Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-26T02:57:24.262Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact sampling from conditional Boolean models with applications to maximum likelihood inference

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

Published online by Cambridge University Press:  01 July 2016

M. N. M. van Lieshout  and
E. W. van Zwet
M. N. M. van Lieshout*
Affiliation:
CWI
E. W. van Zwet*
Affiliation:
University of California, Berkeley
*
∗ Postal address: CWI, PO Box 94079, 1090 GB Amsterdam, The Netherlands. Email address: colette@cwi.nl
∗∗ Postal address: University of California, Department of Statistics, 367 Evans Hall # 3860, Berkeley, CA 94720-3860, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We are interested in estimating the intensity parameter of a Boolean model of discs (the bombing model) from a single realization. To do so, we derive the conditional distribution of the points (germs) of the underlying Poisson process. We demonstrate how to apply coupling from the past to generate samples from this distribution, and use the samples thus obtained to approximate the maximum likelihood estimator of the intensity. We discuss and compare two methods: one based on a Monte Carlo approximation of the likelihood function, the other a stochastic version of the EM algorithm.

Keywords

Boolean modelcoupling from the pastMarkov chain Monte Carlo simulationmaximum likelihood estimationstochastic approximation EM algorithmstochastic EM algorithm

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 62M30: Spatial processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 33 , Issue 2 , June 2001 , pp. 339 - 353
Copyright
Copyright © Applied Probability Trust 2001 

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

Ayala, G., Ferrándiz, J. and Montes, F. (1990). Boolean models: maximum likelihood estimation from circular clumps. Biometrical J. 32, 73–78.Google Scholar
Baddeley, A. J. and Gill, R. D. (1997). Kaplan–Meier estimators for interpoint distance distributions of spatial point processes. Ann. Statist. 25, 263–292.Google Scholar
Baddeley, A. J. and van Lieshout, M. N. M. (1995). Area-interaction point processes. Ann. Inst. Statist. Math. 47, 601–619.Google Scholar
Celeux, G. and Diebolt, Y. (1985). The SEM algorithm: a probabilistic teacher algorithm derived from the EM algorithm for the mixture problem. Comput. Statist. Quart. 2, 73–82.Google Scholar
Delyon, B., Lavielle, M. and Moulines, E. (1999). Convergence of a stochastic approximation version of the EM algorithm. Ann. Statist. 27, 94–128.Google Scholar
Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. R. Statist. Soc. B 39, 2–22.Google Scholar
Dupac, V., (1980). Parameter estimation in the Poisson field of discs. Biometrika 67, 187–190.Google Scholar
Feller, W. (1968). An Introduction to Probability Theory and its Applications, Vol. 2. John Wiley, New York.Google Scholar
Geyer, C. J. (1994). On the convergence of Monte Carlo maximum likelihood calculations. J. R. Statist. Soc. B 56, 261–274.Google Scholar
Geyer, C. J. (1999). Likelihood inference for spatial point processes. In Proc. Sem. Eur. Statist., Stoch. Geometry, Likelihood, Comput., eds Barndorff-Nielsen, O., Kendall, W. S. and van Lieshout, M. N. M. CRC Press/Chapman and Hall, London.Google Scholar
Gilks, W. R., Richardson, S. and Spiegelhalter, D. J. (1996). Markov Chain Monte Carlo in Practice. Chapman and Hall, London.Google Scholar
Häggström, O. and Nelander, K. (1998). Exact sampling from anti-monotone systems. Statist. Neerlandica 52, 360–380.Google Scholar
Hall, P. (1988). Introduction to the Theory of Coverage Processes. John Wiley, New York.Google Scholar
Hanisch, K. H. (1984). Some remarks on estimators of the distribution function of nearest neighbour distance in stationary spatial point processes. Math. Operationsforsch. Statist. Ser. Statist. 15, 409–412.Google Scholar
Kendall, W. S. (1997). On some weighted Boolean models. In Proc. Int. Symp. Adv. Theory Appl. Random Sets, ed. Jeulin, D. World Scientific Publishing, Singapore, pp. 105–120.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. and Thönnes, E. (1999). Perfect simulation in stochastic geometry. Pattern Recognition 32, 1569–1586.Google Scholar
Kolmogorov, A. N. (1937). On the statistical theory of metal crystallisation. Izv. Acad. Sci. USSR Ser. Math. 3, 355–360.Google Scholar
Lantuéjoul, C., (1997). Conditional simulation of object-based models. In Proc. Int. Symp. Adv. Theory Appl. Random Sets, ed. Jeulin, D. World Scientific Publishing, Singapore, pp. 271–288.Google Scholar
Matérn, B., (1960). Spatial Variation (Lecture Notes Statist. 36). Springer, Berlin.Google Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, Chichester.Google Scholar
Molchanov, I. S. (1997). Statistics of the Boolean Model for Practitioners and Mathematicians. John Wiley, Chichester.Google Scholar
Molchanov, I. S. and Stoyan, D. (1994). Asymptotic properties of estimators for parameters of the Boolean model. Adv. Appl. Prob. 26, 301–323.Google Scholar
Moyeed, R. A. and Baddeley, A. J. (1991). Stochastic approximation of the MLE for a spatial point pattern. Scand. J. Statist. 18, 39–50.Google Scholar
Nielsen, S. F. (2000). The stochastic EM algorithm: estimation and asymptotic results. Bernoulli 6, 457–489.Google Scholar
Parzen, E. (1962). Stochastic Processes. Holden-Day, San Francisco.Google Scholar
Penttinen, A. (1984). Modelling Interaction in Spatial Point Patterns: Parameter Estimation by the Maximum Likelihood Method (Jyväskylä Studies Comput. Sci., Econ., Statist. 7). University of Jyväskylä.Google Scholar
Preston, C. J. (1977). Spatial birth-and-death processes. Bull. Int. Statist. Inst. 46, 371–391.Google Scholar
Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9, 223–252.Google Scholar
Ripley, B. D. (1988). Statistical Inference for Spatial Processes. Cambridge University Press.Google Scholar
Schmitt, M. (1991). Estimation of the density in a stationary Boolean model. J. Appl. Prob. 28, 702–709.Google Scholar
Serra, J. (1982). Image Analysis and Mathematical Morphology. Academic Press, London.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Tanner, M. A. and Wei, G. C. G. (1990). A Monte Carlo implementation of the EM-algorithm and the poor man's data augmentation algorithms. J. Amer. Statist. Assoc. 85, 699–704.Google Scholar
Thompson, E. A. and Guo, S. W. (1991). Evaluation of likelihood ratios for complex genetic models. IMA J. Math. Appl. Med. Biol. 8, 149–169.Google Scholar
Thönnes, E., (1998). Perfect and imperfect simulations in stochastic geometry. , University of Warwick.Google Scholar
Thönnes, E., (2000). The conditional Boolean model revisited. Tech. Rept 369, Department of Statistics, University of Warwick.Google Scholar
Van Lieshout, M. N. M. (1997). On likelihoods for Markov random sets and Boolean models. In Proc. Int. Symp. Adv. Theory Appl. Random Sets, ed. Jeulin, D. World Scientific Publishing, Singapore, pp. 121–135.Google Scholar
Weil, W. (1988). Expectation formulas and isoperimetric properties for non-isotropic Boolean models. J. Microscopy 151, 235–245.Google Scholar
Younes, L. (1988). Estimation and annealing for Gibbsian fields. Ann. Inst. H. Poincaré Prob. Statist. 24, 269–294.Google Scholar