Functional Linear Regression Analysis Based on Partial Least Squares and Its Application

  • Huiwen WangEmail author
  • Lele Huang
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 173)


Functional linear model with functional predictors and scalar response is a simple and popular model in the field of functional data analysis. The slope function is usually expanded on some basis functions, such as spline and functional principal component (FPC) basis, and then the model can be converted into a multivariate linear model. The FPC basis can keep most variance information of the functional data, but the correlation with response is not considered. Motivated by this, we use partial least square basis to expand the slope function. Meanwhile, considering the functional predictors are not all significant and variable selection procedure is implemented. In this process, group variable selection is introduced to identify the significant predictors. Then the proposed method is used to analyse the relationship between number of monthly emergency patients and some environmental factors in functional form, and some meaningful results are obtained.


Partial least squares regressions (PLSR) Functional data analysis Basis function 



Wang and Huang’s research was supported by the National Natural Science Foundation of China (No:71031001, 71420107025) and the Innovation Foundation of BUAA for Ph.D. Graduates (YWF-14-YJSY-027). The content is solely the responsibility of the authors and does not necessarily represent the official views of organizations.


  1. Bang, S., Jhun, M.: Simultaneous estimation and factor selection in quantile regression via adaptive sup-norm regularization. Comput. Stat. Data Anal. 56, 813–826 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Berrendero, J.R., Justel, A., Svarc, M.: Principal components for multivariate functional data. Comput. Stat. Data Anal. 55, 2619–2634 (2011)MathSciNetCrossRefGoogle Scholar
  3. Boente, G., Fraiman, R.: Kernel-based functional principal components. Stat. Probab. Lett. 48, 335–345 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Cai, T.T., Hall, P.: Prediction in functional linear regression. Ann. Stat. 34, 2159–2179 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Cai, T.T., Yuan, M.: Minimax and adaptive prediction for functional linear regression. J. Am. Stat. Assoc. 107, 1201–1216 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Crambes, C., Kneip, A., Sarda, P.: Smoothing splines estimators for functional linear regression. Ann. Stat. 37, 35–72 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Delaigle, A., Hall, P.: Methodology and theory for partial least squares applied to functional data. Ann. Stat. 40, 322–352 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Fan, J., Li, R.: Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc. 96, 1348–1360 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Ferraty, F., Vieu, P.: Nonparametric functional data analysis: theory and practice. Springer, New York (2006)zbMATHGoogle Scholar
  10. Hall, P., Horowitz, J.L.: Methodology and convergence rates for functional linear regression. Ann. Stat. 35, 70–91 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Hall, P., Hosseini-Nasab, M.: On properties of functional principal components analysis. J. R. Stat. Soc.: Ser. B (Stat. Methodol.) 68, 109–126 (2005)Google Scholar
  12. Hall, P., Muller, H.G., Wang, J.L.: Properties of principal component methods for functional and longitudinal data analysis. Ann. Stat. 34, 1493–1517 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Jacques, J., Preda, C.: Model-based clustering for multivariate functional data. Comput. Stat. Data Anal. 71, 92–106 (2014)MathSciNetCrossRefGoogle Scholar
  14. Jiang, C.R., Wang, J.L.: Covariate adjusted functional principal components analysis for longitudinal data. Ann. Stat. 38, 1194–1226 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Kneip, A., Utikal, K.J.: Inference for density families using functional principal component analysis. J. Am. Stat. Assoc. 96, 519–542 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Lian, H.: Shrinkage estimation and selection for multiple functional regression. Stat. Sin. 23, 51–74 (2013)MathSciNetzbMATHGoogle Scholar
  17. Matsui, H., Konishi, S.: Variable selection for functional regression models via the regularization. Comput. Stat. Data Anal. 55, 3304–3310 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Preda, C., Saporta, G.: PLS regression on a stochastic process. Comput. Stat. Data Anal. 48, 149–158 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Ramsay, J.O., Dalzell, C.: Some tools for functional data analysis. J. R. Stat. Soc. Ser. B (Methodol.) 53, 539–572 (1991)Google Scholar
  20. Ramsay, J.O., Silverman, B.W.: Functional data analysis. Springer, New York (1997)CrossRefzbMATHGoogle Scholar
  21. Ramsay, J.O., Silverman, B.W.: Applied functional data analysis: methods and case studies, vol. 77. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  22. Schwarz, G., et al.: Estimating the dimension of a model. Ann. Stat. 6, 461–464 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Silverman, B.W.: Smoothed functional principal components analysis by choice of norm. Ann. Stat. 24, 1–24 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B (Methodol.) 58, 267–288 (1996)Google Scholar
  25. Tutz, G., Gertheiss, J.: Feature extraction in signal regression: a boosting technique for functional data regression. J. Comput. Graph. Stat. 19, 154–174 (2010)Google Scholar
  26. Yuan, M.: Gacv for quantile smoothing splines. Comput. Stat. Data Anal. 50 (3), 813–829 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Yuan, M., Cai, T.T.: A reproducing kernel Hilbert space approach to functional linear regression. Ann. Stat. 38, 3412–3444 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Yuan, M., Lin, Y.: Model selection and estimation in regression with grouped variables. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 68, 49–67 (2006)Google Scholar
  29. Zou, H., Hastie, T.: Regularization and variable selection via the elastic net. J. R. Stat. Soc.: Ser. B (Stat. Methodol.) 67, 301–320 (2005)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.School of Economics and ManagementBeihang UniversityBeijingChina

Personalised recommendations