Computational Statistics & Data Analysis 50 (2006) 2422 – 2446 www.elsevier.com/locate/csda Dimension reduction in functional regression with applications U. Amato a , A. Antoniadis b, ∗ , I. De Feis a a Istituto per le Applicazioni del Calcolo ‘M. Picone’ CNR - Sezione di Napoli, Italy b Laboratoire de Modelisation et Calcul (LMC-IMAG), Universite Joseph Fourier,Tour IRMA, B.P. 53, 38041 Grenoble, Cedex 9, France Available online 18 January 2005 Abstract Two dimensional reduction regression methods to predict a scalar response from a discretized sample path of a continuous time covariate process are presented. The methods take into account the functional nature of the predictor and are both based on appropriate wavelet decompositions. Using such decompositions, prediction methods are devised that are similar to minimum average variance estimation (MAVE) or functional sliced inverse regression (FSIR). Their practical implementation is described, together with their application both to simulated and on real data analyzing three calibration examples of near infrared spectra. © 2004 Elsevier B.V.All rights reserved. Keywords: Dimension reduction; Wavelets; MAVE; SIR 1. Introduction In regression or classification problems one of the tasks is to study the structural rela- tionship between a response variable Y and a vector of covariates X = (X 1 ,...,X P ) T via m(x) = E(Y |X = x) where x = (x 1 ,...,x P ) T and m(x) = m (x 1 ,...,x P ). When the number of predictors P is moderate and the number of samples that are observed are large, nonpara- metric statistical techniques are very useful for investigating E(Y |X), especially when an ∗ Corresponding author. Tel.: +33 476514306; fax: +33 476631263. E-mail addresses: u.amato@iac.cnr.it (U. Amato), Anestis.Antoniadis@imag.fr (A. Antoniadis), i.defeis@iac.cnr.it (I. De Feis). 0167-9473/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2004.12.007