Emily Ewers | Searcharxiv

arXiv2025

Bayesian inference for Neyman-Scott point processes with anisotropic clusters

There are few inference methods available to accommodate covariate-dependent anisotropy in point process models. To address this, we propose an extended Bayesian MCMC approach for Neyman-Scott cluster processes. We focus on anisotropy and inhomogeneity in the offspring distribution. Our approach provides parameter estimates as well as significance tests for the covariates and anisotropy through credible intervals, which are determined by the posterior distributions. Additionally, it is possible to test the hypothesis of constant orientation of clusters or constant elongation of clusters. We demonstrate the applicability of this approach through a simulation study for a Thomas-type cluster process.

stat.ME↗

arXiv2024

Rectangular Gilbert Tessellation

A random planar quadrangulation process is introduced as an approximation for certain cellular automata in terms of random growth of rays from a given set of points. This model turns out to be a particular (rectangular) case of the well-known Gilbert tessellation, which originally models the growth of needle-shaped crystals from the initial random points with a Poisson distribution in a plane. From each point the lines grow on both sides of vertical and horizontal directions until they meet another line. This process results in a rectangular tessellation of the plane. The central and still open question is the distribution of the length of line segments in this tessellation. We derive exponential bounds for the tail of this distribution. The correlations between the segments are proved to decay exponentially with the distance between their initial points. Furthermore, the sign of the correlation is investigated for some instructive examples. In the case when the initial set of points is confined in a box $[0,N]^2$, it is proved that the average number of rays reaching the border of the box has a linear order in $N$.

math.PR↗