A notion of depth for sparse functional data

TL;DR

This paper introduces a new method for calculating data depth in sparse functional data that accounts for estimation uncertainty, improving robustness and outlier detection in practical scenarios.

Contribution

It extends existing functional depth notions to incorporate uncertainty from sparse and irregular data observations, enhancing analysis accuracy.

Findings

Incorporating uncertainty improves depth-based analysis.

Method performs well on simulated sparse data.

Application to real medfly data demonstrates practical benefits.

Abstract

Data depth is a well-known and useful nonparametric tool for analyzing functional data. It provides a novel way of ranking a sample of curves from the center outwards and defining robust statistics, such as the median or trimmed means. It has also been used as a building block for functional outlier detection methods and classification. Several notions of depth for functional data were introduced in the literature in the last few decades. These functional depths can only be directly applied to samples of curves measured on a fine and common grid. In practice, this is not always the case, and curves are often observed at sparse and subject dependent grids. In these scenarios the usual approach consists in estimating the trajectories on a common dense grid, and using the estimates in the depth analysis. This approach ignores the uncertainty associated with the curves estimation step. Our goal is to extend the notion of depth so that it takes into account this uncertainty. Using both functional estimates and their associated confidence intervals, we propose a new method that allows the curve estimation uncertainty to be incorporated into the depth analysis. We describe the new approach using the modified band depth although any other functional depth could be used. The performance of the proposed methodology is illustrated using simulated curves in different settings where we control the degree of sparsity. Also a real data set consisting of female medflies egg-laying trajectories is considered. The results show the benefits of using uncertainty when computing depth for sparse functional data.