Residuals and Goodness-of-fit tests for marked Gibbs point processes - - PowerPoint PPT Presentation

Residuals and Goodness-of-fit tests for marked Gibbs point processes

Fr´ ed´ eric Lavancier (Laboratoire Jean Leray, Nantes)

Joint work with J.-F. Coeurjolly (Grenoble)

01/02/2010

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 1 / 32

Outline

1

Type of data of interest

2

Gibbs models Brief background Examples

3

Identification Maximum likelihood Pseudo-likelihood

4

Validations through residuals Residuals for spatial point processes Measures of departures to the true model Asymptotics

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 2 / 32

Outline

1

Type of data of interest

2

Gibbs models Brief background Examples

3

Identification Maximum likelihood Pseudo-likelihood

4

Validations through residuals Residuals for spatial point processes Measures of departures to the true model Asymptotics

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 3 / 32

Type of data

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 4 / 32

Type of data

Scientific questions :

1 Independence or interaction ? regular or clustered distribution ? 2 Spatial variation in the density and age of trees ? 3 interaction in each sub-pattern, between sub-patterns...

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 4 / 32

An example in computer graphics

(Hurtut, Landes, Thollot, Gousseau, Drouilhet, C.’09.)

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 5 / 32

An example with geometrical structures : Voronoi-type Gibbs models

Each cell is associated to a point (the cell nucleus) = ⇒ The tessellation can be viewed as a point process. Is there an interaction between the points, i.e. between the cells ?

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 6 / 32

Outline

1

Type of data of interest

2

Gibbs models Brief background Examples

3

Identification Maximum likelihood Pseudo-likelihood

4

Validations through residuals Residuals for spatial point processes Measures of departures to the true model Asymptotics

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 7 / 32

Marked point processes

State space : S = Rd × ▼ associated to µ = λ ⊗ λ♠. Let xm = (x, m) an element of S, i.e. a marked point.

▼

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 8 / 32

Marked point processes

State space : S = Rd × ▼ associated to µ = λ ⊗ λ♠. Let xm = (x, m) an element of S, i.e. a marked point. Ω is the space of locally finite point configurations ϕ in S : ϕ ∈ Ω ⇐ ⇒ ∃ I ⊂ N, ∃ (xmi

i

)i∈I ∈ SI, ϕ =

• i∈I

δx

mi i .

▼

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 8 / 32

Marked point processes

State space : S = Rd × ▼ associated to µ = λ ⊗ λ♠. Let xm = (x, m) an element of S, i.e. a marked point. Ω is the space of locally finite point configurations ϕ in S : ϕ ∈ Ω ⇐ ⇒ ∃ I ⊂ N, ∃ (xmi

i

)i∈I ∈ SI, ϕ =

• i∈I

δx

mi i .

We write xm ∈ ϕ if ϕ({xm}) = 1. For Λ ∈ B(Rd), ϕΛ is the restriction of ϕ on Λ : ϕΛ =

• xm∈ϕ∩(Λ×▼)

δxm

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 8 / 32

Marked point processes

State space : S = Rd × ▼ associated to µ = λ ⊗ λ♠. Let xm = (x, m) an element of S, i.e. a marked point. Ω is the space of locally finite point configurations ϕ in S : ϕ ∈ Ω ⇐ ⇒ ∃ I ⊂ N, ∃ (xmi

i

)i∈I ∈ SI, ϕ =

• i∈I

δx

mi i .

We write xm ∈ ϕ if ϕ({xm}) = 1. For Λ ∈ B(Rd), ϕΛ is the restriction of ϕ on Λ : ϕΛ =

• xm∈ϕ∩(Λ×▼)

δxm Definition (marked point process) A marked point process is a random variable on Ω.

