Semi-parametric estimation of the Poisson intensity parameter for - PowerPoint PPT Presentation

Semi-parametric estimation of the Poisson intensity parameter for stationary Gibbs point processes Nadia Morsli and Jean-Fran¸ cois Coeurjolly. Laboratory Jean Kuntzmann, Grenoble University, France. nadia.morsli@imag.fr; jean-francois.coeurjolly@upmf-grenoble.fr

Stationary Gibbs point processes on R d ◮ We define a point process X in R d as a locally finite random subset of R d , i.e. N (Λ) = n ( X Λ ) is a finite random variable whenever Λ ⊂ R d is a bounded region. ◮ If the distribution of X is translation invariant, we say that X is stationary. ◮ we are interested in stationary Gibbs point processes on R d which may be defined through of Papangelou conditional intensity λ : R d × N lf − → R + . ◮ The Papangelou conditional intensity has the interpretation that λ ( u , x ) du as the probability that the process X has to send a point in a region of du around a point u which also respects the existing configuration outside of the du .

Examples Gibbs point processes ◮ Strauss point process: λ ( u , x ) = βγ n [0 , R ] ( u , x ) where β > 0 , γ ∈ [0 , 1] , n [0 , R ] ( u , x ) = � v ∈ x Λ 1 ( � v − u � ≤ R ) . ◮ Strauss point process with Hard-Core: ◮ If all points are at distance greater than δ from each other λ ( u , x ) = βγ n [0 , R ] ( u , x ) . ◮ otherwise λ ( u , x ) = 0 . ◮ Piecewise Strauss point process: p � n [ Rj − 1 , Rj ] ( u , x ) λ ( u , x ) = β γ j j =1 where n [ R j − 1 , R j ] ( u , x ) = � v ∈ x Λ 1 ( � v − u � ∈ [ R j − 1 , R j ]) where R 0 = 0 < R 1 < . . . < R p < + ∞ .

Position of the problem We consider Gibbs models such that the Papangelou conditional intensity can be written for u ∈ R d and x ∈ N lf λ ( u , x ; β ⋆ ) = β ⋆ � λ ( u , x ) , where ◮ β ⋆ is the ”Poisson intensity” parameter. λ is a function from R d × N lf to R + . ◮ � ◮ [FR] The Papangelou conditional intensity satisfies λ ( u , x ; β ⋆ ) = λ ( u , x B ( u , R ) ; β ⋆ ) , for any u ∈ R d , x ∈ N lf and such that � λ ( u , ∅ ) = 1. We propose to estimate β ⋆ independently of � λ based on a single observation of a stationary Gibbs point process in R d , � � denoted X , in a domain Λ n R , where (Λ n ) n ≥ 0 is a sequence of increasing cubes and � R ≥ R and we do not assume R known, but only know an upper bound � R .

Definition of the estimator ◮ For all nonnegative measurable functions h on R d × N lf , then �� � �� � R d h ( u , X ) λ ( u , X ; β ⋆ ) du E h ( u , X \ u ) = E (1) u ∈ X (where the left hand side is finite if and only if the right hand side is finite). ◮ With the choice of h defined by � � X ∩ B ( u , � R ) = ∅ [CH] : h ( u , X ) = 1 .

    N Λ n ( X ; � R ) � �� �  � �    1 (( X \ u ) ∩ B ( u , �   E h ( u , X \ u ) = E R ) = ∅ )   u ∈ X Λ n u ∈ X Λ n �� � � � β ⋆ E X ∩ B ( u , � � R ) = ∅ = 1 λ ( u , X ) du Λ n �� � β ⋆ E 1 ( X ∩ B ( u , � R ) = ∅ ) � = λ ( u , X B ( u , R ) ) du Λ n �� � � β ⋆ E = λ ( u , ∅ ) du { u , X ∩ B ( u , � R )= ∅}     �     1 ( X ∩ B ( u , � β ⋆ E = R ) = ∅ ) du   .     Λ n � �� � V Λ n ( X ; � R )

Consistency of the estimator With the ergodic theorem suggests to estimate β ⋆ by the estimator R ) = | Λ n | − 1 N Λ n ( X ; � R ) β n ( X ; � � . | Λ n | − 1 V Λ n ( X ; � R ) Proposition Let X be stationary Gibbs point process, under the assumptions [FR] and [CH] . Then for any fixed 0 < R < � R < + ∞ , the R ) of parameter β ⋆ is strongly consistent. estimator � β n ( X ; �

