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AREA-INTERACTION POINT PROCESSES

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A. J. BADDELEY^{1} AND M. N. M. VAN LIESHOUT^{2}

^{1} *Department of Mathematics, University of Western Australia,*

Nedlands WA 6009, Australia and

Department of Mathematics and Computer Science, University of Leiden, The Netherlands

^{2} *Department of Statistics, University of Warwick, Coventry CV4 7AL, U.K.*
(Received December 6, 1993; revised January 30, 1995)

**Abstract.**
We introduce a new Markov point process that
exhibits a range of clustered, random, and ordered patterns according
to the value of a scalar parameter. In contrast to pairwise
interaction processes, this model has interaction terms of all orders.
The likelihood is closely related to the empty space function *F*,
paralleling the relation between the Strauss process and Ripley's
*K*-function. We show that, in complete analogy with pairwise
interaction processes, the pseudolikelihood equations for this model
are a special case of the Takacs-Fiksel method, and our model is the
limit of a sequence of auto-logistic lattice processes.

*Key words and phrases*:
Clustering, empty space
statistic, Hough transform, inhibition, *K*-function, lattice
process limits, Markov point processes, nearest neighbour distance
distribution, pairwise interaction, penetrable sphere model,
pseudolikelihood, spatial statistics, spherical contact distance
distribution, stationary point process, Strauss model, Takacs-Fiksel
method.

**Source**
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