Rémy Drouilhet

Maximum pseudolikelihood estimator for exponential family models of marked Gibbs point processes

This paper is devoted to the estimation of a vector θ parametrizing an energy function of a Gibbs point process, via the maximum pseudolikelihood method. Strong consistency and asymptotic normality results of this estimator depending on a single realization are presented. In the framework of exponential family models, sufficient conditions are expressed in terms of the local energy function and are verified on a wide variety of examples.

Continuum percolation in the Gabriel graph

In the present study, we establish the existence of site percolation in the Gabriel graph for Poisson and hard-core stationary point processes.

Takacs-Fiksel method for stationary marked Gibbs point processes

This paper studies a method to estimate the parameters governing the distribution of a stationary marked Gibbs point process. This method, known as the Takacs-Fiksel method, is based on the estimation of the left and right hand sides of the Georgii-Nguyen-Zessin formula and leads to a family of estimators due to the possible choices of test functions. We propose several examples illustrating the interest and flexibility of this procedure. We also provide sufficient conditions based on the model and the test functions to derive asymptotic properties (consistency and asymptotic normality) of the resulting estimator. The different assumptions are discussed for exponential family models and for a large class of test functions.

Spatial delaunay gibbs point processes

This paper studies two types of Delaunay Gibbs point processes originally introduced by Baddeley and M⊘ller (1989) by combining stochastic geometry and computational geometry arguments. The energy function of these processes is for a given point pattern φ given by a sum of potentials associated to a subclass of the cliques, called “Delaunay-cliques”, given by the Delaunay graph correponding to the point pattern (empty set, singletons, Delaunay edges, and Delaunay triangles). This restriction is necessary in order that the Delaunay point process becomes Markov in the sense of Baddeley and M⊘ller (1989). We demonstrate that the Markov property is satisfied when interactions are only permitted for “Delaunay–cliques” and to further establish a Hammersley-Clifford type theorem for the Delaunay Gibbs point processes. Furthermore local stability is studied and simulations are given

Existence of Delaunay Pairwise Gibbs Point Process with Superstable Component

The present stuffy deals with the existence of Delaunay pairwise Gibbs point process with superstable component by using the well-known Preston theorem. In particular, we prove the stability, the lower regularity, and the quasilocality properties of the Delaunay model.

Normalized information-based divergences

This paper is devoted to the mathematical study of some divergences based on mutual information which are well suited to categorical random vectors. These divergences are generalizations of the “entropy distance” and “information distance.” Their main characteristic is that they combine a complexity term and the mutual information. We then introduce the notion of (normalized) information-based divergence, propose several examples, and discuss their mathematical properties, in particular, in some prediction framework.

Asymptotic properties of the maximum pseudo-likelihood estimator for stationary Gibbs point processes including the Lennard-Jones model

This paper presents asymptotic properties of the maximum pseudo-likelihood estimator of a vector

R-Local Delaunay Inhibition Model

Vect{theta}

Existence of Gibbsian point processes with geometry-dependent interactions

parameterizing a stationary Gibbs point process. Sufficient conditions, expressed in terms of the local energy function defining a Gibbs point process, to establish strong consistency and asymptotic normality results of this estimator depending on a single realization, are presented.These results are general enough to no longer require the local stability and the linearity in terms of the parameters of the local energy function. We consider characteristic examples of such models, the Lennard-Jones and the finite range Lennard-Jones models. We show that the different assumptions ensuring the consistency are satisfied for both models whereas the assumptions ensuring the asymptotic normality are fulfilled only for the finite range Lennard-Jones model.

EXISTENCE OF 'NEAREST-NEIGHBOUR' SPATIAL GIBBS MODELS

Unlike in the classical framework of Gibbs point processes (usually acting on the complete graph), in the context of nearest-neighbour Gibbs point processes the nonnegativeness of the interaction functions do not ensure the local stability property. This paper introduces domain-wise (but not pointwise) inhibition stationary Gibbs models based on some tailor-made Delaunay subgraphs. All of them are subgraphs of the R-local Delaunay graph, defined as the Delaunay subgraph specifically not containing the edges of Delaunay triangles with circumscribed circles of radii greater than some large positive real value R. The usual relative compactness criterion for point processes needed for the existence result is directly derived from the Ruelle-bound of the correlation functions. Furthermore, assuming only the nonnegativeness of the energy function, we have managed to prove the existence of the existence of R-local Delaunay stationary Gibbs states based on nonnegative interaction functions thanks to the use of the compactness of sublevel sets of the relative entropy.