Gibbs Voronoi tessellations: Modeling, Simulation, Estimation. - - PowerPoint PPT Presentation

Introduction Definitions Simulation Estimation Theory

Gibbs Voronoi tessellations: Modeling, Simulation, Estimation.

Frédéric Lavancier, Laboratoire Jean Leray, Nantes (France)

Work with D. Dereudre (LAMAV, Valenciennes, France).

SPA, Berlin, July 27-31, 2009.

Introduction Definitions Simulation Estimation Theory

Introduction

Voronoï tessellations : applications in Astronomy, Biology, Physics, etc.

Introduction Definitions Simulation Estimation Theory

Introduction

Voronoï tessellations : applications in Astronomy, Biology, Physics, etc. Studied as a random object : based on Poisson point Processes.

Introduction Definitions Simulation Estimation Theory

Introduction

Voronoï tessellations : applications in Astronomy, Biology, Physics, etc. Studied as a random object : based on Poisson point Processes.

Introduction Definitions Simulation Estimation Theory

Introduction

Voronoï tessellations : applications in Astronomy, Biology, Physics, etc. Studied as a random object : based on Poisson point Processes. Drawback : Strong independent structures coming from the Poisson process

Introduction Definitions Simulation Estimation Theory

Introduction

Voronoï tessellations : applications in Astronomy, Biology, Physics, etc. Studied as a random object : based on Poisson point Processes. Drawback : Strong independent structures coming from the Poisson process − → Interactions between the cells ?

Introduction Definitions Simulation Estimation Theory

Introduction

One solution : Gibbs modifications of Poisson Voronoï tessellations.

Introduction Definitions Simulation Estimation Theory

Introduction

One solution : Gibbs modifications of Poisson Voronoï tessellations. Questions : What kind of interactions ?

Introduction Definitions Simulation Estimation Theory

Introduction

One solution : Gibbs modifications of Poisson Voronoï tessellations. Questions : What kind of interactions ? − → Smooth interaction.

Introduction Definitions Simulation Estimation Theory

Introduction

One solution : Gibbs modifications of Poisson Voronoï tessellations. Questions : What kind of interactions ? − → Smooth interaction. − → Hardcore interaction (some Voronoi tessellations are forbidden)

Introduction Definitions Simulation Estimation Theory

Introduction

One solution : Gibbs modifications of Poisson Voronoï tessellations. Questions : What kind of interactions ? − → Smooth interaction. − → Hardcore interaction (some Voronoi tessellations are forbidden) Existence of models.

Introduction Definitions Simulation Estimation Theory

Introduction

One solution : Gibbs modifications of Poisson Voronoï tessellations. Questions : What kind of interactions ? − → Smooth interaction. − → Hardcore interaction (some Voronoi tessellations are forbidden) Existence of models. Unicity of Gibbs measures.

Introduction Definitions Simulation Estimation Theory

Introduction

One solution : Gibbs modifications of Poisson Voronoï tessellations. Questions : What kind of interactions ? − → Smooth interaction. − → Hardcore interaction (some Voronoi tessellations are forbidden) Existence of models. Unicity of Gibbs measures. Simulations.

Introduction Definitions Simulation Estimation Theory

Introduction

One solution : Gibbs modifications of Poisson Voronoï tessellations. Questions : What kind of interactions ? − → Smooth interaction. − → Hardcore interaction (some Voronoi tessellations are forbidden) Existence of models. Unicity of Gibbs measures. Simulations. Parametric estimations.

Introduction Definitions Simulation Estimation Theory

2 Definitions

Introduction Definitions Simulation Estimation Theory

Notations

B(R2) denotes the space of bounded sets in R2.

Introduction Definitions Simulation Estimation Theory

Notations

B(R2) denotes the space of bounded sets in R2. M(R2) is the space of locally finite point configurations γ in R2 : γ ⊂ R2, such that for all Λ ∈ B(R2), card(γ ∩ Λ) < ∞.

Introduction Definitions Simulation Estimation Theory

Notations

B(R2) denotes the space of bounded sets in R2. M(R2) is the space of locally finite point configurations γ in R2 : γ ⊂ R2, such that for all Λ ∈ B(R2), card(γ ∩ Λ) < ∞. γΛ is the restriction of γ on Λ : γΛ = γ ∩ Λ.

