A Bayesian spatio-temporal model for precipitation extremes – STOR team contribution to the EVA2017 challenge. (English) Zbl 1404.62133
This paper outlines one approach which was used in the EVA2017 challenge, whose target was to predict extreme quantiles for rainfall at several sites in the Netherlands.
Letting \(R_{j,m}\) denote the daily rainfall amount at site \(j\) in month \(m=1,\ldots,12\), the authors model the transformed random variable \[ \tilde{R}_{j,m}=\log\left(1+R_{j,m}\right) \] using a Bayesian hierarchical model. Their model assigns a probability \(p_{m,j}\) of observing a non-zero amount of rainfall. The amount of rainfall (for days on which this is positive) is then modelled using an extremal mixture model; given a threshold \(u_{j,m}\), rainfall below this threshold is modelled using a truncated Gamma distribution, and rainfall above this threshold using a generalised Pareto distribution. The prior in the Bayesian model aims to exploit the spatial and seasonal structure by assuming that parameters for neighbouring sites and consecutive months are likely to be similar.
Letting \(R_{j,m}\) denote the daily rainfall amount at site \(j\) in month \(m=1,\ldots,12\), the authors model the transformed random variable \[ \tilde{R}_{j,m}=\log\left(1+R_{j,m}\right) \] using a Bayesian hierarchical model. Their model assigns a probability \(p_{m,j}\) of observing a non-zero amount of rainfall. The amount of rainfall (for days on which this is positive) is then modelled using an extremal mixture model; given a threshold \(u_{j,m}\), rainfall below this threshold is modelled using a truncated Gamma distribution, and rainfall above this threshold using a generalised Pareto distribution. The prior in the Bayesian model aims to exploit the spatial and seasonal structure by assuming that parameters for neighbouring sites and consecutive months are likely to be similar.
Reviewer: Fraser Daly (Edinburgh)
MSC:
| 62P12 | Applications of statistics to environmental and related topics |
| 62G32 | Statistics of extreme values; tail inference |
| 62F15 | Bayesian inference |
| 62M30 | Inference from spatial processes |
| 62E20 | Asymptotic distribution theory in statistics |
Keywords:
Bayesian hierarchical modelling; extreme value analysis; Markov chain Monte Carlo; precipitation extremes; spatio-temporal dependenceReferences:
| [1] | Banerjee, S., Carlin, B.P., Gelfand, A.E.: Hierarchical Modeling and Analysis for Spatial Data. CRC Press, Boca Raton (2004) · Zbl 1053.62105 |
| [2] | Behrens, CN; Lopes, HF; Gamerman, D., Bayesian analysis of extreme events with threshold estimation, Stat. Model., 4, 227-244, (2004) · Zbl 1111.62023 · doi:10.1191/1471082X04st075oa |
| [3] | Brown, BM; Resnick, SI, Extreme values of independent stochastic processes, J. Appl. Probab., 14, 732-739, (1977) · Zbl 0384.60055 · doi:10.2307/3213346 |
| [4] | Coles, S.G.: An Introduction to Statistical Modeling of Extreme Values. Springer Series in Statistics. Springer, London (2001) · Zbl 0980.62043 |
| [5] | Cooley, D.; Nychka, D.; Naveau, P., Bayesian spatial modeling of extreme precipitation return levels, J. Am. Stat. Assoc., 102, 824-840, (2007) · Zbl 1469.62389 · doi:10.1198/016214506000000780 |
| [6] | Davison, AC; Padoan, SA; Ribatet, M., Statistical modeling of spatial extremes, Stat. Sci., 27, 161-186, (2012) · Zbl 1330.86021 · doi:10.1214/11-STS376 |
| [7] | Haan, L., A spectral representation for MAX-stable processes, Ann. Probab., 12, 1194-1204, (1984) · Zbl 0597.60050 · doi:10.1214/aop/1176993148 |
| [8] | Diggle, PJ; Tawn, JA; Moyeed, RA, Model-based geostatistics, J. R. Stat. Soc.: Ser. C: (Appl. Stat.), 47, 299-350, (1998) · Zbl 0904.62119 · doi:10.1111/1467-9876.00113 |
