Kavli Affiliate: Tom Abel
| First 5 Authors: Kwanit Gangopadhyay, Arka Banerjee, Tom Abel, ,
| Summary:
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.
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