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 8 / 32

Example : Poisson point process

For z > 0, the standard (non-marked) poisson point process πz with intensity zλ is defined by

• ∀ Λ,

|πz

Λ| := πz(Λ) ∼ P(zλ(Λ))

∀ Λ, Λ′ with Λ ∩ Λ′ = ∅, πz

Λ and πz Λ′ are independent.

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 9 / 32

Example : Poisson point process

For z > 0, the standard (non-marked) poisson point process πz with intensity zλ is defined by

• ∀ Λ,

|πz

Λ| := πz(Λ) ∼ P(zλ(Λ))

∀ Λ, Λ′ with Λ ∩ Λ′ = ∅, πz

Λ and πz Λ′ are independent.

Example with z = 100 on [0, 1]2 :

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 9 / 32

Example : Poisson point process

For z > 0, the standard (non-marked) poisson point process πz with intensity zλ is defined by

• ∀ Λ,

|πz

Λ| := πz(Λ) ∼ P(zλ(Λ))

∀ Λ, Λ′ with Λ ∩ Λ′ = ∅, πz

Λ and πz Λ′ are independent.

Example with z = 100 on [0, 1]2 : To involve some dependencies between points − → Gibbs modifications.

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 9 / 32

Gibbs measures

Let (VΛ)Λ∈B(Rd) be a family of energies VΛ : Ω − → R ∪ {+∞} ϕ − → VΛ(ϕΛ|ϕΛc) VΛ(ϕ) is the energy of ϕΛ inside Λ knowing the outside configuration ϕΛc

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 10 / 32

Gibbs measures

Let (VΛ)Λ∈B(Rd) be a family of energies VΛ : Ω − → R ∪ {+∞} ϕ − → VΛ(ϕΛ|ϕΛc) VΛ(ϕ) is the energy of ϕΛ inside Λ knowing the outside configuration ϕΛc Definition A probability measure P on Ω is a Gibbs measure (VΛ) if for every bounded set Λ and P-almost every ϕΛc P(dϕΛ|ϕΛc) = e−VΛ(ϕΛ|ϕΛc ) ZΛ(ϕΛc) πΛ(dϕΛ), (1) where ZΛ(ϕΛc) =

• e−VΛ(ϕΛ|ϕΛc )πΛ(dϕΛ).

The equations (1) for all Λ are called DLR (Dobrushin,Landford,Ruelle). ZΛ is the partition function.

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 10 / 32

Assumptions on the model

The choice of (VΛ)Λ∈B(Rd) entirely defines the Gibbs measure P. But, given (VΛ) : Is there exist a Gibbs measure P ? Is it unique (phase transition problem) ?

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 11 / 32

Assumptions on the model

The choice of (VΛ)Λ∈B(Rd) entirely defines the Gibbs measure P. But, given (VΛ) : Is there exist a Gibbs measure P ? Is it unique (phase transition problem) ? We consider parametric families of Gibbs measures Pθ, θ ∈ Θ. We assume : [Mod] : For any θ ∈ Θ ⊂ Rp, (VΛ(.; θ))Λ∈B(R[d]) are invariant by translation, and such that at least one associated Gibbs measure Pθ exists and is stationary. We denote by θ⋆ the true parameter to be estimated, assumed to be in ˚ Θ.

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 11 / 32

Assumptions on the model

The choice of (VΛ)Λ∈B(Rd) entirely defines the Gibbs measure P. But, given (VΛ) : Is there exist a Gibbs measure P ? Is it unique (phase transition problem) ? We consider parametric families of Gibbs measures Pθ, θ ∈ Θ. We assume : [Mod] : For any θ ∈ Θ ⊂ Rp, (VΛ(.; θ))Λ∈B(R[d]) are invariant by translation, and such that at least one associated Gibbs measure Pθ exists and is stationary. We denote by θ⋆ the true parameter to be estimated, assumed to be in ˚ Θ. Remark : General conditions on (VΛ(.; θ)) exist to ensure [Mod]. All the following examples satisfy [Mod]

