Interaction Processes for Unions of Facets, the Asymptotic Behaviour with Increasing Intensity

Abstract

In the series of models with interacting particles in stochastic geometry, a further contribution presents the facet process which is defined in arbitrary Euclidean dimension. In 2D, 3D specially it is a process of interacting segments, flat surfaces, respectively. Its investigation is based on the theory of functionals of finite spatial point processes given by a density with respect to a Poisson process. The methodology based on L 2 expansion of the covariance of functionals of Poisson process is developed for U -statistics of facet intersections which are building blocks of the model. The importance of the concept of correlation functions of arbitrary order is emphasized. Some basic properties of facet processes, such as local stability and repulsivness are shown and a standard simulation algorithm mentioned. Further the situation when the intensity of the process tends to infinity is studied. In the case of Poisson processes a central limit theorem follows from recent results of Wiener-Ito theory. In the case of non-Poisson processes we restrict to models with finitely many orientations. Detailed analysis of correlation functions exhibits various asymptotics for different combination of U -statistics and submodels of the facet process.

Estimation of Parameters in a Planar Segment Process With a Biological Application

“…Particularly we are interested in parameter estimation in models given by a density with respect to the Poisson process (Møller and Waagepetersen, 2004). For systems of objects the disc process case was developed in Møller and Helisova (2008;, the facet process in Večeřa and Beneš (2016); Večeřa (2016), where facets are compact subsets of hyperplanes in R d . Because of problems with normalizing constant the Takacz-Fiksel method or maximum pseudolikelihood method are suitable choices, cf.…”

Planar Segment Processes with Reference Mark Distributions, Modeling and Estimation

“…Interactions enter the model by means of intersections, cf. [19], [18]. An approximation from [3] is used to avoid the problem with unknown moments.…”

Central limit theorem for Gibbsian U-statistics of facet processes

“…In the present paper we use methods developed in [9] to calculate all moments of Gibbsian U -statistics of facets in a bounded window in arbitrary Euclidean dimension. These moments are used to derive the central limit theorem for such statistics.…”

“…As in [9] we call facets some compact subsets of hyperplanes with a given shape, size and orientation. Natural geometrical characteristics of the union of the facets, based on Hausdorff measure of the intersections of pairs, triplets, etc., of facets form U -statistics.…”

“…We restrict ourselves to the facet model with finitely many orientations corresponding to canonical vectors. In [9] basic asymptotic properties of the studied U -statistics are derived. When the order of the submodel is not greater than the order of the observed U -statistic then asymptoticaly the mean value of the U -statistic vanishes.…”