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Poisson-saddlepoint approximation for Gibbs point processes with infinite-order interaction: in memory of Peter Hall

Part of: Distribution theory Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  30 November 2017

Adrian Baddeley  and
Gopalan Nair
Adrian Baddeley*
Affiliation:
Curtin University
Gopalan Nair*
Affiliation:
University of Western Australia
*
* Postal address: Department of Mathematics & Statistics, Curtin University, GPO Box U1987, Perth, WA 6845, Australia. Email address: adrian.baddeley@curtin.edu.au
** Postal address: School of Mathematics & Statistics, University of Western Australia, 35 Stirling Highway, Nedlands, WA 6009, Australia. Email address: gopalan.nair@uwa.edu.au
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Abstract

We develop a computational approximation to the intensity of a Gibbs spatial point process having interactions of any order. Limit theorems from stochastic geometry, and small-sample probabilities estimated once and for all by an extensive simulation study, are combined with scaling properties to form an approximation to the moment generating function of the sufficient statistic under a Poisson process. The approximate intensity is obtained as the solution of a self-consistency equation.

Keywords

Area-interaction processcoverage processGeorgii–Nguyen–Zessin formulaGeyer saturation processGibbs point processPoisson line tessellation

MSC classification

Primary: 60G55: Point processes
Secondary: 60D05: Geometric probability and stochastic geometry 62E17: Approximations to distributions (nonasymptotic)
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 4 , December 2017 , pp. 1008 - 1026
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Anderssen, R. S., Baddeley, A. J., de Hoog, F. R. and Nair, G. M. (2014). Numerical solution of an integral equation from point process theory. J. Integral Equations Appl. 26, 437–453. CrossRefGoogle Scholar
[2] Baddeley, A. (2010). Modeling strategies. In Handbook of Spatial Statistics, CRC, Boca Raton, FL, pp. 339–369. Google Scholar
[3] Baddeley, A. (2017). Local composite likelihood for spatial point processes. To appear in Spatial Statist. Available at https://doi.org/10.1016/j.spasta.2017.03.001. Google Scholar
[4] Baddeley, A. and Møller, J. (1989). Nearest-neighbour Markov point processes and random sets. Internat. Statist. Rev. 57, 89–121. Google Scholar
[5] Baddeley, A. and Nair, G. (2012). Approximating the moments of a spatial point process. Stat 1, 18–30. Google Scholar
[6] Baddeley, A. and Nair, G. (2012). Fast approximation of the intensity of Gibbs point processes. Electron. J. Statist. 6, 1155–1169. Google Scholar
[7] Baddeley, A. J. and van Lieshout, M. N. M. (1995). Area-interaction point processes. Ann. Inst. Statist. Math. 47, 601–619. Google Scholar
[8] Baddeley, A., Rubak, E. and Turner, R. (2015). Spatial Point Patterns: Methodology and Applications with R. Chapman & Hall/CRC, Boca Raton, FL. Google Scholar
[9] Bavaud, F. (1991). Equilibrium properties of the Vlasov functional: the generalized Poisson–Boltzmann–Emden equation. Rev. Modern Physics 63, 129–149. Google Scholar
[10] Bondesson, L. and Fahlén, J. (2003). Mean and variance of vacancy for hard-core disc processes and applications. Scand. J. Statist. 30, 797–816. Google Scholar
[11] Corless, R. M. et al. (1996). On the Lambert W function. Adv. Comput. Math. 5, 329–359. CrossRefGoogle Scholar
[12] Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, Elementary Theory and Methods, 2nd edn. Springer, New York. Google Scholar
[13] Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes, Vol. II, General Theory and Structure, 2nd edn. Springer, New York. Google Scholar
[14] Dereudre, D., Drouilhet, R. and Georgii, H.-O. (2012). Existence of Gibbsian point processes with geometry-dependent interactions. Prob. Theory Relat. Fields 153, 643–670. Google Scholar
[15] Feynman, R. P. (1972). Statistical Mechanics: A Set of Lectures. Benjamin, Reading, MA. Google Scholar
[16] Georgii, H.-O. (1976). Canonical and grand canonical Gibbs states for continuum systems. Commun. Math. Phys. 48, 31–51. Google Scholar
[17] Georgii, H.-O. (1979). Canonical Gibbs Measures (Lecture Notes Math 760). Springer, Berlin. Google Scholar
