Skip to main content

Modeling spatial extremes using normal mean-variance mixtures

Abstract

Classical models for multivariate or spatial extremes are mainly based upon the asymptotically justified max-stable or generalized Pareto processes. These models are suitable when asymptotic dependence is present, i.e., the joint tail decays at the same rate as the marginal tail. However, recent environmental data applications suggest that asymptotic independence is equally important and, unfortunately, existing spatial models in this setting that are both flexible and can be fitted efficiently are scarce. Here, we propose a new spatial copula model based on the generalized hyperbolic distribution, which is a specific normal mean-variance mixture and is very popular in financial modeling. The tail properties of this distribution have been studied in the literature, but with contradictory results. It turns out that the proofs from the literature contain mistakes. We here give a corrected theoretical description of its tail dependence structure and then exploit the model to analyze a simulated dataset from the inverted Brown–Resnick process, hindcast significant wave height data in the North Sea, and wind gust data in the state of Oklahoma, USA. We demonstrate that our proposed model is flexible enough to capture the dependence structure not only in the tail but also in the bulk.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Data availability statement

The hindcast wave height data is kindly provided by Philip Jonathan from Shell Research, and the wind gust data is freely available from mesonet.org.

Code availability

The code can be provided upon request and will be made publicly available later.

References

  • Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th edn. National Bureau of Standards, United States of America (1972)

  • Barndorff-Nielsen, O.E.: Exponentially decreasing distributions for the logarithm of particle size. Proc. R. Soc. Lond. Ser. A 353, 401–419 (1977)

    Article  Google Scholar 

  • Barndorff-Nielsen, O.E.: Normal inverse Gaussian distributions and stochastic volatility modelling. Scand. J. Stat. 24(1), 1–13 (1997)

    MathSciNet  Article  Google Scholar 

  • Blaesild, P., Jensen, J.L.: Multivariate distributions of hyperbolic type. Statistical Distributions in Scientific Work 4, 45–66 (1981)

    MathSciNet  Article  Google Scholar 

  • Bortot, P., Coles, S., Tawn, J.A.: The multivariate gaussian tail model: an application to oceanographic data. J. Royal Stat. Soc. (Series C) 49(1), 31–49 (2000)

    MathSciNet  Article  Google Scholar 

  • Castro Camilo, D., Huser, R.: Local likelihood estimation of complex tail dependence structures, applied to U.S. precipitation extremes. J. Am. Stat. Assoc. 115, 1037–1054 (2020)

    MathSciNet  Article  Google Scholar 

  • Castruccio, S., Huser, R., Genton, M.G.: Higher-order composite likelihood inference for max-stable distributions and processes. J. Comput. Graph. Stat. 25(4), 1212–1229 (2016)

    Article  Google Scholar 

  • Coles, S., Tawn, J.A.: Modelling extreme multivariate events. J. Royal Stat. Soc. (Series B) 53, 377–392 (1991)

    MathSciNet  MATH  Google Scholar 

  • Coles, S., Heffernan, J., Tawn, J.A.: Dependence measures for extreme value analyses. Extremes 2(4), 339–365 (1999)

    Article  Google Scholar 

  • Cooley, D., Thibaud, E., Castillo, F., Wehner, M.F.: A nonparametric method for producing isolines of bivariate exceedance probabilities. Extremes 22, 373–390 (2019)

    MathSciNet  Article  Google Scholar 

  • Dagpunar, J.S.: An easily implemented generalized inverse Gaussian generator. Communications in Statistics - Simulation and Computation 18, 703–710 (1989)

    MathSciNet  Article  Google Scholar 

  • Davison, A.C., Huser, R.: Statistics of extremes. Annual Reviews of Statistics and Its Application 2, 203–235 (2015)

    Article  Google Scholar 

  • Davison, A.C., Padoan, S.A., Ribatet, M.: Statistical modeling of spatial extremes. Stat. Sci. 27(2), 161–186 (2012)