Introduction Definitions Simulation Estimation Theory

Notations

B(R2) denotes the space of bounded sets in R2. M(R2) is the space of locally finite point configurations γ in R2 : γ ⊂ R2, such that for all Λ ∈ B(R2), card(γ ∩ Λ) < ∞. γΛ is the restriction of γ on Λ : γΛ = γ ∩ Λ. Vor(γ) : Voronoï tessellation coming from γ

Introduction Definitions Simulation Estimation Theory

Notations

B(R2) denotes the space of bounded sets in R2. M(R2) is the space of locally finite point configurations γ in R2 : γ ⊂ R2, such that for all Λ ∈ B(R2), card(γ ∩ Λ) < ∞. γΛ is the restriction of γ on Λ : γΛ = γ ∩ Λ. Vor(γ) : Voronoï tessellation coming from γ λ is the Lebesgue measure on R2.

Introduction Definitions Simulation Estimation Theory

Notations

B(R2) denotes the space of bounded sets in R2. M(R2) is the space of locally finite point configurations γ in R2 : γ ⊂ R2, such that for all Λ ∈ B(R2), card(γ ∩ Λ) < ∞. γΛ is the restriction of γ on Λ : γΛ = γ ∩ Λ. Vor(γ) : Voronoï tessellation coming from γ λ is the Lebesgue measure on R2. For z > 0, πz : Poisson point process with intensity zλ

Introduction Definitions Simulation Estimation Theory

Notations

B(R2) denotes the space of bounded sets in R2. M(R2) is the space of locally finite point configurations γ in R2 : γ ⊂ R2, such that for all Λ ∈ B(R2), card(γ ∩ Λ) < ∞. γΛ is the restriction of γ on Λ : γΛ = γ ∩ Λ. Vor(γ) : Voronoï tessellation coming from γ λ is the Lebesgue measure on R2. For z > 0, πz : Poisson point process with intensity zλ πz

Λ : πz restricted on Λ.

Introduction Definitions Simulation Estimation Theory

Gibbs measures

Let (HΛ)Λ∈B(R2) be a family of energies HΛ : M(Λ) × M(Λc) − → R ∪ {+∞} (γΛ, γΛc) − → HΛ(γΛ|γΛc) We suppose that it is compatible. For every Λ ⊂ Λ′ HΛ′(γΛ′|γΛ′c) = HΛ(γΛ|γΛc) + ϕΛ,Λ′(γΛc).

Introduction Definitions Simulation Estimation Theory

Gibbs measures

Let (HΛ)Λ∈B(R2) be a family of energies HΛ : M(Λ) × M(Λc) − → R ∪ {+∞} (γΛ, γΛc) − → HΛ(γΛ|γΛc) We suppose that it is compatible. For every Λ ⊂ Λ′ HΛ′(γΛ′|γΛ′c) = HΛ(γΛ|γΛc) + ϕΛ,Λ′(γΛc). Definition A probability measure P on M(R2) is a Gibbs measure for z > 0 and (HΛ) if for every Λ ∈ B(R2) and P-almost every γΛc P(dγΛ|γΛc) = 1 ZΛ(γΛc)e−HΛ(γΛ|γΛc)πz

Λ(dγΛ),

where ZΛ(γΛc) =

• e−HΛ(γ′

Λ|γΛc)πΛ(dγ′

Λ).

Introduction Definitions Simulation Estimation Theory

A typical energy of a Voronoï tessellation :

HΛ(γΛ|γΛc) =

• C∈ Vor(γ)

C∩Λ=∅

V1(C) +

• C,C′∈ Vor(γ)

C and C′are neighbors (C∪C′)∩Λ=∅

V2(C, C′).

Introduction Definitions Simulation Estimation Theory

A typical energy of a Voronoï tessellation :

HΛ(γΛ|γΛc) =

• C∈ Vor(γ)

C∩Λ=∅

V1(C) +

• C,C′∈ Vor(γ)

C and C′are neighbors (C∪C′)∩Λ=∅

V2(C, C′). Our guiding example : V1(C) =        +∞ if hmin(C) ≤ ε +∞ if hmax(C) ≥ α +∞ if h2

max(C)/Vol(C) ≥ B

• therwise

x hmin hmax C 0 < ε < α, B > 1/2 √ 3 ;

Introduction Definitions Simulation Estimation Theory

A typical energy of a Voronoï tessellation :

HΛ(γΛ|γΛc) =

• C∈ Vor(γ)

C∩Λ=∅

V1(C) +

• C,C′∈ Vor(γ)

C and C′are neighbors (C∪C′)∩Λ=∅

V2(C, C′). Our guiding example : V1(C) =        +∞ if hmin(C) ≤ ε +∞ if hmax(C) ≥ α +∞ if h2

max(C)/Vol(C) ≥ B

• therwise

x hmin hmax C 0 < ε < α, B > 1/2 √ 3 ; V2(C, C′) = θ max(Vol(C), Vol(C′)) min(Vol(C), Vol(C′)) − 1 1

2

, θ ∈ R

Introduction Definitions Simulation Estimation Theory

Existence results

First existence results (bounded interactions) :

Bertin, Billiot and Drouilhet, : 1) Existence of nearest-neighbors spatial Gibbs models , Adv.

