×

Fast approximation of the intensity of Gibbs point processes. (English) Zbl 1268.60063

Using the Georgii-Nguyen-Zessin formula which relates the probability distribution of a point process and its reduced Palm distribution, the authors derive a new approximation for the intensity of the stationary Gibbs point process in a multi-dimensional space. The relation with the standard mean field approximation is clarified and numerical examples are displayed.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
62E17 Approximations to statistical distributions (nonasymptotic)
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] A. Baddeley. Modelling strategies. In A.E. Gelfand, P.J. Diggle, M. Fuentes, and P. Guttorp, editors, Handbook of Spatial Statistics , chapter 20, pages 339-369. CRC Press, Boca Raton, 2010. · doi:10.1201/9781420072884-c20
[2] A. Baddeley and R. Turner. Spatstat: an R package for analyzing spatial point patterns., Journal of Statistical Software , 12(6):1-42, 2005. URL: , ISSN: 1548-7660.
[3] A. Baddeley, R. Turner, J. Møller, and M. Hazelton. Residual analysis for spatial point processes (with discussion)., Journal of the Royal Statistical Society, series B , 67(5):617-666, 2005. · Zbl 1112.62302 · doi:10.1111/j.1467-9868.2005.00519.x
[4] K.K. Berthelsen and J. Møller. A primer on perfect simulation for spatial point processes., Bulletin of the Brazilian Mathematical Society , 33:351-367, 2002. · Zbl 1042.60028 · doi:10.1007/s005740200019
[5] L. Bondesson and J. Fahlén. Mean and variance of vacancy for hard-core disc processes and applications., Scandinavian Journal of Statistics , 30:797-816, 2003. · Zbl 1051.60009 · doi:10.1111/1467-9469.00365
[6] R. Brent. Netlib algorithm zeroin.c. URL, . · Zbl 0886.20023
[7] R. Brent., Algorithms for minimization without derivatives . Prentice-Hall, Englewood Cliffs, New Jersey, 1973. · Zbl 0245.65032
[8] G. Celeux, F. Forbes, and N. Peyrard. EM procedures using mean field-like approximations for Markov model-based image segmentation., Pattern Recognition , 36:131-144, 2003. · Zbl 1010.68158 · doi:10.1016/S0031-3203(02)00027-4
[9] F. Chapeau-Blondeau and A. Monir. Numerical evaluation of the Lambert W function and application to generation of generalized Gaussian noise with exponent 1/2., IEEE Transactions on Signal Processing , 50 :2160-, 2002. · Zbl 1369.33022 · doi:10.1109/TSP.2002.801912
[10] R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, and D.E. Knuth. On the Lambert W function., Computational Mathematics , 5:325-359, 1996. · Zbl 0863.65008 · doi:10.1007/BF02124750
[11] D.J. Daley and D. Vere-Jones., An Introduction to the Theory of Point Processes. Volume I: Elementary Theory and Methods . Springer Verlag, New York, second edition, 2003. · Zbl 1026.60061 · doi:10.1007/b97277
[12] P.J. Diggle., Statistical Analysis of Spatial Point Patterns . Hodder Arnold, London, second edition, 2003. · Zbl 1021.62076
[13] P.J. Diggle, T. Fiksel, P. Grabarnik, Y. Ogata, D. Stoyan, and M. Tanemura. On parameter estimation for pairwise interaction processes., International Statistical Review , 62:99-117, 1994. · Zbl 0829.62093 · doi:10.2307/1403548
[14] R.P. Feynman., Statistical mechanics. A set of lectures . Benjamin, Reading, Mass., 1972. · Zbl 0997.82500
[15] F. Forbes and N. Peyrard. Hidden Markov random field model selection criteria based on mean field-like approximations., IEEE Transactions on Pattern Analysis and Machine Intelligence , 25 :1089-1101, 2003.
[16] M. Galassi. GNU Scientific Library Reference Manual., , 2009. Third edition. ISBN 095612078.
[17] H.-O. Georgii. Canonical and grand canonical Gibbs states for continuum systems., Communications of Mathematical Physics , 48:31-51, 1976. · doi:10.1007/BF01609410
[18] H.-O. Georgii., Gibbs Measures and Phase Transitions . Walter de Gruyter, Berlin, 1988. · Zbl 0657.60122
[19] C.J. Geyer and J. Møller. Simulation procedures and likelihood inference for spatial point processes., Scandinavian Journal of Statistics , 21(4):359-373, 1994. · Zbl 0809.62089
[20] J.A. Gubner, W.B. Chang, and M.M. Hayat. Performance analysis of hypothesis testing for sparse pairwise interaction point processes., IEEE Transactions on Information Theory , 46 :1357-1365, 2000. · Zbl 1011.62085 · doi:10.1109/18.850675
[21] U. Hahn. Scale families in spatial point processes. Conference poster presented at the 12th Workshop on Stochastic Geometry, Stereology and Image Analysis, Prague, Czech Republic, 2003.
