The Annals of Statistics

Covariate adjusted functional principal components analysis for longitudinal data

Ci-Ren Jiang and Jane-Ling Wang

Full-text: Open access


Classical multivariate principal component analysis has been extended to functional data and termed functional principal component analysis (FPCA). Most existing FPCA approaches do not accommodate covariate information, and it is the goal of this paper to develop two methods that do. In the first approach, both the mean and covariance functions depend on the covariate Z and time scale t while in the second approach only the mean function depends on the covariate Z. Both new approaches accommodate additional measurement errors and functional data sampled at regular time grids as well as sparse longitudinal data sampled at irregular time grids. The first approach to fully adjust both the mean and covariance functions adapts more to the data but is computationally more intensive than the approach to adjust the covariate effects on the mean function only. We develop general asymptotic theory for both approaches and compare their performance numerically through simulation studies and a data set.

Article information

Ann. Statist. Volume 38, Number 2 (2010), 1194-1226.

First available in Project Euclid: 19 February 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H25: Factor analysis and principal components; correspondence analysis 62M15: Spectral analysis
Secondary: 62G20: Asymptotic properties

Functional data analysis functional principal components analysis local linear regression longitudinal data analysis smoothing sparse data


Jiang, Ci-Ren; Wang, Jane-Ling. Covariate adjusted functional principal components analysis for longitudinal data. Ann. Statist. 38 (2010), no. 2, 1194--1226. doi:10.1214/09-AOS742.

Export citation


  • Besse, P. and Ramsay, J. (1986). Principal components analysis of sampled functions. Psychometrika 51 285–311.
  • Bhattacharya, P. K. and Müller, H. G. (1993). Asymptotics for nonparametric regression. Sankhyā Ser. A 55 420–441.
  • Boente, G. and Fraiman, R. (2000). Kernel-based functional principal components. Statist. Probab. Lett. 48 335–345.
  • Bosq, D. (2000). Linear Processes in Function Spaces: Theory and Application. Springer, New York.
  • Cardot, H. (2000). Nonparametric estimation of smoothed principal components analysis of sampled noisy functions. J. Nonparametr. Stat. 12 503–538.
  • Cardot, H. (2006). Conditional functional principal components analysis. Scand. J. Statist. 34 317–335.
  • Carey, J. R., Liedo, P., Müller, H. G., Wang, J. L., Sentürk, D. and Harshman, L. (2005). Biodemography of a long-lived tephritid: Reproduction and longevity in a large cohort of female mexican fruit flies, Anastrepha Ludens. Experimental Gerontology 40 793–800.
  • Castro, P. E., Lawton, W. H. and Sylvestre, E. A. (1986). Principal modes of variation for processes with continuous sample curves. Technometrics 28 329–337.
  • Chiou, J.-M., Müller, H.-G. and Wang, J.-L. (2003). Functional quasi-likelihood regression models with smooth random effects. J. R. Stat. Soc. Ser. B Stat. Methodol. 65 405–423.
  • Dauxois, J., Pousse, A. and Romain, Y. (1982). Asymptotic theory for the principal component analysis of a vector random function: Some applications of statistical inference. J. Multivariate Anal. 12 136–154.
  • Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Chapman and Hall, London.
  • Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis: Theory and Practice. Springer, New York.
  • Guo, W. (2002). Funcitonal mixed effects models. Biometrics 58 121–128.
  • Hall, P. and Hosseini-Nasab, M. (2006). On properties of functional principal component analysis. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 109–126.
  • Hall, P., Müller, H.-G. and Wang, J.-L. (2006). Properties of principal component methods for functional and longitudinal data analysis. Ann. Statist. 34 1493–1517.
  • James, G. M., Hastie, T. J. and Suger, C. A. (2000). Principal components models for sparse functional data. Biometrika 87 587–602.
  • Kneip, A. and Utikal, K. (2001). Inference for density families using functional principal component analysis. J. Amer. Statist. Assoc. 96 519–532.
  • Mas, A. and Menneteau, L. (2003). High Dimensional Probability III. 127–134. Birkhäuser, Basel.
  • Paul, D. and Peng, J. (2009). Consistency of restricted maximum likelihood estimators of principal components. Ann. Statist. 37 1229–1271.
  • Peng, J. and Paul, D. (2009). A geometric approach to maximum likelihood estimation of the functional principal components from sparse longitudinal data. J. Comput. Graph. Statist. In press.
  • Ramsay, J. O. and Silverman, B. W. (2002). Applied Functional Data Analysis: Methods and Case Studies. Springer, New York.
  • Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer, New York.
  • Rao, C. R. (1958). Some statistical methods for comparison of growth curves. Biometrics 14 1–17.
  • Rice, J. and Silverman, B. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. J. Roy. Statist. Soc. Ser. B 53 233–243.
  • Rice, J. A. (2004). Functional and longitudinal data analysis: Prospectives on smoothing. Statist. Sinica 14 631–647.
  • Rice, J. A. and Wu, C. (2001). Nonparametric mixed effects models for unequally sampled noisy curves. Biometrics 57 253–259.
  • Shi, M., Weiss, R. and Taylor, J. (1996). An analysis of paediatric cd4 counts for acquired immune deficiency syndrome using flexible random curves. J. Appl. Stat. 45 151–163.
  • Wang, Y. (1998). Mixed-effects smoothing spline anova. J. R. Stat. Soc. Ser. C Stat. Methodol. 60 159–174.
  • Wu, H. and Zhang, J.-T. (2006). Nonparametric Regression Methods for Longitudinal Data Analysis: Mixed-Effects Modeling Approaches. Wiley, Hoboken, NJ.
  • Yao, F. (2007). Asymptotic distributions of nonparametric regression estimators for longitudinal of functional data. J. Multivariate Anal. 98 40–56.
  • Yao, F., Müller, H.-G. and Wang, J.-L. (2005). Functional data analysis for sparse longitudinal data. J. Amer. Statist. Assoc. 100 577–590.
  • Zhang, D., Lin, X., Raz, J. and Sowers, F. (1998). Semiparametric stochastic mixed models for longitudinal data. J. Amer. Statist. Assoc. 93 710–719.