Skip to main content
SpringerLink
Log in

Penalized function-on-function regression

  • Original Paper
  • Published:
Computational Statistics Aims and scope Submit manuscript

Abstract

A general framework for smooth regression of a functional response on one or multiple functional predictors is proposed. Using the mixed model representation of penalized regression expands the scope of function-on-function regression to many realistic scenarios. In particular, the approach can accommodate a densely or sparsely sampled functional response as well as multiple functional predictors that are observed on the same or different domains than the functional response, on a dense or sparse grid, and with or without noise. It also allows for seamless integration of continuous or categorical covariates and provides approximate confidence intervals as a by-product of the mixed model inference. The proposed methods are accompanied by easy to use and robust software implemented in the pffr function of the R package refund. Methodological developments are general, but were inspired by and applied to a diffusion tensor imaging brain tractography dataset.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Aguilera A, Ocaña F, Valderrama M (1999) Forecasting with unequally spaced data by a functional principal component approach. Test 8(1):233–253

    Article  MATH  MathSciNet  Google Scholar 

  • Aneiros-Pérez G, Vieu P (2008) Nonparametric time series prediction: a semi-functional partial linear modeling. J Multivar Anal 99:834–857

    Article  MATH  Google Scholar 

  • Basser P, Mattiello J, LeBihan D (1994) MR diffusion tensor spectroscopy and imaging. Biophys J 66:259–267

    Article  Google Scholar 

  • Basser P, Pajevic S, Pierpaoli C, Duda J (2000) In vivo fiber tractography using DT-MRI data. Magn Reson Med 44:625–632

    Article  Google Scholar 

  • Bunea F, Ivanescu AE, Wegkamp MH (2011) Adaptive inference for the mean of a Gaussian process in functional data. J R Stat Soc Ser B 73(4):531–558

    Article  MATH  MathSciNet  Google Scholar 

  • Claeskens G, Krivobokova T, Opsomer JD (2009) Asymptotic properties of penalized splines estimators. Biometrika 96(3):529–544

    Article  MATH  MathSciNet  Google Scholar 

  • Crainiceanu C, Reiss P, Goldsmith J, Huang L, Huo L, Scheipl F, Swihart B, Greven S, Harezlak J, Kundu M, G, Zhao Y, McLean M, Xiao L (2014) Refund: regression with functional data, Website: http://CRAN.R-project.org/package=refund

  • Crainiceanu CM, Ruppert D (2004) Likelihood ratio tests in linear mixed models with one variance component. J R Stat Soc Ser B 66(1):165–185

    Article  MATH  MathSciNet  Google Scholar 

  • Crainiceanu CM, Staicu A-M, Di C (2009) Generalized multilevel functional regression. J Am Stat Assoc 104(488):177–194

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Evangelou N, Konz D, Esiri MM, Smith S, Palace J, Matthews PM (2000) Regional axonal loss in the corpus callosum correlates with cerebral white matter lesion volume and distribution in multiple sclerosis. Brain 123(9):1845–1849

    Article  Google Scholar 

  • Fan Y, Foutz N, James GM, Jank W (2014) Functional response additive model estimation with online virtual stock markets. Ann Appl Stat, To appear

  • Ferraty F, Laksaci A, Tadj A, Vieu P (2011) Kernel regression with functional response. Electron J Stat 5:159–171

    Article  MATH  MathSciNet  Google Scholar 

  • Ferraty F, Van Keilegom I, Vieu P (2012) Regression when both response and predictor are functions. J Multivar Anal 109:10–28

    Article  MATH  Google Scholar 

  • Ferraty F, Vieu P (2006) Nonparametric functional data analysis. Springer, New York

    MATH  Google Scholar 

  • Ferraty F, Vieu P (2009) Additive prediction and boosting for functional data. Comput Stat Data Anal 53:1400–1413

