First contact distributions for spatial patterns: regularity and estimation

For applications in spatial statistics an important property of a random set X in Rk is its rst contact distribution This is the distribution of the distance from a xed point to the nearest point of X where distance is measured using scalar dilations of a xed test set B We show that if B is convex and contains a neighbourhood of the rst contact distribution function FB is absolutely continuous We give two explicit representations of FB and additional regularity conditions under which FB is continuously dierentiable A KaplanMeier estimator of FB is introduced and its basic properties examine

Extrapolating and interpolating spatial patterns

We discuss issues arising when a spatial pattern is observed within some bounded region of space, and one wishes to predict the process outside of this region (extrapolation) as well as to perform inference on features of the pattern that cannot be observed (interpolation). We focus on spatial cluster analysis. Here the interpolation arises from the fact that the centres of clustering are not observed. We take a Bayesian approach with a repulsive Markov prior, derive the posterior distribution of the complete data, i.e. cluster centres with associated offspring marks, and propose an adaptive coupling from the past algorithm to sample from this posterior. The approach is illustrated by means of the redwood data set (Ripley, 1977)