Given data in $mathbb{R}^{p}$, a Tukey $kappa$-trimmed region is the set of all points that have at least Tukey depth $kappa$ w.r.t. the data. As they are visual, affine equivariant and robust, Tukey regions are useful tools in nonparametric multivar
iate analysis. While these regions are easily defined and interpreted, their practical use in applications has been impeded so far by the lack of efficient computational procedures in dimension $p > 2$. We construct two novel algorithms to compute a Tukey $kappa$-trimmed region, a na{i}ve one and a more sophisticated one that is much faster than known algorithms. Further, a strict bound on the number of facets of a Tukey region is derived. In a large simulation study the novel fast algorithm is compared with the na{i}ve one, which is slower and by construction exact, yielding in every case the same correct results. Finally, the approach is extended to an algorithm that calculates the innermost Tukey region and its barycenter, the Tukey median.
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 fast nonparametric procedure for classifying functional data is introduced. It consists of a two-step transformation of the original data plus a classifier operating on a low-dimensional hypercube. The functional data are first mapped into a finite
-dimensional location-slope space and then transformed by a multivariate depth function into the $DD$-plot, which is a subset of the unit hypercube. This transformation yields a new notion of depth for functional data. Three alternative depth functions are employed for this, as well as two rules for the final classification on $[0,1]^q$. The resulting classifier has to be cross-validated over a small range of parameters only, which is restricted by a Vapnik-Cervonenkis bound. The entire methodology does not involve smoothing techniques, is completely nonparametric and allows to achieve Bayes optimality under standard distributional settings. It is robust, efficiently computable, and has been implemented in an R environment. Applicability of the new approach is demonstrated by simulations as well as a benchmark study.