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 11 / 32

Poisson point process, ▼ = {0} VΛ(ϕ; θ) = θ1|ϕΛ|

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 12 / 32

Multi-type Poisson point process, ▼ = {1, 2} VΛ(ϕ; θ) = θ1

1|ϕ1 Λ| + θ2 1|ϕ2 Λ|

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 13 / 32

Strauss marked point process, ▼ = {1, 2} VΛ(ϕ; θ) =

2

• m=1

θm

1 |ϕm Λ | + 2

• m1,m2=1
• {xm1,y m2}∈P2(ϕΛ)

θm1,m2

2

1[0,Dm1,m2](||y − x||)

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 14 / 32

Gibbs Voronoi tessellation (Dereudre, L.) VΛ(ϕ) =

• C∈ Vor(ϕ)

C∩Λ=∅

V1(C) +

• C,C ′∈ Vor(ϕ)

C and C ′are neighbors (C∪C ′)∩Λ=∅

V2(C, C ′) V1(C) : deals with the shape of the cell and V2(C, C ′) = θ d(vol(C), vol(C ′)).

θ > 0 θ < 0

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 15 / 32

Outline

1

Type of data of interest

2

Gibbs models Brief background Examples

3

Identification Maximum likelihood Pseudo-likelihood

4

Validations through residuals Residuals for spatial point processes Measures of departures to the true model Asymptotics

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 16 / 32

Maximum likelihood method

We observe ϕΛn, a realisation of the point process in Λn. ˆ θMLE

n

= argmax

θ∈Θ

1 Z(θ)e−V (ϕΛn;θ) where Z(θ) is an untractable normalizing constant.

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 17 / 32

Maximum likelihood method

We observe ϕΛn, a realisation of the point process in Λn. ˆ θMLE

n

= argmax

θ∈Θ

1 Z(θ)e−V (ϕΛn;θ) where Z(θ) is an untractable normalizing constant.

Drawbacks Z(θ) needs to be known − → Intensive Monte-Carlo based simulations allow to estimate Z(θ). Almost no theoretical results are available for ˆ θMLE

n

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 17 / 32

Maximum likelihood method

We observe ϕΛn, a realisation of the point process in Λn. ˆ θMLE

n

= argmax

θ∈Θ

1 Z(θ)e−V (ϕΛn;θ) where Z(θ) is an untractable normalizing constant.

Drawbacks Z(θ) needs to be known − → Intensive Monte-Carlo based simulations allow to estimate Z(θ). Almost no theoretical results are available for ˆ θMLE

n

Benefit Seems to be more accurate than other estimation methods (when computable !)

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 17 / 32

Pseudo-likelihood method (Besag (68), Ripley (88), JM (91))

• θMPLE

n

= argmax

θ∈Θ

LPLΛn(ϕ; θ) LPL is the log-pseudolikelihood function, defined by LPLΛn (ϕ; θ) = −

• Λn×▼

e−V (xm|ϕ;θ)µ(dxm) −

• xm∈ϕΛn

V (xm|ϕ \ xm; θ) , where V (xm|ϕ; θ) := VΛ(ϕ ∪ {xm}; θ) − VΛ(ϕ; θ), for any Λ ∋ xm, is the local energy function of xm inside ϕ (independent on Λ).

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 18 / 32

Pseudo-likelihood method (Besag (68), Ripley (88), JM (91))

• θMPLE

n

= argmax

θ∈Θ

LPLΛn(ϕ; θ) LPL is the log-pseudolikelihood function, defined by LPLΛn (ϕ; θ) = −

• Λn×▼

e−V (xm|ϕ;θ)µ(dxm) −

• xm∈ϕΛn

V (xm|ϕ \ xm; θ) , where V (xm|ϕ; θ) := VΛ(ϕ ∪ {xm}; θ) − VΛ(ϕ; θ), for any Λ ∋ xm, is the local energy function of xm inside ϕ (independent on Λ). Benefits Easy and quick computation. Exhaustive asymptotic properties (JM (91), JK (94), Mase (99), BCD (08), DL (09))

• θMPLE

n a.s.

