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Nearest-neighbour Markov point processes on graphs with Euclidean edges

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  29 November 2018

M. N. M. van Lieshout
M. N. M. van Lieshout*
Affiliation:
CWI and University of Twente
*
* Postal address: CWI, PO Box 94079, NL-1090 GB Amsterdam, The Netherlands. Email address: marie-colette.van.lieshout@cwi.nl
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Abstract

We define nearest-neighbour point processes on graphs with Euclidean edges and linear networks. They can be seen as analogues of renewal processes on the real line. We show that the Delaunay neighbourhood relation on a tree satisfies the Baddeley‒Møller consistency conditions and provide a characterisation of Markov functions with respect to this relation. We show that a modified relation defined in terms of the local geometry of the graph satisfies the consistency conditions for all graphs with Euclidean edges that do not contain triangles.

Keywords

Delaunay neighbourgraph with Euclidean edgeslinear networkMarkov point processnearest-neighbour interactionrenewal process

MSC classification

Primary: 60G55: Point processes
Secondary: 60D05: Geometric probability and stochastic geometry
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 4 , December 2018 , pp. 1275 - 1293
Copyright
Copyright © Applied Probability Trust 2018 

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References

[1]Anderes, E.,Møller, J. and Rasmussen, J. G. (2016). Second-order pseudo-stationary random fields and point processes on graphs and their edges.AU Workshop Stoch. Geometry, Stereology, Appl. conf. presentation. Available at http://www.lebesgue.fr/sites/default/files/attach/GraphWithEdgesSHORT.pdf.Google Scholar
[2]Anderes, E.,Møller, J. and Rasmussen, J. G. (2017). Isotropic covariance functions on graphs and their edges. Preprint. Available at http://arxiv.org/abs/1710.01295v1.Google Scholar
[3]Baddeley, A. and Møller, J. (1989).Nearest-neighbour Markov point processes and random sets.Internat. Statist. Rev. 57,89–121.Google Scholar
[4]Baddeley, A. J.,van Lieshout, M. N. M. and Møller, J. (1996).Markov properties of cluster processes.Adv. Appl. Prob. 28,346–355.Google Scholar
[5]Baddeley, A.,Nair, G.,Rakshit, S. and McSwiggan, G. (2017).``Stationary'' point processes are uncommon on linear networks.Stat 6,68–78.Google Scholar
[6]Bondy, A. and Murty, M. R. (2008).Graph Theory.Springer,London.Google Scholar
[7]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
[8]Daley, D. J. and Vere-Jones, D. (2008).An Introduction to the Theory of Point Processes, Vol. II,2nd edn.Springer,New York.Google Scholar
[9]Geman, D.,Geman, S.,Graffigne, C. and Dong, P. (1990).Boundary detection by constrained optimization.IEEE Trans. Pattern Anal. Mach. Intellig. 12,609–628.Google Scholar
[10]Häggström, O.,van Lieshout, M. N. M. and Møller, J. (1999).Characterisation results and Markov chain Monte Carlo algorithms including exact simulation for some spatial point processes.Bernoulli 5,641–658.Google Scholar
[11]Jónsdóttir, K. Ý.,Hahn, U. and Jensen, E. B. V. (2004).Inhomogeneous spatial point processes, with a view to spatio-temporal modelling. In Spatial Point Process Modelling and Its Applications (Lecture Notes Statist. 185), eds A. Baddeley et al.,Springer,New York, pp. 131–136.Google Scholar
[12]Okabe, A. and Sugihara, K. (2012).Spatial Analysis Along Networks. Statistical and Computational Methods.John Wiley,Chichester.Google Scholar
[13]Rakshit, S.,Nair, G. and Baddeley, A. (2017).Second-order analysis of point patterns on a network using any distance metric.Spatial Statist. 22,129–154.Google Scholar
[14]Ripley, B. D. and Kelly, F. P. (1977).Markov point processes.J. London Math. Soc. 15,188–192.Google Scholar
[15]Van Lieshout, M. N. M. (2000).Markov Point Processes and Their Applications.Imperial College Press.Google Scholar