An estimation method of the pair potential function for Gibbs point processes on spheres

Scandinavian Journal of Statistics

Volumn 28, Issue 1, 2001, Pages 185-203

  Billiot, Jean Michel ^c   Goulard, Michel ^a,b  

^a INRA   (France)

^b CEMAGREF   (France)

^c NONE

Author keywords

Asymptotic normality; Consistency; Fourier decomposition; MC likelihood; Parametric bootstrap; Pseudolikelihood

Indexed keywords

EID: 0035285943     PISSN: 03036898     EISSN: None     Source Type: Journal    
DOI: 10.1111/1467-9469.00231     Document Type: Article

References (41)

1. Bertin, E., Billiot, J. M. & Drouilhet, R. (1999). Existence of nearest-neighbours type spatial Gibbs models. Adv. Appl. Probab. 31 (4).
2. Baddeley, A. J., Van Lieshout, M. N. N. & Møller, J. (1996). Markov properties of cluster processes. Adv. Appl. Probab. 28, 346-355.
3. Baddeley, A. J. & Møller, J. (1989). Nearest-neighbours Markov point processes. Internat. Statist. Rev. 2, 89-121.
4. Baddeley, A. J. & Van Lieshout, M. N. N. (1995). Area-interaction point processes. Ann. Inst. Statist. Math. 47, 601-619.
5. Besag, J. (1974). Spatial interaction and the statistical analysis of lattice system. J. Roy. Statist. Soc. Ser. B 26, 192-236.
6. Billiot, J. M. (1995). Estimation dans les modèles spatiaux de Gibbs: synthèse bibliographique. Publ. 1 I.S.U.P 39, 3-33.
7. Carter, D. S. & Prenter, P. M. (1972). Exponential spaces and counting processes. Z. Wahrsch. Verw. Gebiete 21, 1-19.
8. Chadœuf, J., Goulard, M. & Pellerin, S. (1993). A Gibbs point process on finite series of circles: the insertion of the primary roots of maize around the stem. J. Appl. Statist. 20, 177-185.
9. Daley, D. J. & Vere-Jones, D. (1988). An introduction to the theory of point processes. Springer-Verlag, New York.
10. Dacunha-Castelle, D. & Duflo, M. (1983). Probabilités et statistiques tome 2. Problèmes à temps mobile. Masson, Paris.
11. Dieudonné, J. (1972). Eléments d'analyse (tome 1). Gauthier-Villars, Paris.
12. Diggle, P. J. (1983). Statistical analysis of spatial point patterns. Academic Press, London.
13. Diggle, P. J., Gates, D. J. & Stibbard, A. (1987). A non parametric estimator for pairwise-interaction point process. Biometrika 74, 763-770.
14. Diggle, P. J. & Gratton, R. J. (1984). Monte Carlo methods of inference for implicit statistical models (with discussion). J. Roy. Statist. Soc. Ser. B 46, 193-227.
15. Fiksel, T. (1988). Estimation of interaction potentials of Gibbsian point processes. Statistics 19, 77-86.
16. Gates, D. J. & Wescott, M. (1986). Clustering estimates in spatial point processes with stable potentials. Ann. Inst. Statist. Math. 38, 123-135.
17. Geyer, C. J. (1999). Likelihood inference for spatial point processes. In Stochastic geometry: likelihood and computation (eds O. E. Barndorff-Nielsen, W. S. Kendall and M. N. M. van Lieshout). Chapman & Hall, London.
18. Geyer, C. J. & Møller, J. (1994). Simulation procedures and likelihood inference for spatial point processes. Scand. J. Statist. 21, 359-373.
19. Geyer, C. J. & Thompson, E. A. (1992). Constrained Monte Carlo maximum likelihood for dependent data. J. Roy. Statist. Soc. Ser. B 54, 657-699.
20. Girardin, P., Jordan, M-O., Picard, D. & Trendel, R. (1986). Harmonisation des notations concernant la description morphologique d'un pied de maïs (Zea mays L.). Agronomie 7, 853-875.
