Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-17T10:47:34.497Z Has data issue: false hasContentIssue false
  • English
  • Français

A generalization of Matérn hard-core processes with applications to max-stable processes

Part of: Stochastic processes Geophysics Geophysics (general)

Published online by Cambridge University Press:  23 November 2020

Martin Dirrler ,
Christopher Dörr  and
Martin Schlather
Martin Dirrler*
Affiliation:
Landesbank Baden-Württemberg
Martin Schlather*
Affiliation:
University of Mannheim
*
*Postal address: Landesbank Baden-Württemberg, P.O. Box 106049, 70049, Stuttgart, Germany.
*Postal address: Landesbank Baden-Württemberg, P.O. Box 106049, 70049, Stuttgart, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

Matérn hard-core processes are classical examples for point processes obtained by dependent thinning of (marked) Poisson point processes. We present a generalization of the Matérn models which encompasses recent extensions of the original Matérn hard-core processes. It generalizes the underlying point process, the thinning rule, and the marks attached to the original process. Based on our model, we introduce processes with a clear interpretation in the context of max-stable processes. In particular, we prove that one of these processes lies in the max-domain of attraction of a mixed moving maxima process.

Keywords

Matérn hard-core processlog Gaussian Cox processmixed moving maximamax-domain of attractionCox extremal process

MSC classification

Primary: 60G55: Point processes
Secondary: 86A32: Geostatistics 60G70: Extreme value theory; extremal processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 4 , December 2020 , pp. 1298 - 1312
Check for updates
Copyright
© Applied Probability Trust 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andersen, I. T. and Hahn, U. (2016). Matérn thinned Cox processes. Spatial Statist. 15, 1–21.10.1016/j.spasta.2015.10.005CrossRefGoogle Scholar
Chiu, S., Stoyan, D., Kendall, W. S. and Mecke, J. (2013). Stochastic Geometry and its Applications, 3rd edn. John Wiley, Chichester.10.1002/9781118658222CrossRefGoogle Scholar
Coeurjolly, J., Møller, J. and Waagepetersen, R. P. (2017). Palm distributions for log Gaussian Cox processes. Scand. J. Statist. 44, 192–203.10.1111/sjos.12248CrossRefGoogle Scholar
Cox, D. R. (1955). Some statistical models connected with series of events. J. R. Statist. Soc. B 17, 129–164.Google Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, 2nd edn. Springer, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II, 2nd edn. Springer, New York.10.1007/978-0-387-49835-5CrossRefGoogle Scholar
De Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York.10.1007/0-387-34471-3CrossRefGoogle Scholar
Dirrler, M., Schlather, M. and Strokorb, K. (2016). Conditionally max-stable random fields based on log Gaussian Cox processes. Preprint. arXiv:1612.04576.Google Scholar
Dombry, C. and Eyi-Minko, F. (2013). Regular conditional distributions of continuous max-infinitely divisible random fields. Electron. J. Prob. 18, 1–21.10.1214/EJP.v18-1991CrossRefGoogle Scholar
Giné, E., Hahn, M. G. and Vatan, P. (1990). Max-infinitely divisible and max-stable sample continuous processes. Prob. Theory Relat. Fields 87, 139–165.10.1007/BF01198427CrossRefGoogle Scholar
Ibrahim, A. M., ElBatt, T. and El-Keyi, A. (2013). Coverage probability analysis for wireless networks using repulsive point processes. In Proc. 2013 IEEE 24th Ann. Int. Symp. Personal, Indoor, and Mobile Radio Communications (PIMRC), pp. 1002–1007.10.1109/PIMRC.2013.6666284CrossRefGoogle Scholar
Illian, J., Penttinen, A., Stoyan, H. and Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns. John Wiley, Chichester.Google Scholar
Kuronen, M. and Leskelä, L. (2013). Hard-core thinnings of germ-grain models with power-law grain sizes. Adv. Appl. Prob. 45, 595–625.10.1239/aap/1377868531CrossRefGoogle Scholar
Månsson, M. and Rudemo, M. (2002). Random patterns of nonoverlapping convex grains. Adv. Appl. Prob. 34, 718–738.10.1239/aap/1037990950CrossRefGoogle Scholar
Matérn, B. (1960). Spatial Variation: Stochastic Models and Their Application to Some Problems in Forest Surveys and Other Sampling Investigations. Meddelanden Från Statens Skogsforskningsinstitut, Stockholm.Google Scholar
Mecke, J. (1967). Stationäre zufällige Maße auf lokalkompakten Abelschen Gruppen. Z. Wahrscheinlichkeitsth. 9, 36–58.10.1007/BF00535466CrossRefGoogle Scholar
Møller, J. (2003). Shot noise Cox processes. Adv. Appl. Prob. 35, 614–640.10.1239/aap/1059486821CrossRefGoogle Scholar
Møller, J. and Waagepetersen, R. P. (2004). Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall/CRC, Boca Raton.Google Scholar
Møller, J., Syversveen, A. R. and Waagepetersen, R. P. (1998). Log Gaussian Cox processes. Scand. J. Statist. 25, 451–482.10.1111/1467-9469.00115CrossRefGoogle Scholar
Picard, N. (2005). Tree density estimations using a distance method in Mali savanna. Forest Sci. 51, 7–18.Google Scholar
Schlather, M. (2002). Models for stationary max-stable random fields. Extremes 5, 33–44.10.1023/A:1020977924878CrossRefGoogle Scholar
Smith, R. L. (1990). Max-stable processes and spatial extremes. Unpublished manuscript.Google Scholar
Stoev, S. A. (2008). On the ergodicity and mixing of max-stable processes. Stoch. Process. Appl. 118, 1679–1705.10.1016/j.spa.2007.10.013CrossRefGoogle Scholar
Stoyan, D. (1987). Statistical analysis of spatial point processes: A soft-core model and cross-correlations of marks. Biometrical J. 29, 971–980.10.1002/bimj.4710290811CrossRefGoogle Scholar
Stoyan, D. (1988). Thinnings of point processes and their use in the statistical analysis of a settlement pattern with deserted tillages. Statistics 19, 45–56.10.1080/02331888808802069CrossRefGoogle Scholar
Stoyan, D. and Stoyan, H. (1985). On one of Matérn’s hard-core point process models. Math. Nachr. 122, 205–214.10.1002/mana.19851220121CrossRefGoogle Scholar
Teichmann, J., Ballani, F. and van den Boogaart, K. G. (2013). Generalizations of Matérn’s hard-core point processes. Spatial Statist. 3, 33–53.10.1016/j.spasta.2013.02.001CrossRefGoogle Scholar
Westcott, M. (1972). The probability generating functional. J. Austral. Math. Soc. 14, 448–466.10.1017/S1446788700011095CrossRefGoogle Scholar