Extreme Distance Distributions of Poisson Voronoi Cells

TL;DR

This paper investigates a distance measure of Poisson Voronoi tessellations that is emerging in the literature, yet for which its statistical and geometrical properties remain explored only in the asymptotic case when the density of seed points approaches infinity.

Abstract

Poisson point processes provide a versatile framework for modeling the distributions of random points in space. When the space is partitioned into cells, each associated with a single generating point from the Poisson process, there appears a geometric structure known as Poisson Voronoi tessellation. These tessellations find applications in various fields such as biology, material science, and communications, where the statistical properties of the Voronoi cells reveal patterns and structures that hold key insights into the underlying processes generating the observed phenomena. In this paper, we investigate a distance measure of Poisson Voronoi tessellations that is emerging in the literature, yet for which its statistical and geometrical properties remain explored only in the asymptotic case when the density of seed points approaches infinity. Our work, specifically focused on homogeneous Poisson point processes, characterizes the cumulative distribution functions governing the smallest and largest distances between the points generating the Voronoi regions and their respective vertices for an arbitrary density of points in $\mathbb{R}^2$. For that, we conduct a Monte-Carlo type simulation with $10^8$ Voronoi cells and fit the resulting empirical cumulative distribution functions to the Generalized Gamma, Gamma, Log-normal, Rayleigh, and Weibull distributions. Our analysis compares these fits in terms of root mean-squared error and maximum absolute variation, revealing the Generalized Gamma distribution as the best-fit model for characterizing these distances in homogeneous Poisson Voronoi tessellations. Furthermore, we provide estimates for the maximum likelihood and the $95$\% confidence interval of the parameters of the Generalized Gamma distribution along with the algorithm implemented to calculate the maximum and minimum distances.

Figures (3)

• Figure 1: (a) Delanuay triangulation of a sample realization of an homogeneous PPP with intensity parameter $\lambda=0.8$ over an area $A=100$. (b) Poisson Voronoi tessellation of the homogeneous PPP generated in (a). Black dots represent generator seeds in the planar space. Red dots represent the vertices of the Voronoi cells, which correspond to the circumcenters of the triangles generated by the Delanuay triangulation. Gray lines delimit the triangles of the Delanuay triangulation, and orange lines represent the edges of the Voronoi cells. The arrows in blue and magenta exemplify the stochastic quantities of interest in $2$D.
• Figure 2: Poisson Voronoi tessellation of a one-dimensional homogeneous PPP, where $\theta$ represents the distance between generator seed points, and $D$ represents the distances between generator seed points and their edges. The arrows in blue and magenta exemplify the stochastic quantities of interest in $1$D.
• Figure 3: Cumulative distribution functions of the normalized maximum and minimum distances in 2D bounded, homogeneous Poisson Voronoi regions. Straight lines represent the empirical cumulative density function of the maximum and minimum distances that resulted from the Monte-Carlo simulation described in Algorithm \ref{['alg:VoronoiTesselation_2D']} with $\lambda A = 10^8$ Voronoi cells. Dashed lines represent out best-fit approximation to the Generalized Gamma distribution.