Skip to main content
SpringerLink
Log in

Estimation of spatial max-stable models using threshold exceedances

  • Published:
Statistics and Computing Aims and scope Submit manuscript

Abstract

Parametric inference for spatial max-stable processes is difficult since the related likelihoods are unavailable. A composite likelihood approach based on the bivariate distribution of block maxima has been recently proposed. However modeling block maxima is a wasteful approach provided that other information is available. Moreover an approach based on block maxima, typically annual, is unable to take into account the fact that maxima occur or not simultaneously. If time series of, say, daily data are available, then estimation procedures based on exceedances of a high threshold could mitigate such problems. We focus on two approaches for composing likelihoods based on pairs of exceedances. The first one comes from the tail approximation for bivariate distribution proposed by Ledford and Tawn (Biometrika 83:169–187, 1996) when both pairs of observations exceed the fixed threshold. The second one uses the bivariate extension (Rootzén and Tajvidi in Bernoulli 12:917–930, 2006) of the generalized Pareto distribution which allows to model exceedances when at least one of the components is over the threshold. The two approaches are compared through a simulation study where both processes in a domain of attraction of a max-stable process and max-stable processes are successively considered as time replications, according to different degrees of spatial dependency. Results put forward how the nature of the time replications influences the bias of estimations and highlight the choice of each approach regarding to the strength of the spatial dependencies and the threshold choice.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Bacro, J.N., Gaetan, C.: A review on spatial extreme modelling. In: Porcu, E., Montero, J.M., Schlather, M. (eds.) Advances and Challenges in Space-Time Modelling of Natural Events, pp. 103–124. Springer, New York (2012)

    Chapter  Google Scholar 

  • Beirlant, B.J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes: Theory and Applications. Wiley, New York (2004)

    Book  Google Scholar 

  • Bel, L., Bacro, J.N., Lantuéjoul, C.: Assessing extremal dependence of environmental spatial fields. Environmetrics 19, 163–182 (2008)

    Article  MathSciNet  Google Scholar 

  • Bevilacqua, M., Gaetan, C., Mateu, J., Porcu, E.: Estimating space and space-time covariance functions: a weighted composite likelihood approach. J. Am. Stat. Assoc. 107, 268–280 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Buishand, T.A., de Haan, L., Zhou, C.: On spatial extremes: with application to a rainfall problem. Ann. Appl. Stat. 2, 624–642 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Carlstein, A.: The use of subseries values for estimating the variance of a general statistic from a stationary sequence. Ann. Stat. 14, 1171–1179 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  • Casson, E., Coles, S.G.: Spatial regression models for extremes. Extremes 1, 449–468 (1999)

    Article  MATH  Google Scholar 

  • Coles, S.G.: Regional modelling of extreme storms via max-stable processes. J. R. Stat. Soc., Ser. B 55, 797–816 (1993)

    MATH  MathSciNet  Google Scholar 

  • Coles, S.G.: An Introduction to Statistical Modeling of Extreme Value. Springer, Boca Raton (2001)

    Book  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  • Coles, S.G., Tawn, J.A.: Statistical methods models for multivariate extremes: an application to structural design. Appl. Stat. 43, 1–48 (1994)

    Article  MATH  Google Scholar 

  • Coles, S.G., Tawn, J.A.: Modelling extremes of the areal rainfall process. J. R. Stat. Soc., Ser. B 58, 329–347 (1996)

    MATH  MathSciNet  Google Scholar 

  • Davis, R., Yau, C.-Y.: Comments on pairwise likelihood in time series models. Stat. Sin. 21, 255–277 (2011)

    MATH  MathSciNet  Google Scholar 

  • Davison, A.C., Gholamrezaee, M.M.: Geostatistics of extremes. Proc. R. Soc. Lond., Ser. A 468, 581–608 (2012)

    Article  MathSciNet  Google Scholar 

  • Davison, A.C., Smith, R.L.: Models for exceedances over high thresholds. J. R. Stat. Soc., Ser. B 3, 393–442 (1990)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Demarta, S., McNeil, A.J.: The t copula and related copulas. Int. Stat. Rev. 73, 111–129 (2005)

    Article  MATH  Google Scholar 

  • Fawcett, L., Walshaw, D.: Improved estimation for temporally clustered extremes. Environmetrics 18, 173–188 (2007)

    Article  MathSciNet  Google Scholar 

  • Genton, M.G., Ma, Y., Sang, H.: On the likelihood function of Gaussian max-stable processes. Biometrika 98, 481–488 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • Guyon, X.: Random Fields on a Network. Springer, New York (1995)

