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.
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