Classification and redshift estimation by principal component analysis

We show that the first 10 eigencomponents of the Karhunen-Lo\`eve expansion or Principal Component Analysis (PCA) provide a robust classification scheme for the identification of stars, galaxies and quasi-stellar objects from multi-band photometry. To quantify the efficiency of the method, realistic simulations are performed which match the planned Large Zenith Telescope survey. This survey is expected to provide spectral energy distributions with a resolution $R\simeq40$ for $\sim10^6$ galaxies to $R\le23$ ($z\sim 1$), $\sim 10^4$ QSOs, and $\sim 10^5$ stars. We calculate that for a median signal-to-noise ratio of 6, 98% of stars, 100% of galaxies and 93% of QSOs are correctly classified. These values increase to 100% of stars, 100% of galaxies and 100% of QSOs at a median signal-to-noise ratio of 10. The 10-component PCA also allows measurement of redshifts with an accuracy of $\sigma_\mathrm{Res.}\la0.05$ for galaxies with $z\la0.7$, and to $\sigma_\mathrm{Res.}\la0.2$ for QSOs with $z\ga2$, at a median signal-to-noise ratio of 6. At a median signal-to-noise ratio 20, $\sigma_\mathrm{Res.}\la0.02$ for galaxies with $z\la1$ and for QSOs with $z\ga2.5$ (note that for a median $S/N$ ratio of 20, the bluest/reddest objects will have a signal-to-noise ratio of $\la 2$ in their reddest/bluest filters). This redshift accuracy is inherent to the $R\simeq40$ resolution provided by the set of medium-band filters used by the Large Zenith Telescope survey. It provides an accuracy improvement of nearly an order of magnitude over the photometric redshifts obtained from broad-band $BVRI$ photometry.