Determinantal point process models and statistical inference Fr ed - PowerPoint PPT Presentation

Introduction Definition Parametric 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).

Introduction Definition Parametric 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 Conclusion and references

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

Introduction Definition Parametric models Approximation Inference Conclusion Introduction 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?

Introduction Definition Parametric models Approximation Inference Conclusion Background � DPPs were introduced in their general form by O. Macchi in 1975 to model fermions 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. � Statistical models and inference have so far been largely unexplored.

Introduction Definition Parametric 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 Conclusion and references

Introduction Definition Parametric models Approximation Inference Conclusion Notation � X : spatial point process on R d

Introduction Definition Parametric models Approximation Inference Conclusion Notation � X : spatial point process on R d � For any borel set B ⊆ R d , X B = X ∩ B .

Introduction Definition Parametric models Approximation Inference Conclusion Notation � X : spatial point process on R d � For any borel set B ⊆ R d , X B = X ∩ B . � For any integer n > 0, denote ρ ( n ) the n ’th order product density function of X . Intuitively, ρ ( n ) ( x 1 , . . . , x n ) d x 1 · · · d x n is the probability that for each i = 1 , . . . , n , X has a point in a region around x i of volume d x i .

Introduction Definition Parametric models Approximation Inference Conclusion Notation � X : spatial point process on R d � For any borel set B ⊆ R d , X B = X ∩ B . � For any integer n > 0, denote ρ ( n ) the n ’th order product density function of X . Intuitively, ρ ( n ) ( x 1 , . . . , x n ) d x 1 · · · d x n is the probability that for each i = 1 , . . . , n , X has a point in a region around x i of volume d x i . In particular ρ = ρ (1) is the intensity function .

Introduction Definition Parametric models Approximation Inference Conclusion Definition of a determinantal point process Let C be a function from R d × R d → C . Denote [ C ]( x 1 , . . . , x n ) the n × n matrix with entries C ( x i , x j ). � C ( x 1 , x 1 ) � C ( x 1 , x 2 ) Ex : [ C ]( x 1 ) = C ( x 1 , x 1 ) [ C ]( x 1 , x 2 ) = . C ( x 2 , x 1 ) C ( x 2 , x 2 )

Introduction Definition Parametric models Approximation Inference Conclusion Definition of a determinantal point process Let C be a function from R d × R d → C . Denote [ C ]( x 1 , . . . , x n ) the n × n matrix with entries C ( x i , x j ). � C ( x 1 , x 1 ) � C ( x 1 , x 2 ) Ex : [ C ]( x 1 ) = C ( x 1 , x 1 ) [ C ]( x 1 , x 2 ) = . C ( x 2 , x 1 ) C ( x 2 , x 2 ) Definition X is a determinantal point process with kernel C , denoted X ∼ DPP( C ), if its product density functions satisfy ρ ( n ) ( x 1 , . . . , x n ) = det[ C ]( x 1 , . . . , x n ) , n = 1 , 2 , . . . Some conditions on C are necessary for existence (see later), e.g. C must satisfy : for all x 1 , . . . , x n , det[ C ]( x 1 , . . . , x n ) ≥ 0.

Introduction Definition Parametric models Approximation Inference Conclusion First properties (if X ∼ DPP ( C ) exists)

Introduction Definition Parametric models Approximation Inference Conclusion First properties (if X ∼ DPP ( C ) exists) � From the definition, if C is continuous, ρ ( n ) ( x 1 , . . . , x n ) ≈ 0 whenever x i ≈ x j for some i � = j, = ⇒ the points of X repel each other .

Introduction Definition Parametric models Approximation Inference Conclusion First properties (if X ∼ DPP ( C ) exists) � From the definition, if C is continuous, ρ ( n ) ( x 1 , . . . , x n ) ≈ 0 whenever x i ≈ x j for some i � = j, = ⇒ the points of X repel each other . � The intensity of X is ρ ( x ) = C ( x, x ).

Introduction Definition Parametric models Approximation Inference Conclusion First properties (if X ∼ DPP ( C ) exists) � From the definition, if C is continuous, ρ ( n ) ( x 1 , . . . , x n ) ≈ 0 whenever x i ≈ x j for some i � = j, = ⇒ the points of X repel each other . � The intensity of X is ρ ( x ) = C ( x, x ). � The pair correlation function is g ( x, y ) := ρ (2) ( x, y ) C ( x, x ) C ( y, y ) = 1 − C ( x, y ) C ( y, x ) det[ C ]( x, y ) ρ ( x ) ρ ( y ) = C ( x, x ) C ( y, y ) � Thus g ≤ 1 (i.e. repulsiveness) if C is Hermitian.

Introduction Definition Parametric models Approximation Inference Conclusion First properties (if X ∼ DPP ( C ) exists) � From the definition, if C is continuous, ρ ( n ) ( x 1 , . . . , x n ) ≈ 0 whenever x i ≈ x j for some i � = j, = ⇒ the points of X repel each other . � The intensity of X is ρ ( x ) = C ( x, x ). � The pair correlation function is g ( x, y ) := ρ (2) ( x, y ) C ( x, x ) C ( y, y ) = 1 − C ( x, y ) C ( y, x ) det[ C ]( x, y ) ρ ( x ) ρ ( y ) = C ( x, x ) C ( y, y ) � Thus g ≤ 1 (i.e. repulsiveness) if C is Hermitian. � If X ∼ DPP( C ), then X B ∼ DPP( C B )