The Annals of Applied Statistics

Modeling extreme values of processes observed at irregular time steps: Application to significant wave height

Nicolas Raillard, Pierre Ailliot, and Jianfeng Yao

Full-text: Open access

Abstract

This work is motivated by the analysis of the extremal behavior of buoy and satellite data describing wave conditions in the North Atlantic Ocean. The available data sets consist of time series of significant wave height (Hs) with irregular time sampling. In such a situation, the usual statistical methods for analyzing extreme values cannot be used directly. The method proposed in this paper is an extension of the peaks over threshold (POT) method, where the distribution of a process above a high threshold is approximated by a max-stable process whose parameters are estimated by maximizing a composite likelihood function. The efficiency of the proposed method is assessed on an extensive set of simulated data. It is shown, in particular, that the method is able to describe the extremal behavior of several common time series models with regular or irregular time sampling. The method is then used to analyze Hs data in the North Atlantic Ocean. The results indicate that it is possible to derive realistic estimates of the extremal properties of Hs from satellite data, despite its complex space–time sampling.

Article information

Source
Ann. Appl. Stat., Volume 8, Number 1 (2014), 622-647.

Dates
First available in Project Euclid: 8 April 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1396966301

Digital Object Identifier
doi:10.1214/13-AOAS711

Mathematical Reviews number (MathSciNet)
MR3192005

Zentralblatt MATH identifier
06302250

Keywords
Extreme values time series max-stable process composite likelihood irregular time sampling significant wave height satellite data

Citation

Raillard, Nicolas; Ailliot, Pierre; Yao, Jianfeng. Modeling extreme values of processes observed at irregular time steps: Application to significant wave height. Ann. Appl. Stat. 8 (2014), no. 1, 622--647. doi:10.1214/13-AOAS711. https://projecteuclid.org/euclid.aoas/1396966301


