Skip to main content
Log in

Functional variance estimation using penalized splines with principal component analysis

Statistics and Computing Aims and scope Submit manuscript

Abstract

In many fields of empirical research one is faced with observations arising from a functional process. If so, classical multivariate methods are often not feasible or appropriate to explore the data at hand and functional data analysis is prevailing. In this paper we present a method for joint modeling of mean and variance in longitudinal data using penalized splines. Unlike previous approaches we model both components simultaneously via rich spline bases. Estimation as well as smoothing parameter selection is carried out using a mixed model framework. The resulting smooth covariance structures are then used to perform principal component analysis. We illustrate our approach by several simulations and an application to financial interest data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  • Besse, P., Ramsay, J.O.: Principal components analysis of sampled functions. Psychometrika 51, 285–311 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  • Breslow, N.E., Lin, X.: Bias correction in generalized linear mixed models with a single component of dispersion. Biometrika 82, 81–91 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  • Brumback, B.A., Rice, J.A.: Smoothing spline models for the analysis of nested and crossed samples of curves (c/r: P976-994). J. Am. Stat. Assoc. 93, 961–976 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  • Cardot, H.: Conditional functional principal components analysis. Scand. J. Stat. 34, 317–335 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Cardot, H., Chaouch, M., Goga, C., Labrure, C.: Functional principal components analysis with survey data. Technical Report no. 518 of University of Burgundy (http://math.ubourgogne.fr/IMB/IMB2-publication.html) (2007)

  • Chiou, J., Müller, H., Wang, J., Carey, J.: A functional multiplicative effects model for longitudinal data, with application to reproductive histories of female medflies. Stat. Sin. 13, 1119–1133 (2003)

    MATH  Google Scholar 

  • de Boor, C.: A Practical Guide to Splines. Springer, Berlin (1978)

    MATH  Google Scholar 

  • Diebold, F.X., Li, C.: Forecasting the term structure of government bond yields. J. Econom. 130, 337–364 (2006)

    Article  MathSciNet  Google Scholar 

  • Diggle, P., Heagerty, P., Liang, K.Y., Zeger, S.: Analysis of Longitudinal Data. Oxford University Press, Oxford (2002)

    Google Scholar 

  • Duffie, D., Kan, R.: A yield-factor model of interest rates. Math. Finance 6, 379–406 (1996)

    Article  MATH  Google Scholar 

  • Eilers, P.H.C., Marx, B.D.: Flexible smoothing with B splines and penalties. Stat. Sci. 11(2), 89–121 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  • Fan, J., Zhang, J.T.: Two-step estimation of functional linear models with applications to longitudinal data. J. R. Stat. Soc., Ser. B 62, 303–322 (2000)

    Article  MathSciNet  Google Scholar 

  • Ferraty, F., Vieu, P.: Nonparametric Functional Data Analysis. Springer, New York (2006)

    MATH  Google Scholar 

  • Hastie, T., Tibshirani, R.: Varying-coefficient models. J. R. Stat. Soc., Ser. B 55, 757–796 (1993)

    MATH  MathSciNet  Google Scholar 

  • Kauermann, G., Krivobokova, T., Fahrmeir, L.: Some asymptotic results on generalized penalized spline smoothing. J. R. Stat. Soc., Ser. B 487–503 (2009)

  • Li, Y., Ruppert, D.: On the asymptotics of penalized splines. Biometrika 95(2), 415–436 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Lin, X., Carroll, R.J.: Semiparametric regression for clustered data using generalized estimating equations. J. Am. Stat. Assoc. 96, 1045–1056 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Litterman, R., Scheinkman, J.: Common factors affecting bond returns. J. Fixed Income 1, 54–61 (1991)

    Article  Google Scholar 

  • O’Sullivan, F.: A statistical perspective on ill-posed inverse problems (c/r: P519-527). Stat. Sci. 1, 502–518 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  • Pinheiro, J., Bates, D.: Mixed-Effects Models in S and Splus. Springer, New York (2000)

    Book  Google Scholar 

  • R Development Core Team: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2007). ISBN 3-900051-07-0

    Google Scholar 

  • Ramsay, J., Silverman, B.: Functional Data Analysis, 2nd edn. Springer, New York (2005)

    Google Scholar 

  • Rao, C.R.: Some statistical methods for comparison of growth curves. Biometrics 14, 1–17 (1958)

    Article  MATH  Google Scholar 

  • Rice, J.A., Silverman, B.W.: Estimating the mean and covariance structure nonparametrically when the data are curves. J. R. Stat. Soc., Ser. B 53, 233–243 (1991)

    MATH  MathSciNet  Google Scholar 

  • Rice, J.A., Wu, C.O.: Nonparametric mixed effects models for unequally sampled noisy curves. Biometrics 57, 253–259 (2001)

    Article  MathSciNet  Google Scholar 

  • Ruppert, D.: Selecting the number of knots for penalized splines. J. Comput. Graph. Stat. 11, 735–757 (2002)

    Article  MathSciNet  Google Scholar 

  • Ruppert, R., Wand, M., Carroll, R.: Semiparametric Regression. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  • SAS-Institute: SAS/STAT User’s Guide, Version 8. SAS Institute, Inc. (1999)

  • Searle, S., Casella, G., McCulloch, C.: Variance Components. Wiley, New York (1992)

    Book  MATH  Google Scholar 

  • Staniswalis, J., Lee, J.: Nonparametric regression analysis of longitudinal data. J. Am. Stat. Assoc. 93, 1403–1418 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  • Steeley, J.: Modelling the dynamics of the term structure of interest rates. Econ. Soc. Rev. 21, 337–361 (1990)

    Google Scholar 

  • Wager, C., Vaida, F., Kauermann, G.: Model selection for p-spline smoothing using Akaike information criteria. Austr. N. Z. J. Stat. 49(2), 173–190 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Wand, M.: Smoothing and mixed models. Comput. Stat. 18, 223–249 (2003)

    MATH  Google Scholar 

  • Wand, M., Jones, M.: Kernel Smoothing. Chapman & Hall, London (1995)

    MATH  Google Scholar 

  • Wang, N., Carroll, R.J., Lin, X.: Efficient semiparametric marginal estimation for longitudinal/clustered data. J. Am. Stat. Assoc. 100, 147–157 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Wolfinger, R.: Laplace’s approximation for nonlinear mixed models. Biometrika 80, 791–795 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  • Yao, F., Lee, T.C.M.: Penalized spline models for functional principal component analysis. J. R. Stat. Soc., Ser. B 68, 3–25 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Yao, F., Müller, H., Clifford, A.J., Dueker, S.R., Follet, J., Yumei, L., Buchholz, B.A., Vogel, J.S.: Shrinkage estimation for functional principal component scores with application to the population kinetics of plasma folate. Biometrics 57, 253–259 (2003)

    Google Scholar 

  • Yao, F., Müller, H., Wang, J.L.: Functional linear regression analysis for longitudinal data. Ann. Stat. 33, 2873–2903 (2005)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Michael Wegener.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kauermann, G., Wegener, M. Functional variance estimation using penalized splines with principal component analysis. Stat Comput 21, 159–171 (2011). https://doi.org/10.1007/s11222-009-9156-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11222-009-9156-5

Navigation