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Introduction Definitions Campbell Estimation Simulations

Pseudo-likelihood estimation for nonhereditary Gibbs point processes

Frédéric Lavancier,Laboratoire Jean Leray, Nantes, France.

Joint work withDavid Dereudre,

LAMAV, Valenciennes, France.

7th World Congress in Probability and StatisticsSingapore, July 14-19 2008

1 Introduction

Introduction

SettingPseudo-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 workCharacteristics of the non-hereditary interactions.A new equilibrium Campbell equation.Consistency of the Pseudo-likelihood estimator.Some simulations.

Introduction

SettingPseudo-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 workCharacteristics of the non-hereditary interactions.A new equilibrium Campbell equation.Consistency of the Pseudo-likelihood estimator.Some simulations.

2 Gibbs measure and hereditary interactions

Notations

γ denotes a point configuration on Rd (i.e. aninteger-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.

Gibbs measures

(HΛ)Λ denotes a general family of energy functions :

HΛ : (γΛ, γΛc) 7−→ HΛ(γΛ|γΛc)

There are some minimal conditions on (HΛ)Λ.

DefinitionA probability measure µ is a Gibbs measure if for everybounded 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.

Gibbs measures

(HΛ)Λ denotes a general family of energy functions :

HΛ : (γΛ, γΛc) 7−→ HΛ(γΛ|γΛc)

There are some minimal conditions on (HΛ)Λ.

DefinitionA probability measure µ is a Gibbs measure if for everybounded 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.

Hereditary

DefinitionThe 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 forbiddenγ + δx is allowed ⇒ γ is allowed

It is a standard assumption in classical statistical mechanics.Example : The classical hard ball model is hereditary.

Hereditary

HΛ(γ) = +∞⇒ HΛ(γ + δx) = +∞.

γ is forbidden ⇒ γ + δx is forbidden

γ + δx is allowed ⇒ γ is allowed

Hereditary

HΛ(γ) = +∞⇒ HΛ(γ + δx) = +∞.

Hereditary

HΛ(γ) = +∞⇒ HΛ(γ + δx) = +∞.

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 GibbsVoronoi tesselations with geometric hardcore interactions.

Non-hereditary

We are interested in the non hereditary case.Examples :

HΛ(γ) = +∞ HΛ(γ + δx) < +∞

Non-hereditary

We are interested in the non hereditary case.Examples :

HΛ(γ) = +∞ HΛ(γ + δx) < +∞

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,< +∞ otherwise.

HΛ(γ) =∑

V (ver(x1, x2)),

V (ver(x1, x2)) =

{+∞ if ||x1 − x2|| > α,< +∞ otherwise.

HΛ(γ) =∑

V (ver(x1, x2)),

V (ver(x1, x2)) =

{+∞ if ||x1 − x2|| > α,< +∞ otherwise.

HΛ(γ) =∑

V (ver(x1, x2)),

V (ver(x1, x2)) =

{+∞ if ||x1 − x2|| > α,< +∞ otherwise.

HΛ(γ) = +∞ HΛ(γ + δx) < +∞

3 Equilibrium equation

Nguyen-Zessin equilibrium equation

Definition

Let µ be a probability measure onM(Rd). The reduced Campbellmeasure 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 Gibbsmeasure 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.

Nguyen-Zessin equilibrium equation

Definition

Let µ be a probability measure onM(Rd). The reduced Campbellmeasure 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 Gibbsmeasure 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.

Removable points

Definition

Let γ be inM(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.

DefinitionLet x in R(γ). We define the energy of x in γ − δx with thefollowing expression

h(x, γ − δx) = HΛ(γ)−HΛ(γ − δx),

Removable points

Definition

Let γ be inM(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.

DefinitionLet x in R(γ). We define the energy of x in γ − δx with thefollowing expression

h(x, γ − δx) = HΛ(γ)−HΛ(γ − δx),

Equilibrium equations for non-hereditary Gibbs measures

Theorem (Dereudre-Lavancier (2007))Let µ be a Gibbs measure,

1Ix∈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’sequilibrium equation.

- The equation (1) does not characterize the Gibbs measures.

Equilibrium equations for non-hereditary Gibbs measures

Theorem (Dereudre-Lavancier (2007))Let µ be a Gibbs measure,

1Ix∈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’sequilibrium equation.

- The equation (1) does not characterize the Gibbs measures.

4 Pseudo-likelihood estimation

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,θΛ )Λ : the parametric family of energies.- For x in R(γ), hα,θ(x, γ − δx) = Hα,θΛ (γ)−H

α,θΛ (γ − δx).

Let Λn the observation window of γ (e. g. Λn = [−n, n]d).

DefinitionWe define the pseudo likelihood contrast function

PLLΛn(γ, α, θ) =

1Λn

∫Λn

exp(−hα,θ(x, γ)

)dx+

∑x∈Rα,θ(γ)∩Λn

hα,θ(x, γ − δx)

.

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.

DefinitionWe define for µ almost every γ

α̂n(γ) = inf{α > 0, Hα,θΛn (γ)

5 Simulations

Hα,θΛ (γ) =∑

V α,θ(ver(x1, x2)),

V α,θ(ver(x1, x2)) =

{+∞ if ||x1 − x2|| > αθ√

max(V1,V2)min(V1,V2)

− 1 otherwise,

with Vj the volume of cell(xj).

Hα,θΛ (γ) =∑

V α,θ(ver(x1, x2)),

V α,θ(ver(x1, x2)) =

{+∞ if ||x1 − x2|| > αθ√

max(V1,V2)min(V1,V2)

− 1 otherwise,

with Vj the volume of cell(xj).

α = 0.12, θ = 0.5 α = 0.12, θ = −0.5

6/164 removable points 456/634 removable points

α̂ = 0.119, θ̂ = 0.6 α̂ = 0.119, θ̂ = −0.49

α = 0.12, θ = 0.5 α = 0.12, θ = −0.5

6/164 removable points 456/634 removable points

α̂ = 0.119, θ̂ = 0.6 α̂ = 0.119, θ̂ = −0.49

Repartition of α̂n and θ̂n on 200 replicates

α = 0.12, θ = 0.5 sd(α̂n) = 1.7 10−4 sd(θ̂n) = 0.102

α = 0.12, θ = −0.5 sd(α̂n) = 2.3 10−4 sd(θ̂n) = 0.016

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

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) Maximumpseudolikelihood estimator for exponential family models of marked Gibbspoint processes, Electronic Journal of Statistics.

D. Dereudre , (2007) Gibbs Delaunay tessellations with geometric hardcoreconditions, to appear in J.S.P.

D. Dereudre , F. Lavancier, (2007) Pseudo-likelihood estimation fornon-hereditary Gibbs point processes, preprint.

J.L. Jensen and H.R. Künsch, (1994) On asymptotic normality of pseudolikelihood estimates for pairwise interaction process, Ann. Inst. Statist.Math., Vol. 46, 3 :487-7486.

J.L. Jensen and J. Moller (1991) Pseudolikelihood for exponential familymodels of spatial point processes, Ann. Appl. Probab. 1, 445-461.

S. Mase (1995) Consistency of maximum pseudo-likelihood estimator ofcontinuous state space Gibbsian process Ann. Appl. Probab. 5, 603-612.

X.X. Nguyen and H. Zessin, (1979) Integral and differentialcharacterizations of the Gibbs process, Math. Nach. 88 105-115.

Random Tessellation with hardcore interaction

Point processes with forced clustersIntro

Introduction Gibbs measure and hereditary interactionsEquilibrium equationPseudo-likelihood estimationSimulations

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