Workshop - Spatial Statistics and Image Analysis in Biology

Wednesday, May 25, 2016 - 16:05 to 16:30
Jean-François Coeurjolly
Univ. Grenoble, France
Lennard-Jones potential estimation

Among the models of spatial point processes, the class of pairwise interaction point processes is probably the most well-known due to its simple mechanistic interpretation. And among them, the Lennard-Jones model plays a historical and central role. Introduced by physicists in the early 1900's, this model, also known as the 12-6 potential, has pairwise interaction function $\Phi(r)=\frac{\theta_1}{r^{12}}-\frac{\theta_2}{r^6}$. Its main characteristics are the non-boundedness at small distances and its infinite range.

Many probabilistic properties of this process have been established by Ruelle in the 70's. However from a statistical and practical point of view, the infinite range of the process poses additional problems, which had never been considered. In collaboration with F. Lavancier, we have proposed extensions of the pseudolikelihood and logistic regression methods, designed to take into account the infinite range of this process and proved, in an increasing domain setting, that the procedures are consistent and that the corresponding estimators are asymptotically normal. The main contribution is the obtaining of a central limit theorem for triangular arrays of random fields which are almost conditionally centered. We show theoretically and practically that the general methodology and our general results can be applied to the Lennard-Jones model.

Reference: J.-F. Coeurjolly and F. Lavancier Parametric estimation of pairwise Gibbs point processes with infinite range interaction, to appear in Bernoulli, 2016.





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