Open Access
February 2012 Nonlinear manifold representations for functional data
Dong Chen, Hans-Georg Müller
Ann. Statist. 40(1): 1-29 (February 2012). DOI: 10.1214/11-AOS936


For functional data lying on an unknown nonlinear low-dimensional space, we study manifold learning and introduce the notions of manifold mean, manifold modes of functional variation and of functional manifold components. These constitute nonlinear representations of functional data that complement classical linear representations such as eigenfunctions and functional principal components. Our manifold learning procedures borrow ideas from existing nonlinear dimension reduction methods, which we modify to address functional data settings. In simulations and applications, we study examples of functional data which lie on a manifold and validate the superior behavior of manifold mean and functional manifold components over traditional cross-sectional mean and functional principal components. We also include consistency proofs for our estimators under certain assumptions.


Download Citation

Dong Chen. Hans-Georg Müller. "Nonlinear manifold representations for functional data." Ann. Statist. 40 (1) 1 - 29, February 2012.


Published: February 2012
First available in Project Euclid: 15 March 2012

zbMATH: 1246.62146
MathSciNet: MR3013177
Digital Object Identifier: 10.1214/11-AOS936

Primary: 62H25 , 62M09

Keywords: Dimension reduction , Functional data analysis , functional manifold components , modes of functional variation , smoothing

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 1 • February 2012
Back to Top