Jean-Michel Billiot, Jean-François Coeurjolly and Rémy Drouilhet

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1 Colloque International de Statistique Appliquée pour le Développement en Afrique International Conference on Applied Statistics for Development in Africa Sada 07 nn, 1?? 007) MAXIMUM PSEUDO-LIKELIHOOD ESTIMATOR FOR GIBBS POINT PROCESSES Jean-Michel Billiot, Jean-François Coeurjolly and Rémy Drouilhet Abstract. The purpose of this paper is a statistical study of spatial Gibbs point processes. Our framework is general so that it includes a large variety of classical models and some new models based on a nearest-neighbor -type of interaction. We present some method based on the maximization of the pseudo-likelood to estimate some vector θ parametrizing point processes. Finally, we give general conditions under which the consistency and the asymptotic normality hold for such estimates. Keywords: Gibbs point processes, nearest-neighbour Gibbs point processes, pseudo-likelihood 1. Introduction Spatial point pattern data occur frequently in a wide variety of scientific disciplines, including seismology, ecology, forestry, agriculture, geography, spatial epidemiology see for example the classical references [?] or [?] and the references therein. Let us cite a classical example in the domain of forestry: the japanese pines data set e.g. [?]) plotted in Fig.??. The data give the locations of Japanese black pine saplings in a square sampling region in a natural forest. The trees seem not to be uniformly distributed that is that is not the realization of a Poisson process. They preferably seem to aggregate, which could exhibit some special behaviour to environmental conditions or simply competition between trees. A problem that has interested a lot of searchers consisted then in modelling this point process and in estimating the associated parameters in order to attempt to interpret the interaction between points here trees). Figure 1: Japanese Pines data set: the data give the locations of Japanese black pine saplings in a square sampling region in a natural forest. In a general context, the data consist in a spatial point pattern ϕ observed in a bounded region Λ of space R for the sake of simplicity). Thus, ϕ = {x 1,..., x n } where the number of points n 0 is not fixed and each x i is a point in Λ. The data are assumed to be a realization of a point process Φ in R. The typical model is the Poisson process. The class of Gibbs processes is the class for which the probability density function conditionnally on some fixed outside configuration ϕ o with respect to the Poisson process can be written:

2 Jean-Michel Billiot, Jean-François Coeurjolly and Rémy Drouilhet f Φ Λ c =ϕo Λ c Φ Λ ϕ Λ ) exp V ϕ Λ ϕ o Λ c)), where V ϕ Λ ϕ o Λc) measures the energy to insert the configuration ϕ Λ into ϕ o Λ c. We only consider in this paper non-marked point processes but extensions can be obtained with little work. Extensions could also be done by integrating spatial covariates..1. Framework. Description of some Gibbs models The framework of this paper is restricted to stationary Gibbs point processes based on energy function invariant by translation, V ϕ; θ), parametrized by some θ Θ, where Θ is some compact set of R p+1. In its general form, the model is described as V ϕ; θ) = θ 1 ϕ + V ϕ; θ ), 1) with θ = θ 1, θ ). The exponential case is a particular form of??) given by V ϕ; θ) = θ T vϕ), V ϕ; θ ) = θ T v ϕ), ) where vϕ) = v 1 ϕ),..., v p+1 ϕ)) = ϕ, v ϕ)) is the vector of sufficient statistics. We developped the method and obtained asymptotic results for the general form??) but for the sake of simplicity and the clearness of the presentation, we only pay attention on energy function of the form??), see [?] for more details. With such a framework, the local energy, that is the energy required to insert a point x into the configuration ϕ is then expressed as V x ϕ; θ) = θ T v x ϕ) where v x ϕ) = v 1 x ϕ),..., v p+1 x ϕ)) = 1, v x ϕ)) := vϕ {x}) vϕ). The main goal is to obtain some asymptotic results for estimates of θ as the domain grows. Hence the point process has to be defined in R. In [?] are given some sufficient conditions to ensure the existence of an ergodic measure of such a point process: [E1] Locality of the local energy: there exists some fixed range denoted by D such that for any ϕ Ω one has V 0 ϕ; θ) = V 0 ϕ B0, D); θ). [E] Stability of the local energy: there exists K 0 such that for any ϕ Ω: V 0 ϕ; θ) K. Again, this framework may be extended, see [?]... Various examples Classical pairwise interaction point processes e.g. [?]): Poisson process: V ϕ; θ) = θ 1 ϕ. Strauss process: V ϕ; θ) = θ 1 ϕ + θ s [0,D] ϕ) with θ > 0 or with hard-core condition), where s [0,D] ϕ) is the number of [0, D] close pair of points in ϕ that is 1 ξ D). ξ P ϕ)