| [9] | Fawcett, L.; Walshaw, D., A hierarchical model for extreme wind speeds, J. R. Stat. Soc.: Ser. C: (Appl. Stat.), 55, 631-646, (2006) · Zbl 1109.62115 · doi:10.1111/j.1467-9876.2006.00557.x |
| [10] | Fisher, RA; Tippett, LHC, Limiting forms of the frequency distribution of the largest or smallest member of a sample, Proc. Camb. Philos. Soc., 24, 180-190, (1928) · JFM 54.0560.05 · doi:10.1017/S0305004100015681 |
| [11] | Frigessi, A.; Haug, O.; Rue, H., A dynamic mixture model for unsupervised tail estimation without threshold selection, Extremes, 5, 219-235, (2002) · Zbl 1039.62042 · doi:10.1023/A:1024072610684 |
| [12] | Knorr-Held, L., Richardson, S.: Some remarks on Gaussian Markov random field models for disease mapping. In: Green, PJ, Hjort, NL (eds.) Highly Structured Stochastic Systems, pp 203-207. Oxford University Press, Oxford (2003) |
| [13] | Koenker, R.: Quantile Regression. Cambridge University Press, Cambridge (2005) · Zbl 1111.62037 · doi:10.1017/CBO9780511754098 |
| [14] | MacDonald, A.; Scarrott, C.; Lee, D.; Darlow, B.; Reale, M.; Russell, G., A flexible extreme value mixture model, Comput. Stat. Data Anal., 55, 2137-2157, (2011) · Zbl 1328.62296 · doi:10.1016/j.csda.2011.01.005 |
| [15] | Martins, ES; Stedinger, JR; etal., Generalized maximum-likelihood generalized extreme-value quantile estimators for hydrologic data, Water Resour. Res., 36, 737-744, (2000) · doi:10.1029/1999WR900330 |
| [16] | Pickands, J., Statistical inference using extreme order statistics, Ann. Stat., 3, 119-131, (1975) · Zbl 0312.62038 · doi:10.1214/aos/1176343003 |
| [17] | Reich, BJ; Shaby, BA, A hierarchical MAX-stable spatial model for extreme precipitation, Ann. Appl. Stat., 6, 1430-1451, (2012) · Zbl 1257.62120 · doi:10.1214/12-AOAS591 |
| [18] | Roberts, GO; Gelman, A.; Gilks, WR, Weak convergence and optimal scaling of random walk metropolis algorithms, Ann. Appl. Probab., 7, 110-120, (1997) · Zbl 0876.60015 · doi:10.1214/aoap/1034625254 |
| [19] | Sang, H.; Gelfand, AE, Hierarchical modeling for extreme values observed over space and time, Environ. Ecol. Stat., 16, 407-426, (2009) · doi:10.1007/s10651-007-0078-0 |
| [20] | Scarrott, C.; MacDonald, A., A review of extreme value threshold estimation and uncertainty quantification, REVSTAT—Stat. J., 10, 33-60, (2012) · Zbl 1297.62120 |
| [21] | Schlather, M., Models for stationary MAX-stable random fields, Extremes, 5, 33-44, (2002) · Zbl 1035.60054 · doi:10.1023/A:1020977924878 |
| [22] | Smith, R.L.: Max-stable processes and spatial extremes. Unpublished manuscript (1990) |
| [23] | So, BJ; Kwon, HH; Kim, D.; Lee, SO, Modeling of daily rainfall sequence and extremes based on a semiparametric Pareto tail approach at multiple locations, J. Hydrol., 529, 1442-1450, (2015) · doi:10.1016/j.jhydrol.2015.08.037 |
| [24] | Wadsworth, JL; Tawn, JA; Jonathan, P., Accounting for choice of measurement scale in extreme value modeling, Ann. Appl. Stat., 4, 1558-1578, (2010) · Zbl 1202.62066 · doi:10.1214/10-AOAS333 |
| [25] | Wilks, D.S.: Statistical Methods in the Atmospheric Sciences, 2nd edn. International Geophysics Series, vol 59. Elsevier, Amsterdam (2006) |
| [26] | Wintenberger, O.: Editorial: special issue on the Extreme Value Analysis conference challenge ‘Prediction of extremal precipitation’. Extremes (to appear) (2018) · Zbl 1427.00081 |
| [27] | Yunus, RM; Hasan, MM; Razak, NA; Zubairi, YZ; Dunn, PK, Modelling daily rainfall with climatological predictors: Poisson-gamma generalized linear modelling approach, Int. J. Climatol., 37, 1391-1399, (2017) · doi:10.1002/joc.4784 |
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