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Blazars optical emission is generally dominated by relativistic jets, although the host galaxy, accretion disk and broad line region (BLR) may also contribute significantly. Disentangling their contributions has been challenging for years due to the dominance of the jet. To quantify the contributions to the spectral variability, we use the statistical technique for dimensionality reduction Non-Negative Matrix Factorization on a spectroscopic data set of 26 $γ$-ray blazars. This technique allows to model large numbers of spectra in terms of a reduced number of components.We use a priori knowledge to obtain components associated to meaningful physical processes. The sources are classified according to their optical spectrum as host-galaxy dominated BL Lac objects (BL Lacs), BL Lacs, or Flat Spectrum Radio Quasars (FSRQs). Host-galaxy sources show less variability, as expected, and bluer-when-brighter trends, as the other BL Lacs. For FSRQs, more complicated colour-flux behaviours are observed: redder-when-brighter for low states saturating above a certain level and, in some cases, turning to bluer-when-brighter. We are able to reproduce the variability observed during 10 years using only 2 to 4 components, depending on the type. The simplest scenario corresponds to host-galaxy blazars, whose spectra are reconstructed using the stellar population and a power law for the jet. BL Lac spectra are reproduced using from 2 to 4 power laws. Different components can be associated to acceleration/cooling processes taking place in the jet. The reconstruction of FSRQs also incorporates a QSO-like component to account for the BLR, plus a very steep power law, associated to the accretion disk.