Previous |  Up |  Next


Full entry | Fulltext not available (moving wall 24 months)      Feedback
attractiveness; germ-grain model; Markov Chain Monte Carlo simulation; Quermass-interaction process; random set; repulsiveness; Ruelle stability
The paper concerns an extension of random disc Quermass-interaction process, i.e. the model of discs with mutual interactions, to the process of interacting objects of more general shapes. Based on the results for the random disc process and the process with polygonal grains, theoretical results for the generalized process are derived. Further, a simulation method, its advantages and the corresponding complications are described, and some examples are introduced. Finally, a short comparison to the random disc process is given.
[1] Altendorf, H., Latourte, F., Jeulin, D., Faessel, M., Saintyant, L.: 3D reconstruction of a multiscale microstructure by anisotropic tessellation models. Image Anal. Stereol. 33 (2014), 121-130. DOI 10.5566/ias.v33.p121-130
[2] Chiu, S. N., Stoyan, D., Kendall, W. S., Mecke, J.: Stochastic Geometry and Its Applications. Wiley Series in Probability and Statistics John Wiley & Sons, Chichester (2013). MR 3236788 | Zbl 1291.60005
[3] Dereudre, D.: Existence of Quermass processes for non locally stable interaction and non bounded convex grains. Adv. Appl. Probab. 41 (2009), 664-681. DOI 10.1017/S0001867800003517 | MR 2571312
[4] Dereudre, D., Lavancier, F., Helisová, K. Staňková: Estimation of the intensity parameter of the germ-grain Quermass-interaction model when the number of germs is not observed. Scand. J. Stat. 41 (2014), 809-829. DOI 10.1111/sjos.12064 | MR 3249430
[5] Diggle, P. J.: Binary mosaics and the spatial pattern of heather. Biometrics 37 (1981), 531-539. DOI 10.2307/2530566
[6] Geyer, C. J., Møller, J.: Simulation procedures and likelihood inference for spatial point processes. Scand. J. Stat. 21 (1994), 359-373. MR 1310082 | Zbl 0809.62089
[7] Helisová, K.: Modeling, statistical analyses and simulations of random items and behavior on material surfaces. Supplemental UE: TMS 2014 Conference Proceedings, San Diego (2014), 461-468.
[8] Hermann, P., Mrkvička, T., Mattfeldt, T., Minárová, M., Helisová, K., Nicolis, O., Wartner, F., Stehlík, M.: Fractal and stochastic geometry inference for breast cancer: a case study with random fractal models and Quermass-interaction process. Stat. Med. 34 (2015), 2636-2661. DOI 10.1002/sim.6497 | MR 3368407
[9] Kendall, W. S., Lieshout, M. N. M. van, Baddeley, A. J.: Quermass-interaction processes: conditions for stability. Adv. Appl. Probab. 31 (1999), 315-342. DOI 10.1017/S0001867800009137 | MR 1724554
[10] Klazar, M.: Generalised Davenport-Schinzel sequences: results, problems and applications. Integers: The Electronic Journal of Combinatorial Number Theory 2 (2002), A11. MR 1917956
[11] Molchanov, I.: Theory of Random Sets. Probability and Its Applications Springer, London (2005). MR 2132405 | Zbl 1109.60001
[12] Møller, J., Helisová, K.: Power diagrams and interaction processes for unions of discs. Adv. Appl. Probab. 40 (2008), 321-347. DOI 10.1017/S0001867800002548 | MR 2431299 | Zbl 1146.60322
[13] Møller, J., Helisová, K.: Likelihood inference for unions of interacting discs. Scand. J. Stat. 37 (2010), 365-381. DOI 10.1111/j.1467-9469.2009.00660.x | MR 2724503 | Zbl 1226.60016
[14] Møller, J., Waagepetersen, R. P.: Statistical Inference and Simulation for Spatial Point Processes. Monographs on Statistics and Applied Probability 100 Chapman and Hall/CRC, Boca Raton (2004). MR 2004226 | Zbl 1044.62101
[15] Mrkvička, T., Mattfeldt, T.: Testing histological images of mammary tissues on compatibility with the Boolean model of random sets. Image Anal. Stereol. 30 (2011), 11-18. DOI 10.5566/ias.v30.p11-18 | MR 2816303
[16] Mrkvička, T., Rataj, J.: On the estimation of intrinsic volume densities of stationary random closed sets. Stochastic Processes Appl. 118 (2008), 213-231. MR 2376900 | Zbl 1148.62023
[17] Ohser, J., Mücklich, F.: Statistical Analysis of Microstructures in Materials Science. Wiley Series in Statistics in Practice Wiley, Chichester (2000).
[18] Pratt, W. K.: Digital Image Processing. Wiley & Sons, New York (2001).
[19] Team, R Development Core: R: A language and environment for statistical computing. R Found Stat Comp, Vienna. (2010).
[20] Helisová, K. Staňková, Staněk, J.: Dimension reduction in extended Quermass-interaction process. Methodol. Comput. Appl. Probab. 16 (2014), 355-368. DOI 10.1007/s11009-013-9343-x | MR 3199051
[21] Zikmundová, M., Helisová, K. Staňková, Beneš, V.: Spatio-temporal model for a random set given by a union of interacting discs. Methodol. Comput. Appl. Probab. 14 (2012), 883-894. DOI 10.1007/s11009-012-9287-6 | MR 2966326
[22] Zikmundová, M., Helisová, K. Staňková, Beneš, V.: On the use of particle Markov chain Monte Carlo in parameter estimation of space-time interacting discs. Methodol. Comput. Appl. Probab. 16 (2014), 451-463. DOI 10.1007/s11009-013-9367-2 | MR 3199057
Partner of
EuDML logo