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Bayesian analysis of deformed tessellation models

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

Published online by Cambridge University Press:  01 July 2016

Paul G. Blackwell  and
Jesper Møller
Paul G. Blackwell*
Affiliation:
University of Sheffield
Jesper Møller*
Affiliation:
Aalborg University
*
∗ Postal address: Department of Probability and Statistics, University of Sheffield, Sheffield S3 7RH, UK.
∗∗ Postal address: Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg, Denmark. Email address: jm@math.auc.dk
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Abstract

We define a class of tessellation models based on perturbing or deforming standard tessellations such as the Voronoi tessellation. We show how distributions over this class of ‘deformed’ tessellations can be used to define prior distributions for models based on tessellations, and how inference for such models can be carried out using Markov chain Monte Carlo methods; stability properties of the algorithms are investigated. Our approach applies not only to fixed dimension problems, but also to variable dimension problems, in which the number of cells in the tessellation is unknown. We illustrate our methods with two real examples. The first relates to reconstructing animal territories, represented by the individual cells of a tessellation, from observation of an inhomogeneous Poisson point process. The second example involves the analysis of an image of a cross-section through a sample of metal, with the tessellation modelling the micro-crystalline structure of the metal.

Keywords

Random tessellationVoronoi tessellationdeformed templateBayesian inferenceMarkov chain Monte Carloreversible jumpHarris recurrenceuniform ergodicityinhomogeneous Poisson processimage analysis

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 62M30: Spatial processes 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 1 , March 2003 , pp. 4 - 26
Copyright
Copyright © Applied Probability Trust 2003 

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