Electronic Journal of Statistics

Intensity approximation for pairwise interaction Gibbs point processes using determinantal point processes

Jean-François Coeurjolly and Frédéric Lavancier

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The intensity of a Gibbs point process is usually an intractable function of the model parameters. For repulsive pairwise interaction point processes, this intensity can be expressed as the Laplace transform of some particular function. Baddeley and Nair (2012) developped the Poisson-saddlepoint approximation which consists, for basic models, in calculating this Laplace transform with respect to a homogeneous Poisson point process. In this paper, we develop an approximation which consists in calculating the same Laplace transform with respect to a specific determinantal point process. This new approximation is efficiently implemented and turns out to be more accurate than the Poisson-saddlepoint approximation, as demonstrated by some numerical examples.

Article information

Electron. J. Statist., Volume 12, Number 2 (2018), 3181-3203.

Received: December 2017
First available in Project Euclid: 27 September 2018

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Digital Object Identifier

Zentralblatt MATH identifier

Primary: 60G55: Point processes
Secondary: 82B21: Continuum models (systems of particles, etc.)

Determinantal point process Georgii-Nguyen-Zessin formula Gibbs point process Laplace transform

Creative Commons Attribution 4.0 International License.


Coeurjolly, Jean-François; Lavancier, Frédéric. Intensity approximation for pairwise interaction Gibbs point processes using determinantal point processes. Electron. J. Statist. 12 (2018), no. 2, 3181--3203. doi:10.1214/18-EJS1477. https://projecteuclid.org/euclid.ejs/1538013687

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  • A. Baddeley and G. Nair. Fast approximation of the intensity of Gibbs point processes., Electronic Journal of Statistics, 6 :1155–1169, 2012.
  • A. Baddeley, E. Rubak, and R. Turner., Spatial Point Patterns: Methodology and Applications with R. CRC Press, 2015.
  • C. A. N. Biscio and F. Lavancier. Quantifying repulsiveness of determinantal point processes., Bernoulli, 22(4) :2001–2028, 2016.
  • J.-F. Coeurjolly, J. Møller, and R. Waagepetersen. A tutorial on Palm distribution for spatial point processes., to appear in International Statistical Review, 2017.
  • D.J. Daley and D. Vere-Jones., An Introduction to the Theory of Point Processes, Volume I: Elementary Theory and Methods. Springer, New York, second edition, 2003.
  • D. Dereudre. Introduction to the theory of Gibbs point processes. submitted for publication, available at arXiv :1701.08105, 2017.
  • H.-O. Georgii. Canonical and grand canonical Gibbs states for continuum systems., Communications in Mathematical Physics, 48:31–51, 1976.
  • J.B. Hough, M. Krishnapur, Y. Peres, and B. Virag. Zeros of Gaussian Analytic Functions and Determinantal Point Processes., American Mathematical Society, 2009.
  • F. Lavancier, J. Møller, and E. Rubak. Determinantal point process models and statistical inference., Journal of the Royal Statistical Society: Series B (Statistical Methodology), 77(4):853–877, 2015.
  • O. Macchi. The coincidence approach to stochastic point processes., Advances in Applied Probability, 7:83–122, 1975.
  • J. Møller and R. P. Waagepetersen., Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton, 2004.
  • X. X. Nguyen and H. Zessin. Ergodic theorems for spatial processes., Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 48:133–158, 1979.
  • R development core team., A language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2011.
  • F. Riesz and B.S. Nagy., Functional Analysis. Dover Books on Mathematics Series. Dover Publications, 1990.
  • D. Ruelle., Statistical Mechanics: Rigorous Results. W.A. Benjamin, Reading, Massachusetts, 1969.
  • T. Shirai and Y. Takahashi. Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes., Journal of Functional Analysis, 2:414–463, 2003.
  • B. Simon., Operator theory: A comprehensive course in analysis part 4. AMS American Mathematical Society, 2015.
  • K. Stucki and D. Schuhmacher. Bounds for the probability generating functional of a Gibbs point process., Advances in applied probability, 46(1):21–34, 2014.
  • M. N. M. van Lieshout., Markov Point Processes and Their Applications. Imperial College Press, London, 2000.