The Annals of Statistics
- Ann. Statist.
- Volume 47, Number 6 (2019), 3533-3577.
Intrinsic Riemannian functional data analysis
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.
Ann. Statist., Volume 47, Number 6 (2019), 3533-3577.
Received: October 2017
Revised: October 2018
First available in Project Euclid: 31 October 2019
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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