. The asymptotic proofs rely on the field being space-time mixing, with $\alpha$-mixing coefficients decaying at a rate governed by $\min\{2H^\star,1\}$ in the logarithm, so that a uniform strong law of large numbers and a central limit theorem for mixing random fields apply.

What would settle it

Simulate the model with known parameters on the paper's grid, fit it with the two-step procedure, and check whether 95% bootstrap confidence intervals cover the true parameters at the nominal rate; coverage far from 95% would show that the reported inference and forecast-skill comparisons rest on an unproven equivalence.

Watch

Extended reading notes

Core claim

The central claim is that the space-time max-stable field defined by $Z(s,t)=\max\{aZ(s-\tau,t-1),(1-a)W_t(s)\}$, with $(W_t)$ independent copies of a spatial Brown-Resnick field and $a\in(0,1)$, $\tau\in R^2$, is the right building block for forecasting extremes of advective atmospheric variables. The paper proves that the maximum pairwise likelihood estimator of the full parameter vector $\psi=(\kappa,H,\tau,a)$ is strongly consistent, $\hat\psi\overset{\mathrm{a.s.}}{\to}\psi^\star$, and asymptotically normal, $(m^2T)^{1/2}(\hat\psi-\psi^\star)\to N(0,F^{-1}\Sigma(F^{-1})')$, as the spatial grid size $m$ and the number of time points $T$ tend to infinity. The argument runs through explicit formulas for the bivariate exponent measure and through space-time mixing of the field, which yields the necessary strong law and central limit theorem. It then gives a forecasting strategy based on the Markov property that samples exactly from the predictive distribution of $Z(s,t+u)$ given the current field, and demonstrates on ERA5 reanalysis data for northwestern France in December 1999 that the model reproduces the observed asymmetric cross-correlations and produces ensemble forecasts with lower CRPS and RMSE than the DKS space-time Brown-Resnick model for lead times of one to seven hours.

Load-bearing premise

The load-bearing premise is that the two-step estimation used in practice — first the spatial parameters, then the temporal ones — inherits the consistency and asymptotic normality that the paper proves only for the joint pairwise likelihood estimator.

Editorial extensions

If this is right

  • Real-time ensemble nowcasts of wind gust maxima become available from a parsimonious statistical model with explicit uncertainty, complementing numerical weather prediction and AI-based forecast systems.
  • The proven asymptotics justify frequentist confidence intervals and bootstrap-based uncertainty quantification for the model parameters, provided the estimator actually used is covered by the theorems.
  • The advection parameter tracks storm motion, so the model can predict not only the intensity but the spatial displacement of extreme fields over the next few hours.
  • Because the dynamics are explicit and Markovian, the same forecasting strategy transfers to other advective variables such as temperature, rainfall, or pollutant concentration.
  • For space-time lags aligned with the advection direction, the model's theoretical cross-correlations fall within the empirical confidence bands where the symmetric DKS model's do not.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the two-step estimation procedure used in the case study is asymptotically equivalent to the joint estimator analyzed in the theorems — which the paper does not prove — then the reported bootstrap confidence intervals are justified; a simulation study of coverage rates would settle this directly.
  • The explicit conditional distribution (11) also yields closed-form probabilities of exceeding a fixed damage threshold at lead time $u$ along the advection direction, which could be deployed directly in early-warning systems.
  • The constant decay and advection parameters could be replaced by covariates such as pressure fields or weather-regime indicators without destroying the Markov structure, pointing toward regime-dependent and AI-assisted versions of the model.
  • The ratio random field diagnostic of Section D.2 could be developed into a formal hypothesis test for whether the advection vector lies on the estimation grid, turning the paper's informal visual check into a decision rule.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper studies a max-autoregressive space-time max-stable field with advection, originally introduced by Embrechts et al. (2016). The main contributions are: (i) a proof of strong consistency and asymptotic normality of the joint pairwise likelihood estimator as the spatial and temporal domains grow (Theorems 1 and 2); (ii) a forecasting strategy that exploits the Markov property and the recurrence structure to sample exactly from the predictive conditional distribution; and (iii) an application to hourly maxima of 3-second wind gust speeds over northwestern France during the December 1999 storm, in which the proposed model is compared with the DKS space-time Brown-Resnick model using cross-correlation diagnostics and CRPS/RMSE forecast scores. The paper is well written, and the theoretical part is substantial, with proofs in the supplementary material. However, the applied forecast comparison has a major design flaw: the two models are conditioned on different information sets, and the evaluation is in-sample. In addition, the two-step estimation procedure used in practice is not the estimator covered by the asymptotic theorems. These issues are load-bearing for the paper's central applied claims.

