Gibbs Voronoi tessellations: Modeling, Simulation, Estimation.

Transcription

1 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.

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

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

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

5 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

6 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?

7 Introduction One solution : Gibbs modifications of Poisson Voronoï tessellations.

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

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

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

11 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.

12 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.

13 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.

14 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.

15 2 Definitions

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

17 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) <.

18 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.

19 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

20 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.

21 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

22 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 πλ z : πz restricted on Λ.

23 Gibbs measures Let (H Λ ) Λ B(R 2 ) be a family of energies H Λ : M(Λ) M(Λ c ) 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).

24 Gibbs measures Let (H Λ ) Λ B(R 2 ) be a family of energies H Λ : M(Λ) M(Λ c ) 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 P (dγ Λ γ Λ c) = 1 Z Λ (γ Λ c) e H Λ(γ Λ γ Λc ) π z Λ(dγ Λ ), where Z Λ (γ Λ c) = e H Λ(γ Λ γ Λ c ) π Λ (dγ Λ ).

25 A typical energy of a Voronoï tessellation : H Λ (γ Λ γ Λ c) = C Vor(γ) C Λ V 1 (C) + C,C Vor(γ) C and C are neighbors (C C ) Λ V 2 (C, C ).

26 A typical energy of a Voronoï tessellation : H Λ (γ Λ γ Λ c) = C Vor(γ) C Λ V 1 (C) + C,C Vor(γ) C and C are neighbors (C C ) Λ V 2 (C, C ). Our guiding example : + if h min (C) ε + if h V 1 (C) = max (C) α + if h 2 max(c)/vol(c) B 0 otherwise x h min C h max 0 < ε < α, B > 1/2 3 ;

27 A typical energy of a Voronoï tessellation : H Λ (γ Λ γ Λ c) = C Vor(γ) C Λ V 1 (C) + C,C Vor(γ) C and C are neighbors (C C ) Λ V 2 (C, C ). Our guiding example : + if h min (C) ε + if h V 1 (C) = max (C) α + if h 2 max(c)/vol(c) B 0 otherwise x h min C h max 0 < ε < α, B > 1/2 3 ; ( max(vol(c), Vol(C V 2 (C, C ) 1 )) ) = θ min(vol(c), Vol(C )) 1 2, θ R

28 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, ) Existence of Delaunay pairwise Gibbs point process with superstable component, J. Stat. phys. (1999) Vol 95 3/

29 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, ) Existence of Delaunay pairwise Gibbs point process with superstable component, J. Stat. phys. (1999) Vol 95 3/ Existence results with hardcore interactions (B = + ) : Dereudre, Gibbs Delaunay tessellations with geometric hardcore conditions, J. Stat. Phys.,(2008) 131,

30 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, ) Existence of Delaunay pairwise Gibbs point process with superstable component, J. Stat. phys. (1999) Vol 95 3/ Existence results with hardcore interactions (B = + ) : Dereudre, Gibbs Delaunay tessellations with geometric hardcore conditions, J. Stat. Phys.,(2008) 131, In the general case : Dereudre, Drouilhet and Georgii, Existence of Gibbs process with stable cluster interactions, preprint.

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

32 3 Simulation

33 Simulations Strong hardcore interaction

34 Simulations Strong hardcore interaction Rigidity of the tessellation

35 Simulations Strong hardcore interaction Rigidity of the tessellation Several difficulties for the simulations.

36 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, then γ+x γ otherwise "do nothing". (n + 1)f(γ) 3 If 1/3 < a < 2/3 then generate x on γ and if b < nf(γ x), then γ x γ otherwise "do nothing". f(γ)z 4 If a > 2/3 then generate x on γ, y N (x, σ 2 ) and if b < f(γ x + y), then γ x+y γ otherwise "do nothing". f(γ)

37 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

38 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

39 4 Estimation

40 Pseudo-likelihood Estimation The aim : Estimate the parameters of the interaction from one realization γ of the Gibbs measure.

41 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.

42 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.

43 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?

44 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).

45 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).

46 Practical estimation procedures Let Λ n = [ n, n] 2 be the observation window and γ a realization of the Gibbs measure P.

47 Practical estimation procedures Let Λ n = [ n, n] 2 be the observation window and γ a realization of the Gibbs measure P. -Hardcore parameter estimators : ˆε = min{h min (C), C V or(γ) and C Λ n }, ˆα = max{h max (C), C V or(γ) and C Λ n }, ˆB = max{h 2 max(c)/vol(c), C V or(γ) and C Λ n }.