    Article  MATH  MathSciNet  Google Scholar 

  • Goldsmith J, Bobb J, Crainiceanu CM, Caffo BS, Reich D (2011) Penalized functional regression. J Comput Graph Stat 20(4):830–851

    Article  MathSciNet  Google Scholar 

  • Greven S, Crainiceanu CM, Caffo BS, Reich D (2010) Longitudinal functional principal component analysis. Electron J Stat 4:1022–1054

    Article  MATH  MathSciNet  Google Scholar 

  • Greven S, Crainiceanu CM, Küchenhoff H, Peters A (2008) Restricted likelihood ratio testing for zero variance components in linear mixed models. J Comput Graph Stat 17(4):870–891

    Article  Google Scholar 

  • He G, Müller H-G, Wang J-L, Wang W (2010) Functional linear regression via canonical analysis. Bernoulli 16(3):705–729

    Article  MATH  MathSciNet  Google Scholar 

  • Horváth L, Kokoszka P (2012) Inference for functional data with applications. Springer, New York

    Book  MATH  Google Scholar 

  • Huang L, Goldsmith J, Reiss PT, Reich DS, Crainiceanu CM (2013) Bayesian scalar-on-image regression with application to association between intracranial DTI and cognitive outcomes. NeuroImage 83:210–223

    Article  Google Scholar 

  • Kadri H, Preux P, Duflos E, Canu S (2011) Multiple functional regression with both discrete and continuous covariates. In: Ferraty F (ed) Recent advances in functional data analysis and related topics, contributions to statistics. Physica-Verlag, Heidelberg, pp 189–195

    Chapter  Google Scholar 

  • Krivobokova T, Kauermann G (2007) A note on penalized spline smoothing with correlated errors. JASA 102(480):1328–1337

    Article  MATH  MathSciNet  Google Scholar 

  • Lindquist MA (2012) Functional causal mediation analysis with an application to brain connectivity. J Am Stat Assoc 107(500):1297–1309

    Article  MATH  MathSciNet  Google Scholar 

  • Matsui H, Kawano S, Konishi S (2009) Regularized functional regression modeling for functional response and predictors. J Math Ind 1(3):17–25

    MATH  MathSciNet  Google Scholar 

  • McLean MW, Hooker G, Staicu A-M, Scheipl F, Ruppert D (2014) Functional generalized additive models. J Comput Graph Stat 23(1):249–269

    Article  MathSciNet  Google Scholar 

  • Matlab, The MathWorks Inc. (2014) Natick. Massachusetts, United States

  • Nychka D (1988) Confidence intervals for smoothing splines. J Am Stat Assoc 83:1134–1143

    Article  MathSciNet  Google Scholar 

  • Ozturk A, Smith SA, Gordon-Lipkin EM, Harrison DM, Shiee N, Pham DL, Caffo BS, Calabresi PA, Reich DS (2010) MRI of the corpus callosum in multiple sclerosis: association with disability. Mult Scler 16(2):166–177

    Article  Google Scholar 

  • R Development Core Team (2014) R: a language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria. www.R-project.org

  • Ramsay JO, Wickham H, Graves S, Hooker G (2014) FDA: functional data analysis. Website: http://CRAN.R-project.org/package=fda

  • Ramsay JO, Hooker G, Graves S (2009) Functional data analysis with R and Matlab. Springer, New York

    Book  MATH  Google Scholar 

  • Ramsay JO, Silverman BW (2005) Functional data analysis. Springer, New York

    Book  Google Scholar 

  • Reiss P, Ogden T (2009) Smoothing parameter selection for a class of semiparametric linear models. J R Stat Soc Ser B 71(2):505–523

    Article  MATH  MathSciNet  Google Scholar 

  • Ruppert D, Wand MP, Caroll RJ (2003) Semiparametric regression. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Scheipl F, Greven S, Küchenhoff H (2008) Size and power of tests for a zero random effect variance or polynomial regression in additive and linear mixed models. Comput Stat Data Anal 52(7):3283–3299

