The $DDalpha$-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 $DDalpha$-procedure is available as an R-package.
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