Significance. If the theoretical results are correct, they fill a genuine gap: asymptotic theory for pairwise likelihood estimation in a space-time max-stable model with explicit Markovian dynamics and advection. The forecasting strategy is elegant and potentially useful for nowcasting, since it yields exact ensemble forecasts from the recurrence representation. The paper also provides a careful case study with informative cross-correlation diagnostics. The strengths include complete proofs in the supplementary material, a clearly stated model with a parsimonious parameterization, and a reproducible data application. However, the claimed forecast skill advantage over the DKS model is not established by the current experiment, because the conditioning information differs between models and because parameters are estimated and evaluated on the same 105-hour event. The inference gap between the joint estimator in the theorems and the two-step estimator used in practice further weakens the reported uncertainty quantification. The manuscript therefore needs major revision before the applied claims can be accepted.

major comments (4)
  1. [Section 5.4, Figures 8-9] The forecast comparison is asymmetric in conditioning information. For the proposed model, the forecast at (s0, t0+u) is based on conditionally simulating Z(s0 - u*tau_hat, t0) given the four surrounding grid-point values at time t0, then combining that value with an independent innovation through Eq. (14). For the DKS model, the paper states that exact conditional simulation of the three-dimensional Brown-Resnick field is unavailable, so the forecast conditions only on the univariate history at s0 (specifically observations at (s0,t0) and (s0,t0-1)). The proposed model therefore receives contemporaneous spatial information at an upstream location that, under advection, is highly informative for the future value at s0, whereas the DKS model receives only a single-site time series. The reported CRPS and RMSE advantage may reflect this information advantage rather than a better dependence model. To support the claim of 'superior performance', the authors should condition both models on the same information (for example, condition the DKS forecast on the full spatial field at t0 using an approximate conditional simulation, or restrict the proposed model to target-site history only), and include a baseline such as advective persistence, e.g., the observed value at s0 - u*tau_hat at time t0.
  2. [Section 5 / Section 5.4] The forecast evaluation is in-sample. Model parameters, including the per-site GEV margins and the dependence parameters, are estimated from the same 105-hour event that is used to compute the forecast scores. The paper acknowledges this in Section 5 but justifies it by model parsimony. This is not an inherent constraint: one could fit on the first part of the period and evaluate on the remaining hours, or use a rolling-origin scheme. As it stands, the comparison does not rule out overfitting of the proposed model to the specific event, and the claimed forecast skill is not established as a generalizable result. The authors should provide an out-of-sample evaluation, or explicitly discuss and quantify the optimism of the in-sample scores (for example, by comparing against an in-sample-fitted baseline).
  3. [Section 4 (Inference) and Section 5.1] Theorems 1 and 2 concern the joint pairwise likelihood estimator (19) that maximizes (18) over all five parameters simultaneously. In practice, the paper uses a two-step procedure: first estimate kappa and H by maximizing the spatial pairwise likelihood (21), then fix these and estimate tau and a by maximizing (18) with respect to the remaining parameters. No theorem or lemma in the paper establishes consistency or asymptotic normality for this two-step estimator. Section 5.1 nevertheless invokes the asymptotic results to 'guarantee the accuracy of our estimates' and to justify the bootstrap confidence intervals in Table 1. This is a load-bearing gap because the reported inference and the forecast skill both depend on the two-step estimator. The authors should either prove the two-step estimator's consistency and asymptotic normality (e.g., under standard two-step M-estimator theory), or explicitly state that the confidence intervals are heuristic and provide alternative uncertainty quantification, such as a simulation study of the two-step procedure.
  4. [Section 3.2 and Section 5.4] The forecasting strategy relies on conditionally simulating Z(s0 - u*tau_hat, t0) given the four vertices of the grid cell containing s0 - u*tau_hat (Figure 7). The text says 'Empirical evidence suggests that including other sites has a negligible impact on the distribution of the conditional simulation (not shown).' This is an unsupported assertion. For max-stable fields, the conditional distribution at an unobserved location given only the four nearest sites is generally not equal to the conditional distribution given the full observed field, and the discrepancy may grow as s0 - u*tau_hat moves farther from the observation grid. The authors should provide evidence, for example a simulation study comparing the predictive distribution given four sites versus given a larger neighborhood, or at least assess the sensitivity of the forecast scores to this approximation choice.
minor comments (5)
  1. [Section 5.1] The statement that 'the huge number of space-time pairs ... allows us to use our theoretical results' is not a substitute for checking the asymptotic approximation at the actual sample size: the spatial grid has only about 15 by 15 points and T = 105, and Theorems 1 and 2 require both m and T to diverge. A small simulation study of the finite-sample properties of the estimator would strengthen the paper.
  2. [Section 4] The bootstrap procedure resamples time points with replacement, which breaks the temporal dependence structure of the data. Its validity for dependent space-time data is not discussed. A block bootstrap or a parametric bootstrap based on the fitted model would be more standard, and the paper should justify the chosen procedure or state it as a heuristic.
  3. [Section 2.2, Eq. (1)] Equation (1) contains a typographical issue: the expression 'sigma(x)Z(x)^{xi(x)}/xi(x)' and the preceding '-sigma(x)/xi(x)' result in an unusual notation; the intended transformation appears to be (sigma(x)/xi(x))(Z(x)^{xi(x)}-1), and this should be corrected.
  4. [Section 5.3, Figure 5] The x-axis labels in Figure 5, such as '[−2, 2] 8', are unclear because the top and bottom rows are not explicitly labeled as h and u. Please add a clear legend or caption explaining the axis structure.
  5. [Section D.3] The diagnostic in Figure 11 concludes that 'the smoothness of the curves' indicates tau_star is not equal to h/u for the considered lags. This is a qualitative assessment; a quantitative summary, such as the minimum estimated jump size and its uncertainty, would make the diagnostic more rigorous.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the theoretical derivation is self-contained and the empirical limitations are validation concerns, not reductions to inputs.

