AREA-INTERACTION POINT PROCESSES

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

1Department 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.