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.

The strength and range of interpoint interactions in a spatial point process can be quantified by the functionwhere G is the nearest-neighbour distance distribution function and F the empty space function of the process. J(r) is identically equal to 1 for a Poisson process; values of J(r) smaller or larger than 1 indicate clustering or regularity, respectively. We show that, for a large class of point processes, J(r) is constant for distances r greater than the range of spatial interaction. Hence both the range and type of interaction can be inferred from J without parametric model assumptions. It is also possible to evaluate J(r) explicitly for many point process models, so that J is also useful for parameter estimation.Various properties are derived, including the fact that the J function of the superposition of independent point processes is a weighted mean of the J functions of the individual processes. Estimators of J can be constructed from standard estimators of F and G. We compute estimates of J for several standard point pattern datasets and conclude that it is a useful indicator of spatial interaction.

We propose new summary statistics for intensity-reweighted moment stationary point processes, that is, point processes with translation invariant n-point correlation functions for all n ∈ N, that generalise the well known J -, empty space, and spherical Palm contact distribution functions. We represent these statistics in terms of generating functionals and relate the inhomogeneous J -function to the inhomogeneous reduced second moment function. Extensions to space time and marked point processes are briefly discussed.