• Appl. Prob. (SGSA) (1999) 31, 895-909.

2) Existence of Delaunay pairwise Gibbs point process with superstable component , J. Stat. phys. (1999) Vol 95 3/4 719-744.

Introduction Definitions Simulation Estimation Theory

Existence results

First existence results (bounded interactions) :

Bertin, Billiot and Drouilhet, : 1) Existence of nearest-neighbors spatial Gibbs models , Adv.

• Appl. Prob. (SGSA) (1999) 31, 895-909.

2) Existence of Delaunay pairwise Gibbs point process with superstable component , J. Stat. phys. (1999) Vol 95 3/4 719-744.

Existence results with hardcore interactions (B = +∞) :

Dereudre, Gibbs Delaunay tessellations with geometric hardcore conditions, J. Stat. Phys.,(2008) 131, 127-151.

Introduction Definitions Simulation Estimation Theory

Existence results

First existence results (bounded interactions) :

Bertin, Billiot and Drouilhet, : 1) Existence of nearest-neighbors spatial Gibbs models , Adv.

• Appl. Prob. (SGSA) (1999) 31, 895-909.

2) Existence of Delaunay pairwise Gibbs point process with superstable component , J. Stat. phys. (1999) Vol 95 3/4 719-744.

Existence results with hardcore interactions (B = +∞) :

Dereudre, Gibbs Delaunay tessellations with geometric hardcore conditions, J. Stat. Phys.,(2008) 131, 127-151.

In the general case :

Dereudre, Drouilhet and Georgii, Existence of Gibbs process with stable cluster interactions, preprint.

Introduction Definitions Simulation Estimation Theory

Unicity

For the interaction given before : A Gibbs measure exists but we don’t know if it is unique or not (phase transition problem !)

Introduction Definitions Simulation Estimation Theory

3 Simulation

Introduction Definitions Simulation Estimation Theory

Simulations

Strong hardcore interaction

Introduction Definitions Simulation Estimation Theory

Simulations

Strong hardcore interaction ⇒ Rigidity of the tessellation

Introduction Definitions Simulation Estimation Theory

Simulations

Strong hardcore interaction ⇒ Rigidity of the tessellation → Several difficulties for the simulations.

Introduction Definitions Simulation Estimation Theory

Simulations

Strong hardcore interaction ⇒ Rigidity of the tessellation → Several difficulties for the simulations. Birth-death-move MCMC algorithm on [0, 1]2 :

1 Draw independently a and b uniformly on [0, 1]. 2 If a < 1/3 then generate x uniformly on [0, 1]2 and

if b < f(γ + x)z (n + 1)f(γ), then γ+x → γ

• therwise "do nothing".

3 If 1/3 < a < 2/3 then generate x on γ and

if b < nf(γ − x) f(γ)z , then γ−x → γ

• therwise "do nothing".

4 If a > 2/3 then generate x on γ, y ∼ N(x, σ2) and

if b < f(γ − x + y) f(γ) , then γ−x+y → γ

• therwise "do nothing".

Introduction Definitions Simulation Estimation Theory

Examples of simulations

We fix z = 100, ε = 0, α = 0.05 :

B = +∞, θ = 0.5 B = 1, θ = 0.5 B = 0.625, θ = 0.5 B = +∞, θ = −0.5 B = 1, θ = −0.5 B = 0.625, θ = −0.5

Introduction Definitions Simulation Estimation Theory

Monitoring control

B = +∞, θ = 0.5 B = 1, θ = 0.5 B = 0.625, θ = 0.5 B = +∞, θ = −0.5 B = 1, θ = −0.5 B = 0.625, θ = −0.5

Introduction Definitions Simulation Estimation Theory

4 Estimation

Introduction Definitions Simulation Estimation Theory

Pseudo-likelihood Estimation

The aim : Estimate the parameters of the interaction from one realization γ of the Gibbs measure.

