Depth-based classification approaches for directional data

Supervised learning tasks aim to define a data-based rule by which new objects are assigned to one of the given classes. To this end, a training set containing objects with known memberships is exploited. Directional data are points lying on the surface of circles, spheres or hyper-spheres. Given that they lie on a non-linear manifold, directional observations require specific methods to be analyzed. In this thesis, the main interest is to present novel methodologies and to perform reliable inferences for directional data, within the framework of supervised classification. First, a supervised classification procedure for directional data is introduced. The procedure is based on the cumulative distribution of the cosine depth, that is a directional distance-based depth function. The proposed method is compared with the max-depth classifier, a well-known depth-based classifier within the literature, through simulations and a real data example. Second, we study the optimality of the depth distribution and the max-depth classifiers from a theoretical perspective. More specifically, we investigate the necessary conditions under which the classifiers are optimal in the sense of the optimal Bayes rule. Then, we study the robustness of some directional depth-based classifiers in the presence of contaminated data. The performance of the depth distribution classifier, the max-depth classifier and the DD-classifier is evaluated by means of simulations in the presence of both class and attribute noise. Finally, the last part of the thesis is devoted to evaluate the performance of depth-based classifiers on a real directional data set.