, Volume 5, Issue 1, pp 33–44 | Cite as

Models for Stationary Max-Stable Random Fields

  • Martin Schlather


Models for stationary max-stable random fields are revisited and illustrated by two-dimensional simulations. We introduce a new class of models, which are based on stationary Gaussian random fields, and whose realizations are not necessarily semi-continuous functions. The bivariate marginal distributions of these random fields can be calculated, and they form a new class of bivariate extreme value distributions.

bivariate extreme value distribution dependence function Gaussian random field max-stable random field rainfall modeling simulation of max-stable processes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ancona-Navarrete, M.A. and Tawn, J.A., “Diagnostics for extremal dependence in spatial processes,” Submitted to Extremes, 2002.Google Scholar
  2. Barndorff-Nielsen, O.E., Gupta, V.K., Pérez-Abreu, V., and Waymire, E., (eds), Stochastic Methods in Hydrology, World Scientific, Singapore, New Jersey, London, Hong Kong, 1998.Google Scholar
  3. Casson, E. and Coles, S.G., “Spatial regression models for extremes,” Extremes 1, 449–468, (1999).Google Scholar
  4. Casson, E. and Coles, S.G., “Simulation and extremal analysis of hurricane events,” J. R. Statist. Soc., Ser. C 49, 227–245, (2000).Google Scholar
  5. Coles, S.G., “Regional modelling of extreme storms via max-stable processes,” J. R. Statist. Soc., Ser. B. 55, 797–816, (1993).Google Scholar
  6. Coles, S.G. and Tawn, J.A., “Modelling extremes of the areal rainfall process,” J. R. Statist. Soc., Ser. B 58, 329–347, (1996).Google Scholar
  7. Davis, R.A. and Resnick, S.I., “Prediction of stationary max-stable processes,” Ann. Appl. Probab. 3, 497–525, (1993).Google Scholar
  8. de Haan, L., “A spectral representation for max-stable processes,” Ann. Probab. 12, 1194–1204, (1984).Google Scholar
  9. de Haan, L. and Pickands, J., “Stationary min-stable stochastic processes,” Probab. Th. Rel. Fields. 72, 477–492, (1986).Google Scholar
  10. Deheuvels, P., “Point processes and multivariate extreme values,” J. Multivar. Anal. 13, 257–272, (1983).Google Scholar
  11. Giné, E., Hahn, M.G., and Vatan, P., “Max-infinitely divisible and max-stable sample continuous processes,” Probab. Th. Rel. Fields 87, 139–165, (1990).Google Scholar
  12. Heinrich, L. and Molchanov, I.S., “Some limit theorems for extremal and union shot-noise processes,” Math. Nachr. 168, 139–159, (1994).Google Scholar
  13. Ihaka, R. and Gentleman, R., “R: A language for data analysis and graphics,” J. Comput. Graph. Stat. 5(3), 299–314, (1996).Google Scholar
  14. Jeulin, D. and Jeulin, P., “Synthesis of rough surfaces of random morphological functions,” Stereo. Iugosl. 3, 239–246, (1981).Google Scholar
  15. Matheron, G., Random Sets and Integral Geometry, John Wiley & Sons, New York, 1975.Google Scholar
  16. Norberg, T., On the existence and convergence of probabiity measures on continuous semi-lattices. Technical Report 148, Center for Stochastic Processes, University of North Carolina, 1986.Google Scholar
  17. Pickands, J., “Multivariate extreme value distributions,” Bull. Int. Stat. Inst. 49, 859–878, (1981).Google Scholar
  18. Resnick, S.I., Extreme Values, Regular Variation, and Point Processes, Volume 4, Springer, New York, Berlin, 1987.Google Scholar
  19. Schlather, M., “Simulation and analysis of random fields,” R News 1(2), 18–20, (2001).Google Scholar
  20. Smith, R.L., “Max-stable processes and spatial extremes,” Unpublished manuscript, 1990.Google Scholar
  21. Stoyan, D., Kendall, W.S., and Mecke, J., Stochastic Geometry and its Applications (second edition), John Wiley & Sons, Chichester, 1995.Google Scholar
  22. Tawn, J.A., “An extreme-value theory model for dependent observations,” J. Hydrology 101, 227–250, (1988).Google Scholar
  23. Wackernagel, H., Multivariate Geostatistics (second edition), Springer, Berlin, Heidelberg, 1998.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Martin Schlather
    • 1
  1. 1.Soil Physics GroupUniversity of BayreuthBayreuthGermany

Personalised recommendations