Nearest-Neighbor and Contact Distance Distributions for Thomas Cluster Process

TL;DR

This paper derives the statistical distributions of nearest-neighbor and contact distances in the Thomas cluster process, providing new formulas for different reference point scenarios within this Poisson cluster process.

Contribution

It introduces explicit formulas for the CDFs of nearest-neighbor and contact distances in the TCP for various reference point cases, enhancing understanding of spatial point process characteristics.

Findings

Derived the CDF of contact distance for TCP.

Derived the CDFs of nearest-neighbor distances for different reference points.

Provided analytical tools for analyzing spatial clustering in TCP.

Abstract

We characterize the statistics of nearest-neighbor and contact distance distributions for Thomas cluster process (TCP), which is a special case of Poisson cluster process. In particular, we derive the cumulative distribution function (CDF) of the distance to the nearest point of TCP from a reference point for three different cases: (i) reference point is not a part of the point process, (ii) it is chosen uniformly at random from the TCP, and (iii) it is a randomly chosen point from a cluster chosen uniformly at random from the TCP. While the first corresponds to the contact distance distribution, the other two provide two different viewpoints for the nearest-neighbor distance distribution.