Skip to main content

Selecting the derivative of a functional covariate in scalar-on-function regression


This paper presents tests to formally choose between regression models using different derivatives of a functional covariate in scalar-on-function regression. We demonstrate that for linear regression, models using different derivatives can be nested within a model that includes point-impact effects at the end-points of the observed functions. Contrasts can then be employed to test the specification of different derivatives. When nonlinear regression models are employed, we apply a C test to determine the statistical significance of the nonlinear structure between a functional covariate and a scalar response. The finite-sample performance of these methods is verified in simulation, and their practical application is demonstrated using both chemometric and environmental data sets.

This is a preview of subscription content, access via your institution.

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


  • Aguilera, A., Escabias, M., Valderrama, M.: Discussion of different logistic models with functional data. Application to systemic lupus erythematosus. Comput. Statistics and Data Anal. 53(1), 151–163 (2008)

    MathSciNet  Article  Google Scholar 

  • Aneiros-Pérez, G., Vieu, P.: Semi-functional partial linear regression. Statistics & Prob. Lett. 76(11), 1102–1110 (2006)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  • Asencio, M., Hooker, G., Gao, H.O.: Functional convolution models. Stat. Model. 14(4), 315–335 (2014)

    MathSciNet  Article  Google Scholar 

  • Dalzell, C.J., Ramsay, J.O.: Computing reproduing kernels with arbitrary boundary constraints. SIAM J. Sci. Comput. 14(3), 511–518 (1993)

    MathSciNet  Article  Google Scholar 

  • Davidson, R., MacKinnon, J.G.: Several tests for model specification in the presence of alternative hypotheses. Econometrica 49(3), 781–793 (1981)

    MathSciNet  Article  Google Scholar 

  • Febrero-Bande, M., González-Manteiga, W.: Gregression additive models for functional data. TEST 22(2), 278–292 (2013)

    MathSciNet  Article  Google Scholar 

  • Febrero-Bande, M., Oviedo de la Fuente, M.: Statistical computing in functional data analysis: The R package fda.usc. J. Stat. Softw. 51(4), 1–28 (2012)

    Article  Google Scholar 

  • Ferraty, F., Vieu, P.: The functional nonparametric model and application to spectrometric data. Comput. Statistics 17(4), 545–564 (2002)

    MathSciNet  Article  Google Scholar 

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

    MATH  Google Scholar 

  • Ferraty, F., Vieu, P.: Additive prediction and boosting for functional data. Computational Statistics & Data Analysis 53(4), 1400–1413 (2009)

    MathSciNet  Article  Google Scholar 

  • Goldsmith, J., Scheipl, F., Huang, L., Wrobel, J., Di, C., Gellar, J., Harezlak, J., McLean, M. W., Swihart, B., Xiao, L., Crainiceanu, C. and Reiss, P. T.: refund: Regression with Functional Data. R package version 0.1-23. (2020)

  • Hall, P., Hooker, G.: Truncated linear models for functional data. J. Royal Statistical Soc.: Series B (Statistical Methodology) 78(3), 637–653 (2016)

    MathSciNet  Article  Google Scholar 

  • Hastie, T., Mallows, C.: A statistical view of some chemometrics regression tools (discussion). Technometrics 35(2), 140–143 (1993)

    Google Scholar 

  • Hyndman, R.J., Shang, H.L.: Forecasting functional time series (with discussions). J. Korean Statistical Soc. 38(3), 199–211 (2009)

    MathSciNet  Article  Google Scholar 

  • Jarque, C.M.: Sample splitting and applied econometric modeling. J. Bus. & Econom. Statistics 5(2), 267–274 (1987)

    Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  • Müller, H.-G., Stadmüller, U.: Generalized functional linear models. Ann. Stat. 33(2), 774–805 (2005)

    MathSciNet  Article  Google Scholar 

  • Quintela-del-Río, A., Ferraty, F., Vieu, P.: Analysis of time of occurrence of earthquakes: A functional data approach. Math. Geosci. 43(6), 695–719 (2011)

    Article  Google Scholar 

  • Quintela-del-Río, A., Francisco-Fernández, M.: Nonparametric functional data estimation applied to ozone data: Prediction and extreme value analysis. Chemosphere 82(6), 800–808 (2011)

    Article  Google Scholar 

  • Ramsay, J., Hooker, G.: Dynamic Data Analysis: Modeling Data with Differential Equations. Springer, New York (2017)

  • Ramsay, J., Hooker, G., Graves, S.: Functional Data Analysis with R and MATLAB. Springer, Dordrecht (2009)

    Book  Google Scholar 

  • Ramsay, J. O., Graves, S. and Hooker, G.: fda: Functional Data Analysis. R package version (2020)

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

    Book  Google Scholar 

  • Savitzky, A., Golay, M.J.E.: Smoothing and differentiation of data by simplified least squares procedures. Anal. Chem. 36(8), 1627–1639 (1964)

    Article  Google Scholar 

  • Shang, H.L.: Visualizing rate of change: An application to age-specific fertility rates. J. R. Stat. Soc. Ser. A 182(1), 249–262 (2019)

    MathSciNet  Article  Google Scholar 

  • Yao, F., Müller, H.-G.: Functional quadratic regression. Biometrika 97(1), 49–64 (2010)

    MathSciNet  Article  Google Scholar 

  • Yao, F., Müller, H.-G., Wang, J.-L.: Functional data analysis for sparse longitudinal data. J. Am. Stat. Assoc. 100(470), 577–590 (2005)

    MathSciNet  Article  Google Scholar 

Download references


The first author was partially supported by NSF grants DMS-1053252 and DEB-1353039. The second author acknowledges a sabbatical opportunity from the Research School of Finance, Actuarial Studies and Statistics at the Australian National University and the hospitality of the Department of Statistical Science at Cornell University. We would like to thank the associate editor and two anonymous referees for insightful comments that greatly improved the presentation of our results.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Giles Hooker.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hooker, G., Shang, H.L. Selecting the derivative of a functional covariate in scalar-on-function regression. Stat Comput 32, 35 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI:


  • Model selection
  • Variable selection
  • Likelihood ratio test
  • C test