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A new graph related to the directions of nearest neighbours in a point process

By S. N. Chiu and Ilya Molchanov

Abstract

This paper introduces a new graph constructed from a point process. The idea is to connect a point with its nearest neighbour, then to the second nearest and continue this process until the point belongs to the interior of the convex hull of these nearest neighbours. The number of such neighbours is called the degree of a point. We derive the distribution of the degree of the typical point in a Poisson process, prove a central limit theorem for the sum of degrees, and propose an edge-corrected estimator of the distribution of the degree that is unbiased for a stationary Poisson process. Simulation studies show that this degree is a useful concept that allows the separation of clustering and repulsive behaviour of point processes

Topics: 510 Mathematics
Publisher: Applied Probability Trust
Year: 2003
DOI identifier: 10.1239/aap
OAI identifier: oai:boris.unibe.ch:85364
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