Summary
This paper addresses a specific case of regression analysis: the predictor is a random curve and the response is a scalar. We consider three models: the functional linear model, the functional generalized linear model and functional linear regression on quantiles. Spline functions are used to build estimators which minimize a penalized criterion. The method is illustrated by means of real data examples. Then, we give asymptotics results for these estimators.
We would like to thank all the members and participants of the working group on functional data STAPH from Toulouse for helpful discussions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
de Boor, C. (1978) A practical guide to Spline. Springer, New York.
Cardot, H., Crambes, C. and Sarda, P. (1999) Spline estimator of conditional quantiles for functional covariates (in French) Preprint.
Cardot, H., Faivre, R. and M. Goulard (2003) Functional Approaches for predicting land use with the temporal evolution of coarse resolution remote sensing data. Journal of Applied Statistics, 30, 1185–1199.
Cardot, H., Ferraty, F. and Sarda, P. (1999) Functional Linear Model. Statist. & Prob. Letters, 45, 11–22.
Cardot, H., Ferraty, F. and P. Sarda (2003) Spline Estimators for the Functional Linear Model. Statistica Sinica, 13, 571–591.
Cardot, H. and Sarda, P. (2003) Estimation in Generalized Linear Models for Functional Data via Penalized Likelihood. Journal of Multivariate Analysis, to appear.
Dauxois, J. and Pousse, A. (1976) Les analyses factorielles en calcul des probabilités et en statistique. Essai d’étude synthétique (in French) Thèse, Université Paul Sabatier, Toulouse, France.
Dauxois, J., Pousse, A. and Romain, Y. (1982) Asymptotic theory for the principal component analysis of a random vector function: some applications to statistical inference. Journal of Mult. Analysis, 12, 136–154.
Frank, I.E. and Friedman, J.H. (1993) A Statistical View of Some Chemometrics Regression Tools. Technometrics, 35, 109–148.
Hastie, T.J. and Mallows, C, (1993) A discussion of “A statistical view of some chemometrics regression tools” by I.E. Frank and J.H. Friedman. Technometrics, 35, 140–143.
Koenker, R. and Bassett, G. (1978) Regression Quantiles. Econometrica, 46, 33–50.
Kress, R. (1989) Linear Integral Equations, Springer Verlag, New York.
Lejeune, M. and Sarda, P. (1988) Quantile Regression: A Nonparametric Approch. Computational Statistics and Data Analysis, 6, 229–239.
Marx, B.D. and Eilers P.H. (1996) Generalized Linear Regression on Sampled Signals with penalized likelihood. In: Forcina, A., Marchetti, G.M., Hatzinger, R., Galmacci, G. (Eds), Statistical Modelling, Proceedings of the Eleventh International Workshop on Statistical Modelling, Orvietto.
Marx, B.D. and Eilers P.H. (1999) Generalized Linear Regression on Sampled Signals and Curves: A P-Spline Approach. Technometrics, 41, 1–13.
Meynard, J.M. (1997) Which crop models for decision support in crop management: example of the Déciblé system. Quantitative Approaches in System Analysis, 15, 107–112.
Osborne, B. G., Fearn, T., Miller, A. R. and Douglas, S. (1984) Application of near infrared reflectance spectroscopy to the compositinla analysis of biscuits and biscuit dough. J. Sci. Food Agriculture, 35 99–105.
Poiraud-Casanova, S. et Thomas-Agnan, C. (1998) Quantiles Conditionnels. Journal de la Société Française de Statistique, 139, 31–44.
Ramsay, J.O. and Silverman, B.W. (1997) Functional Data Analysis. Springer-Verlag.
Ramsay, J.O. and Silverman, B.W. (2002) Applied Functional Data Analysis: Methods and Case Studies. Springer-Verlag.
Ruppert, D. and Caroll, J. (1988) Transformation and Weighting in Regression. Chapman and Hall.
Stone, C.J. (1986) The dimensionality reduction principle for generalized additive models. Ann. Statist. 14, 590–606.
Tucker, C.J. (1979) Red and Photographic Infrared Linear Combinations for Monitoring Vegetation. Remote Sensing of Environment, 8, 127–150.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Physica-Verlag Heidelberg
About this paper
Cite this paper
Cardot, H., Sarda, P. (2006). Linear Regression Models for Functional Data. In: Sperlich, S., Härdle, W., Aydınlı, G. (eds) The Art of Semiparametrics. Contributions to Statistics. Physica-Verlag HD. https://doi.org/10.1007/3-7908-1701-5_4
Download citation
DOI: https://doi.org/10.1007/3-7908-1701-5_4
Publisher Name: Physica-Verlag HD
Print ISBN: 978-3-7908-1700-3
Online ISBN: 978-3-7908-1701-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)