2009MNRAS.398.2012F

The development of fast and accurate methods of photometric redshift estimation is a vital step towards being able to fully utilize the data of next-generation surveys within precision cosmology. In this paper, we apply a specific approach to spectral connectivity analysis (SCA) called diffusion map. SCA is a class of non-linear techniques for transforming observed data (e.g. photometric colours for each galaxy, where the data lie on a complex subset of p-dimensional space) to a simpler, more natural coordinate system wherein we apply regression to make redshift predictions. In previous applications of SCA to other astronomical problems, we demonstrate its superiority vis-a-vis the principal components analysis, a standard linear technique for transforming data. As SCA relies upon eigen-decomposition, our training set size is limited to ≲10⁴ galaxies; we use the Nyström extension to quickly estimate diffusion coordinates for objects not in the training set. We apply our method to 350738 Sloan Digital Sky Survey (SDSS) main sample galaxies, 29816 SDSS luminous red galaxies and 5223 galaxies from DEEP2 with Canada-France-Hawaii Telescope Legacy Survey ugriz photometry. For all three data sets, we achieve prediction accuracies at par with previous analyses, and find that the use of the Nyström extension leads to a negligible loss of prediction accuracy relative to that achieved with the training sets. As in some previous analyses, we observe that our predictions are generally too high (low) in the low (high) redshift regimes. We demonstrate that this is a manifestation of attenuation bias, wherein measurement error (i.e. uncertainty in diffusion coordinates due to uncertainty in the measured fluxes/magnitudes) reduces the slope of the best-fitting regression line. Mitigation of this bias is necessary if we are to use photometric redshift estimates produced by computationally efficient empirical methods in precision cosmology.