Review Articles

Convergence rate of principal component analysis with local-linear smoother for functional data under a unified weighing scheme

Xingyu Yan ,

a Key Laboratory of Advanced Theory and Application in Statistics, and Data Science – MOE, School of Statistics, East China Normal University, Shanghai, People's Republic of China

Xiaolong Pu ,

a Key Laboratory of Advanced Theory and Application in Statistics, and Data Science – MOE, School of Statistics, East China Normal University, Shanghai, People's Republic of China

Yingchun Zhou ,

a Key Laboratory of Advanced Theory and Application in Statistics, and Data Science – MOE, School of Statistics, East China Normal University, Shanghai, People's Republic of China

yczhou@stat.ecnu.edu.cn

Xiaolei Xun

b School of Data Science, Fudan University, Shanghai, People's Republic of China

xiaolei_xun@fudan.edu.cn

Pages 55-65 | Received 04 Aug. 2018, Accepted 10 Aug. 2019, Published online: 20 Aug. 2019,
  • Abstract
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ABSTRACT

The unified weighing scheme for the local-linear smoother in analysing functional data can deal with data that are dense, sparse or of neither type. In this paper, we focus on the convergence rate of functional principal component analysis using this method. Almost sure asymptotic consistency and rates of convergence for the estimators of eigenvalues and eigenfunctions have been established. We also provide the convergence rate of the variance estimation of the measurement error. Based on the results, the number of observations within each curve can be of any rate relative to the sample size, which is consistent with the earlier conclusions about the asymptotic properties of the mean and covariance estimators.

References

  1. Fan, J. Q., & Gijbels, I. (1996). Local polynomial modelling and its applications. London: Chapman and Hall. [Google Scholar]
  2. Fan, J. Q., & Zhang, W. Y. (2000). Simultaneous confidence bands and hypothesis testing in varying-coefficient models. Scandinavian Journal of Statistics27, 715–731. doi: 10.1111/1467-9469.00218 [Crossref][Web of Science ®], [Google Scholar]
  3. Ferraty, F., & Vieu, P. (2006). Nonparametric functional data analysis: Theory and practice. Berlin: Springer. [Google Scholar]
  4. Greven, S., Crainiceanu, C. M., Caffo, B. S., & Reich, D. (2010). Longitudinal functional principal component analysis. Electronic Journal of Statistics4, 1022–1054. doi: 10.1214/10-EJS575 [Crossref][Web of Science ®], [Google Scholar]
  5. Hall, P., & Hosseini-Nasab, M. (2006). On properties of functional principal components analysis. Journal of the Royal Statistical Society, Series B68, 109–126. doi: 10.1111/j.1467-9868.2005.00535.x [Crossref], [Google Scholar]
  6. Hall, P., Müller, H.-G., & Wang, J.-L. (2006). Properties of principal component methods for functional and longitudinal data analysis. Annals of Statistics34, 1493–1517. doi: 10.1214/009053606000000272 [Crossref][Web of Science ®], [Google Scholar]
  7. James, G. M., Hastie, T. J., & Sugar, C. A. (2000). Principal component models for sparse functional data. Biometrika87, 587–602. doi: 10.1093/biomet/87.3.587 [Crossref][Web of Science ®], [Google Scholar]
  8. Jiang, C. R., & Wang, J. -L. (2010). Covariate adjusted functional principal components analysis for longitudinal data. Annals of Statistics38, 1194–1226. doi: 10.1214/09-AOS742 [Crossref][Web of Science ®], [Google Scholar]
  9. Li, Y. H., & Hsing, T. (2010). Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data. Annals of Statistics38, 3321–3351. doi: 10.1214/10-AOS813 [Crossref][Web of Science ®], [Google Scholar]
  10. Li, Y., Wang, N., & Carroll, R. J. (2013). Selecting the number of principal components in functional data. Journal of the American Statistical Association108, 1284–1294. doi: 10.1080/01621459.2013.788980 [Taylor & Francis Online][Web of Science ®], [Google Scholar]
  11. Ramsay, J. O., & Silverman, B. W. (2005). Functional data analysis. New York: Springer. [Crossref], [Google Scholar]
  12. Rice, J. A., & Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. Journal of the Royal Statistical Society, Series B53, 233–243. [Google Scholar]
  13. Staniswalis, J. G., & Lee, J. J. (1998). Nonparametric regression analysis of longitudinal data. Journal of the American Statistical Association93, 1403–1418. doi: 10.1080/01621459.1998.10473801 [Taylor & Francis Online][Web of Science ®], [Google Scholar]
  14. Yao, F., & Lee, T. C. M. (2006). Penalized spline models for functional principal component analysis. Journal of the Royal Statistical Society, Series B68, 3–25. doi: 10.1111/j.1467-9868.2005.00530.x [Crossref], [Google Scholar]
  15. Yao, F., Müller, H.-G., & Wang, J.-L. (2005). Functional data analysis for sparse longitudinal data. Journal of the American Statistical Association100, 577–590. doi: 10.1198/016214504000001745 [Taylor & Francis Online][Web of Science ®], [Google Scholar]
  16. Zhang, X. K., & Wang, J. -L. (2016). From sparse to dense functional data and beyond. Annals of Statistics44, 2281–2321. doi: 10.1214/16-AOS1446 [Crossref][Web of Science ®], [Google Scholar]
  17. Zhou, L., Lin, H. Z., & Liang, H (2017). Efficient estimation of the nonparametric mean and covariance functions for longitudinal and sparse functional data. Journal of the American Statistical Association. doi: 10.1080/01621459.2017.1356317. [Taylor & Francis Online], [Google Scholar]
  18. Zhu, H. T., Li, R. Z., & Kong, L. L. (2012). Multivariate varying coefficient model for functional responses. Annals of Statistics40, 2634–2666. doi: 10.1214/12-AOS1045 [Crossref][Web of Science ®], [Google Scholar]

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