Geometric Interpretations of the k-Nearest Neighbour Distributions

Abstract

The k-Nearest Neighbour Cumulative Distribution Functions are measures of clustering for discrete datasets that are fast and efficient to compute. They are significantly more informative than the 2-point correlation function. Their connection to N-point correlation functions, void probability functions and Counts-in-Cells is known. However, the connections between the CDFs and geometric and topological summary statistics are yet to be fully explored.This understanding will be crucial to find optimally informative summary statistics to analyse data from stage-4 cosmological surveys. We explore quantitatively the geometric interpretations of the kNN CDF summary statistics. We establish an equivalence between the 1NN CDF at radius r and the volume of spheres with the same radius around data points. We show that higher kNN CDFs represent the volumes of intersections of at least k spheres around data points. We present similar geometric interpretations for the kNN cross-correlation CDFs. We further show that the full shape of the CDFs have information about planar angles, solid angles and arc lengths created at the intersections of spheres around the data points, and can be accessed through the derivatives of the CDF. We show that this information is equivalent to that captured by Germ Grain Minkowski Functionals. Using Fisher analyses we compare the constraining power of various data vectors constructed from kNN CDFs and Minkowski Functionals. We find that the CDFs and their derivatives and the Minkowski Functionals have nearly identical constraining power. However, the CDFs are computationally orders of magnitude faster to evaluate.