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Introduction Definition Stationary models Approximation Inference Conclusion Determinantal point process models and statistical inference Fr ed eric Lavancier , Laboratoire de Math ematiques Jean Leray, Nantes (France) Joint work

  1. Introduction Definition Stationary models Approximation Inference Conclusion Determinantal point process models and statistical inference 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). Some recent results with Christophe Biscio (Nantes).

  2. Introduction Definition Stationary models Approximation Inference Conclusion 1 Introduction 2 Definition, existence and basic properties 3 Parametric stationary models 4 Approximation and modelling of the eigen expansion 5 Inference 6 Examples of real data 7 Conclusion and references

  3. Introduction Definition Stationary models Approximation Inference Conclusion Examples of point pattern datasets ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● (a) Spanish towns (b) Kidney cells of (c) Japanese pines two types (hamster)

  4. Introduction Definition Stationary models Approximation Inference Conclusion First look at DPPs Determinantal point processes (DPPs): a class of repulsive (or regular, or inhibitive) point processes. Some examples : ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● DPP with Poisson DPP stronger repulsion

  5. Introduction Definition Stationary models Approximation Inference Conclusion First look at DPPs Determinantal point processes (DPPs): a class of repulsive (or regular, or inhibitive) point processes. Some examples : ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● DPP with Poisson DPP stronger repulsion Statistical motivation : Do DPPs constitute a tractable and flexible class of models for repulsive point processes?

  6. Introduction Definition Stationary models Approximation Inference Conclusion Gibbs point processes vs DPPs Gibbs point processes : The usual class when modelling repulsiveness (e.g. Strauss model, Area interaction model).

  7. Introduction Definition Stationary models Approximation Inference Conclusion Gibbs point processes vs DPPs Gibbs point processes : The usual class when modelling repulsiveness (e.g. Strauss model, Area interaction model). In general: � moments are not expressible in closed form; � likelihoods involve intractable normalizing constants; � elaborate McMC methods are needed for simulations and approximate likelihood inference; � for infinite Gibbs point processes defined on R d , ‘things’ become rather complicated (existence and uniqueness)

  8. Introduction Definition Stationary models Approximation Inference Conclusion Gibbs point processes vs DPPs DPPs possess a number of appealing properties:

  9. Introduction Definition Stationary models Approximation Inference Conclusion Gibbs point processes vs DPPs DPPs possess a number of appealing properties: (a) simple conditions for existence;

  10. Introduction Definition Stationary models Approximation Inference Conclusion Gibbs point processes vs DPPs DPPs possess a number of appealing properties: (a) simple conditions for existence; (b) all orders of moments are known;

  11. Introduction Definition Stationary models Approximation Inference Conclusion Gibbs point processes vs DPPs DPPs possess a number of appealing properties: (a) simple conditions for existence; (b) all orders of moments are known; (c) the density of the DPP restricted to any compact set S ⊂ R d is expressible on closed form;

  12. Introduction Definition Stationary models Approximation Inference Conclusion Gibbs point processes vs DPPs DPPs possess a number of appealing properties: (a) simple conditions for existence; (b) all orders of moments are known; (c) the density of the DPP restricted to any compact set S ⊂ R d is expressible on closed form; (d) the DPP on any compact set can easily be simulated;

  13. Introduction Definition Stationary models Approximation Inference Conclusion Gibbs point processes vs DPPs DPPs possess a number of appealing properties: (a) simple conditions for existence; (b) all orders of moments are known; (c) the density of the DPP restricted to any compact set S ⊂ R d is expressible on closed form; (d) the DPP on any compact set can easily be simulated; (e) parametric models are available, and inference can be done by MLEs or using the moments.

  14. Introduction Definition Stationary models Approximation Inference Conclusion Background

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