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Power diagrams and interaction processes for unions of discs

Part of: Inference from stochastic processes Special processes Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Jesper Møller  and
Kateřina Helisová
Jesper Møller*
Affiliation:
Aalborg University
Kateřina Helisová*
Affiliation:
Charles University in Prague
*
∗ Postal address: Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg, Denmark. Email address: jm@math.aau.dk
∗∗ Postal address: Department of Probability and Mathematical Statistics, Charles University in Prague, Sokolovská 83, 18675 Praha 8, Czech Republic. Email address: helisova@karlin.mff.cuni.cz
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Abstract

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We study a flexible class of finite-disc process models with interaction between the discs. We let 𝒰 denote the random set given by the union of discs, and use for the disc process an exponential family density with the canonical sufficient statistic depending only on geometric properties of 𝒰 such as the area, perimeter, Euler-Poincaré characteristic, and the number of holes. This includes the quermass-interaction process and the continuum random-cluster model as special cases. Viewing our model as a connected component Markov point process, and thereby establishing local and spatial Markov properties, becomes useful for handling the problem of edge effects when only 𝒰 is observed within a bounded observation window. The power tessellation and its dual graph become major tools when establishing inclusion-exclusion formulae, formulae for computing geometric characteristics of 𝒰, and stability properties of the underlying disc process density. Algorithms for constructing the power tessellation of 𝒰 and for simulating the disc process are discussed, and the software is made public available.

Keywords

Area-interaction processBoolean modeldisc processexponential familygerm-grain modellocal computationslocal stabilityMarkov propertiesinclusion-exclusion formulaeinteractionpoint processpower tessellationsimulationquermass-interaction processrandom closed setRuelle stability

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory 62M30: Spatial processes
Secondary: 68U20: Simulation
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 40 , Issue 2 , June 2008 , pp. 321 - 347
Copyright
Copyright © Applied Probability Trust 2008 

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