Functional Principal Component Regression and Functional Partial Least Squares

• Published in: Journal of the American Statistical Association Vol. 102; no. 479; pp. 984 - 996
• Main Authors: Reiss, Philip T, Ogden, R. Todd
• Format: Journal Article
• Language: English
• Published: Alexandria, VA Taylor & Francis 01.09.2007; American Statistical Association; Taylor & Francis Ltd
• Subjects: Applications; B-splines; Chemicals; Data smoothing; Datasets; Estimation; Estimation methods; Exact sciences and technology; Fines & penalties; Functional linear model; Gasoline; General topics; Least squares; Linear inference, regression; Linear mixed model; Linear regression; Mathematics; Matrices; Multivariate analysis; Multivariate calibration; Oracles; Probability and statistics; Regression analysis; Sciences and techniques of general use; Signal regression; SIMPLS; Spectrum analysis; Statistical methods; Statistical variance; Statistics; Theory and Methods; Wavelengths; B spline; Error estimation; Multivariate calibration; Prediction theory; Cross validation; SIMPLS; Multivariate analysis; Stochastic process; Statistical simulation; Asymptotic convergence; Penalty method; Spline approximation; Numerical approximation; Linear mixed model; Correspondence analysis; Least squares method; B-splines; Discriminant analysis; Smoothing methods; Factor analysis; Asymptotic behavior; Prediction; Statistical estimation; Mean estimation; Functional linear model; Functional; Statistical method; Statistical regression; Function estimation; Dimension reduction; Smoothing; Signal regression; Filtering theory; Mean error; Maximum likelihood; Application; Principal component analysis
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1198/016214507000000527

Regression of a scalar response on signal predictors, such as near-infrared (NIR) spectra of chemical samples, presents a major challenge when, as is typically the case, the dimension of the signals far exceeds their number. Most solutions to this problem reduce the dimension of the predictors either by regressing on components [e.g., principal component regression (PCR) and partial least squares (PLS)] or by smoothing methods, which restrict the coefficient function to the span of a spline basis. This article introduces functional versions of PCR and PLS, which combine both of the foregoing dimension-reduction approaches. Two versions of functional PCR are developed, both using B-splines and roughness penalties. The regularized-components version applies such a penalty to the construction of the principal components (i.e., it uses functional principal components), whereas the regularized-regression version incorporates a penalty in the regression. For the latter form of functional PCR, the penalty parameter may be selected by generalized cross-validation, restricted maximum likelihood (REML), or a minimum mean integrated squared error criterion. Proceeding similarly, we develop two versions of functional PLS. Asymptotic convergence properties of regularized-regression functional PCR are demonstrated. A simulation study and split-sample validation with several NIR spectroscopy data sets indicate that functional PCR and functional PLS, especially the regularized-regression versions with REML, offer advantages over existing methods in terms of both estimation of the coefficient function and prediction of future observations.