The Annals of Statistics

Intrinsic Riemannian functional data analysis

Zhenhua Lin and Fang Yao

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Abstract

In this work we develop a novel and foundational framework for analyzing general Riemannian functional data, in particular a new development of tensor Hilbert spaces along curves on a manifold. Such spaces enable us to derive Karhunen–Loève expansion for Riemannian random processes. This framework also features an approach to compare objects from different tensor Hilbert spaces, which paves the way for asymptotic analysis in Riemannian functional data analysis. Built upon intrinsic geometric concepts such as vector field, Levi-Civita connection and parallel transport on Riemannian manifolds, the developed framework applies to not only Euclidean submanifolds but also manifolds without a natural ambient space. As applications of this framework, we develop intrinsic Riemannian functional principal component analysis (iRFPCA) and intrinsic Riemannian functional linear regression (iRFLR) that are distinct from their traditional and ambient counterparts. We also provide estimation procedures for iRFPCA and iRFLR, and investigate their asymptotic properties within the intrinsic geometry. Numerical performance is illustrated by simulated and real examples.

Article information

Source
Ann. Statist., Volume 47, Number 6 (2019), 3533-3577.

Dates
Received: October 2017
Revised: October 2018
First available in Project Euclid: 31 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.aos/1572487402

Digital Object Identifier
doi:10.1214/18-AOS1787

Mathematical Reviews number (MathSciNet)
MR4025751

Zentralblatt MATH identifier
07151070

Subjects
Primary: 62G05: Estimation 62J05: Linear regression

Keywords
Functional principal component functional linear regression intrinsic Riemannian Karhunen–Loève expansion parallel transport tensor Hilbert space

Citation

Lin, Zhenhua; Yao, Fang. Intrinsic Riemannian functional data analysis. Ann. Statist. 47 (2019), no. 6, 3533--3577. doi:10.1214/18-AOS1787. https://projecteuclid.org/euclid.aos/1572487402


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