full rationale

The paper's central theoretical claims—strong consistency and asymptotic normality of the pairwise likelihood estimator—are proved from stated assumptions (Assumption 1, Lemmas 1–4) via uniform strong laws, mixing bounds, a central limit theorem, and a standard identifiability assumption. These proofs do not assume the conclusions. The model itself is stipulated as a max-autoregressive recurrence, and the forecasting distribution in Equation (11) is a derived consequence of that recurrence; this is a model-based forecast rather than a claim that the recurrence was independently derived. The citation to Embrechts et al. (2016) supplies the model class and the bivariate exponent measure Proposition 1; that is transparent prior published mathematical content, and the current paper provides its own proofs for its new theoretical results, so the self-citation is not used to smuggle in an unverified conclusion. The in-sample evaluation acknowledged in Section 5 ('We assess the various goodness-of-fits in-sample, rather than out-of-sample') and the asymmetric conditioning information in Section 5.4 are genuine limitations of the empirical comparison: the proposed model conditions on an upstream spatial neighbor at t0 while the DKS competitor conditions only on a single-site history, and parameters are estimated on the same storm event used for scoring. However, these are correctness and experimental-design concerns, not circularity: the reported CRPS and RMSE values are not algebraic consequences of the fitted parameters, and the forecast ensemble is not forced by construction to match the observations. No step in the derivation chain reduces by definition or by self-citation to its own inputs.

Assumptions & free parameters 9 free parameters · 6 assumptions · 0 invented entities

The central claims rest on a small number of fitted parameters (kappa, H, tau_1, tau_2, a) plus 216 per-site GEV parameters, on standard Brown-Resnick mixing assumptions, and on the unproven equivalence of the two-step estimator to the joint pairwise MLE. No invented physical or ontological entities are introduced.