→ θ⋆ |Λn|1/2 Σ−1/2( θMPLE

n

− θ⋆)

d

→ N

• 0, Ip
• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 18 / 32

Pseudo-likelihood method (Besag (68), Ripley (88), JM (91))

• θMPLE

n

= argmax

θ∈Θ

LPLΛn(ϕ; θ) LPL is the log-pseudolikelihood function, defined by LPLΛn (ϕ; θ) = −

• Λn×▼

e−V (xm|ϕ;θ)µ(dxm) −

• xm∈ϕΛn

V (xm|ϕ \ xm; θ) , where V (xm|ϕ; θ) := VΛ(ϕ ∪ {xm}; θ) − VΛ(ϕ; θ), for any Λ ∋ xm, is the local energy function of xm inside ϕ (independent on Λ). Benefits Easy and quick computation. Exhaustive asymptotic properties (JM (91), JK (94), Mase (99), BCD (08), DL (09))

• θMPLE

n a.s.

→ θ⋆ |Λn|1/2 Σ−1/2( θMPLE

n

− θ⋆)

d

→ N

• 0, Ip
• Drawback

Seems less accurate than the MLE (when available).

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 18 / 32

Outline

1

Type of data of interest

2

Gibbs models Brief background Examples

3

Identification Maximum likelihood Pseudo-likelihood

4

Validations through residuals Residuals for spatial point processes Measures of departures to the true model Asymptotics

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 19 / 32

Origin of residuals

One-dimesional point processes : I(t, θ⋆) := Nt

• numb. of arriv. times in [0,t]

− t λθ⋆(s)

• parametric instant. arrival rate

ds E(I(t)) = 0 (Karr (1985))= ⇒ I(t, θ) measures the adequacy of the fitted models.

▼ ▼

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 20 / 32

Origin of residuals

One-dimesional point processes : I(t, θ⋆) := Nt

• numb. of arriv. times in [0,t]

− t λθ⋆(s)

• parametric instant. arrival rate

ds E(I(t)) = 0 (Karr (1985))= ⇒ I(t, θ) measures the adequacy of the fitted models. For spatial point processes, the natural counterpart of Nt = ⇒ |ϕΛ|

• numb. of points in Λ

and λθ⋆(t) = ⇒ λθ⋆(xm|ϕ) := e−V (xm|ϕ;θ⋆)

• Papangelou cond. int.

▼ ▼

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 20 / 32

Origin of residuals

One-dimesional point processes : I(t, θ⋆) := Nt

• numb. of arriv. times in [0,t]

− t λθ⋆(s)

• parametric instant. arrival rate

ds E(I(t)) = 0 (Karr (1985))= ⇒ I(t, θ) measures the adequacy of the fitted models. For spatial point processes, the natural counterpart of Nt = ⇒ |ϕΛ|

• numb. of points in Λ

and λθ⋆(t) = ⇒ λθ⋆(xm|ϕ) := e−V (xm|ϕ;θ⋆)

• Papangelou cond. int.

The raw innovation process can be easily defined in the following way IΛ(ϕ, θ⋆) : = |ϕΛ| −

• Λ×▼

e−V (xm|ϕ;θ⋆)µ(dxm) =

• xm∈ϕΛ

1 −

• Λ×▼

e−V (xm|ϕ;θ⋆)µ(dxm).

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 20 / 32

The h-residuals Baddeley, Turner, Moller, Hazelton (05)

IΛ(ϕ, θ⋆) =

xm∈ϕΛ 1 −

• Λ×▼ e−V (xm|ϕ;θ⋆)µ(dxm)

Theorem (Georgii-Nguyen-Zessin) For any h(·, ·, θ) : S × Ω → R, for any θ ∈ Θ, Eθ⋆

xm∈ϕ

h (xm, ϕ \ xm; θ)

• = Eθ⋆
• Rd×▼

h (xm, ϕ; θ) e−V (xm|ϕ;θ⋆)µ(dxm)

• =

⇒ for h = 1, Eθ⋆(IΛ(ϕ, θ⋆)) = 0.

