Depth-weighted Bayes classification

Two procedures for supervised classification are proposed. These are based on data depth and focus on the centre of each class. The classifiers add either a depth or a depth rank term to the objective function of the Bayes classifier. The cost of misclassifying a point depends not only on a class where it belongs, but also on its centrality with respect to this class. The classification of points that are more central is enforced while outliers are downweighted. The proposed objective function can also be used to evaluate the performance of other classifiers instead of the usual average misclassification rate. Use of the depth function increases robustness of the new procedures against the large inclusion of contaminated data that often impede the Bayes classifier. Properties of the new methods are investigated and compared with those of the Bayes classifier. Theoretical results are derived for elliptically symmetric distributions, while comparison for non-symmetric distributions is conducted by means of a simulation study. Comparisons are conducted for both theoretical classifiers and their empirical counterparts. The performance of the newly proposed classifiers is also compared to the performance of several standard methods in some real life situations.