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 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.

arXiv comment21 pages, 15 figures. Published in MNRAS

References in corpus (43)

• Detection of the Baryon Acoustic Peak in the Large-Scale Correlation Function of SDSS Luminous Red Galaxies
• Dark Energy Survey Year 3 Results: Cosmological Constraints from Galaxy Clustering and Weak Lensing
• Cosmological Parameters from Observations of Galaxy Clusters
• Planck 2018 results. V. CMB power spectra and likelihoods
• Why your model parameter confidences might be too optimistic -- unbiased estimation of the inverse covariance matrix
• Dark Energy Survey Year 1 Results: Cosmological Constraints from Cosmic Shear
• Planck 2018 results. VII. Isotropy and Statistics of the CMB
• Cosmological parameters from lensing power spectrum and bispectrum tomography
• The Quijote simulations
• Cosmology and the Bispectrum
• The power spectrum and bispectrum of SDSS DR11 BOSS galaxies I: bias and gravity
• Hyper Suprime-Cam Year 3 Results: Cosmology from Cosmic Shear Power Spectra
• The Kinematic Sunyaev-Zel'dovich Effect with Projected Fields: A Novel Probe of the Baryon Distribution with Planck, WMAP, and WISE Data
• Cosmological Constraints from Galaxy Clustering and the Mass-to-Number Ratio of Galaxy Clusters
• Evolution of One-Point Distributions from Gaussian Initial Fluctuations
• The Topology of the Cosmic Web in Terms of Persistent Betti Numbers
• Nearest Neighbor distributions: new statistical measures for cosmological clustering
• First Detection of Cosmic Structure in the 21-cm Intensity Field
• Direct Minkowski Functional analysis of large redshift surveys: a new high--speed code tested on the luminous red galaxy Sloan Digital Sky Survey-DR7 catalogue
• The Persistence of Large Scale Structures I: Primordial non-Gaussianity
• Cosmological cross-correlations and nearest neighbor distributions
• Statistics of cosmic density profiles from perturbation theory
• Density distribution of the cosmological matter field
• Voronoi volume function: A new probe of cosmology and galaxy evolution
• Probing Galaxy Assembly Bias in BOSS Galaxies Using Void Probabilities
• Prospects of constraining reionization model parameters using Minkowski tensors and Betti numbers
• Morphology of 21cm brightness temperature during the Epoch of Reioinization using Contour Minkowski Tensor
• Modeling Nearest Neighbor distributions of biased tracers using Hybrid Effective Field Theory
• On Minkowski Functionals of CMB polarization
• Probing massive neutrinos with the Minkowski functionals of the galaxy distribution
• Tracer-Field Cross-Correlations with $k$-Nearest Neighbor Distributions
• The HI-halo mass relation at redshift $z\sim 1$ from the Minkowski functionals of 21 cm intensity maps
• Cosmology with Persistent Homology: a Fisher Forecast
• Detection of spatial clustering in the 1000 richest SDSS DR8 redMaPPer clusters with Nearest Neighbor distributions
• 2D k-th nearest neighbor statistics: a highly informative probe of galaxy clustering
• An Introduction into the Theory of Cosmological Structure Formation
• Validating the clustering predictions of empirical models with the FLAMINGO simulations
• Joint Minkowski Functionals and Bispectrum Constraints on Non-Gaussianity in the CMB
• Topological data analysis reveals differences between simulated galaxies and dark matter haloes
• High-energy Neutrino Source Cross-correlations with Nearest-neighbor Distributions
• Constraining cosmology with thermal Sunyaev-Zel'dovich maps: Minkowski functionals, peaks, minima, and moments
• Cosmological constraints from the Minkowski functionals of the BOSS CMASS galaxy sample
• Cosmological information content of Betti curves and $k$-nearest neighbor distributions

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