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Simulation of cluster point processes without edge effects

Part of: Geometric probability and stochastic geometry Multivariate analysis Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Anders Brix  and
Wilfrid S. Kendall
Anders Brix*
Affiliation:
Risk Management Solutions Ltd
Wilfrid S. Kendall*
Affiliation:
University of Warwick
*
∗ Postal address: Risk Management Solutions Ltd, 10 Eastcheap, London EC3N 1AJ, UK. Email address: [email protected]
∗∗ Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK.
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Abstract

The usual direct method of simulation for cluster processes requires the generation of the parent point process over a region larger than the actual observation window, since we have to allow for all possible parents giving rise to observed daughter points, and some of these parents may fall outwith the observation window. When there is no a priori bound on the distance between parent and child then we have to take care to control approximations arising from edge effects. In this paper, we present a simulation method which requires simulation only of those parent points actually giving rise to observed daughter points, thus avoiding edge effect approximation. The idea is to replace the cluster distribution by one which is conditioned to plant at least one daughter point in the observation window, and to modify the parent process to have an inhomogeneous intensity exactly balancing the effect of the conditioning. We furthermore show how the method extends to cases involving infinitely many potential parents, for example gamma-Poisson processes and shot-noise G-Cox processes, allowing us to avoid approximation due to truncation of the parent process.

Keywords

Cox processshot-noise processgamma-Poisson process

MSC classification

Primary: 60G55: Point processes
Secondary: 60D05: Geometric probability and stochastic geometry 62H11: Directional data; spatial statistics
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 34 , Issue 2 , June 2002 , pp. 267 - 280
Copyright
Copyright © Applied Probability Trust 2002 

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References

Bondesson, L. (1982). On simulation from infinitely divisible distributions. Adv. Appl. Prob. 14, 855–869.CrossRefGoogle Scholar
Brix, A. (1999). Generalized gamma measures and shot-noise Cox processes. Adv. Appl. Prob. 31, 929–953.Google Scholar
Brix, A. and Chadœuf, J. (2002). Spatio-temporal modelling of weeds by shot-noise G Cox processes. Biometrical J. 44, 83–99.Google Scholar
Brix, A. and Möller, J. (2001). Space–time multi type log Gaussian Cox processes with a view to modelling weeds. Scand. J. Statist. 28, 471–488.CrossRefGoogle Scholar
Cox, D. R. (1955). Some statistical methods related with series of events (with discussion). J. R. Statist. Soc. B 17, 129–157.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Häggström, O. and Steif, J. E. (2000). Propp–Wilson algorithms and finitary codings for high noise Markov random fields. Comb. Prob. Comput. 9, 425–439.Google Scholar
Kendall, W. S. (1997). Perfect simulation for spatial point processes. In Proc. 51st Session ISI (Istanbul, August 1997) Vol. 3, International Statistical Institute, Voorburg, pp. 163–166.Google Scholar
Neyman, J. (1939). On a class of contagious distributions, applicable in entomology and bacteriology. Ann. Math. Statist. 10, 35–57.Google Scholar
Neyman, J. and Scott, E. L. (1958). Statistical approach to problems of cosmology. J. R. Statist. Soc. B 20, 1–43.Google Scholar
Penttinen, A., Stoyan, D. and Henttonen, H. (1992). Marked point processes in forest statistics. Forest Sci. 38, 806–824.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications. John Wiley, New York.Google Scholar
Van den Berg, J. and Steif, J. E. (1999). On the existence and non-existence of finitary codings for a class of random fields. Ann. Prob. 27, 1501–1522.Google Scholar
Wolpert, R. L. and Ickstadt, K. (1998). Poisson/gamma random field models for spatial statistics. Biometrika 85, 251–267.Google Scholar