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December 2019 Intrinsic Riemannian functional data analysis
Zhenhua Lin, Fang Yao
Ann. Statist. 47(6): 3533-3577 (December 2019). DOI: 10.1214/18-AOS1787

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.

Citation

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Zhenhua Lin. Fang Yao. "Intrinsic Riemannian functional data analysis." Ann. Statist. 47 (6) 3533 - 3577, December 2019. https://doi.org/10.1214/18-AOS1787

Information

Received: 1 October 2017; Revised: 1 October 2018; Published: December 2019
First available in Project Euclid: 31 October 2019

Digital Object Identifier: 10.1214/18-AOS1787

Subjects:
Primary: 62G05, 62J05

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 6 • December 2019
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