▼

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 21 / 32

The h-residuals Baddeley, Turner, Moller, Hazelton (05)

IΛ(ϕ, θ⋆) =

xm∈ϕΛ 1 −

• Λ×▼ e−V (xm|ϕ;θ⋆)µ(dxm)

Theorem (Georgii-Nguyen-Zessin) For any h(·, ·, θ) : S × Ω → R, for any θ ∈ Θ, Eθ⋆

xm∈ϕ

h (xm, ϕ \ xm; θ)

• = Eθ⋆
• Rd×▼

h (xm, ϕ; θ) e−V (xm|ϕ;θ⋆)µ(dxm)

• =

⇒ for h = 1, Eθ⋆(IΛ(ϕ, θ⋆)) = 0. The raw innovations can be generalized h−innovations (in Λ) : IΛ(ϕ; h, θ⋆) :=

• xm∈ϕΛ

h(xm, ϕ \ xm; θ⋆) −

• Λ×▼

h(xm, ϕ; θ⋆)e−V (xm|ϕ;θ⋆)µ(dxm)

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 21 / 32

The h-residuals Baddeley, Turner, Moller, Hazelton (05)

IΛ(ϕ, θ⋆) =

xm∈ϕΛ 1 −

• Λ×▼ e−V (xm|ϕ;θ⋆)µ(dxm)

Theorem (Georgii-Nguyen-Zessin) For any h(·, ·, θ) : S × Ω → R, for any θ ∈ Θ, Eθ⋆

xm∈ϕ

h (xm, ϕ \ xm; θ)

• = Eθ⋆
• Rd×▼

h (xm, ϕ; θ) e−V (xm|ϕ;θ⋆)µ(dxm)

• =

⇒ for h = 1, Eθ⋆(IΛ(ϕ, θ⋆)) = 0. The raw innovations can be generalized h−innovations (in Λ) : IΛ(ϕ; h, θ⋆) :=

• xm∈ϕΛ

h(xm, ϕ \ xm; θ⋆) −

• Λ×▼

h(xm, ϕ; θ⋆)e−V (xm|ϕ;θ⋆)µ(dxm) h−residuals (in Λ) : RΛ(ϕ; h, θ) := IΛ(ϕ; h, θ)

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 21 / 32

The h-residuals Baddeley, Turner, Moller, Hazelton (05)

IΛ(ϕ, θ⋆) =

xm∈ϕΛ 1 −

• Λ×▼ e−V (xm|ϕ;θ⋆)µ(dxm)

Theorem (Georgii-Nguyen-Zessin) For any h(·, ·, θ) : S × Ω → R, for any θ ∈ Θ, Eθ⋆

xm∈ϕ

h (xm, ϕ \ xm; θ)

• = Eθ⋆
• Rd×▼

h (xm, ϕ; θ) e−V (xm|ϕ;θ⋆)µ(dxm)

• =

⇒ for h = 1, Eθ⋆(IΛ(ϕ, θ⋆)) = 0. The raw innovations can be generalized h−innovations (in Λ) : IΛ(ϕ; h, θ⋆) :=

• xm∈ϕΛ

h(xm, ϕ \ xm; θ⋆) −

• Λ×▼

h(xm, ϕ; θ⋆)e−V (xm|ϕ;θ⋆)µ(dxm) h−residuals (in Λ) : RΛ(ϕ; h, θ) := IΛ(ϕ; h, θ) Conjecture from BTMH : RΛ(ϕ; h, θ) ≈ 0 and |Λ|1/2RΛ(ϕ; h, θ) ≈ Gaussian