    MathSciNet  MATH  Google Scholar 

  • de Haan, L.: A spectral representation for max-stable processes. Ann. Probab. 12(4), 1194–1204 (1984)

    MathSciNet  Article  Google Scholar 

  • de Haan, L., Pereira, T.T.: Spatial extremes: models for the stationary case. Ann. Stat. 34(1), 146–168 (2006)

    MathSciNet  MATH  Google Scholar 

  • Di Bernardino, E., Fernández-Ponce, J.M., Palacios-Rodríguez, F., Nolo, M.R.R.G.: On multivariate extensions of the conditional value-at-risk measure. Insurance: Mathematics and Economics 61, 1–16 (2015)

  • Dombry, C., Engelke, S., Oesting, M.: Exact simulation of max-stable process. Biometrika 103(2), 303–317 (2016)

    MathSciNet  Article  Google Scholar 

  • Embrechts, P., McNeil, A., Straumann, D.: Correlation and dependence in risk management: properties and pitfalls. In: Moffatt, H. (ed.) Dempster M, pp. 176–223. Value at Risk and Beyond, Cambrige University Press, Risk Management (2001)

    Google Scholar 

  • Engelke, S., Opitz, T., Wadsworth, J.L.: Extremal dependence of random scale constructions. Extremes 22, 623–666 (2019)

    MathSciNet  Article  Google Scholar 

  • Ferreira, A., de Haan, L.: The generalized Pareto process; with a view towards application and simulation. Bernoulli 20(4), 1717–1737 (2014)

    MathSciNet  Article  Google Scholar 

  • Gong, Y., Huser, R.: Asymmetric tail dependence modeling, with application to cryptocurrency market data. Ann. Appl. Stat. (2021)

  • Heffernan, J.E., Tawn, J.A.: A conditional approach to multivariate extreme values. J. Royal Stat. Soc. (Series B) 66, 497–546 (2004)

    MathSciNet  Article  Google Scholar 

  • Huser, R., Wadsworth, J.L.: Modeling spatial processes with unknown extremal dependence class. J. Am. Stat. Assoc. 114(525), 434–444 (2019)

    MathSciNet  Article  Google Scholar 

  • Huser, R., Wadsworth, J,L.: Advances in statistical modeling of spatial extremes. Wiley Interdisciplinary Reviews: Computational Statistics e1537 (2020)

  • Huser, R., Opitz, T., Thibaud, E.: Bridging asymptotic independence and dependence in spatial extremes using gaussian scale mixtures. Spatial Statistics 21, 166–186 (2017)

    MathSciNet  Article  Google Scholar 

  • Jamalizadeh, A., Balakrishnan, N.: Conditional distributions of multivariate normal mean-variance mixtures. Statist. Probab. Lett. 145, 312–316 (2019)

    MathSciNet  Article  Google Scholar 

  • Kabluchko, Z., Schlather, M., de Haan, L.: Stationary max-stable fields associated to negative definite functions. Ann. Probab. 37(5), 2042–2065 (2009)

    MathSciNet  Article  Google Scholar 

  • Krupskii, P., Huser, R., Genton, M.G.: Factor copula models for replicated spatial data. J. Am. Stat. Assoc. 113(521), 467–479 (2018)

    MathSciNet  Article  Google Scholar 

  • Le, P.D., Davison, A.C., Engelke, S., Leonard, M., Westra, S.: Dependence properties of spatial rainfall extremes and areal reduction factors. J. Hydrol. 565, 711–719 (2018)

    Article  Google Scholar 

  • Ledford, A.W., Tawn, J.A.: Statistics for near independence in multivariate extreme values. Biometrika 83(1), 169–187 (1996)

    MathSciNet  Article  Google Scholar 

  • Manner, H., Segers, J.: Tails of correlation mixtures of elliptical copulas. Insurance: Mathematics and Economics 48, 153–160 (2011)

