Abstract
We introduce a new class of dynamic point process models with simple and intuitive dynamics that are based on the Voronoi tessellations generated by the processes. Under broad conditions, these processes prove to be ergodic and produce, on stabilisation, a wide range of clustering patterns. In the paper, we present results of simulation studies of three statistical measures (Thiel’s redundancy, van Lieshout and Baddeley’s J-function and the empirical distribution of the Voronoi nearest neighbours’ numbers) for inference on these models from the clustering behaviour in the stationary regime. In particular, we make comparisons with the area-interaction processes of Baddeley and van Lieshout.
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Ambler, G.K.: Dominated coupling from the past and some extensions of the area-interaction process. PhD thesis, Department of Mathematics, University of Bristol, UK (2002)
Asmussen, S.: Applied Probability and Queues. Wiley, Chichester (1987)
Baddeley, A., van Lieshout, M.: Area-interaction point processes. Ann. Inst. Stat. Math. 47, 601–619 (1995)
Baddeley, A., van Lieshout, M.: A nonparametric measure of spatial interaction in point patterns. Statistica Neerlandica 50, 344–361 (1996)
Borovkov, K.A., Odell, D.: On spatial thinning-replacement point processes based on Voronoi cells. Adv. Appl. Probab. 39 (2007, in press)
Kendall, W.S.: Perfect simulation for the area-interaction point process. In: Accardi, L., Heyde, C.C. (eds.) Probability Towards 2000, pp. 218–234. Springer, New York (1998)
Lenz, R.: Redundancy as an index of change in point pattern analysis. Geogr. Anal. 11, 374–388
Meyn, S.P., Tweedie, R.L.: Stability of Markovian processes I: Criteria for discrete-time chains. Adv. Appl. Probab. 24, 542–574 (1992)
Okabe, A., Boots, B., Sugihara, K., Chiu, S.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, Chichester (2000)
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This research was supported by the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems.
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Borovkov, K.A., Odell, D.A. Simulation Studies of Some Voronoi Point Processes. Acta Appl Math 96, 87–97 (2007). https://doi.org/10.1007/s10440-007-9093-2
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DOI: https://doi.org/10.1007/s10440-007-9093-2