free parameters (9)
  • kappa (range parameter) = 2.19 (CI 1.83-2.51)
    Range parameter of the Brown-Resnick semivariogram gamma(h) = (||h||/kappa)^{2H}; estimated from data by spatial pairwise likelihood, Section 5.1.
  • H (Hurst parameter) = 0.665 (2H=1.33, CI 1.24-1.41)
    Smoothness parameter of the semivariogram; estimated from data, Section 5.1.
  • tau_1 (advection, east-west component) = 0.35 (CI 0.30-0.38)
    Advection parameter of the recurrence equation; estimated from data via temporal pairwise likelihood, Section 5.1.
  • tau_2 (advection, north-south component) = -0.14 (CI -0.17 to -0.11)
    Advection parameter of the recurrence equation; estimated from data, Section 5.1.
  • a (temporal decay parameter) = 0.97 (CI 0.94-0.99)
    Controls the influence of the past in the recurrence equation; estimated from data, Section 5.1.
  • Per-site GEV marginal parameters (location, scale, shape at each of 216 grid points) = Estimated by maximum likelihood from 105 hourly observations at each site (Section 5)
    Marginal model for wind gust maxima; used to transform data to standard Frechet scale; 648 parameters in total.
  • r (spatial lag threshold) = 21
    Chosen by hand 'to account for all pairs in space' (Section 5.1); affects which pairs contribute to the pairwise likelihood.
  • p (temporal lag order) = 1
    Chosen 'to avoid using too many pairs in time' (Section 5.1).
  • epsilon (parameter space margin) = unspecified
    Introduced in (20) to exclude tau = h/u; the paper says robustness to epsilon should be assessed (Section 4).
assumptions (6)
  • domain assumption The true parameter vector psi* lies in the compact set Psi_epsilon, which excludes tau = h/u for the lags used (Assumption 1)
    Needed for the pairwise likelihood to be well-defined and for identifiability; justified for the data via the ratio random field diagnostic in Section D.3.
  • domain assumption The innovation field W is a spatial Brown-Resnick field with stationary increments and semivariogram gamma(h) = (||h||/kappa)^{2H}, and W is mixing
    Used throughout for the exponent measure, the asymptotic theory, and the spatial dependence structure; standard in max-stable modeling.
  • domain assumption The space-time field Z follows the Markovian recurrence (6), and the aggregated innovation fields fW_{t1}^{t2} are independent and equal in distribution to W
    This is the model construction from Embrechts et al. (2016); the forecasting strategy and conditional distribution (11) rely on it.
  • domain assumption Identifiability of psi from the pairwise densities
    Stated without proof in Section C: 'The elements of Psi_epsilon are identifiable...'; needed for consistency.
  • standard math The asymptotic CLT for the pairwise score applies via Bolthausen (1982), requiring alpha-mixing coefficients that decay fast enough; proven in Lemma 3 under the Brown-Resnick model
    Theorems 1 and 2 rely on these external results.
  • standard math The Dombry et al. (2013) conditional simulation algorithm provides valid samples from the conditional distribution of max-stable fields
    Used in the forecasting strategy to simulate Z(s-u tau, t) from neighboring sites.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Forecasting with Markovian max-stable fields in space and time: An application to wind gust speeds." pith.science (2026). https://pith.science/paper/HPKXDFHY

@misc{pith2026241115511,
  author       = {Pith},
  title        = {Pith review of: Forecasting with Markovian max-stable fields in space and time: An application to wind gust speeds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HPKXDFHY}},
  note         = {Machine review of arXiv:2411.15511}
}
read the original abstract

Hourly maxima of 3-second wind gust speeds are prominent indicators of the severity of wind storms, and accurately forecasting them is thus essential for populations, civil authorities and insurance companies. Space-time max-stable models appear as natural candidates for this, but those explored so far are not suited for forecasting and, more generally, the forecasting literature for max-stable fields is limited. To fill this gap, we consider a specific space-time max-stable model, more precisely a max-autoregressive model with advection, that is well-adapted to model and forecast atmospheric variables. We apply it, as well as our related forecasting strategy, to reanalysis 3-second wind gust data for France in 1999, and show good performance compared to a competitor model. On top of demonstrating the practical relevance of our model, we meticulously study its theoretical properties and show the consistency and asymptotic normality of the space-time pairwise likelihood estimator which is used to calibrate the model.

Figures

Figures reproduced from arXiv: 2411.15511 by the authors.

Figure 1
Figure 1. Considered region (indicated by the shaded rectangle). [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Fields of pointwise hourly maxima of 3-second wind gust on the considered region [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Validation of our model in terms of single-site marginal distributions and spatial [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Points on the diagonal line represent the spatial lags used in Figure 5. The [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]
Figure 5
Figure 5. Figure 5: For each of the space-time lags shown on the x-axis (top: [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]
Figure 6
Figure 6. Figure 6: For each plot (corresponding to a specific spatial lag [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]
Figure 7
Figure 7. Figure 7: In red, the four sites that are used to conditionally simulate the random field at [PITH_FULL_IMAGE:figures/full_fig_p028_7.png]
Figure 8
Figure 8. Figure 8: Scatter plots of observations and forecasts (on the Gumbel scale) from our model [PITH_FULL_IMAGE:figures/full_fig_p030_8.png]
Figure 9
Figure 9. Figure 9: Mean CRPS (left) and RMSE of the mean of the ensemble of 500 independent [PITH_FULL_IMAGE:figures/full_fig_p031_9.png]
Figure 10
Figure 10. Figure 10: Empirical distribution of the margins χh,u(s, t) computed from a single realiza￾tion are shown on [0, 1] for (h, u) such that h ∈ {−2, −1, 0, 1, 2} and u = 1, 2. The four plots correspond to four different parametrizations of our space-time model as indicated below ea…
Figure 11
Figure 11. Figure 11: Empirical distribution functions of the margins of the ratio random fields [PITH_FULL_IMAGE:figures/full_fig_p060_11.png]
Figure 12
Figure 12. Figure 12: Estimated GEV parameters (location, scale and shape from left to right) used [PITH_FULL_IMAGE:figures/full_fig_p060_12.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

