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 Type of data of interest 1 Gibbs models 2 Brief background Examples Identification 3 Maximum likelihood Pseudo-likelihood Validations through residuals 4 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 Type of data of interest 1 Gibbs models 2 Brief background Examples Identification 3 Maximum likelihood Pseudo-likelihood Validations through residuals 4 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 Type of data of interest 1 Gibbs models 2 Brief background Examples Identification 3 Maximum likelihood Pseudo-likelihood Validations through residuals 4 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 = R d × ▼ associated to µ = λ ⊗ λ ♠ . Let x m = ( 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 = R d × ▼ associated to µ = λ ⊗ λ ♠ . Let x m = ( x , m ) an element of S , i.e. a marked point. Ω is the space of locally finite point configurations ϕ in S : � ⇒ ∃ I ⊂ N , ∃ ( x m i ) i ∈I ∈ S I , ϕ = ϕ ∈ Ω ⇐ δ x i . mi i i ∈I F. Lavancier () Residuals and GoF tests for Gibbs pp 01/02/2010 8 / 32

Marked point processes State space : S = R d × ▼ associated to µ = λ ⊗ λ ♠ . Let x m = ( x , m ) an element of S , i.e. a marked point. Ω is the space of locally finite point configurations ϕ in S : � ⇒ ∃ I ⊂ N , ∃ ( x m i ) i ∈I ∈ S I , ϕ = ϕ ∈ Ω ⇐ δ x i . mi i i ∈I We write x m ∈ ϕ if ϕ ( { x m } ) = 1. For Λ ∈ B ( R d ), ϕ Λ is the restriction of ϕ on Λ : � ϕ Λ = δ x m x m ∈ ϕ ∩ (Λ × ▼ ) F. Lavancier () Residuals and GoF tests for Gibbs pp 01/02/2010 8 / 32

Marked point processes State space : S = R d × ▼ associated to µ = λ ⊗ λ ♠ . Let x m = ( x , m ) an element of S , i.e. a marked point. Ω is the space of locally finite point configurations ϕ in S : � ⇒ ∃ I ⊂ N , ∃ ( x m i ) i ∈I ∈ S I , ϕ = ϕ ∈ Ω ⇐ δ x i . mi i i ∈I We write x m ∈ ϕ if ϕ ( { x m } ) = 1. For Λ ∈ B ( R d ), ϕ Λ is the restriction of ϕ on Λ : � ϕ Λ = δ x m x m ∈ ϕ ∩ (Λ × ▼ ) 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 π z with intensity z λ is defined by � | π z Λ | := π z (Λ) ∼ P ( z λ (Λ)) ∀ Λ , ∀ Λ , Λ ′ with Λ ∩ Λ ′ = ∅ , π z Λ and π z 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 π z with intensity z λ is defined by � | π z Λ | := π z (Λ) ∼ P ( z λ (Λ)) ∀ Λ , ∀ Λ , Λ ′ with Λ ∩ Λ ′ = ∅ , π z Λ and π z 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 π z with intensity z λ is defined by � | π z Λ | := π z (Λ) ∼ P ( z λ (Λ)) ∀ Λ , ∀ Λ , Λ ′ with Λ ∩ Λ ′ = ∅ , π z Λ and π z 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 ( V Λ ) Λ ∈B ( R d ) be a family of energies V Λ : Ω − → R ∪ { + ∞} ϕ �− → V Λ ( ϕ Λ | ϕ Λ c ) V Λ ( ϕ ) is the energy of ϕ Λ inside Λ knowing the outside configuration ϕ Λ c F. Lavancier () Residuals and GoF tests for Gibbs pp 01/02/2010 10 / 32

