On prediction rate in partial functional linear regression

We consider a prediction of a scalar variable based on both a function-valued variable and a finite number of real-valued variables. For the estimation of the regression parameters, which include the infinite dimensional function as well as the slope parameters for the real-valued variables, it is inevitable to impose some kind of regularization. We consider two different approaches, which are shown to achieve the same convergence rate of the mean squared prediction error under respective assumptions. One is based on functional principal components regression (FPCR) and the alternative is functional ridge regression (FRR) based on Tikhonov regularization. Also, numerical studies are carried out for a simulation data and a real data.

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

[2]  Frédéric Ferraty,et al.  Nonparametric Functional Data Analysis: Theory and Practice (Springer Series in Statistics) , 2006 .

[3]  P. Sarda,et al.  SPLINE ESTIMATORS FOR THE FUNCTIONAL LINEAR MODEL , 2003 .

[4]  Philippe Vieu,et al.  Maximum ozone concentration forecasting by functional non‐parametric approaches , 2004 .

[5]  Philippe Vieu,et al.  Semi-functional partial linear regression , 2006 .

[6]  P. Hall,et al.  Nonparametric methods for inference in the presence of instrumental variables , 2003, math/0603130.

[7]  Xihong Lin,et al.  Two‐Stage Functional Mixed Models for Evaluating the Effect of Longitudinal Covariate Profiles on a Scalar Outcome , 2007, Biometrics.

[8]  T. Hastie,et al.  [A Statistical View of Some Chemometrics Regression Tools]: Discussion , 1993 .

[9]  Tailen Hsing,et al.  On rates of convergence in functional linear regression , 2007 .

[10]  Paul H. C. Eilers,et al.  Generalized linear regression on sampled signals and curves: a P -spline approach , 1999 .

[11]  M. Yuan,et al.  A Reproducing Kernel Hilbert Space Approach to Functional Linear Regression , 2010, 1211.2607.

[12]  On prediction error in functional linear regression , 2008 .

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

[14]  James O. Ramsay,et al.  Functional Data Analysis , 2005 .

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

[16]  Philippe Vieu,et al.  Nonparametric time series prediction: A semi-functional partial linear modeling , 2008 .

[17]  P. Reiss,et al.  Functional Generalized Linear Models with Images as Predictors , 2010, Biometrics.

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

[19]  Hyejin Shin,et al.  Partial functional linear regression , 2009 .

[20]  P. Vieu,et al.  Nonparametric Functional Data Analysis: Theory and Practice (Springer Series in Statistics) , 2006 .

[21]  Jan Johannes,et al.  Nonparametric estimation in functional linear models with second order stationary regressors. , 2009, 0901.4266.