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 21 / 32

Examples of residuals

RΛ(ϕ; h, θ) :=

• xm∈ϕΛ

h(xm, ϕ \ xm; θ) −

• Λ×▼

h(xm, ϕ; θ)e−V(xm|ϕ;b

θ)µ(dxm)

1 Raw residuals : h(xm, ϕ; θ) := 1 2 Inverse residuals : h(xm, ϕ; θ) := eV (xm|ϕ;θ) 3 Pearson residuals : h(xm, ϕ; θ) := eV (xm|ϕ;θ)/2

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 22 / 32

Examples of residuals

RΛ(ϕ; h, θ) :=

• xm∈ϕΛ

h(xm, ϕ \ xm; θ) −

• Λ×▼

h(xm, ϕ; θ)e−V(xm|ϕ;b

θ)µ(dxm)

1 Raw residuals : h(xm, ϕ; θ) := 1 2 Inverse residuals : h(xm, ϕ; θ) := eV (xm|ϕ;θ) 3 Pearson residuals : h(xm, ϕ; θ) := eV (xm|ϕ;θ)/2 4 Residuals based on the empty space function :

Let d(xm, ϕ) be the distance from xm to the nearest point in ϕ F(r) := P (ϕ ∩ B(0, r) = ∅) = E

• 1[0,r] (d(0m, ϕ))
• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 22 / 32

Examples of residuals

RΛ(ϕ; h, θ) :=

• xm∈ϕΛ

h(xm, ϕ \ xm; θ) −

• Λ×▼

h(xm, ϕ; θ)e−V(xm|ϕ;b

θ)µ(dxm)

1 Raw residuals : h(xm, ϕ; θ) := 1 2 Inverse residuals : h(xm, ϕ; θ) := eV (xm|ϕ;θ) 3 Pearson residuals : h(xm, ϕ; θ) := eV (xm|ϕ;θ)/2 4 Residuals based on the empty space function :

Let d(xm, ϕ) be the distance from xm to the nearest point in ϕ F(r) := P (ϕ ∩ B(0, r) = ∅) = E

• 1[0,r] (d(0m, ϕ))
• Let hr(xm, ϕ, θ) := 1[0,r] (d(xm, ϕ)) eV (xm|ϕ;θ) then

RΛn(ϕ; hr, θ) = FParam(r) − FNonParam(r)

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 22 / 32

Measures of departures to the true model

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 23 / 32

Measures of departures to the true model

1 Objective 1 : fix some test function h (ex : h = 1) and split Λn = ∪j∈J Λj 1 compute the h-residuals in each Λj 2 construct R12 where

R1 :=

• RΛj(ϕ, h,

θn)

• j∈J .
• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 23 / 32

Measures of departures to the true model

1 Objective 1 : fix some test function h (ex : h = 1) and split Λn = ∪j∈J Λj 1 compute the h-residuals in each Λj 2 construct R12 where

R1 :=

• RΛj(ϕ, h,

θn)

• j∈J .

2 Objective 2 : fix h = (h1, . . . , hs)T (ex : hj = hrj , r1 < r2 < · · · < rs)

ˆ θ is based on ϕΛn 8 > > > > > > > > > > < > > > > > > > > > > : RΛn (Φ, h1, b θn) . . . RΛn (Φ, hj , b θn) . . . RΛn (Φ, hs, b θn)

1 compute the s hj-residuals in Λn 2 construct R22 where

R2 :=

• RΛn(ϕ, hj,

θn)

• j=1,...,s .
• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 23 / 32

Asymptotic framework

Aim : asymptotic behavior of RΛn(ϕ, hj, θn), R1, R2 as Λn → Rd. = ⇒ Goodness of Fit Tests.