    MATH  Google Scholar 

  • Heyde, C.: Quasi-Likelihood and Its Application: A General Approach to Optimal Parameter Estimation. Springer, New York (1997)

    Book  MATH  Google Scholar 

  • Huser, R., Davison, A.C.: Space-time modelling of extreme events. Technical report (2012). arXiv:1201.3245

  • Jeon, S., Smith, R.: Dependence structure of spatial extremes using threshold approach. Technical report (2012). arXiv:1209.6344

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

    Article  MATH  MathSciNet  Google Scholar 

  • Kuk, A.: A hybrid pairwise likelihood method. Biometrika 94, 939–952 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Lantuéjoul, C., Bacro, J.-N., Bel, L.: Storm processes and stochastic geometry. Extremes 14, 413–428 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Lindsay, B.: Composite likelihood methods. Contemp. Math. 80, 221–239 (1988)

    Article  MathSciNet  Google Scholar 

  • Madsen, H., Rasmussen, P.F., Rosbjerg, D.: Comparison of annual maximum series and partial duration series methods for modeling extreme hydrologic events: 1. at-site modeling. Water Resour. Res. 33, 747–757 (1997)

    Article  Google Scholar 

  • Nadarajah, S.: Multivariate declustering techniques. Environmetrics 12, 357–365 (2001)

    Article  Google Scholar 

  • Nikoloulopoulos, A., Joe, H., Li, H.: Extreme value properties of multivariate t-copulas. Extreme 12, 129–148 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • Opitz, T.: A spectral construction of the extremal t process. Technical report (2012). arXiv:1207.2296

  • Padoan, S., Bevilacqua, M.: CompRandFld: Composite-Likelihood Based Analysis of Random Fields. R package version 1.0.2 (2012)

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

    Article  MathSciNet  Google Scholar 

  • Pickands, J.: Statistical inference using extreme order statistics. Ann. Stat. 3, 119–131 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  • Rakonczai, P., Tajvidi, N.: On prediction on bivariate extremes. Int. J. Intell. Technol. Appl. Stat. 3, 115–139 (2010)

    Google Scholar 

  • Resnick, S.: Extreme Values, Regular Variation and Point Processes. Springer, New York (1987)

    Book  MATH  Google Scholar 

  • Ribatet, M.: SpatialExtremes: modelling spatial extremes. R package version 1.9-0 (2012)

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Schlather, M., Tawn, J.A.: A dependence measure for multivariate and spatial extreme values: properties and inference. Biometrika 90, 139–156 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Shang, H., Yan, J., Zhang, X.: A two-step composite likelihood approach with data fusion for max-stable processes in spatial extremes modelling. Technical report (2012). arXiv:1204.0286

  • Smith, R.L.: Max-stable processes and spatial extremes. Preprint, University of Surrey (1990a)

  • Smith, R.L.: Regional estimation from spatially dependent data. Preprint, University of North Carolina (1990b)

  • Smith, E.L., Stephenson, A.G.: An extended Gaussian max-stable process model for spatial extremes. J. Stat. Plan. Inference 139, 1266–1275 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • Tawn, J.: Bivariate extreme value theory: models and estimation. Biometrika 75, 397–415 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  • Tawn, J.: Modelling multivariate extreme value distributions. Biometrika 77, 245–253 (1990)

    Article  MATH  Google Scholar 

  • Turkman, K., Amaral Turkman, M., Pereira, J.: Asymptotic models and inference for extremes of spatio-temporal data. Extremes 13, 375–397 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  • Varin, C., Reid, N., Firth, D.: An overview of composite likelihood methods. Stat. Sin. 21, 5–42 (2011)

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

Bacro’s research was partially supported by the ANR-McSim and GICC-Miraccle projects. We are also indebted with the referees for comments that have led to improvements in the article.

Author information

Authors and Affiliations

  1. I3M, Université Montpellier II, 4, Place Eugène Bataillon, 34000, Montpellier, France

    Jean-Noel Bacro

  2. DAIS, Università Ca’ Foscari - Venezia, San Giobbe, Cannaregio 873, 30121, Venezia, Italy

    Carlo Gaetan

Authors
  1. Jean-Noel Bacro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carlo Gaetan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Noel Bacro.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bacro, JN., Gaetan, C. Estimation of spatial max-stable models using threshold exceedances. Stat Comput 24, 651–662 (2014). https://doi.org/10.1007/s11222-013-9394-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11222-013-9394-4

Keywords

Search

Navigation