Introduction Definitions Simulation Estimation Theory

Pseudo-likelihood Estimation

The aim : Estimate the parameters of the interaction from one realization γ of the Gibbs measure. Hardcore parameters : ε, α and B. − → Empirical extremum hardcore parameters.

Introduction Definitions Simulation Estimation Theory

Pseudo-likelihood Estimation

The aim : Estimate the parameters of the interaction from one realization γ of the Gibbs measure. Hardcore parameters : ε, α and B. − → Empirical extremum hardcore parameters. Smooth parameters : z and θ. − → Pseudolikelihood procedure.

Introduction Definitions Simulation Estimation Theory

Pseudo-likelihood Estimation

The aim : Estimate the parameters of the interaction from one realization γ of the Gibbs measure. Hardcore parameters : ε, α and B. − → Empirical extremum hardcore parameters. Smooth parameters : z and θ. − → Pseudolikelihood procedure. Why the pseudo and not the MLE ?

Introduction Definitions Simulation Estimation Theory

Pseudo-likelihood Estimation

The aim : Estimate the parameters of the interaction from one realization γ of the Gibbs measure. Hardcore parameters : ε, α and B. − → Empirical extremum hardcore parameters. Smooth parameters : z and θ. − → Pseudolikelihood procedure. Why the pseudo and not the MLE ? MLE is too time consuming (because of the estimation by simulations of the normalizing constant).

Introduction Definitions Simulation Estimation Theory

Pseudo-likelihood Estimation

The aim : Estimate the parameters of the interaction from one realization γ of the Gibbs measure. Hardcore parameters : ε, α and B. − → Empirical extremum hardcore parameters. Smooth parameters : z and θ. − → Pseudolikelihood procedure. Why the pseudo and not the MLE ? MLE is too time consuming (because of the estimation by simulations of the normalizing constant). Pseudo is proved to be asymptotically consistent and normal in most cases. Bibliography : Besag (1975), Jensen and Moller (1991), Jensen and Kunsch (1994), Mase (1995), Billiot, Coeurjolly and Drouilhet (2008), Dereudre and L. (2009).

Introduction Definitions Simulation Estimation Theory

Practical estimation procedures

Let Λn = [−n, n]2 be the observation window and γ a realization

• f the Gibbs measure P.

Introduction Definitions Simulation Estimation Theory

Practical estimation procedures

Let Λn = [−n, n]2 be the observation window and γ a realization

• f the Gibbs measure P.
• Hardcore parameter estimators :

ˆ ε = min{hmin(C), C ∈ V or(γ) and C ∩ Λn = ∅}, ˆ α = max{hmax(C), C ∈ V or(γ) and C ∩ Λn = ∅}, ˆ B = max{h2

max(C)/Vol(C), C ∈ V or(γ) and C ∩ Λn = ∅}.

Introduction Definitions Simulation Estimation Theory

Practical estimation procedures

Let Λn = [−n, n]2 be the observation window and γ a realization

• f the Gibbs measure P.
• Hardcore parameter estimators :

ˆ ε = min{hmin(C), C ∈ V or(γ) and C ∩ Λn = ∅}, ˆ α = max{hmax(C), C ∈ V or(γ) and C ∩ Λn = ∅}, ˆ B = max{h2

max(C)/Vol(C), C ∈ V or(γ) and C ∩ Λn = ∅}.

• Smooth parameter estimators :

(ˆ z, ˆ θ) = argminz,θPLLΛn(γ, z, θ, ˆ ε, ˆ α, ˆ B),

Introduction Definitions Simulation Estimation Theory

Practical estimation procedures

Let Λn = [−n, n]2 be the observation window and γ a realization

• f the Gibbs measure P.
• Hardcore parameter estimators :

ˆ ε = min{hmin(C), C ∈ V or(γ) and C ∩ Λn = ∅}, ˆ α = max{hmax(C), C ∈ V or(γ) and C ∩ Λn = ∅}, ˆ B = max{h2

max(C)/Vol(C), C ∈ V or(γ) and C ∩ Λn = ∅}.

• Smooth parameter estimators :

(ˆ z, ˆ θ) = argminz,θPLLΛn(γ, z, θ, ˆ ε, ˆ α, ˆ B), with PLLΛn() =

• Λn

z exp (−h(x, γ)) dx+

• x∈γΛn

HΛn(γ−x)<∞

• h(x, γ−x)−ln(z)
• ,

where h(x, γ) = HΛn(γ + x) − HΛn(γ).

