Simultaneous non-parametric regressions of unbalanced longitudinal data. (English) Zbl 0900.62199

Summary: The aim of this paper is to simultaneously estimate n curves corrupted by noise, this means several observations of a random process. The non-parametric estimation of the sampled paths leads to a new kind of functional principal components analysis which simultaneously takes into account a dimensionality and a smoothness constraint. Furthermore, the use of B-spline approximation to estimate the curves allows the study of unbalanced longitudinal data. The relationship between the choice of the smoothing parameter and that of dimensionality is discussed. A simulation study shows good behaviors of this proposed estimate compared to n independent smoothing splines under generalized cross-validation. Finally, the methodology of this paper is illustrated by its application to a real world data set.


62G07 Density estimation
65C99 Probabilistic methods, stochastic differential equations


Full Text: DOI


[1] Becker, R.; Chambers, J.; Wilks, A.: The new S language, a programming environment for data analysis and graphics. (1988) · Zbl 0642.68003
[2] Besse, P.: Pca stability and choice of dimensionality. Statist. probab. Lett. 13, 405-410 (1992) · Zbl 0743.62046
[3] Besse, P.: Models for multivariate analysis. Compstat 94, 271-285 (1994)
[4] Besse, P.; Falguerolles, A. D.: Application of resampling methods to the choice of dimension in principal component analysis. Computer intensive methods in statistics, 167-176 (1993)
[5] Besse, P.; Ferré, L.: Sur l’usage de la validation croisée en analyse en composantes principales. Rev. statist. Appl. 41, 71-76 (1993) · Zbl 0972.62511
[6] Besse, P.; Pousse, A.: Extension des analyses factorielles. Modèles pour l’analyse des données multidimensionnelles, 129-158 (1992)
[7] Besse, P.; Ramsay, J.: Principal component analysis of sampled curves. Psychometrika 51, 285-311 (1986) · Zbl 0623.62048
[8] Boularan, J.; Ferré, L.; Vieu, P.: Growth curves: a two stage nonparametric approach. J. statist. Plann. inference 38, 327-350 (1993) · Zbl 0797.62030
[9] Champely, S.: Analyse de données fonctionnelles, aproximation par LES splines de regression. Ph.d. thesis (1994)
[10] De Boor, C.: A practical guide to splines. (1978) · Zbl 0406.41003
[11] Denby, L.; Mallows, C.: Smooth reduced-rank approximations. I.S.I., 49th session, contributed papers 1, 355-356 (1993)
[12] Ecostat: Banque de données statistiques pour l’enseignement. (1991)
[13] Jolliffe, I.: Principal component analysis. (1986) · Zbl 0584.62009
[14] Jones, M.; Rice, J.: Displaying the important features of large collections of similar curves. Amer. statist. 46, 140-145 (1992)
[15] Kato, T.: Perturbation theory for linear operator. (1966) · Zbl 0148.12601
[16] Kelly, C.; Rice, J.: Monotone smoothing with application to dose-response curves and the assessment of synergism. Biometrics 46, 1071-1085 (1990)
[17] Kneip, A.: Nonparametric estimation of common regressors for similar curve data. Ann. statist. 22, 1386-1427 (1995) · Zbl 0817.62029
[18] Krzanowski, W.: Cross validation choice in principal components analysis. Biometrics 43, 575-584 (1987)
[19] Pezzulli, S.; Silverman, B.: On smoothed principal components analysis. Comput. statist. 8, 1-16 (1993) · Zbl 0775.62146
[20] Ramsay, J.; Dalzell, C.: Some tools for functional data analysis. J. roy. Statist. soc. Ser. B 53, 539-572 (1991) · Zbl 0800.62314
[21] Rice, J.; Silverman, B.: Estimating the mean and covariance structure nonparametrically when the data are curves. J. roy. Statist. soc. Ser. B 53, 233-243 (1991) · Zbl 0800.62214
[22] Silverman, B., Smoothed functional principal components analysis by choice of norm, to be published. · Zbl 0853.62044
[23] Wahba, G.: Spline models for observational data. (1990) · Zbl 0813.62001
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.