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 24 / 32

Asymptotic framework

Aim : asymptotic behavior of RΛn(ϕ, hj, θn), R1, R2 as Λn → Rd. = ⇒ Goodness of Fit Tests. Main assumptions : [Mod] (existence and stationarity of Pθ, ∀θ ∈ Θ) Locality assumption : ∃D > 0, V (xm|ϕ) = V (x|ϕB(0,D)) Some regularity assumptions on V and h For θn an estimate of θ⋆ computed on Λn :

• θn

a.s.

− → θ⋆ |Λn|1/2( θn − θ⋆)

d

− → T for some r.v. T

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 24 / 32

Asymptotic framework

Aim : asymptotic behavior of RΛn(ϕ, hj, θn), R1, R2 as Λn → Rd. = ⇒ Goodness of Fit Tests. Main assumptions : [Mod] (existence and stationarity of Pθ, ∀θ ∈ Θ) Locality assumption : ∃D > 0, V (xm|ϕ) = V (x|ϕB(0,D)) Some regularity assumptions on V and h For θn an estimate of θ⋆ computed on Λn :

• θn

a.s.

− → θ⋆ |Λn|1/2( θn − θ⋆)

d

− → T for some r.v. T Rmk : Fulfilled for the exponentiel family models, i.e. V (xm|ϕ; θ) = θTv(xm|ϕ), with any test function h as above and with θn = θMPLE

n

.

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 24 / 32

Asymptotic control in probability

Let

• θn an estimate of θ⋆ computed on Λn

˜ Λn ⊆ Λn such that |˜ Λn| = O(|Λn|) Proposition (Coeurjolly-L.’09)

R˜

Λn

• ϕ; h,

θn

• = I˜

Λn (ϕ; h, θ⋆) − |˜

Λn|

• θn − θ⋆T

E (h; θ⋆) + oP(|˜ Λn|1/2), where E (h; θ⋆) is a deterministic vector depending on h and V . Xn(ϕ) = oP(wn) means that w −1

n Xn(ϕ) → in probability towards 0 as n → +∞.

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 25 / 32

Framework 1 (h−residuals on subdomains of Λn))

Λn is a cube decomposable as Λn = ∪j∈J Λj,n, where |Λj,n| = O(|Λn|) We compute R1,n :=

• RΛj,n(ϕ, h,

θn)

• j∈J .

Theorem (Coeurjolly-L.’(09)) |Λ0,n|−1R1,n

a.s.

→ 0 and |Λ0,n|−1/2 R1,n

d

→ N(0, Σ1). where Σ1 = λInn I|J | + |J |−1(λRes − λInn) J (J = eeT, e = (1, . . . , 1)T), λInn ≥ 0 is a covariance depending on h and V , λRes ≥ 0 is a covariance depending on h, V and the asympt. behavior of θn.

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 26 / 32

Quadrat-counting based tests

Corollary (C.-Lavancier’09) Test 1 : If λInn > 0 and λRes > 0 then |Λ0,n|−1 Σ−1/2

1

R1,n2

d

→ χ2

|J |,

with Σ−1/2

1

= λ−1/2

Inn I|J | + (

λ−1/2

Res

− λ−1/2

Inn )J.

Test 2 : If λInn > 0 then |Λ0,n|−1 λ−1

Inn × R1,n − R1,n2 d

→ χ2

|J |−1.

where λInn and λRes are“natural”estimates of λInn and λRes. Rmk : from a practical and theoretical point of view, Test 2 >> Test 1.

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 27 / 32

Analogy with the Poisson dispersion test

Let a homogeneous Poisson point process : V (x|ϕ) = θ ⇒ its intensity is e−θ.

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 28 / 32

Analogy with the Poisson dispersion test

Let a homogeneous Poisson point process : V (x|ϕ) = θ ⇒ its intensity is e−θ. Let xj = |ϕΛj,n| be the j-th quadrat count Let x (resp . s2) the sample mean (resp. variance) over the |J | quadrats.