Export citation

References

  • Ailliot, P., Thompson, C. and Thomson, P. (2011). Mixed methods for fitting the GEV distribution. Water Resour. Res. 47 W05551.
  • Ailliot, P., Baxevani, A., Cuzol, A., Monbet, V. and Raillard, N. (2011). Space–time models for moving fields with an application to significant wave height fields. Environmetrics 22 354–369.
  • Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes: Theory and Applications. Wiley, Chichester.
  • Benton, D. and Krishnamoorthy, K. (2002). Performance of the parametric bootstrap method in small sample interval estimates. Adv. Appl. Stat. 2 269–285.
  • Bortot, P. and Gaetan, C. (2014). A latent process model for temporal extremes. Scand. J. Stat. To appear.
  • Caires, S. and Sterl, A. (2005). 100-year return value estimates for ocean wind speed and significant wave height from the ERA-40 data. J. Climate 18 1032–1048.
  • Challenor, P. G., Foale, S. and Webb, D. J. (1990). Seasonal changes in the global wave climate measured by the Geosat altimeter. Int. J. Remote Sens. 11 2205–2213.
  • Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer, London.
  • Cooley, D., Nychka, D. and Naveau, P. (2007). Bayesian spatial modeling of extreme precipitation return levels. J. Amer. Statist. Assoc. 102 824–840.
  • Cox, D. R. and Reid, N. (2004). A note on pseudolikelihood constructed from marginal densities. Biometrika 91 729–737.
  • Davison, A. C. and Smith, R. L. (1990). Models for exceedances over high thresholds. J. R. Stat. Soc. Ser. B Stat. Methodol. 52 393–442.
  • de Haan, L. (1984). A spectral representation for max-stable processes. Ann. Probab. 12 1194–1204.
  • de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York.
  • Drees, H., de Haan, L. and Li, D. (2006). Approximations to the tail empirical distribution function with application to testing extreme value conditions. J. Statist. Plann. Inference 136 3498–3538.
  • Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Applications of Mathematics (New York) 33. Springer, Berlin.
  • Fawcett, L. and Walshaw, D. (2007). Improved estimation for temporally clustered extremes. Environmetrics 18 173–188.
  • Fawcett, L. and Walshaw, D. (2012). Estimating return levels from serially dependent extremes. Environmetrics 23 272–283.
  • Fisher, R. A. and Tippett, L. H. C. (1928). Limiting forms of the frequency distribution of the largest or smallest member of a sample. Math. Proc. Cambridge Philos. Soc. 24 180–190.
  • Huser, R. and Davison, A. C. (2014). Space–time modelling of extreme events. J. R. Stat. Soc. Ser. B Stat. Methodol. 76 439–461.
  • Jeon, S. and Smith, R. L. (2012). Dependence structure of spatial extremes using threshold approach. Preprint. Available at arXiv:1209.6344.
  • Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
  • Lindsay, B. G. (1988). Composite likelihood methods. In Statistical Inference from Stochastic Processes (Ithaca, NY, 1987). Contemp. Math. 80 221–239. Amer. Math. Soc., Providence, RI.
  • Menéndez, M., Méndez, F. J., Losada, I. J. and Graham, N. E. (2008). Variability of extreme wave heights in the northeast Pacific Ocean based on buoy measurements. Geophys. Res. Lett. 35 L22607.
  • Padoan, S. A., Ribatet, M. and Sisson, S. A. (2010). Likelihood-based inference for max-stable processes. J. Amer. Statist. Assoc. 105 263–277.
  • Queffeulou, P. (2004). Long-term validation of wave height measurements from altimeters. Mar. Geod. 27 495–510.
  • Raillard, N., Ailliot, P. and Yao, J. (2014). Supplement to “Modeling extreme values of processes observed at irregular time steps: Application to significant wave height.” DOI:10.1214/13-AOAS711SUPP.
  • Reich, B. J. and Shaby, B. A. (2012). A hierarchical max-stable spatial model for extreme precipitation. Ann. Appl. Stat. 6 1430–1451.
  • Reich, B. J., Shaby, B. A. and Cooley, D. (2013). A hierarchical model for serially-dependent extremes: A study of heat waves in the western US. J. Agric. Biol. Environ. Stat. 1–17.
  • Ribatet, M., Ouarda, T. B. M. J., Sauquet, E. and Gresillon, J. M. (2009). Modeling all exceedances above a threshold using an extremal dependence structure: Inferences on several flood characteristics. Water Resour. Res. 45 W03407.
  • Ribereau, P., Naveau, P. and Guillou, A. (2011). A note of caution when interpreting parameters of the distribution of excesses. Adv. Water Resour. 34 1215–1221.
  • Schlather, M. (2002). Models for stationary max-stable random fields. Extremes 5 33–44.
  • Silva, R. d. S. and Lopes, H. F. (2008). Copula, marginal distributions and model selection: A Bayesian note. Stat. Comput. 18 313–320.
  • Smith, R. L. (1990). Max-stable processes and spatial extremes. Unpublished manuscript.
  • Smith, R. L., Tawn, J. A. and Coles, S. G. (1997). Markov chain models for threshold exceedances. Biometrika 84 249–268.
  • Tournadre, J. and Ezraty, R. (1990). Local climatology of wind and sea state by means of satellite radar altimeter measurements. J. Geophys. Res. 95 18255–18268.
  • Varin, C. (2008). On composite marginal likelihoods. Adv. Stat. Anal. 92 1–28.
  • Varin, C. and Vidoni, P. (2005). A note on composite likelihood inference and model selection. Biometrika 92 519–528.
  • Vinoth, J. and Young, I. R. (2011). Global estimates of extreme wind speed and wave height. J. Climate 24 1647–1665.
  • Wimmer, W., Challenor, P. and Retzler, C. (2006). Extreme wave heights in the North Atlantic from altimeter data. Renewable Energy 31 241–248.

Supplemental materials

  • Supplementary material: Supplementary material: Proof of the consistency of the MPL1E estimates. In the attached supplemental material [Raillard, Ailliot and Yao (2013)], we prove the consistency of the $\mathrm{MPL}_{1}\mathrm{E}$ estimator in an idealized situation with no censoring and known marginal distributions.