Statistics and Computing

, Volume 28, Issue 3, pp 713–723 | Cite as

Supervised functional principal component analysis

  • Yunlong Nie
  • Liangliang Wang
  • Baisen Liu
  • Jiguo Cao


In functional linear regression, one conventional approach is to first perform functional principal component analysis (FPCA) on the functional predictor and then use the first few leading functional principal component (FPC) scores to predict the response variable. The leading FPCs estimated by the conventional FPCA stand for the major source of variation of the functional predictor, but these leading FPCs may not be mostly correlated with the response variable, so the prediction accuracy of the functional linear regression model may not be optimal. In this paper, we propose a supervised version of FPCA by considering the correlation of the functional predictor and response variable. It can automatically estimate leading FPCs, which represent the major source of variation of the functional predictor and are simultaneously correlated with the response variable. Our supervised FPCA method is demonstrated to have a better prediction accuracy than the conventional FPCA method by using one real application on electroencephalography (EEG) data and three carefully designed simulation studies.


Classification Functional data analysis Functional linear model Functional logistic regression 



The authors would like to thank the Editor, the Associated Editor, and two reviewers for their very constructive suggestions and comments on the manuscript. They are extremely helpful for us to improve our work. The authors are also very grateful to Prof. Gen Li and Prof. Xiaoyan Leng for kindly providing us with their data and their computing codes. This research was supported by Nie’s Postgraduate Scholarship-Doctorial (PGS-D) from the Natural Sciences and Engineering Research Council of Canada (NSERC), and the NSERC Discovery grants of Wang and Cao.

Supplementary material

11222_2017_9758_MOESM1_ESM.r (2 kb)
demon_sFPCA.R: This file contains the R codes for demonstration of using the “sFPCA” R package.
11222_2017_9758_MOESM2_ESM.pdf (376 kb)
Supplementary Document: This file contains some additional figures for simulation studies in Section 5, two additional simulation studies with an arbitrary coefficient function and a large number of FPCs related to the outcome, respectively. We also include another real data application analyzing the time course yeast gene expression data. (supplementary.pdf, PDF file), ESM 1 (R 3KB), ESM 2 (PDF 377KB)


  1. Bair, E., Hastie, T., Paul, D., Tibshirani, R.: Prediction by supervised principal components. J. Am. Stat. Assoc. 101(473), 119–137 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. Cardot, H., Faivre, R., Goulard, M.: Functional approaches for predicting land use with the temporal evolution of coarse resolution remote sensing data. J. Appl. Stat. 30(10), 1185–1199 (2003)MathSciNetCrossRefMATHGoogle Scholar
  3. Fukunaga, K., Koontz, W.L.: Representation of random processes using the finite Karhunen–Loeve expansion. Inf. Control 16(1), 85–101 (1970)CrossRefMATHGoogle Scholar
  4. Huang, J.Z., Shen, H., Buja, A.: The analysis of two-way functional data using two-way regularized singular value decompositions. J. Am. Stat. Assoc. 104(488), 1609–1620 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. Li, G., Shen, H., Huang, J.Z.: Supervised sparse and functional principal component analysis. J. Comput. Graph. Stat. 25(3), 859–878 (2016)MathSciNetCrossRefGoogle Scholar
  6. Li, G., Yang, D., Nobel, A.B., Shen, H.: Supervised singular value decomposition and its asymptotic properties. J. Multivar. Anal. 146, 7–17 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. Müller, H.-G., Stadtmüller, U.: Generalized functional linear models. Ann. Stat. 33, 774–805 (2005)Google Scholar
  8. Ramsay, J., Hooker, G., Graves, S.: Functional Data Analysis with R and MATLAB. Use R!. Springer, New York (2009)Google Scholar
  9. Ramsay, J.O., Silverman, B.W.: Applied Functional Data Analysis: Methods and Case Studies, vol. 77. Springer, New York (2002)CrossRefMATHGoogle Scholar
  10. Ramsay, J.O., Silverman, B.W.: Functional Data Analysis, 2nd edn. Springer, New York (2005)MATHGoogle Scholar
  11. Ratcliffe, S.J., Heller, G.Z., Leader, L.R.: Functional data analysis with application to periodically stimulated foetal heart rate data. ii: Functional logistic regression. Stat. Med. 21(8), 1115–1127 (2002)CrossRefGoogle Scholar
  12. Silverman, B.W., et al.: Smoothed functional principal components analysis by choice of norm. Ann. Stat. 24(1), 1–24 (1996)MathSciNetCrossRefMATHGoogle Scholar
  13. Yao, F., Müller, H.-G., Wang, J.-L.: Functional data analysis for sparse longitudinal data. J. Am. Stat. Assoc. 100(470), 577–590 (2005)MathSciNetCrossRefMATHGoogle Scholar
  14. Zhang, X.L., Begleiter, H., Porjesz, B., Wang, W., Litke, A.: Event related potentials during object recognition tasks. Brain Res. Bull. 38(6), 531–538 (1995)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Statistics and Actuarial ScienceSimon Fraser UniversityBurnabyCanada
  2. 2.School of Statistics, Dongbei University of Finance and EconomicsDalianChina

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