Modeling extreme values of processes observed at irregular time steps: application to significant wave height. (English) Zbl 1454.62455

Summary: 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.


62P12 Applications of statistics to environmental and related topics
60G70 Extreme value theory; extremal stochastic processes
62G32 Statistics of extreme values; tail inference


Full Text: DOI arXiv Euclid


[1] Ailliot, P., Thompson, C. and Thomson, P. (2011). Mixed methods for fitting the GEV distribution. Water Resour. Res. 47 W05551.
[2] 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.
[3] Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes : Theory and Applications . Wiley, Chichester. · Zbl 1070.62036
[4] Benton, D. and Krishnamoorthy, K. (2002). Performance of the parametric bootstrap method in small sample interval estimates. Adv. Appl. Stat. 2 269-285. · Zbl 1022.62032
[5] Bortot, P. and Gaetan, C. (2014). A latent process model for temporal extremes. Scand. J. Stat. · Zbl 1309.62090
[6] 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.
[7] 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.
[8] Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values . Springer, London. · Zbl 0980.62043
[9] Cooley, D., Nychka, D. and Naveau, P. (2007). Bayesian spatial modeling of extreme precipitation return levels. J. Amer. Statist. Assoc. 102 824-840. · Zbl 1469.62389
[10] Cox, D. R. and Reid, N. (2004). A note on pseudolikelihood constructed from marginal densities. Biometrika 91 729-737. · Zbl 1162.62365
[11] 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. · Zbl 0706.62039
[12] de Haan, L. (1984). A spectral representation for max-stable processes. Ann. Probab. 12 1194-1204. · Zbl 0597.60050
[13] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory : An Introduction . Springer, New York. · Zbl 1101.62002
[14] 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. · Zbl 1093.62052
[15] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events : For Insurance and Finance. Applications of Mathematics ( New York ) 33 . Springer, Berlin. · Zbl 0873.62116
[16] Fawcett, L. and Walshaw, D. (2007). Improved estimation for temporally clustered extremes. Environmetrics 18 173-188.
[17] Fawcett, L. and Walshaw, D. (2012). Estimating return levels from serially dependent extremes. Environmetrics 23 272-283.
[18] 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. · JFM 54.0560.05
[19] Huser, R. and Davison, A. C. (2014). Space-time modelling of extreme events. J. R. Stat. Soc. Ser. B Stat. Methodol. 76 439-461.
[20] Jeon, S. and Smith, R. L. (2012). Dependence structure of spatial extremes using threshold approach. Preprint. Available at . 1209.6344
[21] Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes . Springer, New York. · Zbl 0518.60021
[22] 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. · Zbl 0672.62069
[23] 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.
[24] Padoan, S. A., Ribatet, M. and Sisson, S. A. (2010). Likelihood-based inference for max-stable processes. J. Amer. Statist. Assoc. 105 263-277. · Zbl 1397.62172
[25] Queffeulou, P. (2004). Long-term validation of wave height measurements from altimeters. Mar. Geod. 27 495-510.
[26] 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.” . · Zbl 1454.62455
[27] Reich, B. J. and Shaby, B. A. (2012). A hierarchical max-stable spatial model for extreme precipitation. Ann. Appl. Stat. 6 1430-1451. · Zbl 1257.62120
[28] 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. · Zbl 1303.62091
[29] 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.
[30] 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.
[31] Schlather, M. (2002). Models for stationary max-stable random fields. Extremes 5 33-44. · Zbl 1035.60054
[32] Silva, R. d. S. and Lopes, H. F. (2008). Copula, marginal distributions and model selection: A Bayesian note. Stat. Comput. 18 313-320.
[33] Smith, R. L. (1990). Max-stable processes and spatial extremes. Unpublished manuscript.
[34] Smith, R. L., Tawn, J. A. and Coles, S. G. (1997). Markov chain models for threshold exceedances. Biometrika 84 249-268. · Zbl 0891.60047
[35] 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.
[36] Varin, C. (2008). On composite marginal likelihoods. Adv. Stat. Anal. 92 1-28. · Zbl 1171.62315
[37] Varin, C. and Vidoni, P. (2005). A note on composite likelihood inference and model selection. Biometrika 92 519-528. · Zbl 1183.62037
[38] Vinoth, J. and Young, I. R. (2011). Global estimates of extreme wind speed and wave height. J. Climate 24 1647-1665.
[39] Wimmer, W., Challenor, P. and Retzler, C. (2006). Extreme wave heights in the North Atlantic from altimeter data. Renewable Energy 31 241-248.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.