41 extracted references · 35 canonical work pages

  1. [1]

    Bauer, P., Thorpe, A., and Brunet, G. (2015). The quiet revolution of numerical weather prediction. Nature , 525:47--55

    work page 2015

  2. [2]

    Bolthausen, E. (1982). On the central limit theorem for stationary mixing random fields. The Annals of Probability , 10:1047--1050

    work page 1982

  3. [3]

    Brown, B. M. and Resnick, S. I. (1977). Extreme values of independent stochastic processes. Journal of Applied Probability , 14(4):732--739

    work page 1977

  4. [4]

    and Kl \"u ppelberg, C

    Buhl, S. and Kl \"u ppelberg, C. (2016). Anisotropic B rown-- R esnick space-time processes: estimation and model assessment. Extremes , 19(4):627--660

    work page 2016

  5. [5]

    A., and Naveau, P

    Cooley, D., Davis, R. A., and Naveau, P. (2007). Prediction for max-stable processes via an approximated conditional density. Colorado State University Department of Statistics Technical Report , 3

    work page 2007

  6. [6]

    A., Kl \"u ppelberg, C., and Steinkohl, C

    Davis, R. A., Kl \"u ppelberg, C., and Steinkohl, C. (2013a). Max-stable processes for modeling extremes observed in space and time. Journal of the Korean Statistical Society , 42(3):399--414

    work page 2013

  7. [7]

    A., Kl \"u ppelberg, C., and Steinkohl, C

    Davis, R. A., Kl \"u ppelberg, C., and Steinkohl, C. (2013b). Statistical inference for max-stable processes in space and time. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 75(5):791--819

    work page 2013

  8. [8]

    A., Kl\"uppelberg, C., and Steinkohl, C

    Davis, R. A., Kl\"uppelberg, C., and Steinkohl, C. (2013c). Statistical inference for max-stable processes in space and time. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 75:791--819

    work page 2013

Show all 41 references
  1. [9]

    Davis, R. A. and Resnick, S. I. (1989). Basic properties and prediction of max- ARMA processes. Advances in Applied Probability , 21(4):781--803

    1989

  2. [10]

    Davis, R. A. and Resnick, S. I. (1993). Prediction of stationary max-stable processes. The Annals of Applied Probability , pages 497--525

    1993

  3. [11]

    C., Padoan, S

    Davison, A. C., Padoan, S. A., and Ribatet, M. (2012). Statistical modeling of spatial extremes. Statistical Science , 27(2):161--186

    2012

  4. [12]

    de Haan, L. (1984). A spectral representation for max-stable processes. The Annals of Probability , 12(4):1194--1204

    1984

  5. [13]

    and Ferreira, A

    de Haan, L. and Ferreira, A. (2006). Extreme V alue T heory: A n I ntroduction . Springer-Verlag New York

    2006

  6. [14]

    and Eyi-Minko, F

    Dombry, C. and Eyi-Minko, F. (2012). Strong mixing properties of max-infinitely divisible random fields. Stochastic Processes and their Applications , 122:3790–3811

    2012

  7. [15]

    Dombry, C., Eyi-Minko, F., and Ribatet, M. (2013). Conditional simulation of max-stable processes. Biometrika , 100(1):111--124

    2013

  8. [16]

    Embrechts, P., Koch, E., and Robert, C. Y. (2016). Space-time max-stable models with spectral separability. Advances in Applied Probability , 48(A):77--97

    2016

  9. [17]

    Gon c alves, A. C. R., Costoya, X., Nieto, R., and Liberato, M. L. R. (2024). Extreme weather events on energy systems: A comprehensive review on impacts, mitigation, and adaptation measures. Sustainable Energy Research , 11(1)

    2024

  10. [18]

    M., Beerli, R., Pfenninger, S., Staffell, I., and Wernli, H

    Grams, C. M., Beerli, R., Pfenninger, S., Staffell, I., and Wernli, H. (2017). Balancing E urope's wind-power output through spatial deployment informed by weather regimes. Nature Climate Change , 7:557--562

