Order Distance in Regular Point Patterns

Abstract

This article examines the kth nearest neighbor distance for three regular point patterns: square, triangular, and hexagonal lattices. The probability density functions of the kth nearest distance and the average kth nearest distances are theoretically derived for k=1, 2, …, 7. As an application of the kth nearest distance, we consider a facility location problem with closing of facilities. The problem is to find the optimal regular pattern that minimizes the average distance to the nearest open facility. Assuming that facilities are closed independently and at random, we show that the triangular lattice is optimal if at least 68% of facilities are open by comparing the upper and lower bounds of the average distances.