  We develop a new framework for use in exploring Type Ia Supernova (SN Ia)
spectra. Combining Principal Component Analysis (PCA) and Partial Least Square
analysis (PLS) we are able to establish correlations between the Principal
Components (PCs) and spectroscopic/photometric SNe Ia features. The technique
was applied to ~120 supernova and ~800 spectra from the Nearby Supernova
Factory. The ability of PCA to group together SNe Ia with similar spectral
features, already explored in previous studies, is greatly enhanced by two
important modifications: (1) the initial data matrix is built using derivatives
of spectra over the wavelength, which increases the weight of weak lines and
discards extinction, and (2) we extract time evolution information through the
use of entire spectral sequences concatenated in each line of the input data
matrix. These allow us to define a stable PC parameter space which can be used
to characterize synthetic SN Ia spectra by means of real SN features. Using
PLS, we demonstrate that the information from important previously known
spectral indicators (namely the pseudo-equivalent width (pEW) of Si II 5972 /
Si II 6355 and the line velocity of S II 5640 / Si II 6355) at a given epoch,
is contained within the PC space and can be determined through a linear
combination of the most important PCs. We also show that the PC space
encompasses photometric features like B or V magnitudes, B-V color and SALT2
parameters c and x1. The observed colors and magnitudes, that are heavily
affected by extinction, cannot be reconstructed using this technique alone. All
the above mentioned applications allowed us to construct a metric space for
comparing synthetic SN Ia spectra with observations.
