General notions of depth for functional data

TL;DR

This paper introduces a unified framework for defining data depth in functional data analysis, proposing the Phi-depth approach that encompasses many existing depths and introduces new ones like location-slope and principal component depths.

Contribution

It formulates postulates for functional data depth, proposes the Phi-depth framework, and introduces new depths including location-slope and principal component depths.

Findings

Phi-depth satisfies desirable properties for functional data.

The framework includes many existing depths as special cases.

New depths like location-slope and principal component depths are introduced.

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.