Power diagrams and interaction processes for unions of discs

Jesper Møller, Katerina Helisova

Research output: Contribution to journalJournal articleResearchpeer-review

21 Citations (Scopus)

Abstract

 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.
Original languageEnglish
JournalAdvances in Applied Probability
Volume40
Issue number2
Pages (from-to)321-347
Number of pages27
ISSN0001-8678
DOIs
Publication statusPublished - 2008

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Power Diagram
Union
Interaction
Tessellation
Random-cluster Model
Statistics
Inclusion-exclusion
Dual Graph
Edge Effects
Markov Property
Sufficient Statistics
Random Sets
Exponential Family
Euler Characteristic
Continuum Model
Perimeter
Point Process
Connected Components
Markov Process
Process Model

Cite this

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title = "Power diagrams and interaction processes for unions of discs",
abstract = " 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-Poincar{\'e} 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.",
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Power diagrams and interaction processes for unions of discs. / Møller, Jesper; Helisova, Katerina.

In: Advances in Applied Probability, Vol. 40, No. 2, 2008, p. 321-347.

Research output: Contribution to journalJournal articleResearchpeer-review

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AB -  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.

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