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 ( R 2 ) denotes the space of bounded sets in R 2 .

Introduction Definitions Simulation Estimation Theory Notations B ( R 2 ) denotes the space of bounded sets in R 2 . M ( R 2 ) is the space of locally finite point configurations γ in R 2 : γ ⊂ R 2 , such that for all Λ ∈ B ( R 2 ) , card ( γ ∩ Λ) < ∞ .

Introduction Definitions Simulation Estimation Theory Notations B ( R 2 ) denotes the space of bounded sets in R 2 . M ( R 2 ) is the space of locally finite point configurations γ in R 2 : γ ⊂ R 2 , such that for all Λ ∈ B ( R 2 ) , card ( γ ∩ Λ) < ∞ . γ Λ is the restriction of γ on Λ : γ Λ = γ ∩ Λ .

Introduction Definitions Simulation Estimation Theory Notations B ( R 2 ) denotes the space of bounded sets in R 2 . M ( R 2 ) is the space of locally finite point configurations γ in R 2 : γ ⊂ R 2 , such that for all Λ ∈ B ( R 2 ) , card ( γ ∩ Λ) < ∞ . γ Λ is the restriction of γ on Λ : γ Λ = γ ∩ Λ . Vor ( γ ) : Voronoï tessellation coming from γ

Introduction Definitions Simulation Estimation Theory Notations B ( R 2 ) denotes the space of bounded sets in R 2 . M ( R 2 ) is the space of locally finite point configurations γ in R 2 : γ ⊂ R 2 , such that for all Λ ∈ B ( R 2 ) , card ( γ ∩ Λ) < ∞ . γ Λ is the restriction of γ on Λ : γ Λ = γ ∩ Λ . Vor ( γ ) : Voronoï tessellation coming from γ λ is the Lebesgue measure on R 2 .

Introduction Definitions Simulation Estimation Theory Notations B ( R 2 ) denotes the space of bounded sets in R 2 . M ( R 2 ) is the space of locally finite point configurations γ in R 2 : γ ⊂ R 2 , such that for all Λ ∈ B ( R 2 ) , card ( γ ∩ Λ) < ∞ . γ Λ is the restriction of γ on Λ : γ Λ = γ ∩ Λ . Vor ( γ ) : Voronoï tessellation coming from γ λ is the Lebesgue measure on R 2 . For z > 0 , π z : Poisson point process with intensity zλ

Introduction Definitions Simulation Estimation Theory Notations B ( R 2 ) denotes the space of bounded sets in R 2 . M ( R 2 ) is the space of locally finite point configurations γ in R 2 : γ ⊂ R 2 , such that for all Λ ∈ B ( R 2 ) , card ( γ ∩ Λ) < ∞ . γ Λ is the restriction of γ on Λ : γ Λ = γ ∩ Λ . Vor ( γ ) : Voronoï tessellation coming from γ λ is the Lebesgue measure on R 2 . For z > 0 , π z : Poisson point process with intensity zλ Λ : π z restricted on Λ . π z

Introduction Definitions Simulation Estimation Theory Gibbs measures Let ( H Λ ) Λ ∈B ( R 2 ) be a family of energies M (Λ) × M (Λ c ) H Λ : − → R ∪ { + ∞} ( γ Λ , γ Λ c ) �− → H Λ ( γ Λ | γ Λ c ) We suppose that it is compatible. For every Λ ⊂ Λ ′ H Λ ′ ( γ Λ ′ | γ Λ ′ c ) = H Λ ( γ Λ | γ Λ c ) + ϕ Λ , Λ ′ ( γ Λ c ) .

Introduction Definitions Simulation Estimation Theory Gibbs measures Let ( H Λ ) Λ ∈B ( R 2 ) be a family of energies M (Λ) × M (Λ c ) H Λ : − → R ∪ { + ∞} ( γ Λ , γ Λ c ) �− → H Λ ( γ Λ | γ Λ c ) We suppose that it is compatible. For every Λ ⊂ Λ ′ H Λ ′ ( γ Λ ′ | γ Λ ′ c ) = H Λ ( γ Λ | γ Λ c ) + ϕ Λ , Λ ′ ( γ Λ c ) . Definition A probability measure P on M ( R 2 ) is a Gibbs measure for z > 0 and ( H Λ ) if for every Λ ∈ B ( R 2 ) and P -almost every γ Λ c 1 Z Λ ( γ Λ c ) e − H Λ ( γ Λ | γ Λ c ) π z P ( dγ Λ | γ Λ c ) = Λ ( dγ Λ ) , e − H Λ ( γ ′ Λ | γ Λ c ) π Λ ( dγ ′ � where Z Λ ( γ Λ c ) = Λ ) .

Introduction Definitions Simulation Estimation Theory A typical energy of a Voronoï tessellation : � � V 2 ( C, C ′ ) . H Λ ( γ Λ | γ Λ c ) = V 1 ( C ) + C ∈ Vor ( γ ) C,C ′ ∈ Vor ( γ ) C ∩ Λ � = ∅ C and C ′ are neighbors ( C ∪ C ′ ) ∩ Λ � = ∅

Introduction Definitions Simulation Estimation Theory A typical energy of a Voronoï tessellation : � � V 2 ( C, C ′ ) . H Λ ( γ Λ | γ Λ c ) = V 1 ( C ) + C ∈ Vor ( γ ) C,C ′ ∈ Vor ( γ ) C ∩ Λ � = ∅ C and C ′ are neighbors ( C ∪ C ′ ) ∩ Λ � = ∅ Our guiding example : C  + ∞ if h min ( C ) ≤ ε  x  + ∞ if h max ( C ) ≥ α  h max V 1 ( C ) = if h 2 + ∞ max ( C ) / V ol ( C ) ≥ B h min   0 otherwise  √ 0 < ε < α , B > 1 / 2 3 ;

Introduction Definitions Simulation Estimation Theory A typical energy of a Voronoï tessellation : � � V 2 ( C, C ′ ) . H Λ ( γ Λ | γ Λ c ) = V 1 ( C ) + C ∈ Vor ( γ ) C,C ′ ∈ Vor ( γ ) C ∩ Λ � = ∅ C and C ′ are neighbors ( C ∪ C ′ ) ∩ Λ � = ∅ Our guiding example : C  + ∞ if h min ( C ) ≤ ε  x  + ∞ if h max ( C ) ≥ α  h max V 1 ( C ) = if h 2 + ∞ max ( C ) / V ol ( C ) ≥ B h min   0 otherwise  √ 0 < ε < α , B > 1 / 2 3 ; � 1 � max( V ol ( C ) , V ol ( C ′ )) 2 V 2 ( C, C ′ ) = θ min( V ol ( C ) , V ol ( C ′ )) − 1 , 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.