Improving the accuracy of estimators for the two-point correlation function

TL;DR

This paper introduces geometrically motivated and quasi-Monte Carlo estimators that significantly improve the accuracy and speed of two-point correlation function estimates in astrophysics, outperforming standard methods.

Contribution

The authors develop and validate new estimators using low-discrepancy sequences, achieving near-linear error scaling and substantial computational speedups over traditional random point methods.

Findings

Error scales as 1/N_q with new estimators, compared to 1/√N_r for standard methods.

Speedup factors range from 50% to several thousand times faster.

Applicable to incomplete galaxy catalogues with maintained accuracy.

Abstract

We show how to increase the accuracy of estimates of the two-point correlation function without sacrificing efficiency. We quantify the error of the pair-counts and of the Landy-Szalay estimator by comparing them with exact reference values. The standard method, using random point sets, is compared to geometrically motivated estimators and estimators using quasi-Monte~Carlo integration. In the standard method, the error scales proportionally to $1/\sqrt{N_r}$, with $N_r$ being the number of random points. In our improved methods, the error scales almost proportionally to $1/N_q$, where $N_q$ is the number of points from a low-discrepancy sequence. We study the run times of the new estimator in comparison to those of the standard estimator, keeping the same level of accuracy. For the considered case, we always see a speedup ranging from 50% up to a factor of several thousand. We also discuss how to apply these improved estimators to incompletely sampled galaxy catalogues.