• Bernoulli
  • Volume 19, Number 3 (2013), 905-930.

Variational estimators for the parameters of Gibbs point process models

Adrian Baddeley and David Dereudre

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This paper proposes a new estimation technique for fitting parametric Gibbs point process models to a spatial point pattern dataset. The technique is a counterpart, for spatial point processes, of the variational estimators for Markov random fields developed by Almeida and Gidas. The estimator does not require the point process density to be hereditary, so it is applicable to models which do not have a conditional intensity, including models which exhibit geometric regularity or rigidity. The disadvantage is that the intensity parameter cannot be estimated: inference is effectively conditional on the observed number of points. The new procedure is faster and more stable than existing techniques, since it does not require simulation, numerical integration or optimization with respect to the parameters.

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Bernoulli, Volume 19, Number 3 (2013), 905-930.

First available in Project Euclid: 26 June 2013

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Campbell measure Gibbs point process non-hereditary interaction pseudolikelihood spatial statistics variational estimator


Baddeley, Adrian; Dereudre, David. Variational estimators for the parameters of Gibbs point process models. Bernoulli 19 (2013), no. 3, 905--930. doi:10.3150/12-BEJ419.

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  • [1] Almeida, M.P. and Gidas, B. (1993). A variational method for estimating the parameters of MRF from complete or incomplete data. Ann. Appl. Probab. 3 103–136.
  • [2] Baddeley, A. and Turner, R. (2000). Practical maximum pseudolikelihood for spatial point patterns (with discussion). Aust. N. Z. J. Stat. 42 283–322.
  • [3] Baddeley, A. and Turner, R. (2005). Spatstat: An R package for analyzing spatial point patterns. J. Statist. Softw. 12 1–42.
  • [4] Baddeley, A.J. (2000). Time-invariance estimating equations. Bernoulli 6 783–808.
  • [5] Barndorff-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory. Chichester: Wiley.
  • [6] Berman, M. and Turner, T.R. (1992). Approximating point process likelihoods with GLIM. Applied Statistics 41 31–38.
  • [7] Besag, J. (1977). Some methods of statistical analysis for spatial data. Bull. Inst. Internat. Statist. 47 77–91, 138–147.
  • [8] Billiot, J.M., Coeurjolly, J.F. and Drouilhet, R. (2008). Maximum pseudolikelihood estimator for exponential family models of marked Gibbs point processes. Electron. J. Stat. 2 234–264.
  • [9] Coeurjolly, J.F., Dereudre, D., Drouilhet, R. and Lavancier, F. (2010). Takacs–Fiksel method for stationary marked Gibbs point processes. Preprint.
  • [10] Coeurjolly, J.F. and Drouilhet, R. (2010). Asymptotic properties of the maximum pseudo-likelihood estimator for stationary Gibbs point processes including the Lennard-Jones model. Electron. J. Stat. 4 677–706.
  • [11] Dereudre, D. (2002). Une caractérisation de champs de Gibbs canoniques sur $\mathbb{R}^{d}$ et $\mathcal{C}([0,1],\mathbb{R}^{d})$. C. R. Math. Acad. Sci. Paris 335 177–182.
  • [12] Dereudre, D. (2008). Gibbs Delaunay tessellations with geometric hardcore conditions. J. Stat. Phys. 131 127–151.
  • [13] Dereudre, D., Drouilhet, R. and Georgii, H.O. (2012). Existence of Gibbsian point processes with geometry-dependent interactions. Probab. Theory Related Fields 153 643–670.
  • [14] Dereudre, D. and Lavancier, F. (2011). Practical simulation and estimation for Gibbs Delaunay–Voronoi tessellations with geometric hardcore interaction. Comput. Statist. Data Anal. 55 498–519.
  • [15] Dereudre, D. and Lavancier, F. (2009). Campbell equilibrium equation and pseudo-likelihood estimation for non-hereditary Gibbs point processes. Bernoulli 15 1368–1396.
  • [16] Fiksel, T. (1984). Estimation of parametrized pair potentials of marked and nonmarked Gibbsian point processes. Elektron. Informationsverarb. Kybernet. 20 270–278.
  • [17] Georgii, H.O. (1979). Canonical Gibbs Measures: Some Extensions of de Finetti’s Representation Theorem for Interacting Particle Systems. Lecture Notes in Math. 760. Berlin: Springer.
  • [18] Georgii, H.O. (1988). Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics 9. Berlin: de Gruyter.
  • [19] Geyer, C. (1999). Likelihood inference for spatial point processes. In Stochastic Geometry (Toulouse, 1996) (O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. van Lieshout, eds.). Monogr. Statist. Appl. Probab. 80 79–140. Chapman & Hall/CRC, Boca Raton, FL.
  • [20] Goulard, M., Särkkä, A. and Grabarnik, P. (1996). Parameter estimation for marked Gibbs point processes through the maximum pseudolikelihood method. Scand. J. Stat. 23 365–379.
  • [21] Jensen, J.L. and Künsch, H.R. (1994). On asymptotic normality of pseudo likelihood estimates for pairwise interaction processes. Ann. Inst. Statist. Math. 46 475–486.
  • [22] Küchler, U. and Sørensen, M. (1997). Exponential Families of Stochastic Processes. Springer Series in Statistics. New York: Springer.
  • [23] Lennard-Jones, J.E. (1924). On the determination of molecular fields. Proc. R. Soc. Lond. Ser. A 106 463–477.
  • [24] Matthes, K., Kerstan, J. and Mecke, J. (1978). Infinitely Divisible Point Processes. Chichester: Wiley.
  • [25] Nguyen, X.X. and Zessin, H. (1979). Integral and differential characterizations of the Gibbs process. Math. Nachr. 88 105–115.
  • [26] Ogata, Y. and Tanemura, M. (1984). Likelihood analysis of spatial point patterns. J. Roy. Statist. Soc. Ser. B 46 496–518.
  • [27] Preston, C. (1976). Random Fields. Lecture Notes in Mathematics 534. Berlin: Springer.
  • [28] Ripley, B.D. (1988). Statistical Inference for Spatial Processes. Cambridge: Cambridge Univ. Press.
  • [29] Ruelle, D. (1970). Superstable interactions in classical statistical mechanics. Comm. Math. Phys. 18 127–159.
  • [30] Takacs, R. (1986). Estimator for the pair-potential of a Gibbsian point process. Statistics 17 429–433.