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