Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-06-02T18:52:18.777Z Has data issue: false hasContentIssue false
  • English
  • Français

Power diagrams and interaction processes for unions of discs

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

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Aurenhammer, F. (1987). Power diagrams: properties, algorithms and applications. SIAM J. Comput. 16, 78–96.Google Scholar
Baddeley, A. and Møller, J. (1989). Nearest-neighbour Markov point processes and random sets. Internat. Statist. Rev. 2, 89–121.Google Scholar
Baddeley, A. J. and van Lieshout, M. N. M. (1995). Area-interaction point processes. Ann. Inst. Statist. Math. 46, 601–619.Google Scholar
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
Barndorff-Nielsen, O. E. (1978). Information and Exponential Families in Statistical Theory. John Wiley, Chichester.Google Scholar
Chan, K. S. and Geyer, C. J. (1994). Discussion of the paper ‘Markov chains for exploring posterior distributions’ by Luke Tierney. Ann. Statist. 22, 1747.Google Scholar
Chin, Y. C. and Baddeley, A. J. (2000). Markov interacting component processes. Adv. Appl. Prob. 32, 597–619.Google Scholar
Cressie, N. A. C. (1993). Statistics for Spatial Data, 2nd edn. John Wiley, New York.Google Scholar
Diggle, P. (1981). Binary mosaics and the spatial pattern of heather. Biometrics 37, 531–539.Google Scholar
Edelsbrunner, H. (1995). The union of balls and its dual shape. Discrete Computational Geometry 13, 415–440.CrossRefGoogle Scholar
Edelsbrunner, H. (2004). Biological applications of computational topology. In Handbook of Discrete and Computational Geometry. eds Goodman, J. and O'Rourke, J., Chapman & Hall/CRC, Boca Raton, FL, pp. 1395–1412.Google Scholar
Georgii, H.-O. (1976). Canonical and grand canonical Gibbs states for continuum systems. Commun. Math. Phys. 48, 31–51.Google Scholar
Geyer, C. J. (1999). Likelihood inference for spatial point processes. In Stochastic Geometry: Likelihood and Computation, eds Barndorff-Nielsen, O. E. et al., Chapman & Hall/CRC, Boca Raton, FL, pp. 79–140.Google Scholar
Geyer, C. J. and Møller, J. (1994). Simulation procedures and likelihood inference for spatial point processes. Scand. J. Statist. 21, 359–373.Google Scholar
Häggström, O., van Lieshout, M. N. M. and Møller, J. (1999). Characterization results and Markov chain Monte Carlo algorithms including exact simulation for some spatial point processes. Bernoulli 5, 641–659.Google Scholar
Hall, P. (1988). Introduction to the Theory of Coverage Processes. John Wiley, New York.Google Scholar
Hanisch, K.-H. (1981). On classes of random sets and point processes. Serdica 7, 160–167.Google Scholar
Kendall, W. S. (1990). A spatial Markov property for nearest-neighbour Markov point processes. J. Appl. Prob. 28, 767–778.Google Scholar
Kendall, W. S. (1998). Perfect simulation for the area-interaction point process. In Probability Towards 2000 (Springer Lecture Notes Statist. 128), eds Accardi, L. and Heyde, C., Springer, New York, pp. 218–234.CrossRefGoogle Scholar
Kendall, W. S. (2004). Geometric ergodicity and perfect simulation. Electron. Commun. Prob. 9, 140–151.Google Scholar
Kendall, W. S. and Møller, J. (2000). Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Adv. Appl. Prob. 32, 844–865.Google Scholar
Kendall, W. S., van Lieshout, M. and Baddeley, A. (1999). Quermass-interaction processes: conditions for stability. Adv. Appl. Prob. 31, 315–342.Google Scholar
Klein, W. (1982). Potts-model formulation of continuum percolation. Phys. Rev. B 26, 2677–2678.Google Scholar
Likos, C., Mecke, K. and Wagner, H. (1995). Statistical morphological of random interfaces in microemulsions. J. Chem. Phys. 102, 9350–9361.Google Scholar
Mecke, K. (1994). Integralgeometrie in der Statistichen Physik (Reine Physik 25). Harri Deutsch, Frankfurt.Google Scholar
Mecke, K. (1996). A morphological model for complex fluids. J. Phys. Condensed Matters 8, 9663–9667.Google Scholar
Molchanov, I. (1997). Statistics of the Boolean Model for Practitioners and Mathematicians. John Wiley, Chichester.Google Scholar
Møller, J. (1994). Contribution to the discussion of N. L. Hjort and H. Omre (1994): Topics in spatial statistics. Scand. J. Statist. 21, 346–349.Google Scholar
Møller, J. (1994). Lectures on Random Voronoi Tessellations (Lecture Notes Statist. 87). Springer, New York.Google Scholar
Møller, J. (1999). Markov chain Monte Carlo and spatial point processes. In Stochastic Geometry: Likelihood and Computation (Monogr. Statist. Appl. Prob. 80), ed. Barndorff-Nielsen, O. E., Chapman & Hall/CRC, Boca Raton, FL, pp. 141–172.Google Scholar
Møller, J. and Helisová, K. (2007). Power diagrams and interaction processes for unions of discs. Res. Rept. R-2007-15, Department of Mathematical Sciences, Aalborg University.Google Scholar
Møller, J. and Helisová, K. (2008). Likelihood inference for unions of interacting discs. In preparation.Google Scholar
Møller, J. and Waagepetersen, R. P. (1998). Markov connected component fields. Adv. Appl. Prob. 30, 1–35.Google Scholar
Møller, J. and Waagepetersen, R. P. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
Nguyen, X. X. and Zessin, H. (1979). Integral and differential characterizations of Gibbs processes. Math. Nachr. 88, 105–115.Google Scholar
Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N. (2000). Spatial Tessellations. Concepts and Applications of Voronoi Diagrams, 2nd edn. John Wiley, Chichester.Google Scholar
Preston, C. J. (1977). Spatial birth-and-death processes. Bull. Internat. Statist. Inst. 46, 371–391.Google Scholar
Ripley, B. D. and Kelly, F. P. (1977). Markov point processes. J. London Math. Soc. 15, 188–192.Google Scholar
Roberts, G. O. and Rosenthal, J. S. (1997). Geometric ergodicity and hybrid Markov chains. Electron. Commun. Prob. 2, 13–25.CrossRefGoogle Scholar
Ruelle, D. (1971). Existence of a phase transition in a continuous classical system. Phys. Rev. Lett. 27, 1040–1041.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Van Lieshout, M. N. M. (2000). Markov Point Processes and Their Applications. Imperial College Press, London.Google Scholar
Weil, W. and Wieacker, J. (1984). Densities for stationary random sets and point processes. Adv. Appl. Prob. 16, 324–346.Google Scholar
Weil, W. and Wieacker, J. (1988). A representation theorem for random sets. Prob. Math. Statist. 9, 147–151.Google Scholar
Widom, B. and Rowlinson, J. S. (1970). a new model for the study of liquid-vapor phase transitions. J. Chem. Phys. 52, 1670–1684.Google Scholar
Wilson, R. (1972). Introduction to Graph Theory. Oliver and Boyd, Edinburgh.Google Scholar