3 Maximum pseudo-likelihood estimator for Gibbs point processes 3 Multi-type Strauss process: V ϕ; θ) = θ 1 ϕ + p+1 i= θ is [Di 1,D i]ϕ) with θ,..., θ p+1 > 0 or with hard-core condition) where D 0 = 0 < D 1 <... < D p+1 are fixed real numbers. Soft core process: V ϕ; θ) = θ 1 ϕ + θ /k ξ P ϕ) ξ /k 1 ξ D). Overlap are process: V ϕ; θ) = θ 1 ϕ + θ ξ P g ϕ) ξ )1 ξ D) with θ > D 0 and g d) = Arcosd/D) d/ D d ). Nearest-neighbour pairwise interaction point processes: the idea is to replace the classical neighborhood relation that is the one based on the complete graph P ϕ)) by a more structured one. Let us cite for example a Nearest-neighbour Multi-type Strauss process: V ϕ; θ) = θ 1 ϕ + p+1 i= θ is [Di 1,D i]ϕ) where s [Di 1,D i]ϕ) is the number of edges of some graph G ϕ) with length comprised between D i 1 and D i. The existence has been proved for example) when e.g. [?]) G ϕ) corresponds to the set of edges of the slightly modified Delaunay triangulation. G ϕ) corresponds to the set of edges of the first-nearest neighbour graph or more generally the k-nearest-neighbour graph k 1). Other point processes e.g. [?]): Extension to interaction on triangles - Geyer s triplet process: V ϕ; θ) = θ 1 ϕ + θ s [0,D] ϕ) + θ 3 t [0,D] ϕ) θ, θ 3 > 0 or θ < 0 and θ 3 > 0) where t [0,D] ϕ) is the number of triangles of ϕ with lengths edges lower than D, that is t [0,D] ϕ) = ξ P 3ϕ) η P ξ) 1 η D). Widow-Rowlinson Area interaction point process): V ϕ; θ) = θ 1 ϕ +θ U ϕ,d where U ϕ,d is the area of the union of discs of radius D centred at the points. Example of Ord s model: V ϕ; θ) = θ 1 ϕ + θ Aϕ) with 0 < D < + ) where Aϕ) = x ϕ 1 V or Dx ϕ) A 0 ), V or D x ϕ) = V orx ϕ) Bx, D) and V orx ϕ) is the Voronoï region associated with x and the set of points ϕ. This model can be viewed as a Strauss type model on the are of voronoï cells Pseudo-likelihood 3. MPLE: presentation and main results The idea of maximum pseudo-likelihood is due to [?] who first introduced the concept for Markov random fields in order to avoid the normalizing constant appearing if one uses the maximum likelihood method. This work was then widely extended and [?] Theorem.) obtained a general expression for Gibbs point processes. With our notation and up to a scalar factor the pseudo-likelihood defined for a configuration ϕ and a domain of observation Λ is denoted by P L Λ ϕ; θ) and given by P L Λ ϕ; θ) = exp exp θ T v x ϕ) ) ) dx exp θ T v x ϕ \ x) ). 3) Λ x ϕ Λ

4 4 Jean-Michel Billiot, Jean-François Coeurjolly and Rémy Drouilhet It is more convenient to define and work with) the log-pseudo-likelihood function, denoted by LP L Λ ϕ; θ). LP L Λ ϕ; θ) = exp θ T v x ϕ) ) dx θ T v x ϕ \ x) 4) Λ x ϕ Λ Our data consist in the realization of a point process with energy function V ; θ ) in a domain Λ R satisfying the existence conditions [E1] and [E]. Thus, θ is the true parameter to be estimated. The Gibbs measure will be denoted by P θ. Moreover the point process is assumed to be observed in a domain Λ n D = x Λn Bx, D), where Λ n R can de decomposed into i In i where for i = i 1, i ) { i = z R, D i j 1 ) z j D i j 1 ) }, j = 1, for some D > 0. As n +, we also assume that Λ n R such that Λ n + and Λ n 0. Finally, we denote by Λ n θ n ϕ) the maximum pseudo-likelihood estimate based on the configuration ϕ, defined as θ n ϕ) = argmax θ Θ LP L Λn ϕ; θ). 3.. Asymptotic results of the MPLE The following results generalize the ones obtained by [?] and [?]. Indeed, [?] prove the consistency of the MPLE for the exponential case of marked) pairwise interaction point processes. And [?] prove the asymptotic normality of MPLE for pairwise interaction point processes with two parameters. [C1] There exists κ, R some positive real numbers and some integer k, such that for all i = 1,..., p + 1 v i 0 ϕ) κ ϕ B0,R) k [C] Identifiability condition : there exists A 1,..., A p+1, p+1 disjoint events of Ω such that P θ A i ) > 0 and such that for all ϕ 1,..., ϕ p+1 A 1 A p+1 the p+1) p+1) matrix with entries v j 0 ϕ i ) is constant and invertible. [C3] The matrix Σ D, θ ) = D i [ D f D ]+1 E θ LPL 1) 0 Φ; θ ) LPL 1) i Φ; θ ) T ) 5) is symmetric and definite positive. The vector LPL 1) i ϕ; θ) denotes the first order derivative vector of the log-pseudo-likelihood function on the domain i. Theorem 1. Under Assumptions [C1] and [C], for P θ almost every ϕ, the maximum pseudo-likelihood estimate θ n ϕ) converges towards θ as n tends to infinity.

