Integrated Depths for Partially Observed Functional Data

Abstract

Partially observed functional data are frequently encountered in applications and are the object of an increasing interest by the literature. We here address the problem of measuring the centrality of a datum in a partially observed functional sample. We propose an integrated functional depth for partially observed functional data, dealing with the very challenging case where partial observability can occur systematically on any observation of the functional dataset. In particular, differently from many techniques for partially observed functional data, we do not request that some functional datum is fully observed, nor we require that a common domain exist, where all of the functional data are recorded. Because of this, our proposal can also be used in those frequent situations where reconstructions methods and other techniques for partially observed functional data are inapplicable. By means of simulation studies, we demonstrate the very good performances of the proposed depth on finite samples. Our proposal enables the use of benchmark methods based on depths, originally introduced for fully observed data, in the case of partially observed functional data. This includes the functional boxplot, the outliergram and the depth versus depth classifiers. We illustrate our proposal on two case studies, the first concerning a problem of outlier detection in German electricity supply functions, the second regarding a classification problem with data obtained from medical imaging. Supplementary materials for this article are available online.

Keywords:

• Classification of partially observed functional data
• Functional boxplot
• Functional depth
• Functional outliers
• Incomplete functional data
• Robustness

Supplementary Materials

Additional results: Extensive simulation results (Section 7). Additional output concerning the analysis of German electricity supply curves and AneuRisk65 dataset (Section 8). (AdditionalResults.pdf)

R-package: R-package fdaPOIFD available at CRAN (Elías et al. 2021) includes the functions to compute the depth, plot the boxplot and the outliergram for partially observed functional data, as well as a vignette to reproduce the simulations.

Notes

1 When data do not come on a common grid, they can be referred to a common grid in a preprocessing step, using standard techniques such as binning. It is important to note that, for each functional datum $X_i,$ the possible evaluation on a common grid is only done over those portions of the domain where the datum is observed (i.e., without extrapolation on missing portions using reconstruction techniques for sparse or partially observed data).

Additional information

Funding

Antonio Elías was supported by the Spanish Ministerio de Educación, Cultura y Deporte under grant FPU15/00625 and the research stay grant EST16/00619. Antonio Elías and Raúl Jiménez are partially supported by the Spanish Ministerio de Economía y Competitividad under grant ECO2015-66593-P and PID2019-109196GB-I00/AEI/10.13039/501100011033.