Nonparametric estimation of interaction functions for two-type pairwise interaction point processes

2002 ◽  
Author(s):  
J.A. Gubner ◽  
Wei-Bin Chang
Keyword(s):  
Nonparametric Estimation ◽  
Point Processes ◽  
Pairwise Interaction ◽  
Interaction Point

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

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

scholarly journals Existence of Gibbs point processes with stable infinite range interaction

2020 ◽  
Vol 57 (3) ◽  
pp. 775-791
Author(s):  
David Dereudre ◽  
Thibaut Vasseur
Keyword(s):  
Point Processes ◽  
Existence Results ◽  
Pairwise Interaction ◽  
Range Interaction ◽  
Pairwise Interactions ◽  
Gibbs Point Processes ◽  
Infinite Range Interactions ◽  
Finite Configuration ◽  
Multi Body ◽  
Infinite Range

AbstractWe provide a new proof of the existence of Gibbs point processes with infinite range interactions, based on the compactness of entropy levels. Our main existence theorem holds under two assumptions. The first one is the standard stability assumption, which means that the energy of any finite configuration is superlinear with respect to the number of points. The second assumption is the so-called intensity regularity, which controls the long range of the interaction via the intensity of the process. This assumption is new and introduced here since it is well adapted to the entropy approach. As a corollary of our main result we improve the existence results by Ruelle (1970) for pairwise interactions by relaxing the superstabilty assumption. Note that our setting is not reduced to pairwise interaction and can contain infinite-range multi-body counterparts.

scholarly journals Area-interaction point processes

1995 ◽  
Vol 47 (4) ◽  
pp. 601-619 ◽  
Author(s):  
A. J. Baddeley ◽  
M. N. M. van Lieshout
Keyword(s):  
Point Processes ◽  
Interaction Point

Markov interacting component processes

2000 ◽  
Vol 32 (03) ◽  
pp. 597-619 ◽  
Author(s):  
Y. C. Chin ◽  
A. J. Baddeley
Keyword(s):  
Density Function ◽  
Point Processes ◽  
Characterization Theorem ◽  
Pairwise Interaction ◽  
Connected Components ◽  
Point Pattern ◽  
Component Process ◽  
Interaction Terms ◽  
Markov Point Processes ◽  
Component Processes

A generalization of Markov point processes is introduced in which interactions occur between connected components of the point pattern. A version of the Hammersley-Clifford characterization theorem is proved which states that a point process is a Markov interacting component process if and only if its density function is a product of interaction terms associated with cliques of connected components. Integrability and superpositional properties of the processes are shown and a pairwise interaction example is used for detailed exploration.

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