×

Prediction in functional linear regression. (English) Zbl 1106.62036

Summary: There has been substantial recent work on methods for estimating the slope function in linear regression for functional data analysis. However, as in the case of more conventional finite-dimensional regression, much of the practical interest in the slope centers on its application for the purpose of prediction, rather than on its significance in its own right. We show that the problems of slope-function estimation, and of prediction from an estimator of the slope function, have very different characteristics. While the former is intrinsically nonparametric, the latter can be either nonparametric or semiparameric.
In particular, the optimal mean-square convergence rate of predictors is \(n^{-1}\), where \(n\) denotes sample size, if the predict and is a sufficiently smooth function. In other cases, convergence occurs at a polynomial rate that is strictly slower than \(n^{-1}\). At the boundary between these two regimes, the mean-square convergence rate is less than \(n^{-1}\) by only a logarithmic factor. More generally, the rate of convergence of the predicted value of the mean response in the regression model, given a particular value of the explanatory variable, is determined by a subtle interaction among the smoothness of the predictand, of the slope function in the model, and of the autocovariance function for the distribution of explanatory variables.

MSC:

62G08 Nonparametric regression and quantile regression
62J05 Linear regression; mixed models
62G20 Asymptotic properties of nonparametric inference

Software:

fda (R)

References:

