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35
Non and SemiParametric Estimation of Interaction in Inhomogeneous Point Patterns
, 2000
"... We develop methods for analysing the `interaction' or dependence between points in a spatial point pattern, when the pattern is spatially inhomogeneous. Completely nonparametric study of interactions is possible using an analogue of the Kfunction. Alternatively one may assume a semiparametr ..."
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Cited by 47 (17 self)
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We develop methods for analysing the `interaction' or dependence between points in a spatial point pattern, when the pattern is spatially inhomogeneous. Completely nonparametric study of interactions is possible using an analogue of the Kfunction. Alternatively one may assume a semiparametric model in which a (parametrically specified) homogeneous Markov point process is subjected to (nonparametric) inhomogeneous independent thinning. The effectiveness of these approaches is tested on datasets representing the positions of trees in forests.
Construction of Diffusions on Configuration Spaces
"... We show that any square field operator on a measurable state space E can be lifted by a natural procedure to a square field operator on the corresponding (multiple) configuration space \Gamma E . We then show the closability of the associated lifted (pre)Dirichlet forms E \Gamma ¯ on L 2 (\Ga ..."
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Cited by 26 (3 self)
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We show that any square field operator on a measurable state space E can be lifted by a natural procedure to a square field operator on the corresponding (multiple) configuration space \Gamma E . We then show the closability of the associated lifted (pre)Dirichlet forms E \Gamma ¯ on L 2 (\Gamma E ; ¯) for a large class of measures ¯ on \Gamma E (without assuming an integration by parts formula) generalizing all corresponding results known so far. Subsequently, we prove that under mild conditions the Dirichlet forms E \Gamma ¯ are quasiregular, and that hence there exist associated diffusions on \Gamma E , provided E is a complete separable metric space and \Gamma E is equipped with a suitable topology, which is the vague topology if E is locally compact. We discuss applications to the case where E is a finite dimensional manifold yielding an existence result on diffusions on \Gamma E which was already announced in [AKR96a, AKR96b], resp. used in [AKR98, AKR97b]. Furthermore...
Generalised Shot noise Cox processes
 ADVANCES IN APPLIED PROBABILITY 35
, 2003
"... We introduce a new class of Cox cluster processes called generalised shotnoise Cox processes (GSNCPs), which extends the definition of shot noise Cox processes (SNCPs) in two directions: the point process which drives the shot noise is not necessarily Poisson, and the kernel of the shot noise can b ..."
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Cited by 23 (6 self)
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We introduce a new class of Cox cluster processes called generalised shotnoise Cox processes (GSNCPs), which extends the definition of shot noise Cox processes (SNCPs) in two directions: the point process which drives the shot noise is not necessarily Poisson, and the kernel of the shot noise can be random. Thereby a very large class of models for aggregated or clustered point patterns is obtained. Due to the structure of GSNCPs, a number of useful results can be established. We focus first on deriving summary statistics for GSNCPs and next on how to make simulation for GSNCPs. Particularly, results for first and second order moment measures, reduced Palm distributions, the Jfunction, simulation with or without edge effects, and conditional simulation of the intensity function driving a GSNCP are given. Our results are exemplified for special important cases of GSNCPs, and we discuss the relation to corresponding results for SNCPs.
Residual analysis for spatial point processes (with discussion
 Journal of the Royal Statistical Society (series B
, 2005
"... [Read before The Royal Statistical Society at a meeting organized by the Research Section on ..."
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Cited by 20 (5 self)
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[Read before The Royal Statistical Society at a meeting organized by the Research Section on
Glauber dynamics of continuous particle systems
"... This paper is devoted to the construction and study of an equilibrium Glaubertype dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space Γ of all locally finite subsets (configurations) in Rd, we fix a Gibbs measure µ cor ..."
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Cited by 16 (7 self)
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This paper is devoted to the construction and study of an equilibrium Glaubertype dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space Γ of all locally finite subsets (configurations) in Rd, we fix a Gibbs measure µ corresponding to a general pair potential φ and activity z> 0. We consider a Dirichlet form E on L2 (Γ,µ) which corresponds to the generator H of the Glauber dynamics. We prove the existence of a Markov process M on Γ that is properly associated with E. In the case of a positive potential φ which satisfies δ: = ∫ Rd(1 − e−φ(x))z dx < 1, we also prove that the generator H has a spectral gap ≥ 1−δ. Furthermore, for any pure Gibbs state µ, we derive a Poincaré inequality. The results about the spectral gap and the Poincaré inequality are a generalization and a refinement of a recent result from [6].
