Hostname: page-component-74656f9884-npj9w Total loading time: 0 Render date: 2026-11-09T13:54:10.134Z Has data issue: false hasContentIssue false
  • English
  • Français

Space‒time max-stable models with spectral separability

Part of: Stochastic processes Inference from stochastic processes

Published online by Cambridge University Press:  25 July 2016

Paul Embrechts ,
Erwan Koch  and
Christian Robert
Paul Embrechts*
Affiliation:
ETH Zürich
Erwan Koch*
Affiliation:
ETH Zürich
Christian Robert*
Affiliation:
Université Lyon 1
*
Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland. Email address: paul.embrechts@math.ethz.ch
Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland. Email address: erwan.koch@math.ethz.ch
ISFA, Université Lyon 1, 50 Avenue Tony Garnier, 69366 Lyon Cedex 07, France. Email address: christian.robert@univ-lyon1.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Natural disasters may have considerable impact on society as well as on the (re-)insurance industry. Max-stable processes are ideally suited for the modelling of the spatial extent of such extreme events, but it is often assumed that there is no temporal dependence. Only a few papers have introduced spatiotemporal max-stable models, extending the Smith, Schlather and Brown‒Resnick spatial processes. These models suffer from two major drawbacks: time plays a similar role to space and the temporal dynamics are not explicit. In order to overcome these defects, we introduce spatiotemporal max-stable models where we partly decouple the influence of time and space in their spectral representations. We introduce both continuous- and discrete-time versions. We then consider particular Markovian cases with a max-autoregressive representation and discuss their properties. Finally, we briefly propose an inference methodology which is tested through a simulation study.

Keywords

Extreme value theoryspatiotemporal max-stable processspectral separabilitytemporal dependence

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G60: Random fields 62M30: Spatial processes

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue A: Probability, Analysis and Number Theory , July 2016 , pp. 77 - 97
Copyright
Copyright © Applied Probability Trust 2016 

References

Beirlant, J.,Goegebeur, Y.,Segers, J. and Teugels, J. (2004).Statistics of Extremes: Theory and Applications.John Wiley,New York.CrossRefGoogle Scholar
Bienvenüe, A. and Robert, C. Y. (2014).Likelihood based inference for high-dimensional extreme value distributions. Preprint. Available at http://arxiv.org/abs/1403.0065v3.Google Scholar
Buhl, S. and Klüppelberg, C. (2015).Anisotropic Brown-Resnick space-time processes: estimation and model assessment. Preprint. Available at http://arxiv.org/abs/1503.06049v3.Google Scholar
Coles, S. (2001).An Introduction to Statistical Modeling of Extreme Values.Springer,London.CrossRefGoogle Scholar
Cooley, D.,Naveau, P. and Poncet, P. (2006).Variograms for spatial max-stable random fields. In Dependence in Probability and Statistics (Lecture Notes Statist. 187), eds P. Bertail, P. Doukhan and P. Soulier,Springer,New York, pp. 373–390.CrossRefGoogle Scholar
Davis, R. A. and Resnick, S. I. (1989).Basic properties and prediction of max-ARMA processes.Adv. Appl. Prob. 21,781–803.CrossRefGoogle Scholar
Davis, R. A.,Klüppelberg, C. and Steinkohl, C. (2013a).Max-stable processes for modeling extremes observed in space and time.J. Korean Statist. Soc. 42,399–414.CrossRefGoogle Scholar
Davis, R. A.,Klüppelberg, C. and Steinkohl, C. (2013b).Statistical inference for max-stable processes in space and time.J. R. Statist. Soc. B 75,791–819.CrossRefGoogle Scholar
Dombry, C. and Eyi-Minko, F. (2014).Stationary max-stable processes with the Markov property.Stoch. Process. Appl. 124,2266–2279.CrossRefGoogle Scholar
Embrechts, P.,Klüppelberg, C. and Mikosch, T. (1997).Modelling Extremal Events for Insurance and Finance.Springer,Berlin.CrossRefGoogle Scholar
Haan, L. de (1984).A spectral representation for max-stable processes.Ann. Prob. 12,1194–1204.CrossRefGoogle Scholar
Haan, L. de and Ferreira, A. (2006).Extreme Value Theory: An Introduction.Springer,New York.CrossRefGoogle Scholar
Haan, L. de and Pickands, J. (1986).Stationary min-stable stochastic processes.Prob. Theory Relat. Fields 72,477–492.CrossRefGoogle Scholar
Hairer, M. (2010).Convergence of Markov processes. Probability at Warwick course. Available at http://www.hairer.org/notes/Convergence.pdf.Google Scholar
Huser, R. and Davison, A. (2014).Space-time modelling of extreme events.J. R. Statist. Soc. B 76,439–461.CrossRefGoogle Scholar
Kabluchko, Z. and Schlather, M. (2010).Ergodic properties of max-infinitely divisible processes.Stoch. Process. Appl. 120,281–295.CrossRefGoogle Scholar
Kabluchko, Z.,Schlather, M. and Haan, L. de (2009).Stationary max-stable fields associated to negative definite functions.Ann. Prob. 37,2042–2065.CrossRefGoogle Scholar
Mardia, K. V. and Jupp, P. E. (1999).Directional Statistics.John Wiley,New York.CrossRefGoogle Scholar
Meinguet, T. (2012).Maxima of moving maxima of continuous functions.Extremes 15,267–297.CrossRefGoogle Scholar
Naveau, P.,Guillou, A.,Cooley, D. and Diebolt, J. (2009).Modelling pairwise dependence of maxima in space.Biometrika 96,1–17.CrossRefGoogle Scholar
Padoan, S. A.,Ribatet, M. and Sisson, S. A. (2010).Likelihood-based inference for max-stable processes.J. Amer. Statist. Assoc. 105,263–277.CrossRefGoogle Scholar
Penrose, M. D. (1992).Semi-min-stable processes.Ann. Prob. 20,1450–1463.CrossRefGoogle Scholar
Resnick, S. I. (1987).Extreme Values, Regular Variation, and Point Processes.Springer,New York.CrossRefGoogle Scholar
Ribatet, M. (2015).SpatialExtremes. R Package, version 2.0–2. Available at http://spatialextremes.r-forge.r-project.org/.Google Scholar
Schlather, M. (2002).Models for stationary max-stable random fields.Extremes 5,33–44.CrossRefGoogle Scholar
Schlather, M. and Tawn, J. A. (2003).A dependence measure for multivariate and spatial extreme values: properties and inference.Biometrika 90,139–156.CrossRefGoogle Scholar
Smith, R. L. (1990).Max-stable processes and spatial extremes. Unpublished manuscript.Google Scholar
Strokorb, K.,Ballani, F. and Schlather, M. (2015).Tail correlation functions of max-stable processes: construction principles, recovery and diversity of some mixing max-stable processes with identical TCF.Extremes 18,241–271.CrossRefGoogle Scholar
Swiss Re., (2014).Natural catastrophes and man-made disasters in 2013: large losses from floods and hail; Haiyan hits the Philippines.Sigma 1/2014,Swiss Re, Zürich.Google Scholar