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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.

MSC:
62G07 Density estimation
65C99 Probabilistic methods, stochastic differential equations
Software:
R
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[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
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