General notions of depth for functional data
Abstract
A data depth measures the centrality of a point with respect to an empirical distribution. Postulates are formulated, which a depth for functional data should satisfy, and a general approach is proposed to construct multivariate data depths in Banach spaces. The new approach, mentioned as Phi-depth, is based on depth infima over a proper set Phi of R^d-valued linear functions. Several desirable properties are established for the Phi-depth and a generalized version of it. The general notions include many new depths as special cases. In particular a location-slope depth and a principal component depth are introduced.
Cite
@article{arxiv.1208.1981,
title = {General notions of depth for functional data},
author = {Karl Mosler and Yulia Polyakova},
journal= {arXiv preprint arXiv:1208.1981},
year = {2018}
}
Comments
The revision of November 2016 introduces the term Tukey graph depth to distinguish it from half-region depth. In the present version (January 2018) I have dropped an erroneous remark on the band depth, thanks to Stanislav Nagy, who pointed out in his dissertation that the band depth is no infimum depth