Point process limits of lattice processes

1982 ◽  
Vol 19 (01) ◽  
pp. 210-216 ◽  
Author(s):  
Julian Besag ◽  
Robin Milne ◽  
Stan Zachary
Keyword(s):  
Point Process ◽  
Pairwise Interaction ◽  
Interaction Point ◽  
Interaction Processes ◽  
Lattice Processes ◽  
Suitable Sequence

Starting from a suitable sequence of auto-Poisson lattice schemes, it is shown that (almost) any purely inhibitory pairwise-interaction point process can be obtained in the limit. Further pairwise-interaction processes are obtained as limits of sequences of auto-logistic lattice schemes.

Point process limits of lattice processes

1982 ◽  
Vol 19 (1) ◽  
pp. 210-216 ◽  
Author(s):  
Julian Besag ◽  
Robin Milne ◽  
Stan Zachary
Keyword(s):  
Point Process ◽  
Pairwise Interaction ◽  
Interaction Point ◽  
Interaction Processes ◽  
Lattice Processes ◽  
Suitable Sequence

Starting from a suitable sequence of auto-Poisson lattice schemes, it is shown that (almost) any purely inhibitory pairwise-interaction point process can be obtained in the limit. Further pairwise-interaction processes are obtained as limits of sequences of auto-logistic lattice schemes.

scholarly journals Bounds for the Probability Generating Functional of a Gibbs Point Process

2014 ◽  
Vol 46 (1) ◽  
pp. 21-34 ◽  
Author(s):  
Kaspar Stucki ◽  
Dominic Schuhmacher
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Upper Bounds ◽  
Pairwise Interaction ◽  
Summary Statistics ◽  
Lower And Upper Bounds ◽  
Generating Functional ◽  
Interaction Processes ◽  
Gibbs Point Process ◽  
Process Approximation

We derive explicit lower and upper bounds for the probability generating functional of a stationary locally stable Gibbs point process, which can be applied to summary statistics such as the F function. For pairwise interaction processes we obtain further estimates for the G and K functions, the intensity, and higher-order correlation functions. The proof of the main result is based on Stein's method for Poisson point process approximation.

scholarly journals Quermass-interaction processes: conditions for stability

1999 ◽  
Vol 31 (02) ◽  
pp. 315-342 ◽  
Author(s):  
W. S. Kendall ◽  
M. N. M. van Lieshout ◽  
A. J. Baddeley
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Convex Set ◽  
Compact Convex ◽  
Main Question ◽  
Interaction Point ◽  
Interaction Processes ◽  
Area Functional ◽  
Compact Convex Set

We consider a class of random point and germ-grain processes, obtained using a rather natural weighting procedure. Given a Poisson point process, on each point one places a grain, a (possibly random) compact convex set. Let Ξ be the union of all grains. One can now construct new processes whose density is derived from an exponential of a linear combination of quermass functionals of Ξ. If only the area functional is used, then the area-interaction point process is recovered. New point processes arise if we include the perimeter length functional, or the Euler functional (number of components minus number of holes). The main question addressed by the paper is that of when the resulting point process is well-defined: geometric arguments are used to establish conditions for the point process to be stable in the sense of Ruelle.

scholarly journals Quermass-interaction processes: conditions for stability

1999 ◽  
Vol 31 (2) ◽  
pp. 315-342 ◽  
Author(s):  
W. S. Kendall ◽  
M. N. M. van Lieshout ◽  
A. J. Baddeley
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Convex Set ◽  
Compact Convex ◽  
Main Question ◽  
Interaction Point ◽  
Interaction Processes ◽  
Area Functional ◽  
Compact Convex Set

We consider a class of random point and germ-grain processes, obtained using a rather natural weighting procedure. Given a Poisson point process, on each point one places a grain, a (possibly random) compact convex set. Let Ξ be the union of all grains. One can now construct new processes whose density is derived from an exponential of a linear combination of quermass functionals of Ξ. If only the area functional is used, then the area-interaction point process is recovered. New point processes arise if we include the perimeter length functional, or the Euler functional (number of components minus number of holes). The main question addressed by the paper is that of when the resulting point process is well-defined: geometric arguments are used to establish conditions for the point process to be stable in the sense of Ruelle.

scholarly journals Bounds for the Probability Generating Functional of a Gibbs Point Process

2014 ◽  
Vol 46 (01) ◽  
pp. 21-34 ◽  
Author(s):  
Kaspar Stucki ◽  
Dominic Schuhmacher
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Upper Bounds ◽  
Pairwise Interaction ◽  
Summary Statistics ◽  
Lower And Upper Bounds ◽  
Generating Functional ◽  
Interaction Processes ◽  
Gibbs Point Process ◽  
Process Approximation

We derive explicit lower and upper bounds for the probability generating functional of a stationary locally stable Gibbs point process, which can be applied to summary statistics such as the F function. For pairwise interaction processes we obtain further estimates for the G and K functions, the intensity, and higher-order correlation functions. The proof of the main result is based on Stein's method for Poisson point process approximation.

Poisson limits and nonparametric estimation for pairwise interaction point processes

2000 ◽  
Vol 37 (1) ◽  
pp. 252-260 ◽  
Author(s):  
Wei-Bin Chang ◽  
John A. Gubner
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Pairwise Interaction ◽  
Nonparametric Estimator ◽  
Interpoint Distance ◽  
Interaction Point ◽  
Interaction Function ◽  
The Mean ◽  
Piecewise Continuous ◽  
Independent Identically Distributed

The distribution of the interpoint distance process of a sequence of pairwise interaction point processes is considered. It is shown that, if the interaction function is piecewise-continuous, then the sequence of interpoint distance processes converges weakly to an inhomogeneous Poisson process under certain sparseness conditions. Convergence of the expectation of the interpoint distance process to the mean of the limiting Poisson process is also established. This suggests a new nonparametric estimator for the interaction function if independent identically distributed samples of the point process are available.

Poisson limits and nonparametric estimation for pairwise interaction point processes

2000 ◽  
Vol 37 (01) ◽  
pp. 252-260 ◽  
Author(s):  
Wei-Bin Chang ◽  
John A. Gubner
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Pairwise Interaction ◽  
Nonparametric Estimator ◽  
Interpoint Distance ◽  
Interaction Point ◽  
Interaction Function ◽  
The Mean ◽  
Piecewise Continuous ◽  
Independent Identically Distributed

The distribution of the interpoint distance process of a sequence of pairwise interaction point processes is considered. It is shown that, if the interaction function is piecewise-continuous, then the sequence of interpoint distance processes converges weakly to an inhomogeneous Poisson process under certain sparseness conditions. Convergence of the expectation of the interpoint distance process to the mean of the limiting Poisson process is also established. This suggests a new nonparametric estimator for the interaction function if independent identically distributed samples of the point process are available.

scholarly journals Poisson limits for pairwise and area interaction point processes

2000 ◽  
Vol 32 (1) ◽  
pp. 75-85 ◽  
Author(s):  
S. Rao Jammalamadaka ◽  
Mathew D. Penrose
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Compact Support ◽  
Point Processes ◽  
Joint Density ◽  
Interaction Point

Suppose n particles xi in a region of the plane (possibly representing biological individuals such as trees or smaller organisms) have a joint density proportional to exp{-∑i<jϕ(n(xi-xj))}, with ℝd; a specified function of compact support. We obtain a Poisson process limit for the collection of rescaled interparticle distances as n becomes large. We give corresponding results for the case of several types of particles, representing different species, and also for the area-interaction (Widom-Rowlinson) point process of interpenetrating spheres.

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