A completely random -tessellation model and Gibbsian extensions
Introduction
Random tessellations are attractive mathematical objects, from both theoretical and practical points of view. The study of the mathematical properties of these objects is still leading to open problems, while the range of applications covers a broad panel of scientific domains such as astronomy, geophysics, image processing or environmental sciences (Lantuéjoul, 2002, Møller and Stoyan, 2007, Le Ber et al., 2009).
When modeling real-world structures, one aims at flexible classes of random models able to represent a wide range of spatial patterns. Gibbsian point processes combined with Voronoï diagrams offer such an attractive approach. The class of Gibbs point processes is enriched continuously (hard-core, Strauss, area-interaction, Quermass-interaction). Random tessellations are for example obtained as Voronoï diagrams of seeds distributed according to such Gibbs processes, see e.g. Dereudre and Lavancier (2011). Of special interest are Gibbs models with interactions based on the Delaunay graph (Baddeley and Møller, 1989, Bertin et al., 1999, Dereudre et al., 2012): Delaunay-neighbor seeds define Voronoi cells with a common edge. Hence a large class of models for random tessellations is made available for applications. Our aim is to sketch a similar theoretical framework for other types of tessellations which are not seed-based. We will focus on -tessellations: tessellations with only -vertices.
An example of a stochastic model for -tessellation is the STIT (STable with respect to ITeration) tessellation model (Nagel and Weiss, 2005). STIT tessellations are obtained by successive splits and rescaling. Analytical results about the distributions of various geometrical features are available (e.g. Mecke et al., 2007, Thäle, 2011, Cowan, 2013, Schreiber and Thäle, 2010, Weiss et al., 2010). Ergodicity and mixing properties of the STIT model have also been investigated (Martínez and Nagel, 2012, Lachièze-Rey, 2011). Recently generalizations of the STIT model involving different splitting procedures have been considered (Cowan, 2010, Schreiber and Thäle, 2013). Such random tessellations are referred to as nested tessellations following the denomination used in Schreiber and Thäle (2013). Realizations of nested-tessellations show a striking feature: any compact convex region is split by a unique tessellation maximal segment (also called I-segment).
Another example of stochastic model is the Gilbert model (Gilbert, 1967, Mackisack and Miles, 1996) based on segments growing until they are blocked by other segments. Analytical results on the distributions of standard geometrical features are more sparse (Mackisack and Miles, 1996, Burridge et al., 2013). Some asymptotic results have been established (Schreiber and Soja, 2011). There are also some restrictions on the geometry of tessellations arising from the Gilbert model. Since segments are born simultaneously and grow at a common speed, the number of other segments a segment can block is limited. Therefore, again, arbitrary -tessellations cannot be obtained by a Gilbert-type construction.
The models discussed in this paper define random -tessellations with realizations from a large class of -tessellations built using three geometrical operators: splits, merges and flips. Our approach is closely related to the one used by Arak, Clifford and Surgailis in their paper (Arak et al., 1993) on a random planar graph model. In particular, a key ingredient is the Poisson line process. As a first step, this paper introduces a new model of random -tessellations which can be considered as a completely random model. Then Gibbs variations are proposed and a general algorithm for simulating them is described.
Throughout the paper, we focus on the case where the domain of interest is bounded. Extension of Gibbs models for -tessellations of the whole plane remains an open problem at this stage.
Section 2 provides definitions, notation and basic results about -tessellations. The completely random -tessellation model is discussed in Section 3. This section also introduces for arbitrary random -tessellations Campbell measures, Papangelou kernels which are widely used in point process theory. Our new model can be considered as a -tessellation analogous to the Poisson point process. This claim is based on Georgii–Nguyen–Zessin type formulae. Therefore the -tessellation model introduced in Section 3 is referred to as a completely random -tessellation. Gibbsian extensions are discussed in Section 4 together with some examples. One example is also a particular case of Arak–Clifford–Surgailis random graph model when its parameters are chosen in order to yield a -tessellation. Formulae of Georgii–Nguyen–Zessin type are provided for hereditary Gibbs models. In Section 5, a simulation algorithm is proposed. The design of the algorithm follows the general principles of Metropolis–Hastings–Green algorithms, already widely used for Gibbs point processes. It involves three types of local operators: split, merge and flip. Conditions ensuring the convergence of the Markov chain to the target distribution are provided.
Below, as a notational convention, bold letters are used for denoting random variables. Detailed proofs of the main results are postponed in appendices.
Section snippets
The space of -tessellations
In this paper, we shall consider only tessellations of a compact domain in . For the sake of simplicity, is supposed to be also convex and polygonal. Let and be respectively the area, the perimeter length, the numbers of edges and vertices of .
A polygonal tessellation of is a finite subdivision of into polygonal sets (called cells) with disjoint interiors. The tessellation vertices are cell vertices. Edges are defined as line segments contained in cell sides,
A completely random -tessellation
Let us start with a formal definition of a random -tessellation. From now on, a -tessellation is considered as a closed set defined as the union of its edges (or segments). The space is equipped with the standard hitting -algebra (see Matheron, 1975) generated by events of the form where runs through the set of compact subsets of . A random -tessellation is a random variable taking values in .
Our candidate of completely random -tessellation is
Gibbsian -tessellations
Although the completely random -tessellation model introduced in the previous section shows appealing features, it may not be appropriate for representing real life structures which may exhibit some kind of regularity. This section is devoted to Gibbsian extensions allowing to control a large spectrum of -tessellation features. Gibbs random -tessellations are defined as follows. Definition 5 Let be a stable non-negative functional on . The Gibbs random -tessellation with unnormalized density is the
A Metropolis–Hastings–Green simulation algorithm
In this section, we derive a simulation algorithm for random Gibbs -tessellations. This algorithm is a special case of the ubiquitous Metropolis–Hastings–Green algorithm, see e.g. Geyer and Møller (1994), Green (1995) and Geyer (1999). It consists of designing a Markov chain with state space and with invariant distribution the target probability measure .
The design of a Metropolis–Hastings–Green algorithm involves two basic ingredients: random proposals of updates and rules for accepting or
Discussion
The main feature of the completely random -tessellation introduced in this paper is that both split and flip Papangelou kernels have very simple expressions showing a kind of lack of spatial dependency. It would be of great interest to further investigate this model. Since analytical probabilistic results are available for the Arak–Clifford–Surgailis (Example 2), it is expected that such results could also be derived for our model. In particular, the following issues are of interest:
- •
Is there a
Acknowledgements
The authors would like to thank the anonymous referees for careful reading and helpful comments. The authors also wish to thank their colleagues who showed interest in this work presented at several workshops since 2006.
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