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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

2

Gibbs measure and hereditary interactions

Introduction Definitions Campbell Estimation Simulations

Notations

γ denotes a point configuration on Rd (i.e. an integer-valued measure) δx denotes the Dirac measure at x. For Λ a subset in Rd, we note γΛ the projection of γ on Λ : γΛ =

• x∈γ∩Λ

δx. M(Rd) = { γ } π is the Poisson process on Rd. πΛ is the Poisson process on Λ. λ is the Lebesgue measure on Rd.

Introduction Definitions Campbell Estimation Simulations

Gibbs measures

(HΛ)Λ denotes a general family of energy functions : HΛ : (γΛ, γΛc) − → HΛ(γΛ|γΛc) There are some minimal conditions on (HΛ)Λ. Definition A probability measure µ is a Gibbs measure if for every bounded Λ and for µ almost every γ µ(dγΛ|γΛc) ∝ e−HΛ(γΛ|γΛc)πΛ(dγΛ).

Introduction Definitions Campbell Estimation Simulations

Gibbs measures

(HΛ)Λ denotes a general family of energy functions : HΛ : (γΛ, γΛc) − → HΛ(γΛ|γΛc) There are some minimal conditions on (HΛ)Λ. Definition A probability measure µ is a Gibbs measure if for every bounded Λ and for µ almost every γ µ(dγΛ|γΛc) ∝ e−HΛ(γΛ|γΛc)πΛ(dγΛ). If HΛ(γ) = +∞ then γ is forbidden µ a.s.

Introduction Definitions Campbell Estimation Simulations

Hereditary

Definition The family of energies (HΛ)Λ is said hereditary if for every Λ, every γ ∈ M(Rd) and every x ∈ Λ HΛ(γ) = +∞ ⇒ HΛ(γ + δx) = +∞.

Introduction Definitions Campbell Estimation Simulations

Hereditary

Definition The family of energies (HΛ)Λ is said hereditary if for every Λ, every γ ∈ M(Rd) and every x ∈ Λ HΛ(γ) = +∞ ⇒ HΛ(γ + δx) = +∞. γ is forbidden ⇒ γ + δx is forbidden

Introduction Definitions Campbell Estimation Simulations

Hereditary

Definition The family of energies (HΛ)Λ is said hereditary if for every Λ, every γ ∈ M(Rd) and every x ∈ Λ HΛ(γ) = +∞ ⇒ HΛ(γ + δx) = +∞. γ is forbidden ⇒ γ + δx is forbidden γ + δx is allowed ⇒ γ is allowed

Introduction Definitions Campbell Estimation Simulations

Hereditary

Definition The family of energies (HΛ)Λ is said hereditary if for every Λ, every γ ∈ M(Rd) and every x ∈ Λ HΛ(γ) = +∞ ⇒ HΛ(γ + δx) = +∞. γ is forbidden ⇒ γ + δx is forbidden γ + δx 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.

HΛ(γ) = +∞ HΛ(γ + δx) < +∞

Introduction Definitions Campbell Estimation Simulations

Non-hereditary

We are interested in the non hereditary case. Examples :

• If the interaction imposes clusters.

HΛ(γ) = +∞ HΛ(γ + δx) < +∞

• 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.

HΛ(γ) =

• {ver(x1,x2),

(x1,x2)∈ Voronoi(γ)}

V (ver(x1, x2)),

Introduction Definitions Campbell Estimation Simulations

Gibbs Voronoi Tessellations.

HΛ(γ) =

• {ver(x1,x2),

(x1,x2)∈ Voronoi(γ)}

V (ver(x1, x2)), where for every vertice ver(x1, x2), V (ver(x1, x2)) =

• +∞

if ||x1 − x2|| > α, < +∞

• therwise.

Introduction Definitions Campbell Estimation Simulations

Introduction Definitions Campbell Estimation Simulations

HΛ(γ) = +∞ HΛ(γ + δx) < +∞

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(Rd). The reduced Campbell measure C!

µ is defined for all test function f from Rd × M(Rd)

into R by C!

µ(f) = Eµ

• x∈γ

f(x, γ − δx)

• .

Introduction Definitions Campbell Estimation Simulations

Nguyen-Zessin equilibrium equation

Definition Let µ be a probability measure on M(Rd). The reduced Campbell measure C!

µ is defined for all test function f from Rd × M(Rd)

into R by C!

µ(f) = Eµ

• x∈γ

f(x, γ − δx)

• .

Theorem (Nguyen-Zessin (1979)) Suppose that the energy (HΛ)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) = HΛ(γ + δx) − HΛ(γ). This theorem is not true in the non-hereditary case.

Introduction Definitions Campbell Estimation Simulations

Removable points

Definition Let γ be in M(Rd) and x be a point of γ. x is said removable from γ if ∃Λ such that x ∈ Λ and HΛ(γ − δx) < +∞. We note R(γ) the set of removable points in γ.

Introduction Definitions Campbell Estimation Simulations

Removable points

Definition Let γ be in M(Rd) and x be a point of γ. x is said removable from γ if ∃Λ such that x ∈ Λ and HΛ(γ − δx) < +∞. 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 − δx with the following expression h(x, γ − δx) = HΛ(γ) − HΛ(γ − δx),

Introduction Definitions Campbell Estimation Simulations

Equilibrium equations for non-hereditary Gibbs measures

Theorem (Dereudre-Lavancier (2007)) Let µ be a Gibbs measure, 1 Ix∈R(γ+δx)C!

