Edge‐corrected estimators of the nearest‐neighbour distance distribution function for three‐dimensional point patterns

In multiphase systems consisting of ‘particles’ embedded in a matrix the three-dimensional spatial distribution of the particles may represent important structural information. In systems where the matrix is transparent or translucent recent developments in microscopy allow the three-dimensional location of particles to be recorded. Using these data a spatial statistical, or second-order stereological, analysis can be carried out. In second-order stereology functions of interparticle distances are used as summary statistics of the spatial distributions. These statistics show whether the particles are randomly arranged or, more commonly, either clustered together or inhibited from close approach to each other. This paper focuses on the estimation of one of these spatial statistics, the nearest-neighbour distance distribution function or G-function. In practice, estimation of the G-function is plagued by an ‘edge-effect’ bias introduced by the sampling process itself. There exist a number of G-function estimators that tackle this edge effect problem; for single sample ‘bricks’ it can be shown that these estimators become increasingly accurate as the brick size increases, i.e. they are consistent. However, in many practical cases the size of a sampling brick is fixed by experimental constraints and in these circumstances the only way to increase sample size is to take replicated sampling regions. In this paper we review a number of existing G-function estimators and propose a new estimator. These estimators are compared using the criterion of how well they overcome the edge-effect when they are applied to replicated samples of a fixed size of brick. These comparisons were made using Monte-Carlo simulation methods; the results show that three existing estimators are clearly unsuitable for estimating the G-function from replicated sample bricks. Of the other estimators the recommended estimator depends upon the number of replicates taken; however, we conclude that if a total of more than about 800 points are analysed then the bias in the pooled estimate of the G-function can be reduced to tolerable levels.