Conditional intensity and Gibbsianness of determinantal point processes
 J. Stat. Phys
, 2005
"... The Papangelou intensities of determinantal (or fermion) point processes are investigated. These exhibit a monotonicity property expressing the repulsive nature of the interaction, and satisfy a bound implying stochastic domination by a Poisson point process. We also show that determinantal point pr ..."
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Cited by 14 (2 self)
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The Papangelou intensities of determinantal (or fermion) point processes are investigated. These exhibit a monotonicity property expressing the repulsive nature of the interaction, and satisfy a bound implying stochastic domination by a Poisson point process. We also show that determinantal point processes satisfy the socalled condition (�λ), which is a general form of Gibbsianness. Under a continuity assumption, the Gibbsian conditional probabilities can be identified explicitly. KEY WORDS: Determinantal point process; fermion point process; Gibbs point process; Papangelou intensity; stochastic domination; percolation.
Indices of Dependence Between Types in Multivariate Point Patterns
 Scandinavian Journal of Statistics
, 1999
"... We propose new summary statistics quantifying several forms of dependence between types in a spatial pattern of points classified into distinct types. These statistics are the multivariate counterparts of the Jfunction for point processes of a single type, introduced in [18]. They are based on comp ..."
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Cited by 10 (3 self)
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We propose new summary statistics quantifying several forms of dependence between types in a spatial pattern of points classified into distinct types. These statistics are the multivariate counterparts of the Jfunction for point processes of a single type, introduced in [18]. They are based on comparing distances from a type i point to either the nearest type j point or to the nearest point in the pattern regardless of type to these distances seen from an arbitrary point in space. Information about the range of interaction can also be inferred. Our statistics can be computed explicitly for a range of wellknown multivariate point process models. Some applications to bivariate data sets are presented as well. Keywords & Phrases: ants' nests, beta cells, empty space function, hamster tumour, J  function, multitype point patterns, myrtle disease, nearestneighbour distance distribution function, point processes, random labelling, spatial interaction, spatial statistics. AMS Mathemat...
Scaling limit of stochastic dynamics in classical continuous systems
, 2002
"... We investigate a scaling limit of gradient stochastic dynamics associated to Gibbs states in classical continuous systems on R d, d ≥ 1. The aim is to derive macroscopic quantities from a given micro or mesoscopic system. The scaling we ..."
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Cited by 5 (5 self)
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We investigate a scaling limit of gradient stochastic dynamics associated to Gibbs states in classical continuous systems on R d, d ≥ 1. The aim is to derive macroscopic quantities from a given micro or mesoscopic system. The scaling we
Infinite interaction diffusion particles I: Equilibrium process and its scaling limit
 Forum Math
"... A stochastic dynamics (X(t))t≥0 of a classical continuous system is a stochastic process which takes values in the space Γ of all locally finite subsets (configurations) in Rd and which has a Gibbs measure µ as an invariant measure. We assume that µ corresponds to a symmetric pair potential φ(x − y) ..."
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Cited by 4 (3 self)
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A stochastic dynamics (X(t))t≥0 of a classical continuous system is a stochastic process which takes values in the space Γ of all locally finite subsets (configurations) in Rd and which has a Gibbs measure µ as an invariant measure. We assume that µ corresponds to a symmetric pair potential φ(x − y). An important class of stochastic dynamics of a classical continuous system is formed by diffusions. Till now, only one type of such dynamics—the socalled gradient stochastic dynamics, or interacting Brownian particles—has been investigated. By using the theory of Dirichlet forms from [27], we construct and investigate a new type of stochastic dynamics, which we call infinite interacting diffusion particles. We introduce a Dirichlet form EΓ µ on L2 (Γ; µ), and under general conditions on the potential φ, prove its closability. For a potential φ having a “weak ” singularity at zero, we also write down an explicit form of the generator of EΓ µ on the set of smooth cylinder functions. We then show that, for any Dirichlet form EΓ µ, there exists a diffusion process that is properly associated with it. Finally, in a way parallel to [17], we study a scaling limit of interacting diffusions in terms of convergence of the corresponding Dirichlet forms, and we also show that these scaled processes are tight in C([0, ∞), D ′), where D ′ is the dual space of D:=C ∞ 0 (Rd).