Bounds for the Probability Generating Functional of a Gibbs Point Process

Stucki, Kaspar; Schuhmacher, Dominic

doi:10.1239/aap/1396360101

Cited by 9 publications

(8 citation statements)

References 14 publications

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“…In particular, inequality (12) implies inequality (1) in the Introduction. Note that better bounds on a constant ν have been obtained in Stucki and Schuhmacher (2014).…”

Section: Total Variation Bounds Between Gibbs Process Distributionsmentioning

confidence: 99%

Gibbs point process approximation: Total variation bounds using Stein’s method

Schuhmacher¹,

Stucki²

2014

Ann. Probab.

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We obtain upper bounds for the total variation distance between the distributions of two Gibbs point processes in a very general setting. Applications are provided to various well-known processes and settings from spatial statistics and statistical physics, including the comparison of two Lennard-Jones processes, hard core approximation of an area interaction process and the approximation of lattice processes by a continuous Gibbs process.Our proof of the main results is based on Stein's method. We construct an explicit coupling between two spatial birth-death processes to obtain Stein factors, and employ the Georgii-Nguyen-Zessin equation for the total bound.

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“…In particular, inequality (12) implies inequality (1) in the Introduction. Note that better bounds on a constant ν have been obtained in Stucki and Schuhmacher (2014).…”

Section: Total Variation Bounds Between Gibbs Process Distributionsmentioning

confidence: 99%

Gibbs point process approximation: Total variation bounds using Stein’s method

Schuhmacher¹,

Stucki²

2014

Ann. Probab.

Self Cite

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“…Such a parameterization tends to create repulsive clusters. When γ 1 = 1 and δ = 0, such a piecewise Strauss model was called annulus model by Stucki and Schuhmacher (2014). This model demonstrates the limitations of our approximation even if when γ 1 is close to zero which means that the model is close to a hard-core process with radius 0.1 our approximation remains satisfactory.…”

Section: Numerical Studymentioning

confidence: 77%

Intensity approximation for pairwise interaction Gibbs point processes using determinantal point processes

Lavancier¹,

Coeurjolly²

2018

Electron. J. Statist.

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The intensity of a Gibbs point process is usually an intractable function of the model parameters. For repulsive pairwise interaction point processes, this intensity can be expressed as the Laplace transform of some particular function. Baddeley and Nair (2012) developped the Poissonsaddlepoint approximation which consists, for basic models, in calculating this Laplace transform with respect to a homogeneous Poisson point process. In this paper, we develop an approximation which consists in calculating the same Laplace transform with respect to a specific determinantal point process. This new approximation is efficiently implemented and turns out to be more accurate than the Poisson-saddlepoint approximation, as demonstrated by some numerical examples.MSC 2010 subject classifications: Primary: 60G55, secondary: 82B21.

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“…In particular, some Gibbs hard-core point processes are expected to be sub-Poisson. Bounds for the probability generating functionals with estimates for Ripley's K-function and the intensity and higher order correlation functions for some stationary locally stable Gibbs point process are given in [46]. Also, some geometric structures on specific Gibbs point processes have already been considered (see e.g.…”

Section: Discussionmentioning

confidence: 99%

Clustering Comparison of Point Processes, with Applications to Random Geometric Models

Błaszczyszyn

Yogeshwaran

2014

Lecture Notes in Mathematics

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In this chapter we review some examples, methods, and recent results involving comparison of clustering properties of point processes. Our approach is founded on some basic observations allowing us to consider void probabilities and moment measures as two complementary tools for capturing clustering phenomena in point processes. As might be expected, smaller values of these characteristics indicate less clustering. Also, various global and local functionals of random geometric models driven by point processes admit more or less explicit bounds involving void probabilities and moment measures, thus aiding the study of impact of clustering of the underlying point process. When stronger tools are needed, directional convex ordering of point processes happens to be an appropriate choice, as well as the notion of (positive or negative) association, when comparison to the Poisson point process is considered. We explain the relations between these tools and provide examples of point processes admitting them. Furthermore, we sketch some recent results obtained using the aforementioned comparison tools, regarding percolation and coverage properties of the Boolean model, the SINR model, subgraph counts in random geometric graphs, and more generally, U-statistics of point processes. We also mention some results on Betti numbers forČech and Vietoris-Rips random complexes generated by stationary point processes. A general observation is that many of the results derived previously for the Poisson point process generalise to some "sub-Poisson" processes, defined as those clustering less than the Poisson process in the sense of void probabilities and moment measures, negative association or dcx-ordering.

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Bounds for the Probability Generating Functional of a Gibbs Point Process

Cited by 9 publications

References 14 publications

Gibbs point process approximation: Total variation bounds using Stein’s method

Gibbs point process approximation: Total variation bounds using Stein’s method

Intensity approximation for pairwise interaction Gibbs point processes using determinantal point processes

Clustering Comparison of Point Processes, with Applications to Random Geometric Models

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