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Weighted lens depth: Some applications to supervised classification
Author(s) -
Cholaquidis Alejandro,
Fraiman Ricardo,
Gamboa Fabrice,
Moreno Leonardo
Publication year - 2023
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.1002/cjs.11724
Subject(s) - geodesic , lens (geology) , manifold (fluid mechanics) , mathematics , metric (unit) , fermat's last theorem , nothing , computer science , artificial intelligence , algorithm , mathematical analysis , pure mathematics , optics , physics , mechanical engineering , philosophy , operations management , epistemology , economics , engineering
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.