  The mid-infrared spectra of ultraluminous infrared galaxies (ULIRGs) contain
a variety of spectral features that can be used as diagnostics to characterise
the spectra. However, such diagnostics are biased by our prior prejudices on
the origin of the features. Moreover, by using only part of the spectrum they
do not utilise the full information content of the spectra. Blind statistical
techniques such as principal component analysis (PCA) consider the whole
spectrum, find correlated features and separate them out into distinct
components.
  We further investigate the principal components (PCs) of ULIRGs derived in
Wang et al.(2011). We quantitatively show that five PCs is optimal for
describing the IRS spectra. These five components (PC1-PC5) and the mean
spectrum provide a template basis set that reproduces spectra of all z<0.35
ULIRGs within the noise. For comparison, the spectra are also modelled with a
combination of radiative transfer models of both starbursts and the dusty torus
surrounding active galactic nuclei. The five PCs typically provide better fits
than the models. We argue that the radiative transfer models require a colder
dust component and have difficulty in modelling strong PAH features.
  Aided by the models we also interpret the physical processes that the
principal components represent. The third principal component is shown to
indicate the nature of the dominant power source, while PC1 is related to the
inclination of the AGN torus.
  Finally, we use the 5 PCs to define a new classification scheme using 5D
Gaussian mixtures modelling and trained on widely used optical classifications.
The five PCs, average spectra for the four classifications and the code to
classify objects are made available at: http://www.phys.susx.ac.uk/~pdh21/PCA/
