Institute of Mathematical Statistics Lecture Notes - Monograph Series

Functional analysis via extensions of the band depth

Rebecka Jornsten and Sara López-Pintado

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The notion of data depth has long been in use to obtain robust location and scale estimates in a multivariate setting. The depth of an observation is a measure of its centrality, with respect to a data set or a distribution. The data depths of a set of multivariate observations translates to a center-outward ordering of the data. Thus, data depth provides a generalization of the median to a multivariate setting (the deepest observation), and can also be used to screen for extreme observations or outliers (the observations with low data depth). Data depth has been used in the development of a wide range of robust and non-parametric methods for multivariate data, such as non-parametric tests of location and scale [Li and Liu (2004)], multivariate rank-tests [Liu and Singh (1993)], non-parametric classification and clustering [Jornsten, (2004)], and robust regression [Rousseeuw and Hubert (1999)].

Many different notions of data depth have been developed for multivariate data. In contrast, data depth measures for functional data have only recently been proposed [Fraiman and Muniz (1999), López-Pintado and Romo (2006a)]. While the definitions of both of these data depth measures are motivated by the functional aspect of the data, the measures themselves are in fact invariant with respect to permutations of the domain (i.e. the compact interval on which the functions are defined). Thus, these measures are equally applicable to multivariate data where there is no explicit ordering of the data dimensions. In this paper we explore some extensions of functional data depths, so as to take the ordering of the data dimensions into account.

Chapter information

Regina Liu, William Strawderman and Cun-Hui Zhang, eds., Complex Datasets and Inverse Problems: Tomography, Networks and Beyond (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2007), 103-120

First available in Project Euclid: 4 December 2007

Permanent link to this document

Digital Object Identifier

Primary: 62G35: Robustness 62G30: Order statistics; empirical distribution functions

data depth functional data band depth robust statistics

Copyright © 2007, Institute of Mathematical Statistics


López-Pintado, Sara; Jornsten, Rebecka. Functional analysis via extensions of the band depth. Complex Datasets and Inverse Problems, 103--120, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2007. doi:10.1214/074921707000000085.

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