kth-order Markov extremal models for assessing heatwave risks

Abstract

Heatwaves are defined as a set of hot days and nights that cause a marked short-term increase in mortality. Obtaining accurate estimates of the probability of an event lasting many days is important. Previous studies of temporal dependence of extremes have assumed either a first-order Markov model or a particularly strong form of extremal dependence, known as asymptotic dependence. Neither of these assumptions is appropriate for the heatwaves that we observe for our data. A first-order Markov assumption does not capture whether the previous temperature values have been increasing or decreasing and asymptotic dependence does not allow for asymptotic independence, a broad class of extremal dependence exhibited by many processes including all non-trivial Gaussian processes. This paper provides a kth-order Markov model framework that can encompass both asymptotic dependence and asymptotic independence structures. It uses a conditional approach developed for multivariate extremes coupled with copula methods for time series. We provide novel methods for the selection of the order of the Markov process that are based upon only the structure of the extreme events. Under this new framework, the observed daily maximum temperatures at Orleans, in central France, are found to be well modelled by an asymptotically independent third-order extremal Markov model. We estimate extremal quantities, such as the probability of a heatwave event lasting as long as the devastating European 2003 heatwave event. Critically our method enables the first reliable assessment of the sensitivity of such estimates to the choice of the order of the Markov process.

References

  1. Bortot, P., Tawn, J.A.: Models for the extremes of Markov chains. Biometrika 85(4), 851–867 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  2. Brockwell, P.J., Davis, R.A.: Introduction to Time Series and Forecasting. Springer (2006)

  3. Chatfield, C.: The Analysis Of Time Series: An Introduction. CRC Press (2003)

  4. Coles, S.G.: An Introduction to Statistical Modeling of Extreme Values. Springer Verlag (2001)

  5. Coles, S.G., Tawn, J.A.: Modelling Extreme Multivariate Events. Journal of the Royal Statistical Society: Series B 53(2), 377–392 (1991)

    MathSciNet  MATH  Google Scholar 

  6. Davison, A.C., Smith, R.L.: Models for exceedances over high thresholds (with discussion). J. R. Stat. Soc. Ser. B 52(3), 393–442 (1990)

    MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  8. de Haan, L., Ferreira, A.: Extreme Value Theory: An Introduction. Springer, New York (2006)

    Google Scholar 

  9. Drees, H., Segers, J., Warchol, M.: Statistics for tail processes of Markov chains. Extremes 18, 369–402 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  10. Dunn, O.J.: Multiple comparisons among means. J. Am. Stat. Assoc. 56(293), 52–64 (1961)

    MathSciNet  Article  MATH  Google Scholar 

  11. Dupuis, D.J.: Modeling waves of extreme temperature: the changing tails of four cities. J. Am. Stat. Assoc. 107(497), 24–39 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  12. Fawcett, L., Walshaw, D.: Markov chain models for extreme wind speeds. Environmetrics 17(8), 795–809 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  13. Ferro, C.A.T., Segers, J.: Inference for clusters of extreme values. J. R. Stat. Soc. Ser. B 65(2), 545–556 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  14. Genest, C., Ghoudi, K., Rivest, L.P.: A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82(3), 543–552 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  15. Heffernan, J.E.: A directory of coefficients of tail dependence. Extremes 3, 279–290 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  16. Heffernan, J.E., Resnick, S.I.: Limit laws for random vectors with an extreme component. Ann. Appl. Probab. 17(2), 537–571 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  17. Heffernan, J.E., Tawn, J.A.: A conditional approach for multivariate extreme values (with discussion). J. R. Stat. Soc. Ser. B 66(3), 497–546 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  18. Hitz, A., Evans, R.: One-component regular variation and graphical modeling of extremes. ArXiv e-prints 1506.03402 (2015)

