Robustness and Explainability in Multivariate Functional Data Analysis

M. Mayrhofer1 U. Radojičić2 P. Filzmoser3
  • 1

    Institute of Statistics and Mathematical Methods in Economics, TU Wien, Vienna, Austria [marcus.mayrhofer(at)tuwien.ac.at]

  • 2

    Institute of Statistics and Mathematical Methods in Economics, TU Wien, Vienna, Austria [una.radojicic(at)tuwien.ac.at]

  • 3

    Institute of Statistics and Mathematical Methods in Economics, TU Wien, Vienna, Austria [peter.filzmoser(at)tuwien.ac.at]

Keywords: Robust Covariance Estimation – Outlier Detection – Shapley Values – Minimum Covariance Determinant

1 Abstract

We propose a method for robust covariance estimation for multivariate functional data by establishing a connection between a stochastic process with a separable covariance structure and the corresponding matrix-variate distribution of its basis representation. Based on this connection we use the Matrix Minimum Covariance Determinant (MMCD) approach [Mayrhofer et al., 2025] in combination with a truncated multivariate functional Mahalanobis semi-distance for robust estimation of mean and covariance.

To go beyond outlier detection, we generalize multivariate outlier explanations based on Shapley values [Mayrhofer and Filzmoser, 2023] to decompose the truncated multivariate functional Mahalanobis semi-distance of individual observations into time-coordinate-specific contributions.

Our approach provides a framework for the generalization of other distance-based functional outlier detection approaches [Berrendero et al., 2020, Oguamalam et al., 2024] and lays the groundwork for future extensions to robust covariance estimation and explainable outlier detection for random surfaces.

References

  • Berrendero et al. [2020] José R Berrendero, Beatriz Bueno-Larraz, and Antonio Cuevas. On Mahalanobis distance in functional settings. Journal of Machine Learning Research, 21(9):1–33, 2020.
  • Mayrhofer and Filzmoser [2023] Marcus Mayrhofer and Peter Filzmoser. Multivariate outlier explanations using Shapley values and Mahalanobis distances. Econometrics and Statistics, 2023. doi: 10.1016/j.ecosta.2023.04.003.
  • Mayrhofer et al. [2025] Marcus Mayrhofer, Una Radojičić, and Peter Filzmoser. Robust covariance estimation and explainable outlier detection for matrix-valued data. Technometrics (accepted), 2025.
  • Oguamalam et al. [2024] Jeremy Oguamalam, Una Radojičić, and Peter Filzmoser. Minimum regularized covariance trace estimator and outlier detection for functional data. Technometrics, pages 1–12, 2024.