Open Access
2024 Transportation-based functional ANOVA and PCA for covariance operators
Valentina Masarotto, Victor M. Panaretos, Yoav Zemel
Author Affiliations +
Electron. J. Statist. 18(1): 1887-1916 (2024). DOI: 10.1214/24-EJS2240


We consider the problem of comparing several samples of stochastic processes with respect to their second-order structure, and describing the main modes of variation in this second order structure, if present. These tasks can be seen as an Analysis of Variance (ANOVA) and a Principal Component Analysis (PCA) of covariance operators, respectively. They arise naturally in functional data analysis, where several populations are to be contrasted relative to the nature of their dispersion around their means, rather than relative to their means themselves. We contribute a novel approach based on optimal (multi)transport, where each covariance can be identified with a a centred Gaussian process of corresponding covariance. By means of constructing the optimal simultaneous coupling of these Gaussian processes, we contrast the (linear) maps that achieve it with the identity with respect to a norm-induced distance. The resulting test statistic, calibrated by permutation, is seen to distinctly outperform the state-of-the-art, and to furnish considerable power even under local alternatives. This effect is seen to be genuinely functional, and is related to the potential for perfect discrimination in infinite dimensions. In the event of a rejection of the null hypothesis stipulating equality, a geometric interpretation of the transport maps allows us to construct a (tangent space) PCA revealing the main modes of variation. As a necessary step to developing our methodology, we prove results on the existence and boundedness of optimal multitransport maps. These are of independent interest in the theory of transport of Gaussian processes. The transportation ANOVA and PCA are illustrated on a variety of simulated and real examples.

Funding Statement

Research supported in part by a Swiss National Science foundation grant to V. M. Panaretos


The authors would like to thank the anonymous referees, an Associate Editor and the Editor for their constructive comments that improved the quality of this paper.


Download Citation

Valentina Masarotto. Victor M. Panaretos. Yoav Zemel. "Transportation-based functional ANOVA and PCA for covariance operators." Electron. J. Statist. 18 (1) 1887 - 1916, 2024.


Received: 1 December 2022; Published: 2024
First available in Project Euclid: 22 April 2024

Digital Object Identifier: 10.1214/24-EJS2240

Primary: 60G15 , 62R10
Secondary: 60H25 , 62J10

Keywords: coupling , Fréchet mean , Functional data analysis , Gaussian measure , multimarginal transport , Optimal transport , Procrustes analysis , tangent space PCA , trace-class operator

Vol.18 • No. 1 • 2024
Back to Top