Local Half-Region Depth for Functional Data

TL;DR

This paper introduces a local half-region depth for functional data, extending existing depth concepts to better capture local features, with theoretical analysis and real data examples demonstrating its effectiveness.

Contribution

It proposes a novel local version of the half-region depth for functional data, analyzing its properties and demonstrating its practical utility.

Findings

The local half-region depth captures local fluctuations in functional data.

Theoretical properties of the new depth are established.

Examples show improved detection of local features in real data.

Abstract

Data depth proves successful in the analysis of multivariate data sets, in particular deriving an overall center and assigning ranks to the observed units. Two key features are: the directions of the ordering, from the center towards the outside, and the recognition of a unique center irrespective of the distribution being unimodal or multimodal. This behaviour is a consequence of the monotonicity of the ranks that decrease along any ray from the deepest point. Recently, a wider framework allowing identification of partial centers was suggested in Agostinelli and Romanazzi [2011]. The corresponding generalized depth functions, called local depth functions are able to record local fluctuations and can be used in mode detection, identification of components in mixture models and in cluster analysis. Functional data [Ramsay and Silverman, 2006] are become common nowadays. Recently, Lopez-Pintado and Romo [2011] has proposed the half-region depth suited for functional data and for high dimensional data. Here, following their work we propose a local version of this data depth, we study its theoretical properties and illustrate its behaviour with examples based on real data sets.