Pseudo-likelihood estimation for non hereditary Gibbs point - PowerPoint PPT Presentation

Introduction Definitions Campbell Estimation Simulations Pseudo-likelihood estimation for non hereditary Gibbs point processes Frédéric Lavancier , Laboratoire Jean Leray, Nantes, France. Joint work with David Dereudre , LAMAV, Valenciennes, France. 7th World Congress in Probability and Statistics Singapore, July 14-19 2008

Introduction Definitions Campbell Estimation Simulations 1 Introduction

Introduction Definitions Campbell Estimation Simulations Introduction Setting Pseudo-likelihood estimation for Gibbs point processes. In the hereditary case : Besag (1975), Jensen and Moller (1991), Jensen and Kunsch (1994), Mase (1995), Billiot, Coeurjolly and Drouilhet (2008) Our aim : generalization to the non hereditary case. Motivation : non hereditary hardcore processes

Introduction Definitions Campbell Estimation Simulations Introduction Setting Pseudo-likelihood estimation for Gibbs point processes. In the hereditary case : Besag (1975), Jensen and Moller (1991), Jensen and Kunsch (1994), Mase (1995), Billiot, Coeurjolly and Drouilhet (2008) Our aim : generalization to the non hereditary case. Motivation : non hereditary hardcore processes Our work Characteristics of the non-hereditary interactions. A new equilibrium Campbell equation. Consistency of the Pseudo-likelihood estimator. Some simulations.

Introduction Definitions Campbell Estimation Simulations Gibbs measure and hereditary interactions 2

Introduction Definitions Campbell Estimation Simulations Notations γ denotes a point configuration on R d (i.e. an integer-valued measure) δ x denotes the Dirac measure at x . For Λ a subset in R d , we note γ Λ the projection of γ on Λ : � γ Λ = δ x . x ∈ γ ∩ Λ M ( R d ) = { γ } π is the Poisson process on R d . π Λ is the Poisson process on Λ . λ is the Lebesgue measure on R d .

Introduction Definitions Campbell Estimation Simulations Gibbs measures ( H Λ ) Λ denotes a general family of energy functions : H Λ : ( γ Λ , γ Λ c ) �− → H Λ ( γ Λ | γ Λ c ) There are some minimal conditions on ( H Λ ) Λ . Definition A probability measure µ is a Gibbs measure if for every bounded Λ and for µ almost every γ µ ( dγ Λ | γ Λ c ) ∝ e − H Λ ( γ Λ | γ Λ c ) π Λ ( dγ Λ ) .

Introduction Definitions Campbell Estimation Simulations Gibbs measures ( H Λ ) Λ denotes a general family of energy functions : H Λ : ( γ Λ , γ Λ c ) �− → H Λ ( γ Λ | γ Λ c ) There are some minimal conditions on ( H Λ ) Λ . Definition A probability measure µ is a Gibbs measure if for every bounded Λ and for µ almost every γ µ ( dγ Λ | γ Λ c ) ∝ e − H Λ ( γ Λ | γ Λ c ) π Λ ( dγ Λ ) . If H Λ ( γ ) = + ∞ then γ is forbidden µ a.s.

Introduction Definitions Campbell Estimation Simulations Hereditary Definition The family of energies ( H Λ ) Λ is said hereditary if for every Λ , every γ ∈ M ( R d ) and every x ∈ Λ H Λ ( γ ) = + ∞ ⇒ H Λ ( γ + δ x ) = + ∞ .

Introduction Definitions Campbell Estimation Simulations Hereditary Definition The family of energies ( H Λ ) Λ is said hereditary if for every Λ , every γ ∈ M ( R d ) and every x ∈ Λ H Λ ( γ ) = + ∞ ⇒ H Λ ( γ + δ x ) = + ∞ . γ is forbidden ⇒ γ + δ x is forbidden

Introduction Definitions Campbell Estimation Simulations Hereditary Definition The family of energies ( H Λ ) Λ is said hereditary if for every Λ , every γ ∈ M ( R d ) and every x ∈ Λ H Λ ( γ ) = + ∞ ⇒ H Λ ( γ + δ x ) = + ∞ . γ is forbidden ⇒ γ + δ x is forbidden γ + δ x is allowed ⇒ γ is allowed

Introduction Definitions Campbell Estimation Simulations Hereditary Definition The family of energies ( H Λ ) Λ is said hereditary if for every Λ , every γ ∈ M ( R d ) and every x ∈ Λ H Λ ( γ ) = + ∞ ⇒ H Λ ( γ + δ x ) = + ∞ . γ is forbidden ⇒ γ + δ x is forbidden γ + δ x is allowed ⇒ γ is allowed It is a standard assumption in classical statistical mechanics. Example : The classical hard ball model is hereditary.

Introduction Definitions Campbell Estimation Simulations Non-hereditary We are interested in the non hereditary case .

Introduction Definitions Campbell Estimation Simulations Non-hereditary We are interested in the non hereditary case . Examples : - If the interaction imposes clusters. H Λ ( γ ) = + ∞ H Λ ( γ + δ x ) < + ∞

Introduction Definitions Campbell Estimation Simulations Non-hereditary We are interested in the non hereditary case . Examples : - If the interaction imposes clusters. H Λ ( γ ) = + ∞ H Λ ( γ + δ x ) < + ∞ - In Dereudre (2007) , the author studies random Gibbs Voronoi tesselations with geometric hardcore interactions.

