June 2022 Intrinsic Riemannian functional data analysis for sparse longitudinal observations
Lingxuan Shao, Zhenhua Lin, Fang Yao
Author Affiliations +
Ann. Statist. 50(3): 1696-1721 (June 2022). DOI: 10.1214/22-AOS2172


A new framework is developed to intrinsically analyze sparsely observed Riemannian functional data. It features four innovative components: a frame-independent covariance function, a smooth vector bundle termed covariance vector bundle, a parallel transport and a smooth bundle metric on the covariance vector bundle. The introduced intrinsic covariance function links estimation of covariance structure to smoothing problems that involve raw covariance observations derived from sparsely observed Riemannian functional data, while the covariance vector bundle provides a rigorous mathematical foundation for formulating such smoothing problems. The parallel transport and the bundle metric together make it possible to measure fidelity of fit to the covariance function. They also play a critical role in quantifying the quality of estimators for the covariance function. As an illustration, based on the proposed framework, we develop a local linear smoothing estimator for the covariance function, analyze its theoretical properties and provide numerical demonstration via simulated and real data sets. The intrinsic feature of the framework makes it applicable to not only Euclidean submanifolds but also manifolds without a canonical ambient space.

Funding Statement

Fang Yao’s research is supported by National Natural Science Foundation of China Grants 11931001 and 11871080, the National Key R&D Program of China Grant 2020YFE0204200, the LMAM and the Key Laboratory of Mathematical Economics and Quantitative Finance (Peking University), Ministry of Education.
Zhenhua Lin’s research is partially supported by NUS startup Grant R-155-000-217-133.
Data collection and sharing for this project was funded by the Alzheimer’s Disease Neuroimaging Initiative (ADNI) (National Institutes of Health Grant U01 AG024904) and DOD ADNI (Department of Defense award number W81XWH-12-2-0012).


For the Alzheimer’s Disease Neuroimaging Initiative


Lingxuan Shao was a visiting student of Zhenhua Lin in National University of Singapore at the time of developing the paper. Lingxuan Shao and Zhenhua Lin are the joint first authors, and Fang Yao is the corresponding author.

Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators can be found at: http://adni.loni.usc.edu/wp-content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf.


Download Citation

Lingxuan Shao. Zhenhua Lin. Fang Yao. "Intrinsic Riemannian functional data analysis for sparse longitudinal observations." Ann. Statist. 50 (3) 1696 - 1721, June 2022. https://doi.org/10.1214/22-AOS2172


Received: 1 April 2021; Revised: 1 January 2022; Published: June 2022
First available in Project Euclid: 16 June 2022

MathSciNet: MR4441137
zbMATH: 07547947
Digital Object Identifier: 10.1214/22-AOS2172

Primary: 62R10
Secondary: 62R30

Keywords: diffusion tensor , Fréchet mean , intrinsic covariance function , parallel transport , smoothing , vector bundle

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 3 • June 2022
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