  19. Joe, H.: Multivariate Models and Dependence Concepts. Chapman and Hall/CRC (1997)

  20. Keef, C., Papastathopoulos, I., Tawn, J.A.: Estimation of the conditional distribution of a multivariate variable given that one of its components is large: Additional constraints for the Heffernan and Tawn model. J. Multivar. Anal. 115, 396–404 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  21. Kulik, R., Soulier, P.: Heavy tailed time series with extremal independence. Extremes 18, 1–27 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  22. Ledford, A.W., Tawn, J.A.: Modelling dependence within joint tail regions. J. R. Stat. Soc. Ser. B 59(2), 475–499 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  23. Ledford, A.W., Tawn, J.A.: Diagnostics for dependence within time series extremes. J. R. Stat. Soc. Ser. B 65(2), 521–543 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  24. Liang, K.L., Self, S.G.: On the asymptotic behaviour of the pseudolikelihood ratio test statistic. J. R. Stat. Soc. Ser. B 58(4), 785–796 (1996)

    MathSciNet  MATH  Google Scholar 

  25. Lugrin, T., Davison, A., Tawn, J.A.: Bayesian uncertainty management in temporal dependence of extremes. Extremes 19, 491–515 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  26. Papastathopoulos, I., Tawn, J.A.: Graphical structures in extreme multivariate events. In: Proceedings of 25th Panhellenic Statistics Conference, pp 315–323 (2013)

  27. Papastathopoulos, I., Strokorb, K., Tawn, J.A., Butler, A.: Extreme events of Markov chains. Adv. Appl. Probab. 49 (2017)

  28. Pickands, J.: The two–dimensional Poisson process and extremal processes. J. Appl. Probab. 8, 745–756 (1971)

    MathSciNet  Article  MATH  Google Scholar 

  29. Reich, B.J., Shaby, B.A., Cooley, D.: A hierarchical model for serially-dependent extremes: A study of heat waves in the western US. J. Agric. Biol. Environ. Stat. 19, 119–135 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  30. Resnick, S.I.: Extreme Values, Regular Variation, and Point Processes. Springer Verlag (1987)

  31. Ribatet, M., Ouarda, T.B.M.J., Sauquet, E., Gresillon, J.M.: Modeling all exceedances above a threshold using an extremal dependence structure: Inferences on several flood characteristics. Water Resour. Res. 45, W03,407 (2009)

  32. Rootzén, H.: Maxima and exceedances of stationary Markov chains. Adv. Appl. Probab. 20, 371–390 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  33. Smith, R.L.: The extremal index for a Markov chain. J. Appl. Probab. 29(1), 37–45 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  34. Smith, R.L., Weissman, I.: Estimating the extremal index. J. R. Stat. Soc. Ser. B 56(3), 515–528 (1994)

    MathSciNet  MATH  Google Scholar 

  35. Smith, R.L., Tawn, J.A., Coles, S.G.: Markov chain models for threshold exceedances. Biometrika 84(2), 249–268 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  36. Wadsworth, J.L., Tawn, J.A.: A new representation for multivariate tail probabilities. Bernoulli 19(5B), 2689–2714 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  37. Winter, H.C., Tawn, J.A.: Modelling heatwaves in central France: A case study in extremal dependence. J. R. Stat. Soc. Ser. C 65(3), 345–365 (2016)

    MathSciNet  Article  Google Scholar 

  38. Winter, H.C., Tawn, J.A., Brown, S.J.: Detecting Changing Behaviour of Heatwaves with Climate Change. Submitted (2016)

  39. Yun, S.: The distributions of cluster functionals of extreme events in a dth-order Markov chain. J. Appl. Probab. 37(1), 29–44 (2000)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hugo C. Winter.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Winter, H.C., Tawn, J.A. kth-order Markov extremal models for assessing heatwave risks. Extremes 20, 393–415 (2017). https://doi.org/10.1007/s10687-016-0275-z

Download citation

Keywords

  • Asymptotic independence
  • Conditional extremes
  • Extremal dependence
  • Heatwaves
  • Markov chain
  • Time-series extremes

AMS 2000 Subject Classifications

  • 60G70
  • 62G32
  • 62P12