The Annals of Applied Statistics

Bayesian inference for the Brown–Resnick process, with an application to extreme low temperatures

Emeric Thibaud, Juha Aalto, Daniel S. Cooley, Anthony C. Davison, and Juha Heikkinen

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Abstract

The Brown–Resnick max-stable process has proven to be well suited for modeling extremes of complex environmental processes, but in many applications its likelihood function is intractable and inference must be based on a composite likelihood, thereby preventing the use of classical Bayesian techniques. In this paper we exploit a case in which the full likelihood of a Brown–Resnick process can be calculated, using componentwise maxima and their partitions in terms of individual events, and we propose two new approaches to inference. The first estimates the partitions using declustering, while the second uses random partitions in a Markov chain Monte Carlo algorithm. We use these approaches to construct a Bayesian hierarchical model for extreme low temperatures in northern Fennoscandia.

Article information

Source
Ann. Appl. Stat., Volume 10, Number 4 (2016), 2303-2324.

Dates
Received: June 2015
Revised: August 2016
First available in Project Euclid: 5 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1483606861

Digital Object Identifier
doi:10.1214/16-AOAS980

Mathematical Reviews number (MathSciNet)
MR3592058

Zentralblatt MATH identifier
06688778

Keywords
Global warming likelihood-based inference max-stable process nonstationary extremes partition space-time declustering

Citation

Thibaud, Emeric; Aalto, Juha; Cooley, Daniel S.; Davison, Anthony C.; Heikkinen, Juha. Bayesian inference for the Brown–Resnick process, with an application to extreme low temperatures. Ann. Appl. Stat. 10 (2016), no. 4, 2303--2324. doi:10.1214/16-AOAS980. https://projecteuclid.org/euclid.aoas/1483606861


