Weighted lens depth: Some applications to supervised classification

Starting with Tukey's pioneering work in the 1970s, the notion of depth in statistics has been widely extended, especially in the last decade. Such extensions include those to high-dimensional data, functional data, and manifold-valued data. In particular, in the learning paradigm, the depth-depth method has become a useful technique. In this article, we extend the lens depth to the case of data in metric spaces and study its main properties. We also introduce, for Riemannian manifolds, the weighted lens depth. The weighted lens depth is nothing more than a lens depth for a weighted version of the Riemannian distance. To build it, we replace the geodesic distance on the manifold with the Fermat distance, which has the important property of taking into account the density of the data together with the geodesic distance. Next, we illustrate our results with some simulations and also in some interesting real datasets, including pattern recognition in phylogenetic trees, using the depth-depth approach.