48 Practical estimation procedures Let Λ n = [ n, n] 2 be the observation window and γ a realization of the Gibbs measure P. -Hardcore parameter estimators : ˆε = min{h min (C), C V or(γ) and C Λ n }, ˆα = max{h max (C), C V or(γ) and C Λ n }, ˆB = max{h 2 max(c)/vol(c), C V or(γ) and C Λ n }. -Smooth parameter estimators : (ẑ, ˆθ) = argmin z,θ P LL Λn (γ, z, θ, ˆε, ˆα, ˆB),

49 Practical estimation procedures Let Λ n = [ n, n] 2 be the observation window and γ a realization of the Gibbs measure P. -Hardcore parameter estimators : ˆε = min{h min (C), C V or(γ) and C Λ n }, ˆα = max{h max (C), C V or(γ) and C Λ n }, ˆB = max{h 2 max(c)/vol(c), C V or(γ) and C Λ n }. -Smooth parameter estimators : (ẑ, ˆθ) = argmin z,θ P LL Λn (γ, z, θ, ˆε, ˆα, ˆB), with P LL Λn () = z exp ( h(x, γ)) dx+ Λ n where h(x, γ) = H Λn (γ + x) H Λn (γ). x γ Λn H Λn (γ x)< ( h(x, γ x) ln(z) ),

50 Theoretical results For the hardcore parameters : Theorem (Dereudre-L. (2009)) For P -almost all γ lim (ˆε, ˆα, ˆB) = (ε, α, B). n

51 Theoretical results For the hardcore parameters : Theorem (Dereudre-L. (2009)) For P -almost all γ For the smooth parameters : Theorem (Dereudre-L. (2009)) For P -almost all γ lim (ˆε, ˆα, ˆB) = (ε, α, B). n lim (ẑ, ˆθ) = (z, θ). n (ẑ, ˆθ) are asymptotic normal if ε, α and B are supposed to be known.

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

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

54 Estimation results The true parameters : ε = 0, α = 0.05, B = 0.625, z = 100 and θ = 0.5. Smooth parameter estimators : ˆθ when z is known ˆθ when z is estimated ẑ

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

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

57 Estimation results The true parameters : ε = 0, α = 0.05, B = 0.625, z = 100 and θ = 0.5. Smooth parameter estimators : ˆθ when z is known ˆθ when z is estimated ẑ

58 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.

59 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.

60 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.

61 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.

62 A. Baddeley, R. Turner, J. Moller and M. Hazelton, Residual analysis for spatial point processes, J. R. Statist. Soc. B, 65, (2005). E. Bertin, J.M. Billiot, R. Drouilhet, (1999)Existence of nearest neighbors spatial Gibbs models, Adv. Appl. Prob. (SGSA) 31, 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, D. Dereudre, (2008) Gibbs Delaunay tessellations with geometric hardcore conditions, J. Stat. Phys., 131, 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, C. Preston, Random Fields, LNM Vol. 534, Springer.

63 5 Some theoretical points

64 The problem of heredity Definition The family of energies (H Λ ) Λ is said hereditary if for every Λ, every γ M(R 2 ) and every x Λ H Λ (γ) = + H Λ (γ + δ x ) = +.

65 The problem of heredity Definition The family of energies (H Λ ) Λ is said hereditary if for every Λ, every γ M(R 2 ) and every x Λ H Λ (γ) = + H Λ (γ + δ x ) = +. γ is forbidden γ + δ x is forbidden

66 The problem of heredity Definition The family of energies (H Λ ) Λ is said hereditary if for every Λ, every γ M(R 2 ) and every x Λ H Λ (γ) = + H Λ (γ + δ x ) = +. γ is forbidden γ + δ x is forbidden γ + δ x is allowed γ is allowed

67 The problem of heredity Definition The family of energies (H Λ ) Λ is said hereditary if for every Λ, every γ M(R 2 ) and every x Λ H Λ (γ) = + H Λ (γ + δ x ) = +. γ is forbidden γ + δ x is forbidden γ + δ x is allowed γ is allowed It is a standard assumption in classical statistical mechanics. (Example : The classical hard ball model is hereditary.)

68 The problem of heredity Definition The family of energies (H Λ ) Λ is said hereditary if for every Λ, every γ M(R 2 ) and every x Λ H Λ (γ) = + H Λ (γ + δ x ) = +. γ is forbidden γ + δ x is forbidden γ + δ x 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.

69 Nguyen-Zessin equilibrium equation Theorem (Nguyen-Zessin (1979), hereditary case) Suppose that the energy (H Λ ) Λ is hereditary. P is Gibbs measure with intensity measure ν if and only if, for every bounded non negative measurable function ψ from R 2 M(R 2 ) to R, ( ) ( ) E P ψ(x, γ x) = E P ψ(x, γ)e h(x,γ) ν(dx), x γ R 2 where h(x, γ) = H Λn (γ + x) H Λn (γ). Proposition (Dereudre, L. (2009), general case) Let P be a Gibbs measure with intensity measure ν, then ( ) E P ψ(x, γ x) = E P ψ(x, γ)e h(x,γ) ν(dx). R 2 x γ Λn H Λn (γ x)<

70 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, ĥ, ˆν = x γ H (γ x)< ψ(x, γ x) ψ(x, γ)e ĥ(x,γ)ˆν(dx), From the equilibrium equation given before, under the true model, ) R (, ψ, ĥ, ˆν 0 ) R (, ψ, ĥ, ˆν is approximatively gaussian. Several diagnostic tools can then be applied when fitting a Gibbs Voronoi model

71 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, ĥ, ˆν = x γ H (γ x)< ψ(x, γ x) ψ(x, γ)e ĥ(x,γ)ˆν(dx), From the equilibrium equation given before, under the true model, ) R (, ψ, ĥ, ˆν 0 ) R (, ψ, ĥ, ˆν 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.