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

Abstract

The analysis of point patterns with linear structures is of interest in many applications. To detect anisotropy in such cases, in particular in 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. 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 is latent. To illustrate the methodologies, we analyze twoand three-dimensional point pattern data sets. The three-dimensional 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 surface of the brain.

Detection and spatial characterization of minicolumnarity in the human cerebral cortex

“…Moreover, under CSR, K u (r, t) = 2πr 2 t. So if K u (r, t) is larger than this value under CSR and tends to be larger in this direction u than in other directions, it may indicate a columnar arrangement with 'main direction' u. As detailed in Møller et al (2015), there is a close relationship between the cylindrical K-function and the space-time (i.e. 2D-space · 1D-time) K-function introduced in Diggle et al (1995).…”

“…Importantly, our novel approach based upon the cylindrical K-function is not assuming the presence of minicolumns but just observe the spatial pattern of the neurons. In Møller et al (2015), we discuss a Bayesian methodology for fitting PLCPPs (extending the model briefly explained in Appendix A.2). This provides a detailed statistical description of the minicolumnar structure, including the direction, diameter and the number of minicolumns, and the number of cells within a minicolumn.…”

“…Apart from a deviation from CSR, detecting anisotropy caused by minicolumnar arrangement of particles is the next step which can be summarized by a directional counterpart of the K-function, the cylindrical Kfunction as developed in Møller et al (2015). The cylindrical Kfunction is denoted K u (r, t) and defined for any direction u in 3D space (i.e.…”

“…Thus, in the present study, the 3D coordinates (x, y, z) of each neuron was recorded through the entire thickness of brain tissue sections. We applied a mathematical tool, the cylindrical K-function (Møller et al, 2015), which is able to detect columnar organization of neurons in human cerebral cortex and also determines the degree of columnarity for each sampling region. Furthermore, our new approach is compared to other techniques in the literature using neuronal somas to estimate minicolumnar structural parameters in the cerebral cortex.…”

A three-dimensional anisotropic point process characterization for pharmaceutical coatings

“…In order to detect and model anisotropy, spatial summary statistics of point pairs need to be functions of both length and orientation of pairwise difference vectors. The K-function, describing the expected number of points within a certain distance, 40 was generalized for the detection of linearities in (Møller et al, 2016). Instead of considering points in a ball, directed cylinders are used as structure elements.…”