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Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes

Part of: Special processes Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Wilfrid S. Kendall  and
Jesper Møller
Wilfrid S. Kendall*
Affiliation:
University of Warwick
Jesper Møller*
Affiliation:
Aalborg University
*
∗ Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address: wsk@stats.warwick.ac.uk
∗∗ Postal address: Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7E, DK-9220 Aalborg, Denmark. Email address: jm@math.auc.dk
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Abstract

In this paper we investigate the application of perfect simulation, in particular Coupling from the Past (CFTP), to the simulation of random point processes. We give a general formulation of the method of dominated CFTP and apply it to the problem of perfect simulation of general locally stable point processes as equilibrium distributions of spatial birth-and-death processes. We then investigate discrete-time Metropolis-Hastings samplers for point processes, and show how a variant which samples systematically from cells can be converted into a perfect version. An application is given to the Strauss point process.

Keywords

Coupling from the Past (CFTP)dominated CFTPexact simulationlocal stabilityMarkov chain Monte CarloMetropolis-Hastingsperfect simulationPapangelou conditional intensityrealizable monotonicityspatial birth-and-death processspatial point processstochastic monotonicityStrauss process

MSC classification

Primary: 62M30: Spatial processes 60G55: Point processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 3 , September 2000 , pp. 844 - 865
Copyright
Copyright © Applied Probability Trust 2000 

