Title: 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 in [2] a functional summary statistic, the cylindrical K-function, which is a directional K-function whose structuring element is a cylinder. Further we introduce in [2] 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 is discussed when the underlying Poisson line process is latent, using either moment based methods or Bayesian inference based on the simulation algorithm in [1]. To illustrate the methodologies, we analyze two and three-dimensional point pattern data sets. The three-dimensional data set is of particular interest as it relates to the minicolumn hypothesis in neuroscience (see [3]), claiming that pyramidal and other brain cells have a columnar arrangement perpendicular to the surface of the brain.
[1] C.J. Geyer and J. Møller (1994). Simulation procedures and likelihood inference for spatial point processes. Scandinavian Journal of Statistics, 21, 359-373.
[2] J. Møller, F. Safavimanesh and J.G. Rasmussen (2016). The cylindrical K-function and Poisson line cluster point processes. Biometrika, 103, 937-954.
[3] A.H. Rafati, F. Safavimanesh, K.-A. Dorph-Petersen, J.G. Rasmussen, J. Møller and J.R. Nyengaard (2016). Detection and spatial characterization of minicolumnarity in the human cerebral cortex. Journal of Microscopy, 261, 115-126.