Distance statistics in random media: high dimension and/or high neighborhood order cases

TL;DR

This paper analytically derives the distance statistics to the k-th nearest neighbor in a d-dimensional Poisson medium, exploring high dimension and neighborhood order limits, with applications in detecting deviations from Poissonian distributions.

Contribution

It provides new analytical results for distance distributions in high-dimensional and high neighborhood order regimes, extending previous theoretical understanding.

Findings

Distance distribution becomes a delta sequence in high dimensions.

Distance statistics in high neighborhood order tend to a Gaussian distribution.

The results can detect deviations from Poisson point distributions.

Abstract

Consider an unlimited homogeneous medium disturbed by points generated via Poisson process. The neighborhood of a point plays an important role in spatial statistics problems. Here, we obtain analytically the distance statistics to $k$th nearest neighbor in a $d$-dimensional media. Next, we focus our attention in high dimensionality and high neighborhood order limits. High dimensionality makes distance distribution behavior as a delta sequence, with mean value equal to Cerf's conjecture. Distance statistics in high neighborhood order converges to a Gaussian distribution. The general distance statistics can be applied to detect departures from Poissonian point distribution hypotheses as proposed by Thompson and generalized here.