The Annals of Statistics

Intrinsic Riemannian functional data analysis

Zhenhua Lin and Fang Yao

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

Primary: 62G05: Estimation 62J05: Linear regression

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


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

Export citation


  • Afsari, B. (2011). Riemannian $L^{p}$ center of mass: Existence, uniqueness, and convexity. Proc. Amer. Math. Soc. 139 655–673.
  • Aldous, D. J. (1976). A characterisation of Hilbert space using the central limit theorem. J. Lond. Math. Soc. (2) 14 376–380.
  • Arsigny, V., Fillard, P., Pennec, X. and Ayache, N. (2006/07). Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 29 328–347.
  • Balakrishnan, A. V. (1960). Estimation and detection theory for multiple stochastic processes. J. Math. Anal. Appl. 1 386–410.
  • Bhattacharya, R. and Patrangenaru, V. (2003). Large sample theory of intrinsic and extrinsic sample means on manifolds. I. Ann. Statist. 31 1–29.
  • Binder, J. R., Frost, J. A., Hammeke, T. A., Cox, R. W., Rao, S. M. and Prieto, T. (1997). Human brain language areas identified by functional magnetic resonance imaging. J. Neurosci. 17 353–362.
  • Bosq, D. (2000). Linear Processes in Function Spaces. Lecture Notes in Statistics 149. Springer, New York.
  • Cardot, H., Ferraty, F. and Sarda, P. (2003). Spline estimators for the functional linear model. Statist. Sinica 13 571–591.
  • Cardot, H., Mas, A. and Sarda, P. (2007). CLT in functional linear regression models. Probab. Theory Related Fields 138 325–361.
  • Chen, D. and Müller, H.-G. (2012). Nonlinear manifold representations for functional data. Ann. Statist. 40 1–29.
  • Cheng, G., Ho, J., Salehian, H. and Vemuri, B. C. (2016). Recursive computation of the Fréchet mean on non-positively curved Riemannian manifolds with applications. In Riemannian Computing in Computer Vision 21–43. Springer, Cham.
  • Cornea, E., Zhu, H., Kim, P. and Ibrahim, J. G. (2017). Regression models on Riemannian symmetric spaces. J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 463–482.
  • Dai, X. and Müller, H.-G. (2018). Principal component analysis for functional data on Riemannian manifolds and spheres. Ann. Statist. 46 3334–3361.
  • Dauxois, J., Pousse, A. and Romain, Y. (1982). Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference. J. Multivariate Anal. 12 136–154.
  • Dayan, E. and Cohen, L. G. (2011). Neuroplasticity subserving motor skill learning. Neuron 72 443–454.
  • Dryden, I. L., Koloydenko, A. and Zhou, D. (2009). Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging. Ann. Appl. Stat. 3 1102–1123.
  • Essen, D. C. V., Smith, S. M., Barch, D. M., Behrens, T. E. J., Yacoub, E., Ugurbil, K. and Consortium, W.-M. H. (2013). The WU-Minn human connectome project: An overview. NeuroImage 80 62–79.
  • Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis: Theory and Practice. Springer Series in Statistics. Springer, New York.
  • Fletcher, P. T. and Joshib, S. (2007). Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Process. 87 250–262.
  • Friston, K. J. (2011). Functional and effective connectivity: A review. Brain Connect. 1 13–36.
  • Hall, P. and Horowitz, J. L. (2005). Nonparametric methods for inference in the presence of instrumental variables. Ann. Statist. 33 2904–2929.
  • Hall, P. and Horowitz, J. L. (2007). Methodology and convergence rates for functional linear regression. Ann. Statist. 35 70–91.
  • Hall, P. and Hosseini-Nasab, M. (2006). On properties of functional principal components analysis. J. R. Stat. Soc. Ser. B. Stat. Methodol. 68 109–126.
  • Happ, C. and Greven, S. (2018). Multivariate functional principal component analysis for data observed on different (dimensional) domains. J. Amer. Statist. Assoc. 113 649–659.
  • Hastie, T., Tibshirani, R. and Friedman, J. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd ed. Springer Series in Statistics. Springer, New York.
  • Hsing, T. and Eubank, R. (2015). Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators. Wiley Series in Probability and Statistics. Wiley, Chichester.
  • Kelly, E. J. and Root, W. L. (1960). A representation of vector-valued random processes. J. Math. Phys. 39 211–216.
  • Kendall, W. S. and Le, H. (2011). Limit theorems for empirical Fréchet means of independent and non-identically distributed manifold-valued random variables. Braz. J. Probab. Stat. 25 323–352.
