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Markov interacting component processes

Part of: Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  19 February 2016

Y. C. Chin  and
A. J. Baddeley
Y. C. Chin*
Affiliation:
The University of Western Australia
A. J. Baddeley*
Affiliation:
The University of Western Australia
*
∗ Postal address: Department of Mathematics and Statistics, The University of Western Australia, Nedlands, WA 6907, Australia.
∗ Postal address: Department of Mathematics and Statistics, The University of Western Australia, Nedlands, WA 6907, Australia.
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Abstract

A generalization of Markov point processes is introduced in which interactions occur between connected components of the point pattern. A version of the Hammersley-Clifford characterization theorem is proved which states that a point process is a Markov interacting component process if and only if its density function is a product of interaction terms associated with cliques of connected components. Integrability and superpositional properties of the processes are shown and a pairwise interaction example is used for detailed exploration.

Keywords

Markov point processesconnected componentssuperpositionspatial statisticsclusteringconditional intensity

MSC classification

Primary: 60G55: Point processes
Secondary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 32 , Issue 3 , September 2000 , pp. 597 - 619
Copyright
Copyright © Applied Probability Trust 2000 

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References

[1] Babu, G. J. and Feigelson, E. D. (1996). Astrostatistics. Chapman & Hall, London.Google Scholar
[2] Baddeley, A. J. and Möller, J. (1989). Nearest neighbour Markov point processes and random sets. Int. Statist. Rev. 57, 89–121.Google Scholar
[3] Baddeley, A. J. and van Lieshout, M. N. M. (1995). Area-interaction point processes. Ann. Inst. Statist. Math. 47, 601–619.Google Scholar
[4] Baddeley, A. J., van Lieshout, M. N. M. and Möller, J. (1996). Markov properties of cluster processes. Adv. Appl. Prob. 28, 346–355.Google Scholar
[5] Barndorff-Nielsen, O. E., Kendall, W. S. and van Lieshout, M. N. M. (eds) (1999). Stochastic Geometry: Likelihood and Computation. No. 80 in Monographs on Statistics and Applied Probability. Chapman & Hall, Boca Raton.Google Scholar
[6] Cameron, P. J. (1994). Combinatorics: Topics, Techniques, Algorithms. Cambridge University Press.Google Scholar
[7] Chin, Y. C. and Baddeley, A. J. (1999). On connected component Markov point processes. Adv. Appl. Prob. 31, 279–282.Google Scholar
[8] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
[9] Diggle, P. J. (1983). Statistical Analysis of Spatial Point Patterns. Academic Press, London.Google Scholar
[10] Everitt, B. (1974). Cluster Analysis. Heinemann, London.Google Scholar
[11] Geyer, C. J. (1999). Likelihood inference for spatial point processes. In barndorff-nielsen:etal:1999 ch. 3, pp. 79–140.Google Scholar
[12] Geyer, C. J. and Möller, J. (1994). Simulation and likelihood inference for spatial point processes. Scand. J. Statist. 21, 359–373.Google Scholar
[13] Grimmett, G. R. (1973). A theorem about random fields. J. London Math. Soc. 5, 81–84.Google Scholar
[14] Häggström, O., van Lieshout, M. N. M. and Möller, J. (1999). Characterisation results and Markov chain Monte Carlo algorithms including exact simulation for some spatial point process. Bernoulli 5, 641–659.Google Scholar
[15] Kelly, F. P. and Ripley, B. D. (1976). A note on Strauss's model for clustering. Biometrika 63, 357–360.Google Scholar
[16] Kendall, W. S., van Lieshout, M. N. M. and Baddeley, A. J. (1999). Quermass-interaction processes: conditions for stability. Adv. Appl. Prob. 31, 315–342.Google Scholar
[17] Klein, W. (1982). Potts-model formulation of continuum percolation. Phys. Rev. B, 26, 2677–2678.Google Scholar
[18] Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. John Wiley, New York.Google Scholar
[19] Möller, J., (1999). Markov chain Monte Carlo and spatial point processes. In barndorff-nielsen:etal:1999, ch. 4, pp. 141–172.Google Scholar
[20] Möller, J. and Waagepetersen, R. P. (1998). Markov connected component fields. Adv. Appl. Prob. 30, 1–35.Google Scholar
[21] Ripley, B. D. (1977). Modelling spatial patterns. J. R. Statist. Soc. Ser. B 39, 177–212.Google Scholar
[22] Ripley, B. D. and Kelly, F. P. (1977). Markov point processes. J. London Math. Soc. 15, 188–192.Google Scholar
[23] Ruelle, D. (1969). Statistical Mechanics. John Wiley, New York.Google Scholar
[24] Sivakumar, K. and Goutsias, J. (1997). Morphological sampling of random closed sets. In Mathematical Morphology and its Applications to Image and Signal Processing, eds Maragos, P., Schafer, R. W. and Butt, M. A.. World Scientific Publishing, Singapore, pp. 73–80.Google Scholar
[25] Sivakumar, K. and Goutsias, J. (1997). Morphologically constrained discrete random sets. In Advances in Theory and Applications of Random Sets, ed. Jeulin, D.. World Scientific Publishing, Singapore, pp. 49–66.Google Scholar
[26] Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and its Applications, 2nd edn. John Wiley, New York.Google Scholar
[27] Strauss, D. J. (1975). A model for clustering. Biometrika 62, 467–475.Google Scholar
[28] Van Lieshout, M. N. M. (1999). Size-biased random closed sets. Pattern Recognition 32, 1631–1644.Google Scholar
[29] Van Lieshout, M. N. M. and Baddeley, A. J. (1996). A nonparametric measure of spatial interaction in point patterns. Statist. Neerl. 50, 344–361.Google Scholar
[30] Widom, B. and Rowlinson, J. S. (1970). A new model for the study of liquid–vapour phase transitions. J. Chem. Phys. 52, 1670–1684.CrossRefGoogle Scholar