Geometric Interpretations of the $k$-Nearest Neighbour Distributions

TL;DR

The paper develops a geometric interpretation of kNN CDFs as volumes and intersections of spheres around data points, linking them to Counts-in-Cells, N-point functions, and Germ-Grain Minkowski Functionals. It shows that derivatives of the 1NN CDF encode surface area, angles, arcs, and the Euler characteristic, establishing a close correspondence with Minkowski Functionals. A Fisher analysis on Quijote simulations demonstrates that the 1NN CDF and its derivatives carry comparable cosmological information to Minkowski Functionals, while offering substantial computational speed advantages; higher kNN CDFs and derivatives also contribute, especially on larger scales. The work provides a unified framework uniting algebraic and geometric clustering statistics and discusses practical directions for data-vector construction and future extensions to cross correlations and topology based analyses for stage 4 cosmological surveys.

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 other geometric and topological spatial summary statistics are yet to be fully explored in the literature. 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 $k$NN CDF summary statistics. We establish an equivalence between the 1NN CDF at radius $r$ and the volume of spheres with the same radius around the data points. We show that higher $k$NN CDFs are equivalent to the volumes of intersections of $\ge k$ spheres around the data points. We present similar geometric interpretations for the $k$NN cross-correlation joint CDFs. We further show that the volume, or the CDFs, have information about the angles and arc lengths created at the intersections of spheres around the data points, which can be accessed through the derivatives of the CDF. We show this information is very similar to that captured by Germ Grain Minkowski Functionals. Using a Fisher analysis we compare the information content and constraining power of various data vectors constructed from the $k$NN CDFs and Minkowski Functionals. We find that the CDFs and their derivatives and the Minkowski Functionals have nearly identical information content. However, $k$NN CDFs are computationally orders of magnitude faster to evaluate. Finally, we find that there is information in the full shape of the CDFs, and therefore caution against using the values of the CDF only at sparsely sampled radii.

Figures (15)

• Figure 1: Shaded are all the query points that have a 1NN distance of $\le r$ from the data points (in black). Thus, the fraction of the total box that is shaded gives us the $\mathrm{CDF}_{1\mathrm{NN}}$ at this particular radius $r$ (equation \ref{['eq:W0_1']}). This is a representative figure with circles in 2D, but the same argument holds with spheres in 3D.
• Figure 2: Shaded in different colours are the volume within exactly one sphere, exactly two spheres and exactly three spheres. These volumes can be obtained by using the $\mathrm{CDF}_{k\mathrm{NN}}$'s using equation \ref{['eq:difference_of_cdfs']}. The symbol $||.||$ denotes the Euclidean distance, so $|| \; 1^{\mathrm{st}}\; \textrm{NN}\; || < r$ means that distance to the first nearest neighbour is less than $r$. The $k$NN CDF corresponds to the fraction of volume within at least$k$ spheres, so it is the sum of the volumes within intersections of exactly $k, k+1, k+2, ...$ spheres. Hence, the 1NN CDF is the proportional to the sum of volumes of the red, green and beige regions; the 2NN CDF proportional to the sum of volumes of the green and beige regions, and the 3NN CDF proportional to the volume of the beige region.
• Figure 3: Geometric interpretation for the joint $\mathrm{CDF}_{k_1,k_2}(r)$.
• Figure 4: This figure shows the intersection circle created by the the intersection of spheres around two points $i$ and $j$. The length of the circle is $\ell_{ij}$. But for intersection of three or more spheres, the circle is not fully on the surface, and we only need to consider the length of the arc that is on the surface.
• Figure 5: The top panel is for the data vector of 1NN CDF and its three derivatives. The four parts are plotted in the four columns, but the Fisher analysis has been done by combining all these into a single data vector. In the first two rows, the light orange curves represent the data vectors from 1000 fiducial realizations of the Quijote simulations, and the dashed purple curve represents the mean of these 1000 fiducial data vectors. The bottom pannel shows the derivatives of the data vectors with respect to the cosmological parameters, normalized by the mean of the 1000 fiducial data vectors.