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References

[1] Baddeley, A. J. and Möller, J. (1989). Nearest neighbour Markov point processes and random sets. Int. Statist. Rev. 57, 89–121.CrossRefGoogle Scholar
[2] Baddeley, A. J. and van Lieshout, M. N. M. (1995). Area-interaction point processes. Ann. Inst. Statist. Math. 47, 601–619.CrossRefGoogle Scholar
[3] Borovkov, A. A. and Foss, S. G. (1992). Stochastically recursive sequences and their generalizations. Siberian Adv. Math. 2, 16–81.Google Scholar
[4] Burdzy, K. and Kendall, W. S. (2000). Efficient Markovian couplings: examples and counterexamples. Ann. Appl. Prob. 10, 362–409.CrossRefGoogle Scholar
[5] Carter, D. S. and Prenter, P. M. (1972). Exponential spaces and counting processes. Z. Wahrscheinlichkeitsth. 21, 1–19.CrossRefGoogle Scholar
[6] Clifford, P. and Nicholls, G. (1994). Comparison of birth-and-death and Metropolis–Hastings Markov chain Monte Carlo for the Strauss process. Available at http://www.math.auckland.ac.nz/simnicholls/.Google Scholar
[7] Corcoran, J. N. and Tweedie, R. L. (1997). Perfect sampling of Harris recurrent Markov chains. Unpublished manuscript.Google Scholar
[8] Darling, R. W. R. (1987). Constructing Nonhomeomorphic Stochastic Flows. Mem. Amer. Math. Soc. 70, No. 376.Google Scholar
[9] Fernández, R., Ferrari, P. A. and Garcia, N. L. (1999). Perfect simulation for interacting point processes, loss networks and Ising models. Preprint.Google Scholar
[10] Fill, J. (1998). An interruptible algorithm for exact sampling via Markov chains. Ann. Appl. Prob. 8, 131–162.CrossRefGoogle Scholar
[11] Foss, S. G. and Tweedie, R. L. (1998). Perfect simulation and backward coupling. Stoch. Models 14, 187–203.CrossRefGoogle Scholar
[12] Geyer, C. J. (1999). Likelihood inference for spatial point processes. Stochastic Geometry: Likelihood and Computation, eds Barndorff-Nielsen, O. E., Kendall, W. S. and van Lieshout, M. N. M.. Chapman & Hall/CRC, Boca Raton, pp. 79–140.Google Scholar
[13] Geyer, C. J. and Möller, J. (1994). Simulation and likelihood inference for spatial point processes. Scand. J. Statist. 21, 359–373.Google Scholar
[14] Green, P. J. and Murdoch, D. J. (1999). Exact sampling for Bayesian inference: towards general purpose algorithms. In Bayesian Statistics 6, eds Bernardo, J. M., Berger, J. O., Dawid, A. P. and Smith, A. F. M.. Oxford University Press.Google Scholar
[15] Häggström, O. and Steif, J. E. (1999). Propp–Wilson algorithms and finitary codings for high noise Markov random fields. To appear in Combin. Prob. Comput. Google Scholar
[16] Häggström, O., van Lieshout, M. N. M. and Möller, J. (1999). Characterization results and Markov chain Monte Carlo algorithms including exact simulation for some spatial point processes. Bernoulli 5, 641–659.Google Scholar
[17] Kelly, F. P. and Ripley, B. D. (1976). A note on Strauss's model for clustering. Biometrika 63, 357–360.CrossRefGoogle Scholar
[18] Kendall, W. S. (1997). On some weighted Boolean models. In Advances in Theory and Applications of Random Sets, ed. Jeulin, D.. World Scientific Publishing Company, Singapore, pp. 105–120.Google Scholar
[19] Kendall, W. S. (1997). Perfect simulation for spatial point processes. In Proc. ISI 51st session, Istanbul (August 1997), Vol. 3, pp. 163–166.Google Scholar
[20] Kendall, W. S. (1998). Perfect simulation for the area-interaction point process. In Probability Towards 2000 eds Accardi, L. and Heyde, C. C.. Springer, New York, pp. 218–234.Google Scholar
[21] Kendall, W. S. and Möller, J. (1999). Perfect implementation of a Metropolis–Hastings simulation of Markov point processes. Res. Rept 348, Department of Statistics, University of Warwick.Google Scholar
[22] Kendall, W. S. and Möller, J. (1999). Perfect Metropolis–Hastings simulation of locally stable point processes. Tech. Rept R-99-2001, Department of Mathematical Sciences, Aalborg University, 1999. Available at http://www.statslab.cam.ac.uk/ mcmc/.Google Scholar
[23] Kendall, W. S. and Thönnes, E. (1999). Perfect simulation in stochastic geometry. Pattern Recognition 32, 1569–1586.Google Scholar
[24] Kendall, W. S., van Lieshout, M. N. M. and Baddeley, A. J. (1999). Quermass-interaction processes: conditions for stability. Adv. Appl. Prob. 31, 315–342.Google Scholar
[25] Klein, W. (1982). Potts-model formulation of continuum percolation. Phys. Rev. B, 26, 2677–2678.CrossRefGoogle Scholar
[26] Lund, R. B. and Wilson, D. B. (1997). Exact sampling algorithms for storage systems. In preparation.Google Scholar
[27] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, New York.Google Scholar
[28] Möller, J., (1989). On the rate of convergence of spatial birth-and-death processes. Ann. Inst. Statist. Math. 41, 565–581.Google Scholar
[29] Möller, J., (1994). Discussion contribution. Scand. J. Statist. 21, 346–349.Google Scholar
[30] Möller, J. (1999). Markov chain Monte Carlo and spatial point processes. In Stochastic Geometry: Likelihood and Computation, eds Barndorff-Nielsen, O. E., Kendall, W. S. and van Lieshout, M. N. M.. Chapman & Hall/CRC, Boca Raton, pp. 141–172.Google Scholar
[31] Möller, J. and Nicholls, G. (1999). Perfect simulation for sample-based inference. Tech. Rept R-99-2011, Department of Mathematical Sciences, Aalborg University.Google Scholar
[32] Möller, J. and Schladitz, K. (1999). Extensions of Fill's algorithm for perfect simulation. J. R. Statist. Soc. Ser. B 61, 955–969.Google Scholar
[33] Möller, J. and Waagepetersen, R. (1998). Markov connected component fields. Adv. Appl. Prob. 30, 1–35.Google Scholar
[34] Murdoch, D. J. and Green, P. J. (1998). Exact sampling from a continuous state space. Scand. J. Statist. 25, 483–502.Google Scholar
[35] Murdoch, D. J. and Rosenthal, J. S. (2000). Efficient use of exact samples. Statist. Comput. 10, 237–243.Google Scholar
[36] Preston, C. J. (1977). Spatial birth-and-death processes. Bull. Inst. Int. Statist. 46, 371–391.Google Scholar
[37] Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures and Algorithms 9, 223–252.Google Scholar
[38] Ripley, B. D. (1977). Modelling spatial patterns (with discussion). J. R. Statist. Soc. Ser. B 39, 172–212.Google Scholar
[39] Ripley, B. D. and Kelly, F. P. (1977). Markov point processes. J. London Math. Soc. 15, 188–192.Google Scholar
[40] Strauss, D. J. (1975). A model for clustering. Biometrika 63, 467–475.CrossRefGoogle Scholar
[41] Thönnes, E., (1999). Perfect simulation of some point processes for the impatient user. Adv. Appl. Prob. 31, 69–87.Google Scholar
[42] Van den Berg, J. and Steif, J. E. (1999). On the existence and non-existence of finitary codings for a class of random fields. Ann. Prob. 27, 1501–1522.Google Scholar
[43] Wilson, D. B. (2000). Layered multishift coupling for use in perfect sampling algorithms (with a primer on CFTP). In Monte Carlo Methods, ed. Madras, N. (Fields Institute Communications 26). AMS, Providence, RI, pp. 141–176.Google Scholar