DOI QR코드

DOI QR Code

Principal component analysis for Hilbertian functional data

  • Kim, Dongwoo (Department of Statistics, Seoul National University) ;
  • Lee, Young Kyung (Department of Information Statistics, Kangwon National University) ;
  • Park, Byeong U. (Department of Statistics, Seoul National University)
  • Received : 2019.11.11
  • Accepted : 2019.12.03
  • Published : 2020.01.31

Abstract

In this paper we extend the functional principal component analysis for real-valued random functions to the case of Hilbert-space-valued functional random objects. For this, we introduce an autocovariance operator acting on the space of real-valued functions. We establish an eigendecomposition of the autocovariance operator and a Karuhnen-Loève expansion. We propose the estimators of the eigenfunctions and the functional principal component scores, and investigate the rates of convergence of the estimators to their targets. We detail the implementation of the methodology for the cases of compositional vectors and density functions, and illustrate the method by analyzing time-varying population composition data. We also discuss an extension of the methodology to multivariate cases and develop the corresponding theory.

Keywords

Acknowledgement

Supported by : National Research Foundation of Korea (NRF)

Research of Young Kyung Lee was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2018R1A2B6001068). Research of Dongwoo Kim and Byeong U. Park was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP) (No. 2019R1A2C3007355).

References

  1. Aitchison J (1986). The Statistical Analysis of Compositional Data, Monographs on Statistics and Applied Probability, Chapman & Hall, London.
  2. Bosq D (2000). Linear Processes in Function Spaces: Theory and Applications, Springer, New York.
  3. Chiou JM, Chen YT, and Yang YF (2014). Multivariate functional principal component analysis: a normalization approach, Statistica Sinica, 24, 1571-1596.
  4. Dai X and Muller HG (2018). Principal component analysis for functional data on Riemannian manifolds and spheres, Annals of Statistics, 46, 3334-3361. https://doi.org/10.1214/17-AOS1660
  5. Hsing T and Eubank R (2015). Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators, Wiley, Singapore.
  6. Jeon JM and Park BU (2020). Additive regression with Hilbertian responses, Annals of Statistics, in print.
  7. Lin Z and Yao F (2019). Intrinsic Riemannian functional data analysis, Annals of Statistics, 47, 3533-3577. https://doi.org/10.1214/18-AOS1787
  8. Ramsay J and Silverman BW (2005). Functional Data Analysis, Springer, New York.
  9. van den Boogaart KG, Egozcue JJ, and Pawlowsky-Glahn V (2014). Bayes Hilbert spaces, Australian and New Zealand Journal of Statistics, 56, 171-194. https://doi.org/10.1111/anzs.12074
  10. Wang H, Shangguan L, Guan R, and Billard L (2015). Principal component analysis for compositional data vectors, Computational Statistics, 30, 1079-1096. https://doi.org/10.1007/s00180-015-0570-1