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Introduction Definition Simulation Parametric models Inference Statistical aspects of determinantal point processes Fr ed eric Lavancier , Laboratoire de Math ematiques Jean Leray, Nantes (France) Joint work with Jesper Mller

  1. Introduction Definition Simulation Parametric models Inference Statistical aspects of determinantal point processes Fr´ ed´ eric Lavancier , Laboratoire de Math´ ematiques Jean Leray, Nantes (France) Joint work with Jesper Møller (Aalborg University, Danemark) and Ege Rubak (Aalborg University, Danemark).

  2. Introduction Definition Simulation Parametric models Inference 1 Introduction 2 Definition, existence and basic properties 3 Simulation 4 Parametric stationary models Parametric families of stationary DPPs Approximation of the eigen expansion Modelling of the eigen expansion 5 Inference

  3. Introduction Definition Simulation Parametric models Inference Type of data ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● Hamster cells Anemones Norwegian pines nuclei (dividing)

  4. Introduction Definition Simulation Parametric models Inference Examples of models realisations Realisations of Poisson point process versus Determinantal point process (DPP) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● DPP with Poisson DPP stronger repulsion

  5. Introduction Definition Simulation Parametric models Inference Introduction � Determinantal point processes (DPP) form a class of repulsive point processes.

  6. Introduction Definition Simulation Parametric models Inference Introduction � Determinantal point processes (DPP) form a class of repulsive point processes. � They were introduced in their general form by O. Macchi in 1975 to model fermions (i.e. particles with repulsion) in quantum mechanics.

  7. Introduction Definition Simulation Parametric models Inference Introduction � Determinantal point processes (DPP) form a class of repulsive point processes. � They were introduced in their general form by O. Macchi in 1975 to model fermions (i.e. particles with repulsion) in quantum mechanics. � Particular cases include the law of the eigenvalues of certain random matrices (Gaussian Unitary Ensemble, Ginibre Ensemble,...)

  8. Introduction Definition Simulation Parametric models Inference Introduction � Determinantal point processes (DPP) form a class of repulsive point processes. � They were introduced in their general form by O. Macchi in 1975 to model fermions (i.e. particles with repulsion) in quantum mechanics. � Particular cases include the law of the eigenvalues of certain random matrices (Gaussian Unitary Ensemble, Ginibre Ensemble,...) � Most theoretical studies have been published in the 2000’s.

  9. Introduction Definition Simulation Parametric models Inference Introduction � Determinantal point processes (DPP) form a class of repulsive point processes. � They were introduced in their general form by O. Macchi in 1975 to model fermions (i.e. particles with repulsion) in quantum mechanics. � Particular cases include the law of the eigenvalues of certain random matrices (Gaussian Unitary Ensemble, Ginibre Ensemble,...) � Most theoretical studies have been published in the 2000’s. � The statistical aspects have so far been largely unexplored.

  10. Introduction Definition Simulation Parametric models Inference Statistical motivation Do DPP’s constitute a tractable and flexible class of models for repulsive point processes?

  11. Introduction Definition Simulation Parametric models Inference Statistical motivation Do DPP’s constitute a tractable and flexible class of models for repulsive point processes? − → Answer: YES .

  12. Introduction Definition Simulation Parametric models Inference Statistical motivation Do DPP’s constitute a tractable and flexible class of models for repulsive point processes? − → Answer: YES . In fact: � DPP’s can be easily and quickly simulated. � There are closed form expressions for the moments. � There is a closed form expression for the density of a DPP on any bounded set. � Inference is feasible, including likelihood inference.

  13. Introduction Definition Simulation Parametric models Inference Statistical motivation Do DPP’s constitute a tractable and flexible class of models for repulsive point processes? − → Answer: YES . In fact: � DPP’s can be easily and quickly simulated. � There are closed form expressions for the moments. � There is a closed form expression for the density of a DPP on any bounded set. � Inference is feasible, including likelihood inference. These properties are unusual for Gibbs point processes which are commonly used to model inhibition (e.g. Strauss process).

  14. Introduction Definition Simulation Parametric models Inference 1 Introduction 2 Definition, existence and basic properties 3 Simulation 4 Parametric stationary models Parametric families of stationary DPPs Approximation of the eigen expansion Modelling of the eigen expansion 5 Inference

  15. Introduction Definition Simulation Parametric models Inference Notation � We view a spatial point process X on R d as a random locally finite subset of R d .

  16. Introduction Definition Simulation Parametric models Inference Notation � We view a spatial point process X on R d as a random locally finite subset of R d . � For any borel set B ⊆ R d , X B = X ∩ B .

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