The cylindrical K-function and Poisson line cluster point processes

TL;DR

This paper introduces the cylindrical K-function for analyzing anisotropic point patterns with linear structures and proposes Poisson line cluster point processes, with applications in neuroscience and geography.

Contribution

It presents a new directional K-function based on cylinders and a novel class of anisotropic Cox point processes called Poisson line cluster processes.

Findings

The cylindrical K-function effectively detects anisotropy in point patterns.

Poisson line cluster processes model linear structures with clustered points.

Application to neuroscience data supports the model's relevance.

Abstract

Analyzing point patterns with linear structures has recently been of interest in e.g. neuroscience and geography. To detect anisotropy in such cases, we introduce a functional summary statistic, called the cylindrical $K$-function, since it is a directional $K$-function whose structuring element is a cylinder. Further we introduce a class of anisotropic Cox point processes, called Poisson line cluster point processes. The points of such a process are random displacements of Poisson point processes defined on the lines of a Poisson line process. Parameter estimation based on moment methods or Bayesian inference for this model is discussed when the underlying Poisson line process and the cluster memberships are treated as hidden processes. To illustrate the methodologies, we analyze a two and a three-dimensional point pattern data set. The 3D data set is of particular interest as it relates to the minicolumn hypothesis in neuroscience, claiming that pyramidal and other brain cells have a columnar arrangement perpendicular to the pial surface of the brain.