Asymptotic normality of the estimator Proposition Let X be stationary (ergodic) Gibbs point process, under the assumptions [FR] and [CH] . Then we have, for any fixed 0 < R ≤ � R < + ∞ , as n → + ∞ and as | Λ n | → + ∞ , � R ) − β ⋆ � � β n ( X ; � � d → N (0 , σ 2 ( β ⋆ )) , | Λ n | − where � � � I ∆ 0 ( R ) ( X , h ; β ⋆ ) I ∆ j ( R ) ( X , h ; β ⋆ ) E j ∈ B (0 , 1) ◮ σ 2 ( β ⋆ ) = , 2 R d ( 1 − F ( � R ) ) � � ⋆ ) = ◮ I ∆ k ( R ) ( X , h , β h ( u , X \ u ) − h ( u , X ) λ ( u , X ) du . ∆ k ( R ) u ∈ X ∆ k ( R )

By using the results of JF.Coeurjolly and E. Rubak (2012), we can calculate the value σ 2 ( β ⋆ ) differently as follows: Proposition Let X be stationary Gibbs point process. Under the assumptions [FR] and [CH] , we have for any fixed 0 < R < � R < + ∞ , R )) + β ⋆ 2 � β ⋆ (1 − F ( � (1 − F 0 , v ( � R )) dv B (0 , � R ) σ 2 ( β ⋆ ) = , � � 2 1 − F ( � R ) where ◮ F ( � R ) = P β ⋆ ( X ∩ B (0 , � R ) � = ∅ ) . ◮ F 0 , v ( � R ) = P β ⋆ ( X ∩ B (0 , � R ) � = ∅ , X ∩ B ( v , � R ) � = ∅ ) .

Simulation study Strauss point process: λ ( u , x ) = βγ n [0 , R ]( u , x ) ◮ s1 : β = 200, γ = 0 . 2, R = ϕ . ◮ s2 : β = 200, γ = 0 . 5, R = ϕ . ◮ s3 : β = 200, γ = 0 . 8, R = ϕ . 400 L=1 ● L=2 L=1 ● ● L=2 L=1 ● L=2 350 ● ● 400 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 350 ● ● ● ● ● ● ● ● 350 ● ● ● ● 300 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 300 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 300 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 250 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 250 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 250 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 200 ● 200 ● ● ● ● ● 200 ● ● ● ● ● ● 150 150 ● ● 150 ● ● ● ● ● ● ● ● ● ● 100 Model s1 Model s2 Model s3 100 ● 100 0.8 0.9 1 1.1 1.2 0.8 0.9 1 1.1 1.2 0.8 0.9 1 1.1 1.2 0.8 0.9 1 1.1 1.2 0.8 0.9 1 1.1 1.2 0.8 0.9 1 1.1 1.2 Factor of the finite range Factor of the finite range Factor of the finite range Boxplots of the Poisson intensity parameter estimates for different parameters � R from 0 . 8 to 1 . 2 times the finite range parameter ϕ , from 500 replications of the models s1,s2,s3 generated on the window [0 , L ] 2 ⊕ 1 . 2 ϕ and estimated on the window [0 , L ] 2 for L = 1 , 2.

Strauss point process with Hard-Core: λ ( u , x ) = βγ n [0 , R ]( u , x ) ◮ shc1 : β = 200, γ = 0 . 2, δ = ϕ/ 2, R = ϕ . ◮ shc2 : β = 200, γ = 0 . 5, δ = ϕ/ 2, R = ϕ . ◮ shc3 : β = 200, γ = 0 . 8, δ = ϕ/ 2, R = ϕ . 400 500 L=1 ● L=2 L=1 ● L=2 L=1 ● L=2 ● ● ● ● ● ● ● ● ● 350 300 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 400 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 300 ● ● ● ● ● ● ● ● ● ● ● ● 250 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 250 ● ● ● ● ● ● ● ● ● ● ● 300 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 200 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 200 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 200 150 150 ● ● ● ● ● ● ● ● ● 100 ● 100 Model shc1 Model shc2 100 Model shc3 ● ● 0.8 0.9 1 1.1 1.2 0.8 0.9 1 1.1 1.2 0.8 0.9 1 1.1 1.2 0.8 0.9 1 1.1 1.2 0.8 0.9 1 1.1 1.2 0.8 0.9 1 1.1 1.2 Factor of the finite range Factor of the finite range Factor of the finite range Boxplots of the Poisson intensity parameter estimates for different parameters � R from 0 . 8 to 1 . 2 times the finite range parameter ϕ , from 500 replications of the models shc1,shc2,shc3 generated on the window [0 , L ] 2 ⊕ 1 . 2 ϕ and estimated on the window [0 , L ] 2 for L = 1 , 2.