ON THE ESTIMATION OF DISTANCE DISTRIBUTION FUNCTIONS FOR POINT PROCESSES AND RANDOM SETS

This paper discusses various estimators for the nearest neighbour distance distribution function D of a stationary point process and for the quadratic contact distribution function Hq of a stationary random closed set. It recommends the use of Hanisch's estimator of D, which is of Horvitz-Thompson type, and the minussampling estimator of Hq. This recommendation is based on simulations for Poisson processes and Boolean models.

A non-parametric measure of spatial interaction in point patterns

The strength and range of interpoint interactions in a spatial point process can be quantified by the function J = (1 - G)/(1 - F), where 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 very 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 interpoint interaction may be inferred from J without parametric model assumptions. We evaluate J(r) explicitly for a variety of point processes. The J function of the superposition of independent point processes is a weighted mean of the J functions of the individual processes.