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A Review on Spatial Extreme Modelling

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Book cover Advances and Challenges in Space-time Modelling of Natural Events

Part of the book series: Lecture Notes in Statistics ((LNSP,volume 207))

Abstract

In this chapter we review recent advances in modelling spatial extremes. After a brief illustration of the extreme value theory for univariate and multivariate values, we concentrate on spatial max-stable processes. Statistical inference and simulation for these processes are subject of a close examination. Max-stable processes are also contrasted with spatial hierarchical models. The review ends with summarizing some open problems.

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References

  1. Bacro, J.N., Gaetan, C.: Composite likelihood estimation for spatial max-stable processes using threshold exceedances. Submitted for publication (2011)

    Google Scholar 

  2. Banerjee, S., Carlin, B.P., Gelfand, A.E.: Hierarchical Modeling and Analysis for Spatial Data. Chapman & Hall/CRC Press, Boca Raton (2004)

    MATH  Google Scholar 

  3. Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes: Theory and Applications. John Wiley & Sons, New York (2004)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Besag, J.: Spatial interaction and the statistical analysis of lattice systems (with discussion). J. Roy. Stat. Soc. B. 36, 192–236 (1974)

    MathSciNet  MATH  Google Scholar 

  6. Brown, B., Resnick, S.: Extremes values of independent stochastic processes. J. Appl. Probab. 14, 732–739 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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  MathSciNet  MATH  Google Scholar 

  8. Capéraà, P., Fougères, A.L., Genest, C.: A non parametric estimation procedure for bivariate extreme value copulas. Biometrika. 84, 567–577 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  12. 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 

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

    MathSciNet  MATH  Google Scholar 

  14. Coles, S.G., Walshaw, D.: Directional modelling of extreme wind speed. J. Appl. Stat. 43, 139–157 (1994)

    Article  MATH  Google Scholar 

  15. Cooley, D, Naveau P., Poncet, P.: Variograms for spatial max-stable random fields. In: Bickel P, Diggle P, Fienberg S, Gather U, Olkin I, Zeger S, Bertail P, Soulier P, Doukhan P (eds) Dependence in Probability and Statistics. Lect. Notes Stat. 187, 373–390, Springer, New York (2006)

    Google Scholar 

  16. Cooley, D., Nychka, D., Naveau, P.: Bayesian spatial modeling of extreme precipitation return levels. J. Am. Stat. Assoc. 102, 824–840 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Davis, R.A., Mikosch, T.: Extreme value theory for space-time processes with heavy-tailed distributions. Stoch. Proc. Appl. 118, 560–584 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  19. Davison, A., Padoan, S., Ribatet, M.: Statistical modelling of spatial extremes. Accepted for publication, Statistical Science (2011)

    Google Scholar 

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

    Google Scholar 

  21. de Haan, L., Ferreira, A.: Extreme Value Theory: an Introduction. Springer, New-York (2006)

    MATH  Google Scholar 

  22. de Haan, L., Pereira, T.T.: Spatial extremes: models for the stationary case. Ann. Stat. 34146–168 (2006)

    Article  MATH  Google Scholar 

  23. Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events: for Insurance and Finance, corrected edn. Springer, New York (1997)

    Book  MATH  Google Scholar 

  24. Ferro, C.A.T., Segers, J.: Inference for clusters of extreme values. J. Roy. Stat. Soc. B. 65, 545–556 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Fisher, R.A., Tippett, L.H.C.: Limiting forms of the frequency distribution of the largest or smallest member of a sample. P. Camb. Philos. Soc. 24, 180–190 (1928)

    Article  MATH  Google Scholar 

  26. Fougères, A., Nolan, J.P., Rootzén, H.: Models for dependent extremes using stable mixtures. Scand. J. Stat.: 36, 42–59 (2009)

    Google Scholar 

  27. Fougères, A.L.: Multivariate extremes. In: Finkenstädt B, Rootzén H (eds) Extreme Values in Finance, Telecommunications, and the Environment. Chapman & Hall/CRC, Boca Raton, 373–388 (2004)

    Google Scholar 

  28. Friederichs, P.: Statistical downscaling of extreme precipitation events using extreme value theory. Extremes. 13, 109–132 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gaetan, C., Grigoletto, M.: A hierarchical model for the analysis of spatial rainfall extremes. J. Agr. Biol. Envir. St. 12434–449 (2007)

    Article  MathSciNet  Google Scholar 

  30. Galambos, J.: The Asymptotic Theory of Extreme Order Statistics, 2nd edn. Wiley, New York (1987)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Ghosh, S., Mallick, B.K.: A hierarchical Bayesian spatio-temporal model for extreme precipitation events. Environmetrics 22, 192–204 (2011)

    Article  MathSciNet  Google Scholar 

  33. Gnedenko, B.V.: Sur la distribution limite du terme maximum d’une série aléatoire. Ann. Math. 44, 423–453 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  34. Gumbel, E.: Distribution de valeurs extremes en plusieurs dimensions. Publications de l’Institut de Statistique de l’Université de Paris. 9, 171–173 (1960)

    MathSciNet  MATH  Google Scholar 

  35. Hüsler, J., Reiss, R.: Maxima of normal random vectors: between independence to complete dependence. Stat. Probabil. Lett. 7, 283–286 (1989)

    Article  MATH  Google Scholar 

  36. Jenkinson, A.F.: The frequency distribution of the annual maximum (or minimum) values of meteorological elements. Q. J. Roy. Meteor. Soc. 81, 158–171 (1955)

    Article  Google Scholar 

  37. Joe, H.: Families of min-stable multivariate exponential and multivariate extreme value distributions. Stat. Probabil. Lett. 9, 75–81 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  40. Oesting, M., Kabluchko, Z., Schlather, M. : Simulation of Brown-Resnick processes. Extremes. To appear, doi: 10.1007/s10687-011-0128-8 (2011).

