On the Concept of Depth for Functional Data

Abstract

The statistical analysis of functional data is a growing need in many research areas. We propose a new depth notion for functional observations based on the graphic representation of the curves. Given a collection of functions, it allows to establish the centrality of a function and provides a natural center-outward ordering of the sample curves. Robust statistics such as the median function or a trimmed mean function can be defined from this depth definition. Its finite-dimensional version provides a new depth for multivariate data that is computationally very fast and turns out to be convenient to study high-dimensional observations. The natural properties are established for the new depth and the uniform consistency of the sample depth is proved. Simulation results show that the trimmed mean presents a better behavior than the mean for contaminated models. Several real data sets are considered to illustrate this new concept of depth. Finally, we use this new depth to generalize to functions the Wilcoxon rank sum test. It allows to decide whether two groups of curves come from the same population. This functional rank test is applied to girls and boys growth curves concluding that they present different growth patterns.

A half-region depth for functional data

“…We first consider four models (M1-M4) with what we call ''magnitude'' contamination. These models were analyzed by Fraiman and Muniz (2001) and López-Pintado and Romo (2009). They all consist in adding some outliers to an elementary model M0 defined as…”

“…This curve can be used to illustrate the representative pattern within the sample of curves. In López-Pintado and Romo (2007) different inference tools (such as the central region, the scale curve, and a rank test) were introduced based on the ordering provided by the band depth (see López-Pintado and Romo, 2009 for more details on this depth). All these methods can also be applied using the half-region depth presented here.…”

“…The raw trajectories are observed on sparse and unevenly spaced points and therefore a preliminary smoothing step is required. The original data were smoothed using a spline basis (see López-Pintado and Romo, 2009, for details about the data). Since the number of observations per tree is very irregular, in those curves with fewer observations the smoothness procedure is less effective as shown in Fig.…”

“…Most of these multivariate depths are not adequate for high-dimensional data, therefore their applicability is restricted to low-dimensional vector observations. Recently, alternative notions of depth for functional data have been introduced which can be adapted to high-dimensional data without a large computational burden (see Fraiman and Muniz, 2001;Cuevas et al, 2006Cuevas et al, , 2007Cuesta-Albertos and Nieto-Reyes, 2008;Jörnsten, 2007 andRomo, 2009). In this paper we propose an alternative graph-based notion of depth which is simple, computationally fast, and can be easily adapted to high-dimensional data.…”

Visualization tools for uncertainty and sensitivity analyses on thermal-hydraulic transients

“…A "depth" is associated with each element of the sample, which allows classifying and thus finding the concepts of median and outliers. For functional data, Lopez-Pintado and Romo 5) introduced a notion of band depth. This allows classifying a set of curves and thus defining functional quantiles, to identify the most central (median) curves and outliers curves Each curve is associated with a real that is the band depth.…”