The Annals of Statistics

Functional linear regression with points of impact

Alois Kneip, Dominik Poß, and Pascal Sarda

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The paper considers functional linear regression, where scalar responses $Y_{1},\ldots,Y_{n}$ are modeled in dependence of i.i.d. random functions $X_{1},\ldots,X_{n}$. We study a generalization of the classical functional linear regression model. It is assumed that there exists an unknown number of “points of impact,” that is, discrete observation times where the corresponding functional values possess significant influences on the response variable. In addition to estimating a functional slope parameter, the problem then is to determine the number and locations of points of impact as well as corresponding regression coefficients. Identifiability of the generalized model is considered in detail. It is shown that points of impact are identifiable if the underlying process generating $X_{1},\ldots,X_{n}$ possesses “specific local variation.” Examples are well-known processes like the Brownian motion, fractional Brownian motion or the Ornstein–Uhlenbeck process. The paper then proposes an easily implementable method for estimating the number and locations of points of impact. It is shown that this number can be estimated consistently. Furthermore, rates of convergence for location estimates, regression coefficients and the slope parameter are derived. Finally, some simulation results as well as a real data application are presented.

Article information

Ann. Statist., Volume 44, Number 1 (2016), 1-30.

Received: July 2014
Revised: February 2015
First available in Project Euclid: 10 December 2015

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression 62M99: None of the above, but in this section
Secondary: 62J05: Linear regression

Functional linear regression model selection stochastic processes nonstandard asymptotics


Kneip, Alois; Poß, Dominik; Sarda, Pascal. Functional linear regression with points of impact. Ann. Statist. 44 (2016), no. 1, 1--30. doi:10.1214/15-AOS1323.

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  • Bickel, P. J., Ritov, Y. and Tsybakov, A. B. (2009). Simultaneous analysis of lasso and Dantzig selector. Ann. Statist. 37 1705–1732.
  • Bosq, D. (2000). Linear Processes in Function Spaces: Theory and Applications. Lecture Notes in Statistics 149. Springer, New York.
  • Cai, T. T. and Hall, P. (2006). Prediction in functional linear regression. Ann. Statist. 34 2159–2179.
  • Cardot, H., Ferraty, F. and Sarda, P. (1999). Functional linear model. Statist. Probab. Lett. 45 11–22.
  • Cardot, H. and Johannes, J. (2010). Thresholding projection estimators in functional linear models. J. Multivariate Anal. 101 395–408.
  • Cardot, H., Mas, A. and Sarda, P. (2007). CLT in functional linear regression models. Probab. Theory Related Fields 138 325–361.
  • Comte, F. and Johannes, J. (2012). Adaptive functional linear regression. Ann. Statist. 40 2765–2797.
  • Crambes, C., Kneip, A. and Sarda, P. (2009). Smoothing splines estimators for functional linear regression. Ann. Statist. 37 35–72.
  • Delaigle, A. and Hall, P. (2012). Methodology and theory for partial least squares applied to functional data. Ann. Statist. 40 322–352.
  • Ferraty, F., Hall, P. and Vieu, P. (2010). Most-predictive design points for functional data predictors. Biometrika 97 807–824.
  • Frank, I. E. and Friedman, J. H. (1993). A statistical view of some chemometrics regression tools. Technometrics 35 109–135.
  • Gillespie, D. T. (1996). Exact numerical simulation of the Ornstein–Uhlenbeck process and its integral. Phys. Rev. E (3) 54 2084–2091.
  • Hall, P. and Horowitz, J. L. (2007). Methodology and convergence rates for functional linear regression. Ann. Statist. 35 70–91.
  • Hall, P. and Hosseini-Nasab, M. (2006). On properties of functional principal components analysis. J. R. Stat. Soc. Ser. B. Stat. Methodol. 68 109–126.
  • He, G., Müller, H. G. and Wang, J. L. (2000). Extending correlation and regression from multivariate to functional data. In Asymptotics in Statistics and Probability 301–315. VSP, Leiden.
  • Hsing, T. and Ren, H. (2009). An RKHS formulation of the inverse regression dimension-reduction problem. Ann. Statist. 37 726–755.
  • James, G. M., Wang, J. and Zhu, J. (2009). Functional linear regression that’s interpretable. Ann. Statist. 37 2083–2108.
  • Kneip, A., Poß, D. and Sarda, P. (2015). Supplement to “Functional linear regression with points of impact.” DOI:10.1214/15-AOS1323SUPP.
  • Kneip, A. and Sarda, P. (2011). Factor models and variable selection in high-dimensional regression analysis. Ann. Statist. 39 2410–2447.
  • McKeague, I. W. and Sen, B. (2010). Fractals with point impact in functional linear regression. Ann. Statist. 38 2559–2586.
  • Müller, H.-G. and Stadtmüller, U. (2005). Generalized functional linear models. Ann. Statist. 33 774–805.
  • van de Geer, S. and Lederer, J. (2013). The Bernstein–Orlicz norm and deviation inequalities. Probab. Theory Related Fields 157 225–250.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes with Applications to Statistics. Springer, New York.
  • Zhou, S., Lafferty, J. and Wasserman, L. (2008). Time varying undirected graphs. In Proceedings of the $21$st Annual Conference on Computational Learning Theory (COLT 2008). Available at
  • Zhou, S., van de Geer, S. and Bühlmann, P. (2009). Adaptive lasso for high dimensional regression and Gaussian graphical modeling. Available at

Supplemental materials

  • Supplement to “Functional linear regression with points of impact”. The supplementary document by Kneip, Poss and Sarda (2015) contains three Appendices. An application to NIR data can be found in Appendix A. In Appendix B, it is shown that the eigenfunctions of a Brownian motion satisfy assertion 2.5 in Theorem 2. Appendix C provides the proofs of Theorem 4 and Propositions 1 and 2.