The cylindrical $K$-function and Poisson line cluster point processes

The analysis of point patterns with linear structures is of interest in many applications. To detect anisotropy in such patterns, in particular in the case of a columnar structure, we introduce a functional summary statistic, the cylindrical $K$-function, which is a directional $K$-function whose structuring element is a cylinder. We further 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 for this model based on moment methods or Bayesian inference is discussed in the case where the underlying Poisson line process is latent. To illustrate the proposed methods, we analyse two- and three-dimensional point pattern datasets. The three-dimensional dataset is of particular interest as it relates to the minicolumn hypothesis in neuroscience, which claims that pyramidal and other brain cells have a columnar arrangement perpendicular to the surface of the brain.