Estimating the J function without edge correction

Abstract

The interaction between points in a spatial point process can be measured by its empty space function F, its nearest-neighbour distance distribution function G, and by combinations such as the J-function $J = (1-G)/(1-F)$. The estimation of these functions is hampered by edge effects: the uncorrected, empirical distributions of distances observed in a bounded sampling window W give severely biased estimates of F and G. However, in this paper we show that the corresponding {\em uncorrected} estimator of the function $J=(1-G)/(1-F)$ is approximately unbiased for the Poisson case, and is useful as a summary statistic. Specifically, consider the estimate $\hat{J}_W$ of J computed from uncorrected estimates of F and G. The function $J_W(r)$, estimated by $\hat{J}_W$, possesses similar properties to the J function, for example $J_W(r)$ is identically 1 for Poisson processes. This enables direct interpretation of uncorrected estimates of J, something not possible with uncorrected estimates of either F, G or K. We propose a Monte Carlo test for complete spatial randomness based on testing whether $J_W(r)(r)\equiv 1$. Computer simulations suggest this test is at least as powerful as tests based on edge corrected estimators of J.