Statistical aspects of determinantal point processes

English

Séminaire Probabilités & Statistique

20/09/2012 - 14:00 Mr Jesper Moller (Aalborg University) Salle 1 - Tour IRMA

Determinantal point processes are largely unexplored in statistics, though they possess a number of appealing properties and have been studied in mathematical physics, combinatorics, and random matrix theory. In this talk we consider statistical aspects of determinantal point processes defined on R^d, with a focus on d = 2.

Determinantal point processes are defined by a function C satisfying certain regularity conditions and they possess the following properties:

    (a)  Determinantal point processes are flexible models for repulsive interaction.

    (b)  All orders of moments of a determinantal point process are described by certain determinants of matrices with entries given in terms of C.

    (c)  A one-to-one smooth transformation or an independent thinning of a determinantal point process is also a determinantal point process.

    (d)  A determinantal point process can easily be simulated, since it is a mixture of ‘determinantal projection processes’.

    (e)  A determinantal point process restricted to a compact set has a density (with respect to a Poisson process) which can be expressed in closed form including the normalizing constant.

In contrast Gibbs point processes, which constitute another flexible class of models for repulsive interaction, do not in general have moments that are expressible in closed form, the density involves an intractable normalizing constant, and rather time consuming Markov chain Monte Carlo methods are needed for simulations and approximate likelihood inference.

In the talk we describe how to simulate determinantal point processes in practice and investigate how to construct parametric models. Furthermore, different inferential approaches based on both moments and the likelihood are studied.

The work has been carried out in collaboration with Ege Rubak, Aalborg University, and Frédéric Lavancier, University of Nantes.