[18] Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. De Gruyter, Berlin. Google Scholar
[19] Georgii, H.-O. (2011). Gibbs Measures and Phase Transitions, 2nd edn. De Gruyter, Berlin. CrossRefGoogle Scholar
[20] Geyer, C. (1999). Likelihood inference for spatial point processes. In Stochastic Geometry (Monogr. Statist. Appl. Prob 80), Chapman & Hall/CRC, Boca Raton, FL, pp. 79–140. Google Scholar
[21] Geyer, C. J. and Møller, J. (1994). Simulation procedures and likelihood inference for spatial point processes. Scand. J. Statist. 21, 359–373. Google Scholar
[22] Gilbert, E. N. (1965). The probability of covering a sphere with N circular caps. Biometrika 52, 323–330. Google Scholar
[23] Hahn, U. (2003). Scale families in spatial point processes. Conference poster presented at the 12th Workshop on Stochastic Geometry, Stereology and Image Analysis, Prague. Google Scholar
[24] Hall, P. (1985). Distribution of size, structure and number of vacant regions in a high-intensity mosaic. Z. Wahrscheinlichkeitsth. 70, 237–261. CrossRefGoogle Scholar
[25] Hall, P. (1988). Introduction to the Theory of Coverage Processes. John Wiley, New York. Google Scholar
[26] Kallenberg, O. (1984). An informal guide to the theory of conditioning in point processes. Internat. Statist. Rev. 52, 151–164. Google Scholar
[27] Kelly, F. P. and Ripley, B. D. (1976). A note on Strauss's model for clustering. Biometrika 63, 357–360. Google Scholar
[28] Kirkwood, J. G. and Monroe, E. (1941). Statistical mechanics of fusion. J. Chem. Phys. 9, 514–526. Google Scholar
[29] Kozlov, O. K. (1976). Gibbsian description of point random fields. Theory Prob. Appl. 21, 339–356. Google Scholar
[30] Leontovitch, M. A. (1992). On computing the mean Brownian area. In Selected works of A. N. Kolmogorov, Springer, Dordrecht, pp. 128–138. Google Scholar
[31] Mase, S. (1990). Mean characteristics of Gibbsian point processes. Ann. Inst. Statist. Math. 42, 203–220. Google Scholar
[32] Miles, R. E. (1973). The various aggregates of random polygons determined by random lines in a plane. Adv. Math. 10, 256–290. Google Scholar
[33] Møller, J. and Waagepetersen, R. P. (2004). Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall/CRC, Boca Raton, FL. Google Scholar
[34] Moran, P. A. P. (1973). The random volume of interpenetrating spheres in space. J. Appl. Prob. 10, 483–490. Google Scholar
[35] Nguyen, X.-X. and Zessin, H. (1976). Punktprozesse mit Wechselwirkung. Z. Wahrscheinlichkeitsth. 37, 91–126. Google Scholar
[36] Nguyen, X.-X. and Zessin, H. (1979). Integral and differential characterizations of the Gibbs process. Math. Nachr. 88, 105–115. Google Scholar
[37] Papangelou, F. (1974). The conditional intensity of general point processes and an application to line processes. Z. Wahscheinlichkeitsth. 28, 207–226. Google Scholar
[38] Percus, J. K. and Yevick, G. J. (1958). Analysis of classical statistical mechanics by means of collective coordinates. Phys. Rev. (2) 110, 1–13. CrossRefGoogle Scholar
[39] Robbins, H. E. (1944). On the measure of a random set. Ann. Math. Statist. 15, 70–74. Google Scholar
[40] Robbins, H. E. (1945). On the measure of a random set. II. Ann. Math. Statist. 16, 342–347. CrossRefGoogle Scholar
[41] Ruelle, D. (1969). Statistical Mechanics: Rigorous Results. Benjamin, Reading, MA. Google Scholar
[42] Schuhmacher, D. and Stucki, K. (2014). Gibbs point process approximation: total variation bounds using Stein's method. Ann. Prob. 42, 1911–1951. Google Scholar
[43] Strauss, D. J. (1975). A model for clustering. Biometrika 62, 467–475. CrossRefGoogle Scholar
[44] Stucki, K. and Schuhmacher, D. (2014). Bounds for the probability generating functional of a Gibbs point process. Adv. Appl. Prob. 46, 21–34. Google Scholar
[45] Westcott, M. (1982). Approximations to hard-core models and their application to statistical analysis. In Essays in Statistical Science (J. Appl. Prob. Spec. Vol. 19A), Applied Probability Trust, Sheffield, pp. 281–292. Google Scholar
[46] Widom, B. and Rowlinson, J. S. (1970). New model for the study of liquid-vapor phase transitions. J. Chem. Phys. 52, 1670–1683. CrossRefGoogle Scholar
[47] Zhang, J. (1992). The mean field theory in EM procedures for Markov random fields. IEEE Trans. Signal Process. 40, 2570–2583. Google Scholar