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 sp
ace 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.
A new procedure, called DDa-procedure, is developed to solve the problem of classifying d-dimensional objects into q >= 2 classes. The procedure is completely nonparametric; it uses q-dimensional depth plots and a very efficient algorithm for discrim
ination analysis in the depth space [0,1]^q. Specifically, the depth is the zonoid depth, and the algorithm is the alpha-procedure. In case of more than two classes several binary classifications are performed and a majority rule is applied. Special treatments are discussed for outsiders, that is, data having zero depth vector. The DDa-classifier is applied to simulated as well as real data, and the results are compared with those of similar procedures that have been recently proposed. In most cases the new procedure has comparable error rates, but is much faster than other classification approaches, including the SVM.
