David Dereudre

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.

Practical simulation and estimation for Gibbs Delaunay-Voronoi tessellations with geometric hardcore interaction

General models of Gibbs Delaunay-Voronoi tessellations, which can be viewed as extensions of Ords process, are considered. The interaction may occur on each cell of the tessellation and between neighbour cells. The tessellation may also be subjected to a geometric hardcore interaction, forcing the cells not to be too large, too small, or too flat. This setting, natural for applications, introduces some theoretical difficulties since the interaction is not necessarily hereditary. Mathematical results available for studying these models are reviewed and further outcomes are provided. They concern the existence, the simulation and the estimation of such tessellations. Based on these results, tools to handle these objects in practice are presented: how to simulate them, estimate their parameters and validate the fitted model. Some examples of simulated tessellations are studied in detail.

Campbell equilibrium equation and pseudo-likelihood estimation for non-hereditary Gibbs point processes

In this paper, we study Gibbs point processes involving a hardcore interaction which is not necessarily hereditary. We first extend the famous Campbell equilibrium equation, initially proposed by Nguyen and Zessin [Math. Nachr. 88 (1979) 105-115], to the non-hereditary setting and consequently introduce the new concept of removable points. A modified version of the pseudo-likelihood estimator is then proposed, which involves these removable points. We consider the following two-step estimation procedure: first estimate the hardcore parameter, then estimate the smooth interaction parameter by pseudo-likelihood, where the hard core parameter estimator is plugged in. We prove the consistency of this procedure in both the hereditary and non-hereditary settings.

Variational estimators for the parameters of Gibbs point process models

This paper proposes a new estimation technique for fitting parametric Gibbs point process models to a spatial point pattern dataset. The technique is a counterpart, for spatial point processes, of the variational estimators for Markov random fields developed by Almeida and Gidas. The estimator does not require the point process density to be hereditary, so it is applicable to models which do not have a conditional intensity, including models which exhibit geometric regularity or rigidity. The disadvantage is that the intensity parameter cannot be estimated: inference is effectively conditional on the observed number of points. The new procedure is faster and more stable than existing techniques, since it does not require simulation, numerical integration or optimization with respect to the parameters.

Variational principle for Gibbs point processes with finite range interaction

The variational principle for Gibbs point processes with general finite range interaction is proved. Namely, the Gibbs point processes are identified as the minimizers of the free excess energy equals to the sum of the specific entropy and the mean energy. The interaction is very general and includes superstable pairwise potential, finite or infinite multibody potential, geometrical interaction, hardcore interaction. The only restrictive assumption involves the finite range property.

Consistency of likelihood estimation for Gibbs point processes

Strong consistency of the maximum likelihood estimator (MLE) for parametric Gibbs point process models is established. The setting is very general. It includes pairwise pair potentials, finite and infinite multibody interactions and geometrical interactions, where the range can be finite or infinite. The Gibbs interaction may depend linearly or non-linearly on the parameters, a particular case being hardcore parameters and interaction range parameters. As important examples, we deduce the consistency of the MLE for all parameters of the Strauss model, the hardcore Strauss model, the Lennard-Jones model and the area-interaction model.

An explicit representation of the transition densities of the skew Brownian motion with drift and two semipermeable barriers

Abstract In this paper, we obtain an explicit representation of the transition density of the one-dimensional skew Brownian motion with (a constant drift and) two semipermeable barriers. Moreover, we propose a rejection sampling method to simulate this density in an exact way.

Exact simulation of Brownian diffusions with drift admitting jumps

In this paper, using an algorithm based on the retrospective rejection sampling scheme, we propose an exact simulation of a Brownian diffusion whose drift admits several jumps. We treat explicitly and extensively the case of two jumps, providing numerical simulations. Our main contribution is to manage the technical diffculty due to the presence of two jumps thanks to a new explicit expression of the transition density of the skew Brownian motion with two semipermeable barriers and a constant drift.

Phase transition for continuum Widom-Rowlinson model with random radii

In this paper we study the phase transition of continuum Widom–Rowlinson measures in