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AN APPROXIMATION FOR THE KTH NEAREST DISTANCE AND ITS APPLICATION TO LOCATIONAL ANALYSIS

In this paper, we provide a simple approximation for the distance from an arbitrary location to the kth nearest point. Distance is measured as the Euclidean and the rectilinear distances on a continuous plane. The approximation demonstrates that the kth nearest distance is proportional to the square root of k and inversely proportional to the square root of the density of points. The accuracy of the approximation is assessed for regular and random point patterns. Comparing the approximation with road network distances shows that the approximation on a continuous plane can be used for estimating the kth nearest distance on actual road networks. As an application of the approximation to locational analysis, we obtain the average distance to the nearest open facility when some of the existing facilities are closed.