Abstract
The paper deals with random marked sets in \({\mathbb R}^d\) which have integer dimension smaller than d. Statistical analysis is developed which involves the random-field model test and estimation of first and second-order characteristics. Special models are presented based on tessellations and solutions of stochastic differential equations (SDE). The simulation of these sets makes use of marking by means of Gaussian random fields. A space-time nature of the model based on SDE is taken into account. Numerical results of the estimation and testing are discussed. Real data analysis from the materials research investigating a grain microstructure with disorientations of faces as marks is presented.
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References
Baddeley A, Møller J, Waagepetersen R (2000) Non- and semiparametric estimation of interaction in inhomogeneous point patterns. Stat Neerl 54:329–350
Ballani F, Kabluchko Z, Schlather M (2012) Random marked sets. Adv Appl Probab 44(3):603–616
Beneš V, Rataj J (2004) Stochastic geometry—selected topics. Kluwer, Dordrecht
Fréchet M (1951) Sur les tableaux de corrélation dont les marges sont données. Annals de l’Université de Lyon 9, sec. A:53–77
Grabarnik P, Myllymaki M, Stoyan D (2011) Correct testing of mark independence for mark point patterns. Ecol Model 222:3888–3894
Hellmund G, Prokešová M, Vedel-Jensen E (2008) Lévy-based Cox point processes. Adv Appl Probab SGSA 40(3):603–629
Illian J, Penttinen A, Stoyan H, Stoyan D (2008) Statistical analysis and modelling of spatial point patterns. Wiley, New York
Ilucová L, Saxl I, Svoboda M, Sklenička V, Král P (2007) Structure of ECAP Aluminium after different number of passes. Image Anal Stereol 26:37–43
Molchanov I (1984) Labelled random sets. Theory Probab Math Stat 29:113–119
Pawlas Z (2009) Empirical distributions in marked point processes. Stoch Process Appl 119:4194–4209
Randle V, Engler O (2000) Introduction to texture analysis—macrotexture, microtexture and orientation mapping. CRC Press, Boca Raton
Saxl I, Sklenička V, Ilucová L, Svoboda M, Dvořák J, Král P (2009) The link between microstructure and creep in aluminium procesed by equal-channel angular dressing. Mater Sci Eng A 503:82
Schlather M, Ribeiro PJ, Diggle PJ (2004) On the second-order characteristics of marked point processes. Detecting dependence between marks and locations of marked point processes. J. R. Stat. Soc. B 66:79–93
Stoyan D, Kendall WS, Mecke J (1995) Stochastic geometry and its applications. Wiley & Sons, Chichester
Šedivý O, Staněk J, Kratochvílová B, Beneš V (2013) Sliced inverse regression for dimension reduction in random marked sets with covariates. Adv Appl Probab 45(3)
Zähle M (1982) Random processes of Hausdorff rectifiabble closed SETS. Math Nachr 108:49–72
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Staněk, J., Šedivý, O. & Beneš, V. On Random Marked Sets with a Smaller Integer Dimension. Methodol Comput Appl Probab 16, 397–410 (2014). https://doi.org/10.1007/s11009-013-9335-x
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DOI: https://doi.org/10.1007/s11009-013-9335-x