  • Kleffe, J. (1973). Principal components of random variables with values in a separable Hilbert space. Statistics 4 391–406.
  • Kokoszka, P. and Reimherr, M. (2017). Introduction to Functional Data Analysis. Texts in Statistical Science Series. CRC Press, Boca Raton, FL.
  • Kong, D., Xue, K., Yao, F. and Zhang, H. H. (2016). Partially functional linear regression in high dimensions. Biometrika 103 147–159.
  • Lang, S. (1995). Differential and Riemannian Manifolds, 3rd ed. Graduate Texts in Mathematics 160. Springer, New York.
  • Lang, S. (1999). Fundamentals of Differential Geometry. Graduate Texts in Mathematics 191. Springer, New York.
  • Lee, J. M. (1997). Riemannian Manifolds: An Introduction to Curvature. Graduate Texts in Mathematics 176. Springer, New York.
  • Lee, J. M. (2013). Introduction to Smooth Manifolds, 2nd ed. Graduate Texts in Mathematics 218. Springer, New York.
  • Lila, E., Aston, J. A. D. and Sangalli, L. M. (2016). Smooth principal component analysis over two-dimensional manifolds with an application to neuroimaging. Ann. Appl. Stat. 10 1854–1879.
  • Lin, Z. and Yao, F. (2019). Functional regression with unknown manifold structures. Available at arXiv:1704.03005.
  • Moakher, M. (2005). A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 26 735–747.
  • Park, J. E., Jung, S. C., Ryu, K. H., Oh, J. Y., Kim, H. S., Choi, C. G., Kim, S. J. and Shim, W. H. (2017). Differences in dynamic and static functional connectivity between young and elderly healthy adults. Neuroradiol. 59 781–789.
  • Petersen, A. and Müller, H.-G. (2019). Fréchet regression for random objects with Euclidean predictors. Ann. Statist. 47 691–719.
  • Phana, K. L., Wager, T., Taylor, S. F. and Liberzon, I. (2002). Functional neuroanatomy of emotion: A meta-analysis of emotion activation studies in PET and fMRI. NeuroImage 16 331–348.
  • Raichlen, D. A., Bharadwaj, P. K., Fitzhugh, M. C., Haws, K. A., Torre, G.-A., Trouard, T. P. and Alexander, G. E. (2016). Differences in resting state functional connectivity between young adult endurance athletes and healthy controls. Front. Human Neurosci. 10 610.
  • Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer Series in Statistics. Springer, New York.
  • Rao, C. R. (1958). Some statistical methods for comparison of growth curves. Biometrics 14 1–17.
  • Salehian, H., Chakraborty, R., Ofori, E., Vaillancourt, D. and Vemuri, B. C. (2015). An efficient recursive estimator of the Fréchet mean on hypersphere with applications to Medical Image Analysis. In 5th MICCAI Workshop on Mathematical Foundations of Computational Anatomy (MFCA).
  • Sasaki, S. (1958). On the differential geometry of tangent bundles of Riemannian manifolds. Tôhoku Math. J. (2) 10 338–354.
  • Shi, X., Styner, M., Lieberman, J., Ibrahim, J. G., Lin, W. and Zhu, H. (2009). Intrinsic regression models for manifold-valued data. In Medical Image Computing and Computer-Assisted Intervention—MICCAI 12 192–199.
  • Silverman, B. W. (1996). Smoothed functional principal components analysis by choice of norm. Ann. Statist. 24 1–24.
  • Steinke, F., Hein, M. and Schölkopf, B. (2010). Nonparametric regression between general Riemannian manifolds. SIAM J. Imaging Sci. 3 527–563.
  • Wang, L. (2008). Karhunen–Loeve expansions and their applications Ph.D. thesis The London School of Economics and Political Science.
  • Wang, J.-L., Chiou, J.-M. and Müller, H.-G. (2016). Review of functional data analysis. Ann. Rev. Statist. Appl. 3 257–295.
  • Wang, H. and Marron, J. S. (2007). Object oriented data analysis: Sets of trees. Ann. Statist. 35 1849–1873.
  • Yao, F., Müller, H.-G. and Wang, J.-L. (2005a). Functional data analysis for sparse longitudinal data. J. Amer. Statist. Assoc. 100 577–590.
  • Yao, F., Müller, H.-G. and Wang, J.-L. (2005b). Functional linear regression analysis for longitudinal data. Ann. Statist. 33 2873–2903.
  • Yuan, M. and Cai, T. T. (2010). A reproducing kernel Hilbert space approach to functional linear regression. Ann. Statist. 38 3412–3444.
  • Yuan, Y., Zhu, H., Lin, W. and Marron, J. S. (2012). Local polynomial regression for symmetric positive definite matrices. J. R. Stat. Soc. Ser. B. Stat. Methodol. 74 697–719.
  • Zhang, X. and Wang, J.-L. (2016). From sparse to dense functional data and beyond. Ann. Statist. 44 2281–2321.