    Article  MATH  Google Scholar 

  • Scheipl F, Staicu A-M, Greven S (2014) Functional additive mixed models. J Comput Graph Stat. doi:10.1080/10618600.2014.901914

  • Song SK, Sun SW, Ramsbottom MJ, Chang C, Russell J, Cross AH (2002) Dysmyelination revealed through MRI as increased radial (but unchanged axial) diffusion of water. Neuroimage 17(3):1429–1436

    Article  Google Scholar 

  • Staicu A-M, Crainiceanu CM, Reich DS, Ruppert D (2012) Modeling functional data with spatially heterogeneous shape characteristics. Biometrics 68(2):331–343

    Article  MATH  MathSciNet  Google Scholar 

  • Tievsky AL, Ptak T, Farkas J (1999) Investigation of apparent diffusion coefficient and diffusion tensor anisotropy in acute and chronic multiple sclerosis lesions. Am J Neuroradiol 20(8):1491–1499

    Google Scholar 

  • Valderrama MJ, Ocaña FA, Aguilera AM, Ocaña-Peinado FM (2010) Forecasting pollen concentration by a two-step functional model. Biometrics 66(2):578–585

  • Wahba G (1983) Bayesian ‘confidence intervals’ for the cross-validated smoothing spline. J R Stat Soc Ser B 45:133–150

    MATH  MathSciNet  Google Scholar 

  • Wood SN (2006) Generalized additive models: an introduction with R. Chapman & Hall/CRC, New York

    Google Scholar 

  • Wood SN (2011) Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. J R Stat Soc Ser B 73(1):3–36

    Article  MathSciNet  Google Scholar 

  • Wood SN (2014) MGCV: mixed GAM computation vehicle with GCV/AIC/REML smoothness estimation. Website: http://CRAN.R-project.org/package=mgcv

  • Wu Y, Fan J, Müller H-G (2010) Varying-coefficient functional linear regression. Bernoulli 16(3):730–758

  • Yao F, Müller H-G, Wang J-L (2005a) Functional data analysis for sparse longitudinal data. J Am Stat Assoc 100(740):577–590

    Article  MATH  Google Scholar 

  • Yao F, Müller H-G, Wang J-L (2005b) Functional linear regression analysis for longitudinal data. Ann Stat 33(6):2873–2903

    Article  MATH  Google Scholar 

  • Zhang JT, Chen J (2007) Statistical inferences for functional data. Ann Stat 35(3):1052–1079

    Article  MATH  Google Scholar 

Download references

Acknowledgments

Staicu’s research was supported by U.S. National Science Foundation Grant Number DMS 1007466 and by the NCSU Faculty Research and Professional Development Grant. Fabian Scheipl and Sonja Greven were funded by Emmy Noether Grant GR 3793/1-1 from the German Research Foundation. We thank Daniel Reich and Peter Calabresi for the DTI tractography data. We would like to thank Ciprian Crainiceanu for helpful discussions and comments on earlier versions of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Montclair State University, Montclair, NJ, USA

    Andrada E. Ivanescu

  2. Department of Statistics, North Carolina State University, Raleigh, NC, USA

    Ana-Maria Staicu

  3. Department of Statistics, Ludwig-Maximilians-University Munich, Munich, Germany

    Fabian Scheipl & Sonja Greven

Authors
  1. Andrada E. Ivanescu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ana-Maria Staicu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fabian Scheipl
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sonja Greven
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrada E. Ivanescu.

Additional information

The refund package is available from CRAN at the following website: http://CRAN.R-project.org/package=refund.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 2377 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ivanescu, A.E., Staicu, AM., Scheipl, F. et al. Penalized function-on-function regression. Comput Stat 30, 539–568 (2015). https://doi.org/10.1007/s00180-014-0548-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00180-014-0548-4

Keywords

Search

Navigation