    Google Scholar 

  41. Lantuéjoul, C.: Geostatistical Simulation. Springer, Berlin, Heidelberg (2002)

    MATH  Google Scholar 

  42. Lantuéjoul, C., Bacro, J.N., Bel, L. : Storm processes and stochastic geometry. Extremes. To appear, doi: 10.1007/s10687-010-0121-7 (2011)

    Google Scholar 

  43. Ledford, A.W., Tawn, J.A.: Diagnostics for dependence within time series extremes. J. Roy. Stat. Soc. B.: 65, 521–543 (2003)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  45. Mannshardt-Shamseldin, E., Smith, R., SR Sain, L.M., Cooley, D.: Downscaling extremes: a comparison of extreme value distributions in point-source and gridded presipitation data. Ann. Appl. Stat. 4, 484–502 (2010)

    Google Scholar 

  46. Mikosch, T.: Copulas: tales and facts (with discussion). Extremes 9:3–62 (2006)

    Article  MathSciNet  Google Scholar 

  47. Naveau, P., Guillou, A., Cooley, D., Diebolt, J.: Modelling pairwise dependence of maxima in space. Biometrika, 96, 1–17 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  48. Oesting, M., Kabluchko, Z., Schlather, M.: Simulation of Brown-Resnick processes. Extremes. To appear, doi: 10.1007/s10687-011-0128-8 (2011)

    Google Scholar 

  49. Padoan, S.: ExtremalProc: modelling and analysis of extremal processes. R package version 0.1-2 (2010)

    Google Scholar 

  50. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  52. Pickands, J.: Multivariate extreme value distributions. In: Proceedings of the 43rd Session ISI, Buenos Aires, 859–878 (1981)

    Google Scholar 

  53. Resnick, S.: Extreme values, Regular variation and Point Processes. Springer Verlag, New York (1987)

    MATH  Google Scholar 

  54. Ribatet, M.: Geostatistics for extremes. Manuscript, Université Montpellier II, Montpellier (2010a)

    Google Scholar 

  55. Ribatet, M.: SpatialExtremes: Modelling Spatial Extremes. R package version 1.6-0 (2010b)

    Google Scholar 

  56. Sang, H., Gelfand, A.: Continuous spatial process models for spatial extreme values. J. Agr. Biol. Envir. St. 15, 49–65 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  58. Schlather, M.: RandomFields: Simulation and Analysis of Random Fields. R package version 1.3.41 (2009)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  62. Smith, R.L.: Extreme value analysis of environmental time series: an application to trend detection in ground-level ozone. Stat. Sci. 4, 367–393 (1989)

    Article  MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  65. Smith, R.L., Weissman, L.: Characterization and estimation of the multivariate extremal index. Preprint University of North Carolina, Chapel Hill (1996)

    Google Scholar 

  66. Stephenson, A.G., Tawn, J.: Bayesian inference for extremes: accounting for the three extremal types. Extremes. 7, 291–307 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  67. Stoev, S.A., Taqqu, M.S.: Extremal stochastic integrals: a parrallel between max-stable and alpha-stable processes. Extremes 8, 237–266 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  73. Wang, Y., Stoev, S.A.: On the association of sum- and max-stable processes. Stat. Probabil. Lett. 80, 480–488 (2010a)

    Article  MathSciNet  MATH  Google Scholar 

  74. Wang, Y., Stoev, S.A.: On the structure and representations of max-stable processes. Adv. Appl. Probab. 42, 855–877 (2010b)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  76. Zhang, Z., Smith., R.L.: The behavior of multivariate maxima of moving maxima processes. J. Appl. Probab. 41, 1113–1123 (2004)

    Google Scholar 

  77. Zhang, Z., Smith, R.L.: On the estimation and application of max-stable processes. J. Stat. Plan. Infer. 12, 1135–1153 (2010)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Gaetan’s research was partially supported by MIUR grant 2008MRFM2H. Bacro’s research was partially supported by the ANR-McSim and GICC-Miraccle projects.

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Authors and Affiliations

  1. I3M, Université Montpellier II, 34095, Montpellier, Cedex 5, France

    Jean-Noël Bacro

  2. Dipartimento di Scienze Ambientali, Informatica e Statistica, Università Ca’ Foscari-Venezia, I-30121, Venezia, San Giobbe, Cannaregio 873, Italy

    Carlo Gaetan

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  1. Jean-Noël Bacro
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  2. Carlo Gaetan
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Correspondence to Jean-Noël Bacro .

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Editors and Affiliations

  1. Facultad de Derecho y Ciencias Sociales, Área de Estadística, Universidad de Castilla la Mancha, Avenida Camilo Jose Cela 14, Ciudad Real, 13005, Spain

    Emilio Porcu

  2. , Statistics, Universidad de Castilla la Mancha, Cobertizo San Pedro Martir, Toledo, 45071, Spain

    José–María Montero

  3. , Mathematics Stochastics, Universität Göttingen, Goldschmidtstrasse 7, Göttingen, 37077, Germany

    Martin Schlather

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Bacro, JN., Gaetan, C. (2012). A Review on Spatial Extreme Modelling. In: Porcu, E., Montero, J., Schlather, M. (eds) Advances and Challenges in Space-time Modelling of Natural Events. Lecture Notes in Statistics(), vol 207. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17086-7_5

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