    2017

  11. [19]

    and Davison, A

    Huser, R. and Davison, A. C. (2014). Space-time modelling of extreme events. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 76(2):439--461

    2014

  12. [20]

    and Genton, M

    Huser, R. and Genton, M. G. (2016). Non-stationary dependence structures for spatial extremes. Journal of Agricultural, Biological, and Environmental Statistics , 21(3):470--491

    2016

  13. [21]

    Jordan, A., Kr\"uger, F., and Lerch, S. (2019). Evaluating probabilistic forecasts with scoringRules . Journal of Statistical Software , 90(12):1--37

    2019

  14. [22]

    Kabluchko, Z. (2009). Spectral representations of sum- and max-stable processes. Extremes , 12(4):401--424

    2009

  15. [23]

    and Schlather, M

    Kabluchko, Z. and Schlather, M. (2010). Ergodic properties of max-infinitely divisible processes. Stochastic Processes and their Applications , 120(3):281--295

    2010

  16. [24]

    Kabluchko, Z., Schlather, M., and de Haan, L. (2009). Stationary max-stable fields associated to negative definite functions. The Annals of Probability , 37(5):2042--2065

    2009

  17. [25]

    Koh, J., Koch, E., and Davison, A. C. (2024). Space-time extremes of severe US thunderstorm environments. Journal of the American Statistical Association (to appear)

    2024

  18. [26]

    Künsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations. The Annals of Statistics , 17(3):1217--1241

    1989

  19. [27]

    Lebedev, A. (2009). Nonlinear prediction in max-autoregressive processes. Mathematical Notes , 85

    2009

  20. [28]

    Matheron, G. (1963). Principles of geostatistics. Economic Geology , 58(8):1246--1266

    1963

  21. [29]

    Matheson, J. E. and Winkler, R. L. (1976). Scoring rules for continuous probability distributions. Management Science , 22(10):1087--1096

    1976

  22. [30]

    M., Brown, T., and Neumann, F

    Mockert, F., Grams, C. M., Brown, T., and Neumann, F. (2023). Meteorological conditions during periods of low wind speed and insolation in G ermany: T he role of weather regimes. Meteorological Applications , 30(4):e2141

    2023

  23. [31]

    Opitz, T. (2013). Extremal t processes: Elliptical domain of attraction and a spectral representation. Journal of Multivariate Analysis , 122:409--413

    2013

  24. [32]

    A., Ribatet, M., and Sisson, S

    Padoan, S. A., Ribatet, M., and Sisson, S. A. (2010). Likelihood-based inference for max-stable processes. Journal of the American Statistical Association , 105(489):263--277

    2010

  25. [33]

    and Li, Q

    Qian, L. and Li, Q. (2022). A class of max-INAR (1) processes with explanatory variables . Journal of Statistical Computation and Simulation , 92(9):1898--1919

    2022

  26. [34]

    Rasp, S., Hoyer, S., Merose, A., Langmore, I., Battaglia, P., Russell, T., Sanchez-Gonzalez, A., Yang, V., Carver, R., Agrawal, S., et al. (2024). Weatherbench 2: A benchmark for the next generation of data-driven global weather models. Journal of Advances in Modeling Earth Sy...

    2024

  27. [35]

    Ribatet, M. (2022). Spatial E xtremes: M odelling S patial E xtremes. R package version 2.1-0

    2022

  28. [36]

    Schlather, M. (2002). Models for stationary max-stable random fields. Extremes , 5(1):33--44

    2002

  29. [37]

    Smith, R. L. (1990). Max-stable processes and spatial extremes. Unpublished manuscript , University of Surrey

    1990

  30. [38]

    and Mikosch, T

    Straumann, D. and Mikosch, T. (2006). Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equations approach. The Annals of Statistics , 34:2449–2495

    2006

  31. [39]

    Tang, H. et al. (2021). Uncertain max-autoregressive model with imprecise observations. Journal of Intelligent & Fuzzy Systems , 41(6):6915--6922

    2021

  32. [40]

    and Stoev, S

    Wang, Y. and Stoev, S. A. (2011). Conditional sampling for spectrally discrete max-stable random fields. Advances in Applied Probability , 43(2):461--483

    2011

  33. [41]

    Weber, R. O. and Kaufmann, P. (1998). Relationship of synoptic winds and complex terrain flows during the MISTRAL field experiment. Journal of Applied Meteorology , 37(11):1486--1496

    1998

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.