Introduction Definitions Campbell Estimation Simulations Gibbs Voronoi Tessellations.

Introduction Definitions Campbell Estimation Simulations Gibbs Voronoi Tessellations.

Introduction Definitions Campbell Estimation Simulations Gibbs Voronoi Tessellations. � H Λ ( γ ) = V ( ver ( x 1 , x 2 )) , { ver ( x 1 ,x 2 ) , ( x 1 ,x 2 ) ∈ Voronoi ( γ ) }

Introduction Definitions Campbell Estimation Simulations Gibbs Voronoi Tessellations. � H Λ ( γ ) = V ( ver ( x 1 , x 2 )) , { ver ( x 1 ,x 2 ) , ( x 1 ,x 2 ) ∈ Voronoi ( γ ) } where for every vertice ver ( x 1 , x 2 ) , � + ∞ if || x 1 − x 2 || > α, V ( ver ( x 1 , x 2 )) = < + ∞ otherwise.

Introduction Definitions Campbell Estimation Simulations

Introduction Definitions Campbell Estimation Simulations H Λ ( γ ) = + ∞ H Λ ( γ + δ x ) < + ∞

Introduction Definitions Campbell Estimation Simulations 3 Equilibrium equation

Introduction Definitions Campbell Estimation Simulations Nguyen-Zessin equilibrium equation Definition Let µ be a probability measure on M ( R d ) . The reduced Campbell µ is defined for all test function f from R d × M ( R d ) measure C ! into R by �� � C ! µ ( f ) = E µ f ( x, γ − δ x ) . x ∈ γ

Introduction Definitions Campbell Estimation Simulations Nguyen-Zessin equilibrium equation Definition Let µ be a probability measure on M ( R d ) . The reduced Campbell µ is defined for all test function f from R d × M ( R d ) measure C ! into R by �� � C ! µ ( f ) = E µ f ( x, γ − δ x ) . x ∈ γ Theorem ( Nguyen-Zessin (1979)) Suppose that the energy ( H Λ ) Λ is hereditary. µ is a Gibbs measure if and only if C ! µ ( dx, dγ ) = e − h ( x,γ ) λ ⊗ µ ( dx, dγ ) . where h ( x, γ ) = H Λ ( γ + δ x ) − H Λ ( γ ) . This theorem is not true in the non-hereditary case.

Introduction Definitions Campbell Estimation Simulations Removable points Definition Let γ be in M ( R d ) and x be a point of γ . x is said removable from γ if ∃ Λ such that x ∈ Λ and H Λ ( γ − δ x ) < + ∞ . We note R ( γ ) the set of removable points in γ .

Introduction Definitions Campbell Estimation Simulations Removable points Definition Let γ be in M ( R d ) and x be a point of γ . x is said removable from γ if ∃ Λ such that x ∈ Λ and H Λ ( γ − δ x ) < + ∞ . We note R ( γ ) the set of removable points in γ . Definition Let x in R ( γ ) . We define the energy of x in γ − δ x with the following expression h ( x, γ − δ x ) = H Λ ( γ ) − H Λ ( γ − δ x ) ,

Introduction Definitions Campbell Estimation Simulations Equilibrium equations for non-hereditary Gibbs measures Theorem ( Dereudre-Lavancier (2007)) Let µ be a Gibbs measure, I x ∈R ( γ + δ x ) C ! µ ( dx, dγ ) = e − h ( x,γ ) λ ⊗ µ ( dx, dγ ) . 1 (1)

Introduction Definitions Campbell Estimation Simulations Equilibrium equations for non-hereditary Gibbs measures Theorem ( Dereudre-Lavancier (2007)) Let µ be a Gibbs measure, I x ∈R ( γ + δ x ) C ! µ ( dx, dγ ) = e − h ( x,γ ) λ ⊗ µ ( dx, dγ ) . 1 (1) Remark - If ( H Λ ) Λ is hereditary, x is always in R ( γ + δ x ) . So, (1) becomes equivalent to the Nguyen-Zessin’s equilibrium equation. - The equation (1) does not characterize the Gibbs measures.

Introduction Definitions Campbell Estimation Simulations 4 Pseudo-likelihood estimation

Introduction Definitions Campbell Estimation Simulations The pseudo likelihood contrast function Let Θ be a bounded open set in R p . - θ in Θ : the smooth parameter of the energy. - α in R + : the hardcore support parameter. - ( H α,θ Λ ) Λ : the parametric family of energies. - For x in R ( γ ) , h α,θ ( x, γ − δ x ) = H α,θ Λ ( γ ) − H α,θ Λ ( γ − δ x ) . Let Λ n the observation window of γ (e. g. Λ n = [ − n, n ] d ). Definition We define the pseudo likelihood contrast function PLL Λ n ( γ, α, θ ) =   1 � � � � − h α,θ ( x, γ ) h α,θ ( x, γ − δ x )  . exp dx +  Λ n Λ n x ∈R α,θ ( γ ) ∩ Λ n

Introduction Definitions Campbell Estimation Simulations Estimation of both α and θ Let µ be a stationary Gibbs measure for the parameters α ∗ , θ ∗ . α ∗ and θ ∗ have to be estimated.