Smoothing splines estimators for functional linear regression. (English) Zbl 1169.62027

Summary: The paper considers functional linear regression, where scalar responses \(Y_{1}, \dots , Y_n\) are modeled in dependence of random functions \(X_{1}, \dots , X_n\). We propose a smoothing splines estimator for the functional slope parameter based on a slight modification of the usual penalty. Theoretical analysis concentrates on the error in an out-of-sample prediction of the response for a new random function \(X_{n+1}\). It is shown that rates of convergence of the prediction error depend on the smoothness of the slope function and on the structure of the predictors. We then prove that these rates are optimal in the sense that they are minimax over large classes of possible slope functions and distributions of the predictive curves. For the case of models with errors-in-variables the smoothing spline estimator is modified by using a denoising correction of the covariance matrix of discretized curves. The methodology is then applied to a real case study where the aim is to predict the maximum of the concentration of ozone by using the curve of this concentration measured the preceding day.


62G08 Nonparametric regression and quantile regression
62J05 Linear regression; mixed models
62G20 Asymptotic properties of nonparametric inference
62G05 Nonparametric estimation
62M20 Inference from stochastic processes and prediction


VanHuffel; fda (R)
Full Text: DOI arXiv


[1] Aneiros-Perez, G., Cardot, H., Estevez-Perez, G. and Vieu, P. (2004). Maximum ozone concentration forecasting by functional nonparametric approaches. Environmetrics 15 675-685.
[2] Bosq, D. (2000). Linear Processes in Function Spaces. Lecture Notes in Statist. 149 . Springer, New York. · Zbl 0962.60004
[3] Cardot, H. (2000). Nonparametric estimation of smoothed principal components analysis of sampled noisy functions. J. Nonparametr. Statist. 12 503-538. · Zbl 0951.62030
[4] Cai, T. T. and Hall, P. (2006). Prediction in functional linear regression. Ann. Statist. 34 2159-2179. · Zbl 1106.62036
[5] Cardot, H., Crambes, C., Kneip, A. and Sarda, P. (2007). Smoothing splines estimators in functional linear regression with errors-in-variables. Comput. Statist. Data Anal. 51 4832-4848. · Zbl 1162.62333
[6] Cardot, H., Crambes, C. and Sarda, P. (2007). Ozone pollution forecasting. In Statistical Methods for Biostatistics and Related Fields (W. Härdle, Y. Mori and P. Vieu, eds.) 221-244. Springer, New York.
[7] Cardot, H., Ferraty, F. and Sarda, P. (2003). Spline estimators for the functional linear model. Statist. Sinica 13 571-591. · Zbl 1050.62041
[8] Cardot, H., Mas, A. and Sarda, P. (2007). CLT in functional linear regression models. Probab. Theory Related Fields 138 325-361. · Zbl 1113.60025
[9] Chiou, J. M., Müller, H. G. and Wang, J. L. (2003). Functional quasi-likelihood regression models with smoothed random effects. J. Roy. Statist. Soc. Ser. B 65 405-423. · Zbl 1065.62065
[10] Cuevas, A., Febrero, M. and Fraiman, R. (2002). Linear functional regression: The case of a fixed design and functional response. Canadian J. Statistics 30 285-300. JSTOR: · Zbl 1012.62039
[11] Demmel, J. (1992). The componentwise distance to the nearest singular matrix. SIAM J. Matrix Anal. Appl. 13 10-19. · Zbl 0749.65031
[12] Eilers, P. H. and Marx, B. D. (1996). Flexible smoothing with B-splines and penalties. Statist. Sci. 11 89-102. · Zbl 0955.62562
[13] Eubank, R. L. (1988). Spline Smoothing and Nonparametric Regression . Dekker, New York. · Zbl 0702.62036
[14] Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis: Methods, Theory, Applications and Implementations . Springer, London. · Zbl 1119.62046
[15] Fuller, W. A. (1987). Measurement Error Models . Wiley, New York. · Zbl 0800.62413
[16] Gasser, T., Sroka, L. and Jennen-Steinmetz, C. (1986). Residual variance and residual pattern in nonlinear regression. Biometrika 3 625-633. JSTOR: · Zbl 0649.62035
[17] Golub, G. H. and Van Loan, C. F. (1980). An analysis of the total least squares problem. SIAM J. Numer. Anal. 17 883-893. JSTOR: · Zbl 0468.65011
[18] Hall, P. and Horowitz, J. L. (2007). Methodology and convergence rates for functional linear regression. Ann. Statist. · Zbl 1114.62048
[19] 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 (M. L. Puri, ed.) 301-315. VSP, Leiden.
[20] Kneip, A. (1994). Ordered linear smoothers. Ann. Statist. 22 835-866. · Zbl 0815.62022
[21] Li, Y. and Hsing, T. (2006). On rates of convergence in functional linear regression. J. Mulitivariate Anal. Published online DOI: 10.1016/j.jmva.2006.10.004. · Zbl 1130.62035
[22] Marx, B. D. and Eilers, P. H. (1999). Generalized linear regression on sampled signals and curves: A P -spline approach. Technometrics 41 1-13.
[23] Müller, H.-G. and Stadtmüller, U. (2005). Generalized functional linear models. Annn. Statist. 33 774-805. · Zbl 1068.62048
[24] Ramsay, J. O. and Dalzell, C. J. (1991). Some tools for functional data analysis. J. Roy. Statist. Soc. Ser. B 53 539-572. JSTOR: · Zbl 0800.62314
[25] Ramsay, J. O. and Silverman, B. W. (2002). Applied Functional Data Analysis . Springer, New York. · Zbl 1011.62002
[26] Ramsay, J. O. and Silverman, B. W. (2005). Applied Functional Data Analysis , 2nd ed. Springer, New York. · Zbl 1079.62006
[27] Stone, C. J. (1982). Optimal global rates of convergence for nonparametric regression. Ann. Statist. 10 1040-1053. · Zbl 0511.62048
[28] Utreras, F. (1983). Natural spline functions, their associated eigenvalue problem. Numer. Math. 42 107-117. · Zbl 0522.41011
[29] Van Huffel, S. and Vandewalle, J. (1991). The Total Least Squares Problem: Computational Aspects and Analysis . SIAM, Philadelphia. · Zbl 0789.62054
[30] Wahba, G. (1977). Practical approximate solutions to linear operator equations when the data are noisy. SIAM J. Numer. Anal. 14 651-667. JSTOR: · Zbl 0402.65032
[31] Wahba, G. (1990). Spline Models for Observational Data . SIAM, Philadelphia. · Zbl 0813.62001
[32] Yao, F., Müller, H.-G. and Wang, J. L. (2005). Functional data analysis for sparse longitudinal data. J. Amer. Statist. Assoc. 100 577-590. · Zbl 1117.62451
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.