Introduction Definitions Simulation Estimation Theory

Theoretical results

For the hardcore parameters : Theorem (Dereudre-L. (2009)) For P-almost all γ lim

n→∞(ˆ

ε, ˆ α, ˆ B) = (ε, α, B).

Introduction Definitions Simulation Estimation Theory

Theoretical results

For the hardcore parameters : Theorem (Dereudre-L. (2009)) For P-almost all γ lim

n→∞(ˆ

ε, ˆ α, ˆ B) = (ε, α, B). For the smooth parameters : Theorem (Dereudre-L. (2009)) For P-almost all γ lim

n→∞(ˆ

z, ˆ θ) = (z, θ). (ˆ z, ˆ θ) are asymptotic normal if ε, α and B are supposed to be known.

Introduction Definitions Simulation Estimation Theory

Estimation results

The true parameters : ε = 0, α = 0.05, B = 0.625, z = 100 and θ = −0.5.

Introduction Definitions Simulation Estimation Theory

Estimation results

The true parameters : ε = 0, α = 0.05, B = 0.625, z = 100 and θ = −0.5. Typical tessellation : Hardcore parameter estimators :

ˆ α ˆ B

Introduction Definitions Simulation Estimation Theory

Estimation results

The true parameters : ε = 0, α = 0.05, B = 0.625, z = 100 and θ = −0.5 . Smooth parameter estimators :

ˆ θ when z is known ˆ θ when z is estimated ˆ z

Introduction Definitions Simulation Estimation Theory

Estimation results

The true parameters : ε = 0, α = 0.05, B = 0.625, z = 100 and θ = 0.5.

Introduction Definitions Simulation Estimation Theory

Estimation results

The true parameters : ε = 0, α = 0.05, B = 0.625, z = 100 and θ = 0.5. Typical tessellation : Hardcore parameter estimators :

ˆ α ˆ B

Introduction Definitions Simulation Estimation Theory

Estimation results

The true parameters : ε = 0, α = 0.05, B = 0.625, z = 100 and θ = 0.5. Smooth parameter estimators :

ˆ θ when z is known ˆ θ when z is estimated ˆ z

Introduction Definitions Simulation Estimation Theory

Conclusion

Our Gibbs Voronoi model : forces the shape and the maximal size of the cells provides some repulsive or attractive interaction between two neighbour cells.

Introduction Definitions Simulation Estimation Theory

Conclusion

Our Gibbs Voronoi model : forces the shape and the maximal size of the cells provides some repulsive or attractive interaction between two neighbour cells. The simulation can be achieved by a Birth-Death-Move MCMC algorithm − → very time consuming because of the hardcore interactions.

Introduction Definitions Simulation Estimation Theory

Conclusion

Our Gibbs Voronoi model : forces the shape and the maximal size of the cells provides some repulsive or attractive interaction between two neighbour cells. The simulation can be achieved by a Birth-Death-Move MCMC algorithm − → very time consuming because of the hardcore interactions. A two-step estimation procedure can be applied

1 the hardcore parameters are estimated in a natural way, 2 the smooth parameters are estimated by pseudo-likelihood

where the the hardcore parameters are plugged in.

Introduction Definitions Simulation Estimation Theory

Conclusion

Our Gibbs Voronoi model : forces the shape and the maximal size of the cells provides some repulsive or attractive interaction between two neighbour cells. The simulation can be achieved by a Birth-Death-Move MCMC algorithm − → very time consuming because of the hardcore interactions. A two-step estimation procedure can be applied

1 the hardcore parameters are estimated in a natural way, 2 the smooth parameters are estimated by pseudo-likelihood

where the the hardcore parameters are plugged in. This is consistent and allows to distinguish between the repulsive and the attractive case in a non-trivial situation.

Introduction Definitions Simulation Estimation Theory

• A. Baddeley, R. Turner, J. Moller and M. Hazelton, Residual analysis

for spatial point processes, J. R. Statist. Soc. B, 65, 617-666 (2005).

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

neighbors spatial Gibbs models , Adv. Appl. Prob. (SGSA) 31, 895-909. J.-M. Billiot, J.-F. Coeurjolly and R. Drouilhet, (2008) Maximum pseudolikelihood estimator for exponential family models of marked Gibbs point processes, Electronic J. Statistics, 2, 234-254.

• D. Dereudre , (2008) Gibbs Delaunay tessellations with geometric hardcore

conditions, J. Stat. Phys., 131, 127-151.

• D. Dereudre, R. Drouilhet and H.-O. Georgii, (2009) Existence of Gibbs

process with stable cluster interactions, preprint.

