The Annals of Statistics

Covariate adjusted functional principal components analysis for longitudinal data

Ci-Ren Jiang and Jane-Ling Wang

Full-text: Open access

Enhanced PDF (495 KB)

Abstract

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

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

Dates
First available in Project Euclid: 19 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.aos/1266586627

Digital Object Identifier
doi:10.1214/09-AOS742

Mathematical Reviews number (MathSciNet)
MR2604710

Zentralblatt MATH identifier
1183.62102

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

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

Citation

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. https://projecteuclid.org/euclid.aos/1266586627


Export citation

References

  • Besse, P. and Ramsay, J. (1986). Principal components analysis of sampled functions. Psychometrika 51 285–311.
    Mathematical Reviews (MathSciNet): MR848110
    Digital Object Identifier: doi:10.1007/BF02293986
  • 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.
    Mathematical Reviews (MathSciNet): MR1771495
  • Bosq, D. (2000). Linear Processes in Function Spaces: Theory and Application. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1783138
    Zentralblatt MATH: 0962.60004
  • Cardot, H. (2000). Nonparametric estimation of smoothed principal components analysis of sampled noisy functions. J. Nonparametr. Stat. 12 503–538.
    Mathematical Reviews (MathSciNet): MR1785396
    Zentralblatt MATH: 0951.62030
    Digital Object Identifier: doi:10.1080/10485250008832820
  • Cardot, H. (2006). Conditional functional principal components analysis. Scand. J. Statist. 34 317–335.
    Mathematical Reviews (MathSciNet): MR2346642
    Digital Object Identifier: doi:10.1111/j.1467-9469.2006.00521.x
  • 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.
    Mathematical Reviews (MathSciNet): MR650934
    Zentralblatt MATH: 0539.62064
    Digital Object Identifier: doi:10.1016/0047-259X(82)90088-4
  • Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Chapman and Hall, London.
    Mathematical Reviews (MathSciNet): MR1383587
    Zentralblatt MATH: 0873.62037
  • Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis: Theory and Practice. Springer, New York.
    Mathematical Reviews (MathSciNet): MR2229687
    Zentralblatt MATH: 1119.62046
  • Guo, W. (2002). Funcitonal mixed effects models. Biometrics 58 121–128.
    Mathematical Reviews (MathSciNet): MR1891050
    Digital Object Identifier: doi:10.1111/j.0006-341X.2002.00121.x
    JSTOR: links.jstor.org
  • 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.
    Mathematical Reviews (MathSciNet): MR2212577
    Zentralblatt MATH: 1141.62048
    Digital Object Identifier: doi:10.1111/j.1467-9868.2005.00535.x
  • 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.
    Mathematical Reviews (MathSciNet): MR1789811
    Zentralblatt MATH: 0962.62056
    Digital Object Identifier: doi:10.1093/biomet/87.3.587
    JSTOR: links.jstor.org
  • Kneip, A. and Utikal, K. (2001). Inference for density families using functional principal component analysis. J. Amer. Statist. Assoc. 96 519–532.
    Mathematical Reviews (MathSciNet): MR1946423
    Zentralblatt MATH: 1019.62060
    Digital Object Identifier: doi:10.1198/016214501753168235
    JSTOR: links.jstor.org
  • Mas, A. and Menneteau, L. (2003). High Dimensional Probability III. 127–134. Birkhäuser, Basel.
    Mathematical Reviews (MathSciNet): MR2033885
    Zentralblatt MATH: 1053.60002
  • Paul, D. and Peng, J. (2009). Consistency of restricted maximum likelihood estimators of principal components. Ann. Statist. 37 1229–1271.
    Mathematical Reviews (MathSciNet): MR2509073
    Zentralblatt MATH: 1161.62032
    Digital Object Identifier: doi:10.1214/08-AOS608
    Project Euclid: euclid.aos/1239369021
  • 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.
    Mathematical Reviews (MathSciNet): MR1910407
    Zentralblatt MATH: 1011.62002
  • Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer, New York.
    Mathematical Reviews (MathSciNet): MR2168993
  • 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.
    Mathematical Reviews (MathSciNet): MR1094283
    JSTOR: links.jstor.org
  • Rice, J. A. (2004). Functional and longitudinal data analysis: Prospectives on smoothing. Statist. Sinica 14 631–647.
    Mathematical Reviews (MathSciNet): MR2087966
    Zentralblatt MATH: 1073.62033
  • Rice, J. A. and Wu, C. (2001). Nonparametric mixed effects models for unequally sampled noisy curves. Biometrics 57 253–259.
    Mathematical Reviews (MathSciNet): MR1833314
    Digital Object Identifier: doi:10.1111/j.0006-341X.2001.00253.x
    JSTOR: links.jstor.org
  • 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.
    Mathematical Reviews (MathSciNet): MR1625640
    Zentralblatt MATH: 0909.62034
    Digital Object Identifier: doi:10.1111/1467-9868.00115
    JSTOR: links.jstor.org
  • Wu, H. and Zhang, J.-T. (2006). Nonparametric Regression Methods for Longitudinal Data Analysis: Mixed-Effects Modeling Approaches. Wiley, Hoboken, NJ.
    Mathematical Reviews (MathSciNet): MR2216899
    Zentralblatt MATH: 1127.62041
  • Yao, F. (2007). Asymptotic distributions of nonparametric regression estimators for longitudinal of functional data. J. Multivariate Anal. 98 40–56.
    Mathematical Reviews (MathSciNet): MR2292916
    Zentralblatt MATH: 1102.62040
    Digital Object Identifier: doi:10.1016/j.jmva.2006.08.007
  • 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.
    Mathematical Reviews (MathSciNet): MR1631369
    Zentralblatt MATH: 0918.62039
    Digital Object Identifier: doi:10.2307/2670121
    JSTOR: links.jstor.org

The Institute of Mathematical Statistics

The Annals of Statistics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more