Extinction law classification and lens redshift estimate by means of the principal component analysis

Aims. We propose a method based on the Principal Component Analysis (PCA) to classify and estimate the redshift of an extinction law in a distant gravitational lens galaxy. Such extinction laws are very poorly known and an efficient method to characterize them is badly needed. Methods. We first compute the principal axes of an exhaustive collection of redshifted theoretical extinction laws. Then, we project on these new axes the extinction law we wish to classify. The position of its projection among those redshifted extinction laws from the collection allows us to characterize it and to estimate its redshift. Results. Monte Carlo simulations show that the method is efficient and relatively precise for reasonably good signal-to-noise ratio data. The application of the method to a real case, the gravitational lens system SBS 0909+532, leads to very encouraging results.