µ(dx, dγ) = e−h(x,γ)λ ⊗ µ(dx, dγ).

(1)

Introduction Definitions Campbell Estimation Simulations

Equilibrium equations for non-hereditary Gibbs measures

Theorem (Dereudre-Lavancier (2007)) Let µ be a Gibbs measure, 1 Ix∈R(γ+δx)C!

µ(dx, dγ) = e−h(x,γ)λ ⊗ µ(dx, dγ).

(1) Remark

• If (HΛ)Λ is hereditary, x is always in R(γ + δx).

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 Rp.

• θ in Θ : the smooth parameter of the energy.
• α in R+ : the hardcore support parameter.
• (Hα,θ

Λ )Λ : the parametric family of energies.

• For x in R(γ), hα,θ(x, γ − δx) = Hα,θ

Λ (γ) − Hα,θ Λ (γ − δx).

Let Λn the observation window of γ (e. g. Λn = [−n, n]d). Definition We define the pseudo likelihood contrast function PLLΛn(γ, α, θ) = 1 Λn  

• Λn

exp

• −hα,θ(x, γ)
• dx +
• x∈Rα,θ(γ)∩Λn

hα,θ(x, γ − δx)   .

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.

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. Definition We define for µ almost every γ ˆ αn(γ) = inf

• α > 0, Hα,θ

Λn (γ) < ∞

• .

ˆ θn(γ) = argminθ∈ΘPLLΛn(γ, ˆ αn(γ), θ).

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. Definition We define for µ almost every γ ˆ αn(γ) = inf

• α > 0, Hα,θ

Λn (γ) < ∞

• .

ˆ θn(γ) = argminθ∈ΘPLLΛn(γ, ˆ αn(γ), θ). Theorem (Dereudre-Lavancier (2007)) For µ almost every γ lim

n→∞

• ˆ

αn(γ), ˆ θn(γ)

• = (α∗, θ∗)

Introduction Definitions Campbell Estimation Simulations

5 Simulations

Introduction Definitions Campbell Estimation Simulations

Gibbs Voronoi Tessellations.

Hα,θ

Λ (γ) =

• {ver(x1,x2),

(x1,x2)∈ Voronoi(γ)}

V α,θ(ver(x1, x2)),

Introduction Definitions Campbell Estimation Simulations

Gibbs Voronoi Tessellations.

Hα,θ

Λ (γ) =

• {ver(x1,x2),

(x1,x2)∈ Voronoi(γ)}

V α,θ(ver(x1, x2)), where for every vertice ver(x1, x2), V α,θ(ver(x1, x2)) =

• +∞

if ||x1 − x2|| > α θ

• max(V1,V2)

min(V1,V2) − 1

• therwise,

with Vj the volume of cell(xj).

Introduction Definitions Campbell Estimation Simulations

α = 0.12, θ = 0.5 α = 0.12, θ = −0.5

Introduction Definitions Campbell Estimation Simulations

α = 0.12, θ = 0.5 α = 0.12, θ = −0.5 6/164 removable points 456/634 removable points

Introduction Definitions Campbell Estimation Simulations

α = 0.12, θ = 0.5 α = 0.12, θ = −0.5 6/164 removable points 456/634 removable points ˆ α = 0.119, ˆ θ = 0.6 ˆ α = 0.119, ˆ θ = −0.49

Introduction Definitions Campbell Estimation Simulations

Repartition of ˆ αn and ˆ θn on 200 replicates

α = 0.12, θ = 0.5 sd(ˆ αn) = 1.7 10−4 sd(ˆ θn) = 0.102 α = 0.12, θ = −0.5 sd(ˆ αn) = 2.3 10−4 sd(ˆ θn) = 0.016

Asymptotic normality of ˆ θn ? − → If α is known : ok. − → Otherwise... ?

Introduction Definitions Campbell Estimation Simulations

• E. Bertin, J.M. Billiot, R. Drouilhet, (1999)Existence of nearest-

neighbours spatial Gibbs models , Adv. Appl. Prob. (SGSA) 31, 895-909.

• J. Besag , (1975). Statistical analysis of non-lattice data, The statistician,

24 192-236. J.-M. Billiot, , J.-F. Coeurjolly, and R. Drouilhet, (2008) Maximum pseudolikelihood estimator for exponential family models of marked Gibbs point processes, Electronic Journal of Statistics.

• D. Dereudre , (2007) Gibbs Delaunay tessellations with geometric hardcore

conditions, to appear in J.S.P.

• D. Dereudre , F. Lavancier, (2007) Pseudo-likelihood estimation for

non-hereditary Gibbs point processes, preprint. J.L. Jensen and H.R. Künsch, (1994) On asymptotic normality of pseudo likelihood estimates for pairwise interaction process, Ann. Inst. Statist. Math., Vol. 46, 3 :487-7486. J.L. Jensen and J. Moller (1991) Pseudolikelihood for exponential family models of spatial point processes, Ann. Appl. Probab. 1, 445-461.

• S. Mase (1995) Consistency of maximum pseudo-likelihood estimator of

continuous state space Gibbsian process Ann. Appl. Probab. 5, 603-612. X.X. Nguyen and H. Zessin, (1979) Integral and differential characterizations of the Gibbs process, Math. Nach. 88 105-115.

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Random Tessellation with hardcore interaction Point processes with forced clusters Intro