zbMATH — the first resource for mathematics

Practical simulation and estimation for Gibbs Delaunay-Voronoi tessellations with geometric hardcore interaction. (English) Zbl 1247.65015
Summary: General models of Gibbs Delaunay-Voronoi tessellations, which can be viewed as extensions of Ord’s process, are considered. The interaction may occur on each cell of the tessellation and between neighbour cells. The tessellation may also be subjected to a geometric hardcore interaction, forcing the cells not to be too large, too small, or too flat. This setting, natural for applications, introduces some theoretical difficulties since the interaction is not necessarily hereditary. Mathematical results available for studying these models are reviewed and further outcomes are provided. They concern the existence, the simulation and the estimation of such tessellations. Based on these results, tools to handle these objects in practice are presented: how to simulate them, estimate their parameters and validate the fitted model. Some examples of simulated tessellations are studied in detail.

65D17 Computer-aided design (modeling of curves and surfaces)
62-07 Data analysis (statistics) (MSC2010)
Full Text: DOI
[1] Baddeley, A.; Møller, J., Nearest-neighbor Markov point processes and random sets, Int. stat. rev., 2, 89-121, (1989) · Zbl 0721.60010
[2] Baddeley, A.; Turner, R.; Møller, J.; Hazelton, M., Residual analysis for spatial point processes, J. R. stat. soc. ser. B, 65, 617-666, (2005) · Zbl 1112.62302
[3] Bertin, E.; Billiot, J.M.; Drouilhet, R., Existence of “nearest-neighbour” spatial Gibbs models, Adv. appl. probab. (SGSA), 31, 895-909, (1999) · Zbl 0955.60008
[4] Bertin, E.; Billiot, J.M.; Drouilhet, R., Existence of Delaunay pairwise Gibbs point processes with superstable component, J. stat. phys., 95, 719-744, (1999) · Zbl 0933.60048
[5] Bertin, E.; Billiot, J.M.; Drouilhet, R., Spatial Delaunay Gibbs point processes, Comm. statist. stochastic models, 15, 2, 181-199, (1999) · Zbl 0936.60043
[6] Bertin, E.; Billiot, J.M.; Drouilhet, R., Phase transition in the nearest-neighbor continuum Potts models, J. stat. phys., 114, 1-2, 79-100, (2004) · Zbl 1060.82017
[7] Billiot, J.-M.; Coeurjolly, J.-F.; Drouilhet, R., Maximum pseudolikelihood estimator for exponential family models of marked Gibbs point processes, Electron. J. stat., 2, 234-254, (2008) · Zbl 1135.62364
[8] Dereudre, D., Gibbs Delaunay tessellations with geometric hardcore conditions, J. stat. phys., 131, 127-151, (2008) · Zbl 1151.82015
[9] Dereudre, D., Drouilhet, R., Georgii, H.-O., Existence of Gibbsian point processes with geometry-dependent interactions. Preprint. arXiv:1003.2875. · Zbl 1256.60036
[10] Dereudre, D.; Lavancier, F., Campbell equilibrium equation and pseudo-likelihood estimation for non-hereditary Gibbs point processes, Bernoulli, 15, 4, 1368-1396, (2009) · Zbl 1200.62023
[11] Eglen, S.J.; Willshaw, D.J., Influence of cell fate mechanisms upon retinal mosaic formation: a modelling study, Development, 129, 23, 5399-5408, (2002)
[12] Emily, M.; François, O., A statistical approach to estimating the strength of cell – cell interactions under the differential adhesion hypothesis, Theor. biol. med. modelling, 4, 37, 1-13, (2007)
[13] Farhadifar, R.; Röper, J.-C.; Aigouy, B.; Eaton, S.; Jülicher, F., The influence of cell mechanics, cell – cell interactions, and proliferation on epithelial packing, Curr. biol., 17, 24, 2095-2104, (2007)
[14] Geyer, C.J.; Thompson, E.A., Constrained Monte Carlo maximum likelihood for dependent data, J. R. stat. soc. ser. B, 54, 657-699, (1992)
[15] Honda, H., Description of cellular patterns by Dirichlet domains: the two-dimensional case, J. theoret. biol., 72, 523-543, (1978)
[16] Icke, V.; Van de Weygaert, R., Fragmenting the universe, Astron. astrophys., 184, 1-2, 16-32, (1987)
[17] Icke, V.; Van de Weygaert, R., Fragmenting the universe. II—voronoi vertices as Abell clusters, Astron. astrophys., 213, 1-2, 1-9, (1989)
[18] Lautensack, C.; Sych, T., 3D image analysis of open foams using random tessellations, Image anal. stereol., 25, 87-93, (2006)
[19] Matsuda, T.; Shima, E., Topology of supercluster-void structure, Progr. theoret. phys., 71, 4, 855-858, (1984) · Zbl 1046.85500
[20] Meyn, S.P.; Tweedie, R.L., Markov chains and stochastic stability, (1993), Springer-Verlag London · Zbl 0925.60001
[21] Møller, J., Lectures on random Voronoi tessellations, (1994), Springer New York · Zbl 0812.60016
[22] Møller, J.; Waagepetersen, R., Statistical inference and simulation for spatial point processes, (2003), Chapman and Hall/CRC Boca Raton
[23] Nguyen, X.X.; Zessin, H., Integral and differential characterizations of the Gibbs process, Math. nachr., 88, 105-115, (1979) · Zbl 0444.60040
[24] Poupon, A., Voronoi and Voronoi-related tessellations in studies of protein structure and interaction, Curr. opin. struct. biol., 2, 233-241, (2004)
[25] Preston, C., Random fields, () · Zbl 0357.60052
[26] Propp, J.G.; Wilson, D.B., Exact sampling with coupled Markov chains and applications to statistical mechanics, Random structures algorithms, 9, 223-252, (1996) · Zbl 0859.60067
[27] Ripley, B.D., Modelling spatial patterns (with discussion), J. R. stat. soc. ser. B, 39, 172-212, (1977) · Zbl 0369.60061
[28] Ruelle, D., Superstable interactions in classical statistical mechanics, Comm. math. phys., 18, 127-159, (1970) · Zbl 0198.31101
[29] Stoyan, D.; Kendall, W.S.; Mecke, J., Stochastic geometry and its applications, (1995), Wiley New York · Zbl 0838.60002
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.