  The $DD\alpha$-classifier, a nonparametric fast and very robust procedure, is
described and applied to fifty classification problems regarding a broad
spectrum of real-world data. The procedure first transforms the data from their
original property space into a depth space, which is a low-dimensional unit
cube, and then separates them by a projective invariant procedure, called
$\alpha$-procedure. To each data point the transformation assigns its depth
values with respect to the given classes. Several alternative depth notions
(spatial depth, Mahalanobis depth, projection depth, and Tukey depth, the
latter two being approximated by univariate projections) are used in the
procedure, and compared regarding their average error rates. With the Tukey
depth, which fits the distributions' shape best and is most robust,
`outsiders', that is data points having zero depth in all classes, need an
additional treatment for classification. Evidence is also given about the
dimension of the extended feature space needed for linear separation. The
$DD\alpha$-procedure is available as an R-package.