  • McNeil, A.J., Frey, R., Embrechts, P.: Quantitative Risk Management: Concepts. Princeton University Press, Techniques and Tools (2005)

    MATH  Google Scholar 

  • Morris, S.A., Reich, B.J., Thibaud, E., Cooley, D.: A space-time skew-\(t\) model for threshold exceedances. Biometrics 73, 749–758 (2017)

    MathSciNet  Article  Google Scholar 

  • Murphy-Barltrop, C.J.R., Wadsworth, J.L., Eastoe, E.F.: On the estimation of bivariate return curves for extreme values. (2021) https://arxiv.org/abs/2107.01942

  • Nolde, N.: Geometric interpretation of the residual dependence coefficient. J. Multivar. Anal. 123, 85–95 (2014)

    MathSciNet  Article  Google Scholar 

  • Nolde, N., Wadsworth, J.L.: Linking representations for multivariate extremes via a limit set. Adv. Appl. Prob. (2021)

  • Padoan, S.A., Ribatet, M., Sisson, S.A.: Likelihood-based inference for max-stable processes. J. Am. Stat. Assoc. 105(489), 263–277 (2010)

    MathSciNet  Article  Google Scholar 

  • Prause, K.: The generalized hyperbolic model: Estimation, financial derivatives, and risk measures. PhD thesis, Albert-Ludwigs-Universität Freiburg. (1999)

  • R Core Team: R: A language and environment for statistical computing, R Foundation for Statistical Computing. Austria, Vienna (2020). (https://www.R-project.org/)

    Google Scholar 

  • Rootzén, H., Tajvidi, N.: Multivariate generalized Pareto distributions. Bernoulli 12(5), 917–930 (2006)

    MathSciNet  Article  Google Scholar 

  • Schlueter, S., Fischer, M.: The weak tail dependence coefficient of the elliptical generalized hyperbolic distribution. Extremes 15, 159–174 (2012)

    MathSciNet  Article  Google Scholar 

  • Sklar, A.: Functions de répartition à n dimensions et leurs marges. Publications de l’institut de Statistique de l’Université de Paris 8, 229–231 (1959)

    MathSciNet  MATH  Google Scholar 

  • von Hammerstein, E.A.: Tail behavior and tail dependence of generalized hyperbolic distributions. In: Kallsen, J., Papapantoleon, A. (eds.) Advanced Modelling in Mathematical Finance, pp. 3–40. Springer, Cham (2016)

    Chapter  Google Scholar 

  • Wadsworth, J.L., Tawn, J.A.: Dependence modelling for spatial extremes. Biometrika 99(2), 253–272 (2012)

    MathSciNet  Article  Google Scholar 

  • Wadsworth, J.L, Tawn, J.A: Higher-dimensional spatial extremes via single-site conditioning. (2019). https://arxiv.org/abs/1912.06560

  • Zscheischler, J., Orth, R., Seneviratne, S.I.: Bivariate return periods of temperature and precipitation explain a large fraction of European crop yields. Biogeosciences 14, 3309–3320 (2017)

    Article  Google Scholar 

Download references

Acknowledgements

We are grateful to a referee and an Associate Editor for careful reading and constructive comments that improved our paper. We gratefully acknowledge Philip Jonathan of Shell Research for providing the wave height data analysed in Section 4.1.

Funding

This publication is based upon work supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Awards No. OSR-CRG2017-3434 and OSR-CRG2020-4394.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Raphaël Huser.

Ethics declarations

Conflicts of interest

Not applicable.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhang, Z., Huser, R., Opitz, T. et al. Modeling spatial extremes using normal mean-variance mixtures. Extremes 25, 175–197 (2022). https://doi.org/10.1007/s10687-021-00434-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10687-021-00434-2

Keywords

  • Asymptotic independence
  • Copula model
  • Generalized hyperbolic distribution
  • Normal mean-variance mixtures
  • Spatial extremes

AMS 2000 Subject Classifications

  • 60G70
  • 60E07
  • 62H20