• D. Dereudre , F. Lavancier, (2009) Campbell equilibrium equation and

pseudo-likelihood estimation for non-hereditary Gibbs point processes, to appear in Bernoulli.

• D. Dereudre , F. Lavancier, (2009) Practical simulation and estimation

for Gibbs Delaunay-Voronoi tessellations with geometric hardcore interaction, preprint. J.L. Jensen and J. Moller (1991) Pseudolikelihood for exponential family models of spatial point processes, Ann. Appl. Probab. 1, 445-461.

• C. Preston, Random Fields, LNM Vol. 534, Springer.

Introduction Definitions Simulation Estimation Theory

5 Some theoretical points

Introduction Definitions Simulation Estimation Theory

The problem of heredity

Definition The family of energies (HΛ)Λ is said hereditary if for every Λ, every γ ∈ M(R2) and every x ∈ Λ HΛ(γ) = +∞ ⇒ HΛ(γ + δx) = +∞.

Introduction Definitions Simulation Estimation Theory

The problem of heredity

Definition The family of energies (HΛ)Λ is said hereditary if for every Λ, every γ ∈ M(R2) and every x ∈ Λ HΛ(γ) = +∞ ⇒ HΛ(γ + δx) = +∞. γ is forbidden ⇒ γ + δx is forbidden

Introduction Definitions Simulation Estimation Theory

The problem of heredity

Definition The family of energies (HΛ)Λ is said hereditary if for every Λ, every γ ∈ M(R2) 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 Simulation Estimation Theory

The problem of heredity

Definition The family of energies (HΛ)Λ is said hereditary if for every Λ, every γ ∈ M(R2) 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 Simulation Estimation Theory

The problem of heredity

Definition The family of energies (HΛ)Λ is said hereditary if for every Λ, every γ ∈ M(R2) 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.) The Gibbs Voronoi Tessellations are not hereditary. − → When one adds a point in a too large cell, the new tessellation may be allowed.

Introduction Definitions Simulation Estimation Theory

Nguyen-Zessin equilibrium equation

Theorem (Nguyen-Zessin (1979), hereditary case) Suppose that the energy (HΛ)Λ is hereditary. P is Gibbs measure with intensity measure ν if and only if, for every bounded non negative measurable function ψ from R2 × M(R2) to R, EP

• x∈γ

ψ(x, γ − x)

• = EP
• R2 ψ(x, γ)e−h(x,γ)ν(dx)
• ,

where h(x, γ) = HΛn(γ + x) − HΛn(γ). Proposition (Dereudre, L. (2009), general case) Let P be a Gibbs measure with intensity measure ν, then EP    

• x∈γΛn

HΛn(γ−x)<∞

ψ(x, γ − x)     = EP

• R2 ψ(x, γ)e−h(x,γ)ν(dx)
• .

Introduction Definitions Simulation Estimation Theory

Validation : residuals process

We can extend the concept of residuals (see Baddeley et al., 2005) to the non-hereditary setting. The residuals process on a set ∆ is defined for any function ψ by R

• ∆, ψ, ˆ

h, ˆ ν

• =
• x∈γ∆

H∆(γ−x)<∞

ψ(x, γ − x) −

• ∆

ψ(x, γ)e−ˆ

h(x,γ)ˆ

ν(dx), From the equilibrium equation given before, under the true model, R

• ∆, ψ, ˆ

h, ˆ ν

• ≈ 0

R

• ∆, ψ, ˆ

h, ˆ ν

• is approximatively gaussian.

− → Several diagnostic tools can then be applied when fitting a Gibbs Voronoi model

Introduction Definitions Simulation Estimation Theory

Validation : residuals process

We can extend the concept of residuals (see Baddeley et al., 2005) to the non-hereditary setting. The residuals process on a set ∆ is defined for any function ψ by R

• ∆, ψ, ˆ

h, ˆ ν

• =
• x∈γ∆

H∆(γ−x)<∞

ψ(x, γ − x) −

• ∆

ψ(x, γ)e−ˆ

h(x,γ)ˆ

ν(dx), From the equilibrium equation given before, under the true model, R

• ∆, ψ, ˆ

h, ˆ ν

• ≈ 0

R

• ∆, ψ, ˆ

h, ˆ ν

• is approximatively gaussian.

− → Several diagnostic tools can then be applied when fitting a Gibbs Voronoi model For further asymptotic results on the residuals process R : − → See the talk of J.-F. Coeurjolly on Friday morning.