Latent Process Modelling of Threshold Exceedances in Hourly Rainfall Series



Two features are often observed in analyses of both daily and hourly rainfall series. One is the tendency for the strength of temporal dependence to decrease when looking at the series above increasing thresholds. The other is the empirical evidence for rainfall extremes to approach independence at high enough levels. To account for these features, Bortot and Gaetan (Scand J Stat 41:606–621, 2014) focus on rainfall exceedances above a fixed high threshold and model their dynamics through a hierarchical approach that allows for changes in the temporal dependence properties when moving further into the right tail. It is found that this modelling procedure performs generally well in analyses of daily rainfalls, but has some inherent theoretical limitations that affect its goodness of fit in the context of hourly data. In order to overcome this drawback, we develop here a modification of the Bortot and Gaetan model derived from a copula-type technique. Application of both model versions to rainfall series recorded in Camborne, England, shows that they provide similar results when studying daily data, but in the analysis of hourly data the modified version is superior.


Asymptotic independence Exceedance Extreme values Generalized Pareto distribution Hourly rainfall Hierarchical model Latent process 

Supplementary material

13253_2016_254_MOESM1_ESM.pdf (27 kb)
Supplementary material 1 (pdf 27 KB)


  1. Bortot, P., and Gaetan, C. (2014), “A latent process model for temporal extremes,” Scandinavian Journal of Statistics, 41, 606–621.Google Scholar
  2. Bortot, P., and Tawn, J. (1998), “Models for the extremes of Markov chains,” Biometrika, 85, 851–867.Google Scholar
  3. Casciani, M. (2015), Analisi dei valori estremi di serie storiche: un approccio bayesiano,, Master’s thesis, Facoltà di Economia, Università degli Studi di Roma “La Sapienza”, Rome, Italy.Google Scholar
  4. Chavez-Demoulin, V., and Davison, A. C. ( 2012), “Modelling time series extremes,” REVSTAT - Statistical Journal, 10, 109–133.Google Scholar
  5. Coles, S. G. (2001), An Introduction to Statistical Modeling of Extreme Values, New York: Springer.Google Scholar
  6. Coles, S. G., Tawn, J. A., and Smith, R. L. ( 1994), “A seasonal Markov model for extremely low temperatures,” Environmetrics, 5, 221–239.Google Scholar
  7. Davis, R., and Yau, C.-Y. (2011), “Comments on pairwise likelihood in time series models,” Statistica Sinica, 21, 255–277.Google Scholar
  8. Davison, A. C., and Smith, R. L. ( 1990), “Models for exceedances over high thresholds,” Journal of Royal Statistical Society: Series B, 3, 393–442.Google Scholar
  9. de Haan, L. (1984), “A spectral representation for max-stable processes,” The Annals of Probability, 12, 1194–1204.Google Scholar
  10. Eastoe, E. F., and Tawn, J. A. (2009) , “Modelling non-stationary extremes with application to surface level ozone,” Journal of the Royal Statistical Society: Series C (Applied Statistics), 58, 25–45.Google Scholar
  11. Ferro, C. A. T., and Segers, J. ( 2003), “Inference for clusters of extreme values,” Journal of Royal Statistical Society: Series B, 65, 545–556.Google Scholar
  12. Gaver, D., and Lewis, P. (1980), “First-order autoregressive Gamma sequences and point processes,” Advances in Applied Probability, 12, 727–745.Google Scholar
  13. Huser, R., and Davison, A. C. (2014) , “Space-time modelling of extreme events,” Journal of the Royal Statistical Society: Series B, 76, 439–461.Google Scholar
  14. Jonathan, P., and Ewans, K. (2013), “Statistical modelling of extreme ocean environments for marine design: a review,” Ocean Engineering, 62, 91–109.Google Scholar
  15. Katz, R. W., Parlange, M. B., and Naveau, P. ( 2002), “Statistics of extremes in hydrology,” Advances in Water Resources, 25, 1287–1304.Google Scholar
  16. Koutsoyiannis, D. (2004), “Statistics of extremes and estimation of extreme rainfall: II. Empirical investigation of long rainfall records / Statistiques de valeurs extrêmes et estimation de précipitations extrêmes: II. Recherche empirique sur de longues séries de précipitations,” Hydrological Sciences Journal, 49, 591–610.Google Scholar
  17. Leadbetter, K. R., Lindgren, G., and Rootzén, H. ( 1983), Extremes and Related Properties of Random Sequences and Processes, Berlin: Springer.Google Scholar
  18. Ledford, A. W., and Tawn, J.A. ( 1997), “Modelling dependence within joint tail regions,” Journal of the Royal Statistical Society: Series B, 59, 475–499.Google Scholar
  19. —— 2003. “Diagnostics for dependence within time series extremes,” Journal of Royal Statistical Society: Series B, 65, 521–543.Google Scholar
  20. Lindsay, B. (1988), “Composite likelihood methods,” Contemporary Mathematics, 80, 221–239.Google Scholar
  21. Marin, J.-M., Pudlo, P., Robert, C. P., and Ryder, R. J. (2011), “Approximate Bayesian computational methods,” Statistics and Computing, 22, 1167–1180.Google Scholar
  22. Pickands, J. (1975), “Statistical inference using extreme order statistics,” The Annals of Statistics, 3, 119–131.Google Scholar
  23. Raillard, N., Ailliot, P., and Yao, J. ( 2014), “Modeling extreme values of processes observed at irregular time steps: Application to significant wave height,” The Annals of Applied Statistics, 8, 622–647.Google Scholar
  24. Reich, B. J., Shaby, B. A., and Cooley, D. ( 2014), “A hierarchical model for serially-dependent extremes: a study of heat waves in the Western US,” Journal of Agricultural, Biological, and Environmental Statistics, 19, 119–135.Google Scholar
  25. Reiss, R., and Thomas, M. (2007), Statistical Analysis of Extreme Values, third edn, Basel: Birkhäuser.Google Scholar
  26. Robinson, M. E., and Tawn, J. A. ( 2000), “Extremal analysis of processes sampled at different frequencies,” Journal of the Royal Statistical Society. Series B (Statistical Methodology), 62, 117–135.Google Scholar
  27. Smith, R. L. (1990), “Regional estimation from spatially dependent data,” Preprint, University of North Carolina.Google Scholar
  28. Smith, R., Tawn, J. A., and Coles, S. ( 1997), “Markov chain models for threshold exceedances,” Biometrika, 84, 249–268.Google Scholar
  29. Varin, C., and Vidoni, P. (2005), “A note on composite likelihood inference and model selection,” Biometrika, 52, 519–528.Google Scholar
  30. Walker, S. (2000), “A note on the innovation distribution of a Gamma distributed autoregressive process,” Scandinavian Journal of Statistics, 27, 575–576.Google Scholar
  31. Warren, D. (1992), “A multivariate Gamma distribution arising from a Markov model,” Stochastic Hydrology and Hydraulics, 6, 183–190.Google Scholar

Copyright information

© International Biometric Society 2016

Authors and Affiliations

  1. 1.Dipartimento di Scienze StatisticheUniversità di BolognaBolognaItaly
  2. 2. Dipartimento di Scienze Ambientali, Informatica e StatisticaUniversità Ca’ Foscari - VeneziaVeniceItaly

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