[22] S.R. Jammalamadaka and M.D. Penrose. Poisson limits for pairwise and area-interaction processes., Advances in Applied Probability , 32:75-85, 2000. · Zbl 0960.60048 · doi:10.1239/aap/1013540023
[23] O. Kallenberg. An informal guide to the theory of conditioning in point processes., International Statistical Review , 52:151-164, 1984. · Zbl 0552.60041 · doi:10.2307/1403098
[24] F.P. Kelly and B.D. Ripley. A note on Strauss’s model for clustering., Biometrika , 63:357-360, 1976. · Zbl 0332.60034 · doi:10.1093/biomet/63.2.357
[25] O.K. Kozlov. Gibbsian description of point random fields., Theory of probability and its applications , 21:339-355, 1976. · Zbl 0364.60086 · doi:10.1137/1121038
[26] Y.A. Kutoyants., Statistical Inference for Spatial Poisson Processes . Number 134 in Lecture Notes in Statistics. Springer, New York, 1998. · Zbl 0904.62108
[27] J.E. Lennard-Jones. On the determination of molecular fields., Proc. Royal Soc. London A , 106:463-477, 1924.
[28] S. Mase. Mean characteristics of Gibbsian point processes., Annals of the Institute of Statistical Mathematics , 42:203-220, 1990. · Zbl 0723.60118 · doi:10.1007/BF00050833
[29] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller. Equation of state calculations by fast computing machines., Journal of Chemical Physics , 21 :1087-1092, 1953.
[30] I. Molchanov and S. Zuyev. Variational analysis of functionals of Poisson processes., Mathematics of Operations Research , 25:485-508, 2000. · Zbl 1018.49022 · doi:10.1287/moor.25.3.485.12217
[31] J. Møller and R.P. Waagepetersen., Statistical Inference and Simulation for Spatial Point Processes . Chapman and Hall/CRC, Boca Raton, 2004.
[32] X.X. Nguyen and H. Zessin. Punktprozesse mit Wechselwirkung., Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete , 37:91-126, 1976. · Zbl 0328.60057 · doi:10.1007/BF00536775
[33] X.X. Nguyen and H. Zessin. Integral and differential characterizations of the Gibbs process., Mathematische Nachrichten , 88:105-115, 1979. · Zbl 0444.60040 · doi:10.1002/mana.19790880109
[34] Y. Ogata and M. Tanemura. Likelihood analysis of spatial point patterns., Journal of the Royal Statistical Society, series B , 46:496-518, 1984. · Zbl 0579.62087
[35] L.S. Ornstein and F. Zernike. Accidental deviations of density and opalesence at the critical point of a single substance., Proceedings of the Section of Sciences, Royal Academy of Sciences/ Koninklijke Akademie van Wetenschappen, Amsterdam , 17:793-806, 1914.
[36] F. Papangelou. The conditional intensity of general point processes and an application to line processes., Zeitschrift fuer Wahscheinlichkeitstheorie und verwandte Gebiete , 28:207-226, 1974. · Zbl 0265.60047 · doi:10.1007/BF00533242
[37] J.K. Percus and G.J. Yevick. Analysis of classical statistical mechanics by means of collective coordinates., Physical Review , 110:1-13, 1958. · Zbl 0096.23105 · doi:10.1103/PhysRev.110.1
[38] R Development Core Team., R: A Language and Environment for Statistical Computing . R Foundation for Statistical Computing, Vienna, Austria, 2011. ISBN 3-900051-07-0.
[39] B.D. Ripley., Statistical Inference for Spatial Processes . Cambridge University Press, 1988. · Zbl 0716.62100
[40] R. Roy and F.W.J. Olver. Lambert W function. In F.W.J. Olver, D.M. Lozier, and R.F. Boisvert, editors, NIST Handbook of Mathematical Functions . Cambridge University Press, 2010.
[41] D. Ruelle., Statistical Mechanics: Rigorous Results . W.A. Benjamin, Reading, Mass., 1969. · Zbl 0177.57301
[42] R. Saunders, R.J. Kryscio, and G.M. Funk. Poisson limits for a hard-core clustering model., Stochastic Processes and their Applications , 12:97-106, 1982. · Zbl 0465.60019 · doi:10.1016/0304-4149(81)90014-4
[43] D. Stoyan, W.S. Kendall, and J. Mecke., Stochastic Geometry and its Applications . John Wiley and Sons, Chichester, second edition, 1995. · Zbl 0838.60002
[44] D.J. Strauss. A model for clustering., Biometrika , 63:467-475, 1975. · Zbl 0313.62044 · doi:10.1093/biomet/62.2.467
[45] M.N.M. van Lieshout., Markov Point Processes and their Applications . Imperial College Press, London, 2000. · Zbl 0968.60005
[46] M. Westcott. Approximations to hard-core models and their application to statistical analysis., Journal of Applied Probability , 19A:281-292, 1982. · Zbl 0492.62083 · doi:10.2307/3213567
[47] J. Zhang. The mean field theory in EM procedures for Markov random fields., IEEE Transactions on Signal Processing , 40(10) :2570-2583, 1992. · Zbl 0850.93772 · doi:10.1109/78.157297
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.