Ridge regression for the functional concurrent model. (English) Zbl 1473.62403

Summary: The aim of this paper is to propose estimators of the unknown functional coefficients in the Functional Concurrent Model (FCM). We extend the Ridge Regression method developed in the classical linear case to the functional data framework. Two distinct penalized estimators are obtained: one with a constant regularization parameter and the other with a functional one. We prove the probability convergence of these estimators with rate. Then we study the practical choice of both regularization parameters. Additionally, we present some simulations that show the accuracy of these estimators despite a very low signal-to-noise ratio.


62R10 Functional data analysis
62J05 Linear regression; mixed models
62J07 Ridge regression; shrinkage estimators (Lasso)
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference


fda.usc; fda (R)
Full Text: DOI Euclid


[1] Bosq, D. (2000)., Linear Processes in Function Spaces: Theory and Applications, volume 149 of Lectures Notes in Statistics. Springer-Verlag, New York. · Zbl 0962.60004
[2] Brezis, H. (2010)., Functional analysis, Sobolev spaces and partial differential equations. Springer Science & Business Media.
[3] Cornillon, P.-A. and Matzner-Lober, E. (2010)., Régression avec R. Springer Science & Business Media. · Zbl 1233.62137
[4] Febrero-Bande, M. and Oviedo de la Fuente, M. (2012). Statistical computing in functional data analysis: the r package fda. usc., Journal of Statistical Software, 51(4):1-28.
[5] Green, P. J. and Silverman, B. W. (1994)., Nonparametric regression and generalized linear models: a roughness penalty approach. Chapman & Hall / CRC Press. · Zbl 0832.62032
[6] Hall, P., Horowitz, J. L., et al. (2007). Methodology and convergence rates for functional linear regression., The Annals of Statistics, 35(1):70-91. · Zbl 1114.62048
[7] Hall, P. and Hosseini-Nasab, M. (2006). On properties of functional principal components analysis., Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68(1):109-126. · Zbl 1141.62048
[8] Hoerl, A. E. (1962). Application of ridge analysis to regression problems., Chemical Engineering Progress, 58(3):54-59.
[9] Hoerl, A. E. and Kennard, R. W. (1970). Ridge regression: Biased estimation for nonorthogonal problems., Technometrics, 12(1):55-67. · Zbl 0202.17205
[10] Horváth, L. and Kokoszka, P. (2012)., Inference for Functional Data with Applications, volume 200 of Springer Series in Statistics. Springer, New York.
[11] Hsing, T. and Eubank, R. (2015)., Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators. Wiley Series in Probability and Statistics. John Wiley & Sons, Ltd., Chichester. · Zbl 1338.62009
[12] Huh, M.-H. and Olkin, I. (1995). Asymptotic aspects of ordinary ridge regression., American Journal of Mathematical and Management Sciences, 15(3-4):239-254. · Zbl 0855.62058
[13] Ledoux, M. and Talagrand, M. (1991)., Probability in Banach Spaces, Isoperimetry and Processes, volume 23 of A Series of Modern Surveys in Mathematics Series. Springer-Verlag, Berlin. · Zbl 0748.60004
[14] Ramsay, J. O., Hooker, G., and Graves, S. (2009)., Functional data analysis with R and MATLAB. Springer Science & Business Media. · Zbl 1179.62006
[15] Ramsay, J. O. and Silverman, B. W. (2005)., Functional data analysis. Springer, New York, second edition. · Zbl 1079.62006
[16] Şentürk, D. and Müller, H.-G. (2010). Functional varying coefficient models for longitudinal data., Journal of the American Statistical Association, 105(491):1256-1264. · Zbl 1390.62135
[17] Stone, C. J. (1982). Optimal global rates of convergence for nonparametric regression., Annals of Statistics, 10(4):1040-1053. · Zbl 0511.62048
[18] Wu, C. O., Chiang, C.-T., and Hoover, D. R. (1998). Asymptotic confidence regions for kernel smoothing of a varying-coefficient model with longitudinal data., Journal of the American statistical Association, 93(444):1388-1402. · Zbl 1064.62523
[19] Yao, F., Müller, H.-G., and Jane-Ling, W. (2005a). Functional linear regression analysis for longitudinal data., The Annals of Statistics, 33(6):2873-2903. · Zbl 1084.62096
[20] Yao, F., Müller, H.-G., and Wang, J.-L. (2005b). Functional data analysis for sparse longitudinal data.
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.