5 Maximum pseudo-likelihood estimator for Gibbs point processes 5 Define for θ Θ, U ) θ) = E Pθ )) v j 0 Φ)v k 0 Φ) exp θ T v 0 Φ). Theorem. Under Assumptions [C1] and [C], we have, for any fixed D, the following convergence in distribution as n + Λ n 1/ U ) θ ) θn θ ) N 0, Σ D, ) θ ), 6) In addition under Assumption [C3] Λ n 1/ Σn D, θ n ) 1/ U n ) θ n ) θn θ ) N 0, I p+1 ), 7) where for some θ and some finite configuration ϕ, the matrices Σ n D, θ) and U n ) θ) are respectively defined by Σ n D, θ) = Λ n 1 D LPL 1) i ϕ; θ) LPL 1) j ϕ; θ) T i In j i [ D D f]+1,j I n and for all j, k = 1,..., p + 1 ) U ) n θ) = 1 j,k Λ n Λ n v j 0 Φ)v k 0 Φ) exp ) θ T v 0 Φ) dx Both results can be proved using a central limit theorem obtained by [?] and results concerning minimum contrast estimators obtained by [?]. Assumptions [C1] and [C] are satisfied for all the examples presented. Assumption [C3] needs more attention but is satisfied for various models, see [?]. Acknowledgements This work was support by a grant from IMAG Project AlpB. References [1] BADDELEY, A.J., AND TURNER, J.. spatstat: An R Package for Analyzing Spatial Point Patterns. J. Stat. Soft., 16) 005), 1 4. [] BERTIN, E., BILLIOT, J.-M., AND DROUILHET, R. Existence of "Nearest-Neighbour" Gibbs Point Models. Adv. Appl. Prob., ), [3] BILLIOT, J.-M., COEURJOLLY, J.-F., AND DROUILHET, R. Maximum pseudolikelihood estimator for nearest-neighbours Gibbs point processes E-print arxiv http ://fr.arxiv.org/abs/math.st/ ), 1 9. [4] BESAG, J. Statistical analysis of non-lattice data. The statistician, ), [5] GUYON, X. Champs aléatoires sur un réseau, Masson, Paris, 199.

6 6 Jean-Michel Billiot, Jean-François Coeurjolly and Rémy Drouilhet [6] JENSEN, J.L., AND KUNSCH, H.R. On asymptotic normality of pseudolikelihood estimates of pairwise interaction processes.ann. Inst. Statist. Math., ), [7] JENSEN, J.L., AND MØLLER, J. Pseudolikelihood for exponential family models of spatial point processes. Ann. Appl. Probab., ), [8] MØLLER, J., AND WAGGEPETERSEN R. Statistical Inference and Simulation for Spatial Point Processes, Chapman and Hall/CRC, Boca Raton, 003. [9] STOYAN, D., KENDALL W.S., AND MECKE, J. Stochastic Geometry and its Applications, Wiley, Chichester, second edition, Jean-Michel Billiot, Jean-François Coeurjolly and Rémy Drouilhet SAGAG Team, Department of Statistics, University of Grenoble 151 Av. Centrale BP GRENOBLE CEDEX 9 Jean-Michel.Billiot@upmf-grenoble.fr, Jean-Francois.Coeurjolly@upmf-grenoble.fr and Remy.Drouilhet@upmf-grenoble.fr