Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-28T07:05:22.588Z Has data issue: false hasContentIssue false
  • English
  • Français

Bayesian analysis of deformed tessellation models

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

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Blackwell, P. G. (2000). Inference for generalized tessellation processes. Res. Rep. 504/00, Department of Probability and Statistics, University of Sheffield.Google Scholar
Blackwell, P. G. (2001). Bayesian inference for a random tessellation process. Biometrics 57, 502–507.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Chiu, S. N., van de Weygaert, R. and Stoyan, D. (1996). The sectional Poisson Voronoi tessellation is not a Voronoi tessellation. Adv. Appl. Prob. 28, 356–376.CrossRefGoogle Scholar
Cleveland, W. S. (1979). Robust locally weighted regression and smoothing scatterplots. J. Amer. Statist. Assoc. 74, 829–836.CrossRefGoogle Scholar
Dagpunar, J. S. (1989). An easily implemented generalized inverse Gaussian generator. Commun. Statist. B: Simul. Comput. 18, 703–710.CrossRefGoogle Scholar
Denison, D. G. T., Holmes, C. C., Mallick, B. K. and Smith, A. F. M. (2002). Bayesian Methods for Nonlinear Classification and Regression. John Wiley, New York.Google Scholar
Gelman, A. and Meng, X.-L. (1998). Simulating normalizing constants: from importance sampling to bridge sampling to path sampling. Statist. Sci. 13, 163–185.CrossRefGoogle Scholar
Geyer, C. J. (1999). Likelihood inference for spatial point processes. In Stochastic Geometry: Likelihood and Computation (Monogr. Statist. Appl. Prob. 80), eds Barndorff-Nielsen, O. E., Kendall, W. S. and van Lieshout, M. N. M., Chapman and Hall/CRC, Boca Raton, FL, pp. 79–140.Google Scholar
Geyer, C. J. and Möller, J. (1994). Simulation procedures and likelihood inference for spatial point processes. Scand. J. Statist. 21, 359–373.Google Scholar
Gilks, W. R., Richardson, S. and Spiegelhalter, D. J. (1996). Markov Chain Monte Carlo in Practice. Chapman and Hall, London.Google Scholar
Green, P. J. (1995). Reversible jump MCMC computation and Bayesian model determination. Biometrika 82, 711–732.Google Scholar
Grenander, U. (1993). General Pattern Theory. Oxford University Press.Google Scholar
Grenander, U. and Miller, M. I. (1994). Representation of knowledge in complex systems (with discussion). J. R. Statist. Soc. B 56, 549–603.Google Scholar
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions, Vol. 1, 2nd edn. John Wiley, New York.Google Scholar
Kruuk, H. (1978). Spatial organization and territorial behaviour of the European badger, Meles meles . J. Zoology (London) 184, 1–19.CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.CrossRefGoogle Scholar
Möller, J., (1989). Random tessellations in Rd . Adv. Appl. Prob. 21, 37–73.CrossRefGoogle Scholar
Möller, J., (1992). Random Johnson–Mehl tessellations. Adv. Appl. Prob. 24, 814–844.CrossRefGoogle Scholar
Möller, J., (1994). Lectures on Random Voronoi Tessellations (Lecture Notes Statist. 87). Springer, New York.Google Scholar
Möller, J., (1995). Generation of Johnson–Mehl crystals and comparative analysis of models for random nucleation. Adv. Appl. Prob. 27, 367–383.CrossRefGoogle Scholar
Möller, J., (1999). Topics in Voronoi and Johnson–Mehl tessellations. In Stochastic Geometry: Likelihood and Computation (Monogr. Statist. Appl. Prob. 80), eds Barndorff-Nielsen, O. E., Kendall, W. S. and van Lieshout, M. N. M., Chapman and Hall/CRC, Boca Raton, FL, pp. 173–198.Google Scholar
Möller, J. and Skare, Ø. (2001). Bayesian image analysis with coloured Voronoi tessellations and a view to applications in reservoir modelling. Statist. Modelling 1, 213–232.Google Scholar
Nicholls, G. (1998). Bayesian image analysis with Markov chain Monte Carlo and colored continuum triangulation mosaics. J. R. Statist. Soc. B 60, 643–659.CrossRefGoogle 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.CrossRefGoogle Scholar
Robert, C. P. and Casella, G. (1999). Monte Carlo Statistical Methods. Springer, New York.CrossRefGoogle Scholar
Roberts, G. O. and Rosenthal, J. S. (1997). Geometric ergodicity and hybrid Markov chains. Electron. Commun. Prob. 2, 13–25.CrossRefGoogle Scholar
Roberts, G. O. and Rosenthal, J. S. (1998). Two convergence properties of hybrid samplers. Ann. Appl. Prob. 8, 397–407.CrossRefGoogle Scholar
Roberts, G. O., Gelman, A. and Gilks, W. R. (1997). Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Prob. 7, 110–120.Google Scholar
Snethlage, M., Martinez, V. J., Stoyan, D. and Saar, E. (2002). Point field models for the galaxy point pattern. Astron. Astrophys. 388, 758–765.CrossRefGoogle Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar