Skip to main content
SpringerLink
Log in
Book cover

Stochastic Geometry pp 181–229Cite as

Introduction to the Theory of Gibbs Point Processes

  • Chapter
  • First Online:

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2237))

Abstract

The Gibbs point processes (GPP) constitute a large class of point processes with interaction between the points. The interaction can be attractive, repulsive, depending on geometrical features whereas the null interaction is associated with the so-called Poisson point process. In a first part of this mini-course, we present several aspects of finite volume GPP defined on a bounded window in \(\mathbb {R}^d\). In a second part, we introduce the more complicated formalism of infinite volume GPP defined on the full space \(\mathbb {R}^d\). Existence, uniqueness and non-uniqueness of GPP are non-trivial questions which we treat here with completely self-contained proofs. The DLR equations, the GNZ equations and the variational principle are presented as well. Finally we investigate the estimation of parameters. The main standard estimators (MLE, MPLE, Takacs-Fiksel and variational estimators) are presented and we prove their consistency. For sake of simplicity, during all the mini-course, we consider only the case of finite range interaction and the setting of marked points is not presented.

This is a preview of subscription content, log in via an institution.

eBook
USD   19.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   29.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. A. Baddeley, D. Dereudre, Variational estimators for the parameters of Gibbs point process models. Bernoulli 19(3), 905–930 (2013)

    Article  MathSciNet  Google Scholar 

  2. A. Baddeley, R. Turner, Practical maximum pseudolikelihood for spatial point patterns (with discussion). Aust. N. Z. J. Stat. 42(3), 283–322 (2000)

    Article  MathSciNet  Google Scholar 

  3. A.J. Baddeley, M.N.M. van Lieshout, Area-interaction point processes. Ann. Inst. Stat. Math. 47(4), 601–619 (1995)

    Article  MathSciNet  Google Scholar 

  4. A. Baddeley, P. Gregori, J. Mateu, R. Stoica, D. Stoyan, Case Studies in Spatial Point Process Models. Lecture Notes in Statistics, vol. 185 (Springer, New York, 2005)

    Google Scholar 

  5. J. Besag, Spatial interaction and the statistical analysis of lattice systems. J. R. Stat. Soc. Ser. B 36, 192–236 (1974). With discussion by D. R. Cox, A. G. Hawkes, P. Clifford, P. Whittle, K. Ord, R. Mead, J. M. Hammersley, and M. S. Bartlett and with a reply by the author

    Google Scholar 

  6. J.-M. Billiot, J.-F. Coeurjolly, R. Drouilhet, Maximum pseudolikelihood estimator for exponential family models of marked Gibbs point processes. Electron. J. Stat. 2, 234–264 (2008)

    Article  MathSciNet  Google Scholar 

  7. J.T. Chayes, L. Chayes, R. Kotecký, The analysis of the Widom-Rowlinson model by stochastic geometric methods. Commun. Math. Phys. 172(3), 551–569 (1995)

    Article  MathSciNet  Google Scholar 

  8. S.N. Chiu, D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and Its Applications, 3rd edn. (Wiley, Chichester, 2013)

    Book  Google Scholar 

  9. J.-F. Coeurjolly, D. Dereudre, R. Drouilhet, F. Lavancier, Takacs-Fiksel method for stationary marked Gibbs point processes. Scand. J. Stat. 39(3), 416–443 (2012)

    Article  MathSciNet  Google Scholar 

  10. D.J. Daley, D. Vere-Jones, An Introduction to the Theory of Point Processes. Vol. I. Elementary Theory and Methods. Probability and Its Applications (New York), 2nd edn. (Springer, New York, 2003).

    Google Scholar 

  11. D. Dereudre, Diffusion infini-dimensionnelles et champs de Gibbs sur l’espace des trajectoires continues. PhD, Ecole polytechnique Palaiseau (2002)

    Google Scholar 

  12. D. Dereudre, The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains. Adv. Appl. Probab. 41(3), 664–681 (2009)

    Article  MathSciNet  Google Scholar 

  13. D. Dereudre, Variational principle for Gibbs point processes with finite range interaction. Electron. Commun. Probab. 21, Paper No. 10, 11 (2016)

    Google Scholar 

  14. D. Dereudre, P. Houdebert, Infinite volume continuum random cluster model. Electron. J. Probab. 20(125), 24 (2015)

    Google Scholar 

  15. D. Dereudre, F. Lavancier, Consistency of likelihood estimation for Gibbs point processes. Ann. Stat. 45(2), 744–770 (2017)

    Article  MathSciNet  Google Scholar 

  16. D. Dereudre, R. Drouilhet, H.-O. Georgii, Existence of Gibbsian point processes with geometry-dependent interactions. Probab. Theory Relat. Fields 153(3–4), 643–670 (2012)

    Article  MathSciNet  Google Scholar 

  17. D. Dereudre, F. Lavancier, K. Staňková Helisová, Estimation of the intensity parameter of the germ-grain quermass-interaction model when the number of germs is not observed. Scand. J. Stat. 41(3), 809–829 (2014)

    Article  MathSciNet  Google Scholar 

  18. R.L. Dobrushin, E.A. Pecherski, A criterion of the uniqueness of Gibbsian fields in the noncompact case, in Probability Theory and Mathematical Statistics (Tbilisi, 1982). Lecture Notes in Mathematics, vol. 1021 (Springer, Berlin, 1983), pp. 97–110

    Google Scholar 

  19. T. Fiksel, Estimation of parametrized pair potentials of marked and nonmarked Gibbsian point processes. Elektron. Informationsverarb. Kybernet. 20(5–6), 270–278 (1984)

    MathSciNet  MATH  Google Scholar 

  20. H.-O. Georgii, Canonical Gibbs Measures. Lecture Notes in Mathematics, vol. 760 (Springer, Berlin, 1979). Some extensions of de Finetti’s representation theorem for interacting particle systems

    Google Scholar 

  21. H.-O. Georgii, Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction. Probab. Theory Relat. Fields 99(2), 171–195 (1994)

    Article  MathSciNet  Google Scholar 

  22. H.-O. Georgii, Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics, vol. 9, 2nd edn. (Walter de Gruyter, Berlin, 2011)

    Google Scholar 

  23. H.-O. Georgii, T. Küneth, Stochastic comparison of point random fields. J. Appl. Probab. 34(4), 868–881 (1997)

    Article  MathSciNet  Google Scholar 

  24. H.-O. Georgii, H.J. Yoo, Conditional intensity and Gibbsianness of determinantal point processes. J. Stat. Phys. 118(1–2), 55–84 (2005)

    Article  MathSciNet  Google Scholar 

  25. H.-O. Georgii, H. Zessin, Large deviations and the maximum entropy principle for marked point random fields. Probab. Theory Relat. Fields 96(2), 177–204 (1993)

    Article  MathSciNet  Google Scholar 

  26. C.J. Geyer, J. Møller, Simulation procedures and likelihood inference for spatial point processes. Scand. J. Stat. 21(4), 359–373 (1994)

    MathSciNet  MATH  Google Scholar 

  27. X. Guyon, Random Fields on a Network. Probability and Its Applications (New York) (Springer, New York, 1995). Modeling, statistics, and applications, Translated from the 1992 French original by Carenne Ludeña

    Google Scholar 

  28. P. Hall, On continuum percolation. Ann. Probab. 13(4), 1250–1266 (1985)

    Article  MathSciNet  Google Scholar 

  29. J.L. Jensen, Asymptotic normality of estimates in spatial point processes. Scand. J. Stat. 20(2), 97–109 (1993)

    MathSciNet  MATH  Google Scholar 

  30. J.L. Jensen, H.R. Künsch, On asymptotic normality of pseudo likelihood estimates for pairwise interaction processes. Ann. Inst. Stat. Math. 46(3), 475–486 (1994)

    MathSciNet  MATH  Google Scholar 

  31. J.L. Jensen, J. Møller, Pseudolikelihood for exponential family models of spatial point processes. Ann. Appl. Probab. 1(3), 445–461 (1991)

    Article  MathSciNet  Google Scholar 

  32. W.S. Kendall, J. Møller, Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Adv. Appl. Probab. 32(3), 844–865 (2000)

    Article  MathSciNet  Google Scholar 

  33. O.K. Kozlov, Description of a point random field by means of the Gibbs potential. Uspehi Mat. Nauk 30(6(186)), 175–176 (1975)

    Google Scholar 

  34. J.L. Lebowitz, A. Mazel, E. Presutti, Liquid-vapor phase transitions for systems with finite-range interactions. J. Stat. Phys. 94(5–6), 955–1025 (1999)

    Article  MathSciNet  Google Scholar 

  35. S. Mase, Uniform LAN condition of planar Gibbsian point processes and optimality of maximum likelihood estimators of soft-core potential functions. Probab. Theory Relat. Fields 92(1), 51–67 (1992)

    Article  MathSciNet  Google Scholar 

  36. K. Matthes, J. Kerstan, J. Mecke, Infinitely Divisible Point Processes (Wiley, Chichester, 1978). Translated from the German by B. Simon, Wiley Series in Probability and Mathematical Statistics

    Google Scholar 

  37. J. Mayer, E. Montroll, Molecular distributions. J. Chem. Phys. 9, 2–16 (1941)

    Article  Google Scholar 

  38. R. Meester, R. Roy, Continuum Percolation. Cambridge Tracts in Mathematics, vol. 119 (Cambridge University Press, Cambridge, 1996)

    Google Scholar 

  39. I.S. Molchanov, Consistent estimation of the parameters of Boolean models of random closed sets. Teor. Veroyatnost. i Primenen. 36(3), 580–587 (1991)

    MathSciNet  MATH  Google Scholar 

  40. J. Møller, Lectures on random Voronoı̆ tessellations. Lecture Notes in Statistics, vol. 87 (Springer, New York, 1994)

    Google Scholar 

  41. J. Møller, K. Helisová, Likelihood inference for unions of interacting discs. Scand. J. Stat. 37(3), 365–381 (2010)

    Article  MathSciNet  Google Scholar 

  42. J. Møller, R.P. Waagepetersen, Statistical Inference and Simulation for Spatial Point Processes. Monographs on Statistics and Applied Probability, vol. 100 (Chapman & Hall/CRC, Boca Raton, 2004)

    Google Scholar 

  43. X. Nguyen, H. Zessin, Integral and differential characterizations Gibbs processes. Mathematische Nachrichten, 88(1), 105–115 (1979)

    Article  MathSciNet  Google Scholar 

  44. Y. Ogata, M. Tanemura, Likelihood analysis of spatial point patterns. J. R. Stat. Soc. Ser. B 46(3), 496–518 (1984)

    MathSciNet  MATH  Google Scholar 

  45. S. Poghosyan, D. Ueltschi, Abstract cluster expansion with applications to statistical mechanical systems. J. Math. Phys. 50(5), 053509, 17 (2009)

    Article  MathSciNet  Google Scholar 

  46. C. Preston, Random fields. Lecture Notes in Mathematics, vol. 534 (Springer, Berlin, 1976)

    Book  Google Scholar 

  47. D. Ruelle, Statistical Mechanics: Rigorous Results (W. A. Benjamin, Inc., New York, 1969)

    MATH  Google Scholar 

  48. D. Ruelle, Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127–159 (1970)

    Article  MathSciNet  Google Scholar 

  49. R. Takacs, Estimator for the pair-potential of a Gibbsian point process. Statistics, 17(3), 429–433 (1986)

    Article  MathSciNet  Google Scholar 

  50. J. van den Berg, C. Maes, Disagreement percolation in the study of Markov fields. Ann. Probab. 22(2), 749–763 (1994)

    Article  MathSciNet  Google Scholar 

  51. M.N.M. van Lieshout, Markov Point Processes and Their Applications (Imperial College Press, London, 2000)

    Book  Google Scholar 

  52. B. Widom, J.S. Rowlinson, New model for the study of liquid-vapor phase transitions. J. Chem. Phys. 52, 1670–1684 (1970)

    Article  Google Scholar 

Download references

Acknowledgements

The author thanks P. Houdebert, A. Zass and the anonymous referees for the careful reading and the interesting comments. This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01), the CNRS GdR 3477 GeoSto and the ANR project PPP (ANR-16-CE40-0016).

Author information

Authors and Affiliations

  1. University Lille, Villeneuve-d’Ascq, France

    David Dereudre

Authors
  1. David Dereudre
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Dereudre .

Editor information

Editors and Affiliations

  1. LAMAV, Université Polytechnique des Hauts de France (UPHF), Valenciennes, France

    David Coupier

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Dereudre, D. (2019). Introduction to the Theory of Gibbs Point Processes. In: Coupier, D. (eds) Stochastic Geometry. Lecture Notes in Mathematics, vol 2237. Springer, Cham. https://doi.org/10.1007/978-3-030-13547-8_5

Download citation

Publish with us

Policies and ethics

Search

Navigation