Improving Correlation Function Fitting with Ridge Regression: Application to Cross-Correlation Reconstruction

TL;DR

This paper enhances correlation function fitting for redshift distribution reconstruction by incorporating full covariance and applying ridge regression, significantly reducing errors in photometric redshift calibration.

Contribution

It introduces a ridge regression-based method for conditioning covariance matrices, improving the stability and accuracy of correlation function fits in large-scale structure analysis.

Findings

Reduces redshift distribution reconstruction error by up to 40%.

Demonstrates improved correlation function parameter estimation.

Provides a publicly available IDL code for power-law fitting with ridge regression.

Abstract

Cross-correlation techniques provide a promising avenue for calibrating photometric redshifts and determining redshift distributions using spectroscopy which is systematically incomplete (e.g., current deep spectroscopic surveys fail to obtain secure redshifts for 30-50% or more of the galaxies targeted). In this paper we improve on the redshift distribution reconstruction methods presented in Matthews & Newman (2010) by incorporating full covariance information into our correlation function fits. Correlation function measurements are strongly covariant between angular or spatial bins, and accounting for this in fitting can yield substantial reduction in errors. However, frequently the covariance matrices used in these calculations are determined from a relatively small set (dozens rather than hundreds) of subsamples or mock catalogs, resulting in noisy covariance matrices whose inversion is ill-conditioned and numerically unstable. We present here a method of conditioning the covariance matrix known as ridge regression which results in a more well behaved inversion than other techniques common in large-scale structure studies. We demonstrate that ridge regression significantly improves the determination of correlation function parameters. We then apply these improved techniques to the problem of reconstructing redshift distributions. By incorporating full covariance information, applying ridge regression, and changing the weighting of fields in obtaining average correlation functions, we obtain reductions in the mean redshift distribution reconstruction error of as much as ~40% compared to previous methods. In an appendix, we provide a description of POWERFIT, an IDL code for performing power-law fits to correlation functions with ridge regression conditioning that we are making publicly available.