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References

  • Aalto, J., le Roux, P. C. and Luoto, M. (2014). The meso-scale drivers of temperature extremes in high-latitude Fennoscandia. Climate Dynamics 42 237–252.
  • Andrieu, C. and Roberts, G. O. (2009). The pseudo-marginal approach for efficient Monte Carlo computations. Ann. Statist. 37 697–725.
  • Asadi, P., Davison, A. C. and Engelke, S. (2015). Extremes on river networks. Ann. Appl. Stat. 9 2023–2050.
  • Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes: Theory and Applications. Wiley, Chichester. With contributions from Daniel De Waal and Chris Ferro.
  • Brown, B. M. and Resnick, S. I. (1977). Extreme values of independent stochastic processes. J. Appl. Probab. 14 732–739.
  • Buhl, S. and Klüppelberg, C. (2016). Anisotropic Brown–Resnick space-time processes: Estimation and model assessment. Extremes 19 627–660.
  • Casson, E. and Coles, S. (1999). Spatial regression models for extremes. Extremes 1 449–468.
  • Castruccio, S., Huser, R. and Genton, M. G. (2015). High-order composite likelihood inference for max-stable distributions and processes. J. Comput. Graph. Statist. To appear. DOI:10.1080/10618600.2015.1086656.
  • Chavez-Demoulin, V. and Davison, A. C. (2012). Modelling time series extremes. REVSTAT 10 109–133.
  • Coles, S. G. and Tawn, J. A. (1991). Modelling extreme multivariate events. J. Roy. Statist. Soc. Ser. B 53 377–392.
  • Cooley, D., Nychka, D. and Naveau, P. (2007). Bayesian spatial modeling of extreme precipitation return levels. J. Amer. Statist. Assoc. 102 824–840.
  • Davison, A. C. and Gholamrezaee, M. M. (2012). Geostatistics of extremes. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468 581–608.
  • Davison, A. C., Huser, R. and Thibaud, E. (2013). Geostatistics of dependent and asymptotically independent extremes. Math. Geosci. 45 511–529.
  • Davison, A. C., Padoan, S. A. and Ribatet, M. (2012). Statistical modeling of spatial extremes. Statist. Sci. 27 161–186.
  • de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York.
  • Dieker, A. B. and Mikosch, T. (2015). Exact simulation of Brown–Resnick random fields at a finite number of locations. Extremes 18 301–314.
  • Dombry, C., Éyi-Minko, F. and Ribatet, M. (2013). Conditional simulation of max-stable processes. Biometrika 100 111–124.
  • Engelke, S., Malinowski, A., Kabluchko, Z. and Schlather, M. (2015). Estimation of Hüsler–Reiss distributions and Brown–Resnick processes. J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 239–265.
  • Fuentes, M., Henry, J. and Reich, B. (2013). Nonparametric spatial models for extremes: Application to extreme temperature data. Extremes 16 75–101.
  • Huser, R. and Davison, A. C. (2013). Composite likelihood estimation for the Brown–Resnick process. Biometrika 100 511–518.
  • Huser, R. and Genton, M. G. (2016). Non-stationary dependence structures for spatial extremes. J. Agric. Biol. Environ. Stat. 21 470–491.
  • IPCC (2013). Summary for policymakers. In Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change (T. F. Stocker, D. Qin, G. K. Plattner, M. Tignor, S. K. Allen, J. Boschung, A. Nauels, Y. Xia, V. Bex and P. M. Midgley, eds.) 3–29. Cambridge Univ. Press, New York.
  • Kabluchko, Z., Schlather, M. and de Haan, L. (2009). Stationary max-stable fields associated to negative definite functions. Ann. Probab. 37 2042–2065.
  • Nadarajah, S. (2001). Multivariate declustering techniques. Environmetrics 12 357–365.
  • Nikoloulopoulos, A. K., Joe, H. and Li, H. (2009). Extreme value properties of multivariate $t$ copulas. Extremes 12 129–148.
  • Opitz, T. (2013). Extremal $t$ processes: Elliptical domain of attraction and a spectral representation. J. Multivariate Anal. 122 409–413.
  • Padoan, S. A., Ribatet, M. and Sisson, S. A. (2010). Likelihood-based inference for max-stable processes. J. Amer. Statist. Assoc. 105 263–277.
  • Rand, W. M. (1971). Objective criteria for the evaluation of clustering methods. J. Amer. Statist. Assoc. 66 846–850.
  • Ribatet, M. (2013). Spatial extremes: Max-stable processes at work. J. SFdS 154 156–177.
  • Ribatet, M. (2015). SpatialExtremes: Modelling spatial extremes. R package version 2.0-2.
  • Ribatet, M., Cooley, D. and Davison, A. C. (2012). Bayesian inference from composite likelihoods, with an application to spatial extremes. Statist. Sinica 22 813–845.
  • Sang, H. and Gelfand, A. E. (2009). Hierarchical modeling for extreme values observed over space and time. Environ. Ecol. Stat. 16 407–426.
  • Sang, H. and Gelfand, A. E. (2010). Continuous spatial process models for spatial extreme values. J. Agric. Biol. Environ. Stat. 15 49–65.
  • Schlather, M. (2002). Models for stationary max-stable random fields. Extremes 5 33–44.
  • Shaby, B. A. (2014). The open-faced sandwich adjustment for MCMC using estimating functions. J. Comput. Graph. Statist. 23 853–876.
  • Shaby, B. A. and Reich, B. J. (2012). Bayesian spatial extreme value analysis to assess the changing risk of concurrent high temperatures across large portions of European cropland. Environmetrics 23 638–648.
  • Smith, R. L. (1990). Max-stable processes and spatial extremes. Unpublished manuscript, Univ. Surrey. Available at http://www.stat.unc.edu/postscript/rs/spatex.pdf.
  • Stephenson, A. G. (2009). High-dimensional parametric modelling of multivariate extreme events. Aust. N. Z. J. Stat. 51 77–88.
  • Stephenson, A. and Tawn, J. (2005). Exploiting occurrence times in likelihood inference for componentwise maxima. Biometrika 92 213–227.
  • Thibaud, E. and Opitz, T. (2015). Efficient inference and simulation for elliptical Pareto processes. Biometrika 102 855–870.
  • Thibaud, E., Aalto, J., Cooley, D. S., Davison, A. C. and Heikkinen, J. (2016). Supplement to “Bayesian inference for the Brown–Resnick process, with an application to extreme low temperatures.” DOI:10.1214/16-AOAS980SUPP.
  • Virtanen, T., Neuvonen, S. and Nikula, A. (1998). Modelling topoclimatic patterns of egg mortality of Epirrita autumnata (Lepidoptera: Geometridae) with a Geographical Information System: Predictions for current climate and warmer climate scenarios. Journal of Applied Ecology 35 311–322.
  • Wadsworth, J. L. (2015). On the occurrence times of componentwise maxima and bias in likelihood inference for multivariate max-stable distributions. Biometrika 102 705–711.
  • Wadsworth, J. L. and Tawn, J. A. (2014). Efficient inference for spatial extreme value processes associated to log-Gaussian random functions. Biometrika 101 1–15.

Supplemental materials

  • Supplementary Material for “Bayesian inference for the Brown–Resnick process, with an application to extreme low temperatures”. The online supplement provides additional results for the exploratory analysis and the MCMC algorithm used to fit the two Brown–Resnick models, and reports additional diagnostics for the fit of the three models considered in the manuscript.