21. Glötzl, E. & Rauschenschwandtner, B. (1981). On statistics of Gibbsian point processes. The first Pannonian symposium on mathematical statistics, Lectures Notes in Statistics 8, 83-93, Springer-Verlag, New-York.
22. Guyon, X. (1992). Champs aléatoires sur réseau, modélisations, statistique et applications. Masson, Paris.
23. Guyon, X. & Hardouin, C. (1992). The chi-square coding test for nested Markov random field hypotheses. In Stochastic models, statistical methods and algorithms in image analysis, (eds P. Barone, A. Frigesi, M. Piccioni), Lecture Notes in Statistics 74, 165-176. Springer-Verlag, New York.
24. Hanisch, K. H. & Stoyan, D. (1983). Remarks on statistical inference and prediction for a hard-core clustering model, Math. Operationsforsch. Statist. Ser. Statist. 14, 559-567.
25. Heikkinen, J. & Penttinen, A. (1994). Bayesian smoothing in the estimation of the pair potential function for Gibbsian point processes. Preprints from the Department of Statistics 17, University of Jyväskylä.
26. Heinrich, L. (1992). Minimum contrast estimates for parameters of spatial ergodic point processes. In: Transactions of the 11th Prague conference on random processes, information theory and statistical decision functions 479-492, Academia Publishing House, Prague.
27. Heinrich, L. (1993). Asymptotic properties of minimum contrast estimators for parameters of Boolean models. Metrika 40, 67-94.
28. Jensen, J. L. & Møller, J. (1991). Pseudolikelihood for exponential family of spatial processes. Ann. Appl. Probab. 1, 445-461.
29. Kendall, W. S., Lieshout, M. N. M. van & Baddeley, A. J. (1999). Quermass-interaction processes: conditions for stability. Ann. Appl. Probab. 31, 315-342.
30. Klein, D. (1984). Convergence of grand canonical Gibbs measures. Comm. Math. Physics 92, 295-308.
31. Møller, J. (1998) Aspects of spatial statistics, stochastic geometry and Markov chain Monte Carlo methods. Doctoral Thesis, Aalborg University, March 1998.
32. Ogata, Y. & Tanemura, M. (1981). Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure. Ann. Inst. Statist. Math. B 33, 315-338.
33. Pellerin, S., Trendel, R. & Duparque, A. (1990). Relation entre quelques caractères morphologiques et la sensibilité à la verse en végétation de maïs (Zea mays L.). Agronomie 10, 439-446.
34. Pellerin, S., Tricot, F. & Chadœuf, J. (1989). Disposition des racines adventives autour de la tige de maïs (Zea mays L.). Agronomie 9, 859-866.
35. Penttinen, A. (1984). Modelling interactions in spatial point patterns: parameter estimation by the maximum likelihood method. Jyväskylä Studies in Computer Science, Economics and Statistics, Vol. 7 Jyväskylä University, Jyväskylä.
36. Ripley, B. D. (1977). Modelling spatial patterns (with discussion). J. Roy. Statist. Soc. Ser. B 39, 172-212.
37. Ripley, B. D. (1979). Simulating spatial patterns: dependent samples from multivariate density. Algorithm AS 137. Appl. Statist. 28, 109-112.
38. Ruelle, D. (1969). Statistical mechanics: rigorous results. W. A. Benjamin, New York.
39. Senoussi, R. (1990). Statistique asymptotique presque-sûre de modèles statistiques convexes. Ann. Inst. Poincaré 26, 19-44.
40. Stoyan, D. & Grabarnik, P. (1991). Second-order characteristics for stochastic structures connected with Gibbs point processes. Math. Nachr. 151, 95-100.
41. Stoyan, D., Kendall, W. S. & Mecke, J. (1995). Stochastic geometry and its applications, 2nd edn. Wiley, Chichester.