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Functional variance estimation using penalized splines with principal component analysis

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.

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References

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

    MATH  Article  MathSciNet  Google Scholar 

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

    MATH  Article  MathSciNet  Google Scholar 

  3. 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)

    MATH  Article  MathSciNet  Google Scholar 

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

    MATH  Article  MathSciNet  Google Scholar 

  5. 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)

  6. 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 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    MATH  Article  Google Scholar 

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

    MATH  Article  MathSciNet  Google Scholar 

  12. 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 

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

    MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

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

    MATH  Article  MathSciNet  Google Scholar 

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

    MATH  Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    MATH  Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  21. 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 

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

    Google Scholar 

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

    MATH  Article  Google Scholar 

  24. 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 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  Book  Google Scholar 

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

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

    MATH  Book  Google Scholar 

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

    MATH  Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  32. 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)

    MATH  Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Article  MathSciNet  Google Scholar 

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

    MATH  Article  MathSciNet  Google Scholar 

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

    MATH  Article  MathSciNet  Google Scholar 

  38. 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 

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

    MATH  Article  Google Scholar 

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Correspondence to Michael Wegener.

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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

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  • Functional data analysis
  • Principal components
  • Penalized splines
  • Mixed models