Asymptotics of prediction in functional linear regression with functional outputs

We study prediction in the functional linear model with functional outputs : $Y=SX+\epsilon $ where the covariates $X$ and $Y$ belong to some functional space and $S$ is a linear operator. We provide the asymptotic mean square prediction error with exact constants for our estimator which is based on functional PCA of the input and has a classical form. As a consequence we derive the optimal choice of the dimension $k_{n}$ of the projection space. The rates we obtain are optimal in minimax sense and generalize those found when the output is real. Our main results hold with no prior assumptions on the rate of decay of the eigenvalues of the input. This allows to consider a wide class of parameters and inputs $X(\cdot) $ that may be either very irregular or very smooth. We also prove a central limit theorem for the predictor which improves results by Cardot, Mas and Sarda (2007) in the simpler model with scalar outputs. We show that, due to the underlying inverse problem, the bare estimate cannot converge in distribution for the norm of the function space

[1]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[2]  Hervé Cardot,et al.  Thresholding projection estimators in functional linear models , 2008, J. Multivar. Anal..

[3]  André Mas.  Weak convergence in the functional autoregressive model , 2005, math/0509256.

[4]  Jane-ling Wang,et al.  Functional linear regression analysis for longitudinal data , 2005, math/0603132.

[5]  T. Tony Cai,et al.  Prediction in functional linear regression , 2006 .

[6]  D. Bosq.  Linear Processes in Function Spaces: Theory And Applications , 2000 .

[7]  M. Lifshits.  Gaussian Random Functions , 1995 .

[8]  J. Ramsay,et al.  The historical functional linear model , 2003 .

[9]  P. Sarda,et al.  Smoothing splines estimators for functional linear regression , 2009, 0902.4344.

[10]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[11]  A. Acosta.  Existence and convergence of probability measures in Banach spaces. , 1970 .

[12]  Jaromír Antoch,et al.  Electricity consumption prediction with functional linear regression using spline estimators , 2010 .

[13]  H. Müller,et al.  FUNCTIONAL RESPONSE MODELS , 2004 .

[14]  P. Sarda,et al.  CLT in functional linear regression models , 2005, math/0508073.

[15]  C. J. Stone,et al.  Optimal Global Rates of Convergence for Nonparametric Regression , 1982 .

[16]  Joel L. Horowitz,et al.  Methodology and convergence rates for functional linear regression , 2007, 0708.0466.

[17]  J. Friedman,et al.  [A Statistical View of Some Chemometrics Regression Tools]: Response , 1993 .

[18]  P. Sarda,et al.  Functional Linear Regression with Functional Response: Application to Prediction of Electricity Consumption , 2008 .

[19]  J. Ramsay,et al.  Some Tools for Functional Data Analysis , 1991 .

[20]  J. Friedman,et al.  A Statistical View of Some Chemometrics Regression Tools , 1993 .

[21]  Julian J. Faraway,et al.  An F test for linear models with functional responses , 2004 .

[22]  M. A. Kaashoek,et al.  Classes of Linear Operators Vol. I , 1990 .

[23]  Z. Q. John Lu,et al.  Nonparametric Functional Data Analysis: Theory And Practice , 2007, Technometrics.

[24]  R. Cooke.  Real and Complex Analysis , 2011 .

[25]  Henry W. Altland,et al.  Applied Functional Data Analysis , 2003, Technometrics.

[26]  D. Aldous.  The Central Limit Theorem for Real and Banach Valued Random Variables , 1981 .

[27]  A. Tsybakov,et al.  Introduction à l'estimation non-paramétrique , 2003 .

[28]  W. Rudin.  Real and complex analysis, 3rd ed. , 1987 .

[29]  Hans-Georg Müller,et al.  Functional Data Analysis , 2016 .

[30]  C. W. Groetsch,et al.  Inverse Problems in the Mathematical Sciences , 1993 .

[31]  Pascal Sarda,et al.  Smoothing splines estimators in functional linear regression with errors-in-variables , 2007, Comput. Stat. Data Anal..

[32]  Ana M. Aguilera,et al.  Estimation of Functional Regression Models for Functional Responses by Wavelet Approximation , 2008 .

[33]  J. Weidmann.  Linear Operators in Hilbert Spaces , 1980 .

[34]  J. Faraway.  Regression analysis for a functional response , 1997 .

[35]  A. Cuevas,et al.  Linear functional regression: The case of fixed design and functional response , 2002 .