[1] Besse, P. and Ramsay, J. O. (1986). Principal components analysis of sampled functions. Psychometrika 51 285–311. · Zbl 0623.62048 · doi:10.1007/BF02293986
[2] Bhatia, R., Davis, C. and McIntosh, A. (1983). Perturbation of spectral subspaces and solution of linear operator equations. Linear Algebra Appl. 52 / 53 45–67. · Zbl 0518.47013 · doi:10.1016/0024-3795(83)80007-X
[3] Boente, G. and Fraiman, R. (2000). Kernel-based functional principal components. Statist. Probab. Lett. 48 335–345. · Zbl 0997.62024 · doi:10.1016/S0167-7152(00)00014-6
[4] Brown, L. D. and Low, M. G. (1996). A constrained risk inequality with applications to nonparametric functional estimation. Ann. Statist. 24 2524–2535. · Zbl 0867.62023 · doi:10.1214/aos/1032181166
[5] Brumback, B. A. and Rice, J. A. (1998). Smoothing spline models for the analysis of nested and crossed samples of curves (with discussion). J. Amer. Statist. Assoc. 93 961–994. JSTOR: · Zbl 1064.62515 · doi:10.2307/2669837
[6] Cai, T. T. and Hall, P. (2005). Prediction in functional linear regression. Technical report. Available at stat.wharton.upenn.edu/\~tcai/paper/FLR-Tech-Report.pdf. · Zbl 1106.62036 · doi:10.1214/009053606000000830
[7] Cardot, H. (2000). Nonparametric estimation of smoothed principal components analysis of sampled noisy functions. J. Nonparametr. Statist. 12 503–538. · Zbl 0951.62030 · doi:10.1080/10485250008832820
[8] Cardot, H., Ferraty, F. and Sarda, P. (1999). Functional linear model. Statist. Probab. Lett. 45 11–22. · Zbl 0962.62081 · doi:10.1016/S0167-7152(99)00036-X
[9] Cardot, H., Ferraty, F. and Sarda, P. (2000). Étude asymptotique d’un estimateur spline hybride pour le modèle linéaire fonctionnel. C. R. Acad. Sci. Paris Sér. I Math. 330 501–504. · Zbl 0944.62040 · doi:10.1016/S0764-4442(00)00207-X
[10] Cardot, H., Ferraty, F. and Sarda, P. (2003). Spline estimators for the functional linear model. Statist. Sinica 13 571–591. · Zbl 1050.62041
[11] Cardot, H. and Sarda, P. (2003). Linear regression models for functional data. Unpublished manuscript. · Zbl 1271.62145
[12] Cardot, H. and Sarda, P. (2005). Estimation in generalized linear models for functional data via penalized likelihood. J. Multivariate Anal. 92 24–41. · Zbl 1065.62127 · doi:10.1016/j.jmva.2003.08.008
[13] Cuevas, A., Febrero, M. and Fraiman, R. (2002). Linear functional regression: The case of fixed design and functional response. Canad. J. Statist. 30 285–300. JSTOR: · Zbl 1012.62039 · doi:10.2307/3315952
[14] Escabias, M., Aguilera, A. M. and Valderrama, M. J. (2005). Modeling environmental data by functional principal component logistic regression. Environmetrics 16 95–107. · doi:10.1002/env.696
[15] Ferraty, F. and Vieu, P. (2000). Dimension fractale et estimation de la régression dans des espaces vectoriels semi-normés. C. R. Acad. Sci. Paris Sér. I Math. 330 139–142. · Zbl 0942.62045 · doi:10.1016/S0764-4442(00)00140-3
[16] Ferraty, F. and Vieu, P. (2002). The functional nonparametric model and application to spectrometric data. Comput. Statist. 17 545–564. · Zbl 1037.62032 · doi:10.1007/s001800200126
[17] Ferraty, F. and Vieu, P. (2004). Nonparametric models for functional data, with application in regression, time-series prediction and curve discrimination. J. Nonparametr. Statist. 16 111–125. · Zbl 1049.62039 · doi:10.1080/10485250310001622686
[18] Ferré, L. and Yao, A. F. (2003). Functional sliced inverse regression analysis. Statistics 37 475–488. · Zbl 1032.62052 · doi:10.1080/0233188031000112845
[19] Girard, S. (2000). A nonlinear PCA based on manifold approximation. Comput. Statist. 15 145–167. · Zbl 0976.62056 · doi:10.1007/s001800000025
[20] Hall, P. and Horowitz, J. L. (2004). Methodology and convergence rates for functional linear regression. Unpublished manuscript. · Zbl 1114.62048
[21] He, G., Müller, H.-G. and Wang, J.-L. (2003). Functional canonical analysis for square integrable stochastic processes. J. Multivariate Anal. 85 54–77. · Zbl 1014.62070 · doi:10.1016/S0047-259X(02)00056-8
[22] James, G. M. (2002). Generalized linear models with functional predictors. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 411–432. JSTOR: · Zbl 1090.62070 · doi:10.1111/1467-9868.00342
[23] James, G. M., Hastie, T. J. and Sugar, C. A. (2000). Principal component models for sparse functional data. Biometrika 87 587–602. JSTOR: · Zbl 0962.62056 · doi:10.1093/biomet/87.3.587
[24] Masry, E. (2005). Nonparametric regression estimation for dependent functional data: Asymptotic normality. Stochastic Process. Appl. 115 155–177. · Zbl 1101.62031 · doi:10.1016/j.spa.2004.07.006
[25] Müller, H.-G. and Stadtmüller, U. (2005). Generalized functional linear models. Ann. Statist. 33 774–805. · Zbl 1068.62048 · doi:10.1214/009053604000001156
[26] Preda, C. and Saporta, G. (2004). PLS approach for clusterwise linear regression on functional data. In Classification , Clustering , and Data Mining Applications (D. Banks, L. House, F. R. McMorris, P. Arabie and W. Gaul, eds.) 167–176. Springer, Berlin.
[27] Ramsay, J. O. and Dalzell, C. J. (1991). Some tools for functional data analysis (with discussion). J. Roy. Statist. Soc. Ser. B 53 539–572. JSTOR: · Zbl 0800.62314
[28] Ramsay, J. O. and Silverman, B. W. (1997). Functional Data Analysis. Springer, New York. · Zbl 0882.62002
[29] Ramsay, J. O. and Silverman, B. W. (2002). Applied Functional Data Analysis : Methods and Case Studies. Springer, New York. · Zbl 1011.62002 · doi:10.1007/b98886
[30] Ratcliffe, S. J., Heller, G. Z. and Leader, L. R. (2002). Functional data analysis with application to periodically stimulated foetal heart rate data. II. Functional logistic regression. Statistics in Medicine 21 1115–1127.
[31] Rice, J. A. and Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. J. Roy. Statist. Soc. Ser. B 53 233–243. JSTOR: · Zbl 0800.62214
[32] Silverman, B. W. (1995). Incorporating parametric effects into functional principal components analysis. J. Roy. Statist. Soc. Ser. B 57 673–689. JSTOR: · Zbl 0827.62051
[33] Silverman, B. W. (1996). Smoothed functional principal components analysis by choice of norm. Ann. Statist. 24 1–24. · Zbl 0853.62044 · doi:10.1214/aos/1033066196
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.