Power diagrams and interaction processes for unions of discs

Abstract

We study a flexible class of finite-disc process models with interaction between the discs. We let 𝒰 denote the random set given by the union of discs, and use for the disc process an exponential family density with the canonical sufficient statistic depending only on geometric properties of 𝒰 such as the area, perimeter, Euler-Poincaré characteristic, and the number of holes. This includes the quermass-interaction process and the continuum random-cluster model as special cases. Viewing our model as a connected component Markov point process, and thereby establishing local and spatial Markov properties, becomes useful for handling the problem of edge effects when only 𝒰 is observed within a bounded observation window. The power tessellation and its dual graph become major tools when establishing inclusion-exclusion formulae, formulae for computing geometric characteristics of 𝒰, and stability properties of the underlying disc process density. Algorithms for constructing the power tessellation of 𝒰 and for simulating the disc process are discussed, and the software is made public available.

Power diagrams and interaction processes for unions of discs

“…For models with two or more parameters, the interpretation is more complicated and depends not only on the signs of the parameters but also on how large the parameters are, and it is, for example, possible to obtain rather similar realizations for different combinations of the parameters. See Figures 5-8 in our research report [31]. As an illustration, Figure 3 …”

“…The algorithm for simulating from the conditional processes with densities (28) and (30) is the same except that we replace λ θ (x, v) in (34) by the Papangelou conditional intensities in (29)- (31), and that the state space has of course to be in accordance with (28) and (30). The convergence properties and computations are therefore similar to those discussed above.…”

The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains

“…In [9], the authors showed that the minimal assumption on Q, to define kernels (2.2) in the following definition, is assumption (H) with the constant α equal to −θ 1 π (if θ 1 < 0). In our paper we need more to prove the existence of infinite Gibbs measures.…”

“…Let us point out that important problems Existence of quermass-interaction processes SGSA • 681 may appear if the interaction is not local. We have tried to prove the existence of a Gibbs germgrain model for nonlocal interaction, proposed by Møller and Helisova [9], which is equal to the number of connected components. But, in this case, some dependence with 'infinity' appears if there is percolation, and so we are not able to prove that the limiting measureμ satisfies the DLR equations.…”

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.…”