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 28 / 32

Analogy with the Poisson dispersion test

Let a homogeneous Poisson point process : V (x|ϕ) = θ ⇒ its intensity is e−θ. Let xj = |ϕΛj,n| be the j-th quadrat count Let x (resp . s2) the sample mean (resp. variance) over the |J | quadrats. For h = 1, we get R1,n − R1,n = (x − xj)j∈J and λInn = e−θ which can be estimated by

• λInn = |ϕΛn|

|Λn| =

• j∈J |ϕΛj,n|

|Λ0,n| × |J | = |Λ0,n|−1 x

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 28 / 32

Analogy with the Poisson dispersion test

Let a homogeneous Poisson point process : V (x|ϕ) = θ ⇒ its intensity is e−θ. Let xj = |ϕΛj,n| be the j-th quadrat count Let x (resp . s2) the sample mean (resp. variance) over the |J | quadrats. For h = 1, we get R1,n − R1,n = (x − xj)j∈J and λInn = e−θ which can be estimated by

• λInn = |ϕΛn|

|Λn| =

• j∈J |ϕΛj,n|

|Λ0,n| × |J | = |Λ0,n|−1 x Therefore |Λ0,n|−1 λ−1

Inn × R1,n(ϕ; h) − R1,n(ϕ; h)2 = (|J | − 1)s2/x

is the dispersion statistic test. Test 2 is a generalization of the quadrat counting test to (marked) Gibbs p.p.

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 28 / 32

Framework 2 ((h1, . . . , hs)−residuals on Λn))

8 > > > > > > > > > > < > > > > > > > > > > : RΛn (Φ, h1, b θn) . . . RΛn (Φ, hj , b θn) . . . RΛn (Φ, hs, b θn)

Take s different test-functions hj Compute each hj-residuals from Λn Construct R2,n :=

• RΛn(ϕ, hj,

θn)

• j=1,...,s .

Theorem (Coeurjolly-L.’(09)) |Λn|−1R2,n(ϕ; h)

a.s.

→ 0 and |Λn|−1/2 R2,n(ϕ; h)

d

→ N(0, Σ2), where Σ2 is a matrix depending on h, V and the asymptotic behavior of θn

Corollary (Empty space function type test) Test 3 : Under some conditions ensuring Σ2 definite-positive |Λn|−1 Σ−1/2

2

R2,n(ϕ; h)2

d

→ χ2

|J |,

where Σ2 is a ” natural”estimate of Σ2

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 29 / 32

Theoretical Ingredient

Theorem (Coeurjolly-L.’(09)) Let Xn,i, n ∈ N, i ∈ Zd, be a triangular array field in a measurable space S. For n ∈ N, let Kn ⊂ Zd and for k ∈ Kn, assume Zn,k = fn,k (Xn,k+i, i ∈ I0) , where I0 = {i ∈ Zd, |i| ≤ 1} and fn,k : SI0 → Rp. Let Sn =

k∈Kn Zn,k. If

(i) c3 := supn∈N supk∈Kn E|Zn,k|3 < ∞, (ii) ∀n ∈ N, ∀k ∈ Kn, E(Zn,k|Xn,j, j = k) = 0, (iii) |Kn| → +∞ as n → ∞, (iv) There exists a symmetric matrix Σ ≥ 0 such that E

• |Kn|−1

k∈Kn

• j∈❇k(1)∩Kn

Zn,kZn,j

T − Σ

• → 0,

then |Kn|−1/2Sn

d

− → N (0, Σ) as n → ∞.

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 30 / 32

Conclusion

h-residuals can be computed for a wide class of Gibbs models They yield some diagnostic tools to check the quality of the model The asymptotic law of the h-residuals is obtained in a large setting This is the basis to construct several goodness of fit tests, including a generalization of the quadrat counting test. Perspectives Improve the estimation of the covariance matrices Simulation study to assess the power of tests and compare them

• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 31 / 32

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• F. Lavancier ()

Residuals and GoF tests for Gibbs pp 01/02/2010 32 / 32