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.

A nonparametric estimator for pairwise-interaction point processes

Biometrika ◽  
1987 ◽  
Vol 74 (4) ◽  
pp. 763-770 ◽  
Author(s):  
PETER J. DIGGLE ◽  
DAVID J. GATES ◽  
ALYSON STIBBARD
Keyword(s):  
Point Processes ◽  
Pairwise Interaction ◽  
Nonparametric Estimator ◽  
Interaction Point

Asymptotic independence of counts in isotropic planar point processes of phase-type

2000 ◽  
Vol 32 (2) ◽  
pp. 363-375
Author(s):  
Marie-Ange Remiche
Keyword(s):  
Poisson Process ◽  
Arbitrary Shape ◽  
Point Processes ◽  
The Other ◽  
Asymptotic Independence ◽  
Planar Point ◽  
Phase Type ◽  
Disjoint Sets ◽  
The Mean ◽  
The One

The isotropic planar point processes of phase-type are natural generalizations of the Poisson process on the plane. On the one hand, those processes are isotropic and stationary for the mean count, as in the case of the Poisson process. On the other hand, they exhibit dependence of counts in disjoint sets. In a recent paper, we have proved that the number of points in a square window has a Poisson distribution asymptotically as the window is located far away from the origin of the process. We extend our work to the case of a window of arbitrary shape.

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.

Dynamical systems defined on point processes

1998 ◽  
Vol 35 (03) ◽  
pp. 581-588
Author(s):  
Laurence A. Baxter
Keyword(s):  
Stochastic Process ◽  
Poisson Process ◽  
Point Processes ◽  
Linear Boltzmann Equation ◽  
Semidynamical System ◽  
The Mean ◽  
Mean Function ◽  
Necessary And Sufficient ◽  
The Relationship

This paper introduces a new stochastic process in which the iterates of a dynamical system evolving in discrete time coincide with the events of a Poisson process. The autocovariance function of the stochastic process is studied and a necessary and sufficient condition for it to vanish is deduced. It is shown that the mean function of this process comprises a continuous-time semidynamical system if the underlying dynamical map is linear. The flow of probability density functions generated by the stochastic process is analysed in detail, and the relationship between the flow and the solutions of the linear Boltzmann equation is investigated. It is shown that the flow is a semigroup if and only if the point process defining the stochastic process is Poisson, thereby providing a new characterization of the Poisson process.

On Parameter Estimation for Pairwise Interaction Point Processes

1994 ◽  
Vol 62 (1) ◽  
pp. 99 ◽  
Author(s):  
Peter J. Diggle ◽  
Thomas Fiksel ◽  
Pavel Grabarnik ◽  
Yosihiko Ogata ◽  
Dietrich Stoyan ◽  
...  
Keyword(s):  
Parameter Estimation ◽  
Point Processes ◽  
Pairwise Interaction ◽  
Interaction Point
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