Gibbs measures Let ( V Λ ) Λ ∈B ( R d ) be a family of energies V Λ : Ω − → R ∪ { + ∞} ϕ �− → V Λ ( ϕ Λ | ϕ Λ c ) V Λ ( ϕ ) is the energy of ϕ Λ inside Λ knowing the outside configuration ϕ Λ c Definition A probability measure P on Ω is a Gibbs measure ( V Λ ) if for every bounded set Λ and P -almost every ϕ Λ c P ( d ϕ Λ | ϕ Λ c ) = e − V Λ ( ϕ Λ | ϕ Λ c ) π Λ ( d ϕ Λ ) , (1) Z Λ ( ϕ Λ c ) � e − V Λ ( ϕ Λ | ϕ Λ c ) π Λ ( d ϕ Λ ). where Z Λ ( ϕ Λ c ) = The equations (1) for all Λ are called DLR (Dobrushin,Landford,Ruelle). Z Λ 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 ( V Λ ) Λ ∈B ( R d ) entirely defines the Gibbs measure P . But, given ( V Λ ) : 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 ( V Λ ) Λ ∈B ( R d ) entirely defines the Gibbs measure P . But, given ( V Λ ) : Is there exist a Gibbs measure P ? Is it unique (phase transition problem) ? We consider parametric families of Gibbs measures P θ , θ ∈ Θ. We assume : [Mod] : For any θ ∈ Θ ⊂ R p , ( V Λ ( . ; θ )) Λ ∈B ( R [ d ]) are invariant by translation, and such that at least one associated Gibbs measure P θ 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 ( V Λ ) Λ ∈B ( R d ) entirely defines the Gibbs measure P . But, given ( V Λ ) : Is there exist a Gibbs measure P ? Is it unique (phase transition problem) ? We consider parametric families of Gibbs measures P θ , θ ∈ Θ. We assume : [Mod] : For any θ ∈ Θ ⊂ R p , ( V Λ ( . ; θ )) Λ ∈B ( R [ d ]) are invariant by translation, and such that at least one associated Gibbs measure P θ exists and is stationary. We denote by θ ⋆ the true parameter to be estimated, assumed to be in ˚ Θ. Remark : General conditions on ( V Λ ( . ; 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 } V Λ ( ϕ ; θ ) = θ 1 | ϕ Λ | F. Lavancier () Residuals and GoF tests for Gibbs pp 01/02/2010 12 / 32

Multi-type Poisson point process, ▼ = { 1 , 2 } V Λ ( ϕ ; θ ) = θ 1 1 | ϕ 1 Λ | + θ 2 1 | ϕ 2 Λ | F. Lavancier () Residuals and GoF tests for Gibbs pp 01/02/2010 13 / 32

Strauss marked point process, ▼ = { 1 , 2 } 2 2 � � � θ m 1 | ϕ m θ m 1 , m 2 V Λ ( ϕ ; θ ) = Λ | + 1 [0 , D m 1 , m 2 ] ( || y − x || ) 2 m =1 m 1 , m 2 =1 { x m 1 , y m 2 }∈P 2 ( ϕ Λ ) F. Lavancier () Residuals and GoF tests for Gibbs pp 01/02/2010 14 / 32

Gibbs Voronoi tessellation (Dereudre, L.) � � V 2 ( C , C ′ ) V Λ ( ϕ ) = V 1 ( C ) + C ∈ Vor( ϕ ) C , C ′ ∈ Vor( ϕ ) C ∩ Λ � = ∅ C and C ′ are neighbors ( C ∪ C ′ ) ∩ Λ � = ∅ V 1 ( C ) : deals with the shape of the cell and V 2 ( 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 Type of data of interest 1 Gibbs models 2 Brief background Examples Identification 3 Maximum likelihood Pseudo-likelihood Validations through residuals 4 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 ϕ Λ n , a realisation of the point process in Λ n . 1 θ MLE ˆ Z ( θ ) e − V ( ϕ Λ n ; θ ) = argmax n θ ∈ Θ where Z ( θ ) is an untractable normalizing constant. F. Lavancier () Residuals and GoF tests for Gibbs pp 01/02/2010 17 / 32