Annals of Statistics

Methodology and theory for partial least squares applied to functional data

Aurore Delaigle and Peter Hall

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The partial least squares procedure was originally developed to estimate the slope parameter in multivariate parametric models. More recently it has gained popularity in the functional data literature. There, the partial least squares estimator of slope is either used to construct linear predictive models, or as a tool to project the data onto a one-dimensional quantity that is employed for further statistical analysis. Although the partial least squares approach is often viewed as an attractive alternative to projections onto the principal component basis, its properties are less well known than those of the latter, mainly because of its iterative nature. We develop an explicit formulation of partial least squares for functional data, which leads to insightful results and motivates new theory, demonstrating consistency and establishing convergence rates.

Article information

Ann. Statist., Volume 40, Number 1 (2012), 322-352.

First available in Project Euclid: 4 April 2012

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Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression

Central limit theorem computational algorithm consistency convergence rates functional linear models generalized Fourier basis principal components projection stochastic expansion


Delaigle, Aurore; Hall, Peter. Methodology and theory for partial least squares applied to functional data. Ann. Statist. 40 (2012), no. 1, 322--352. doi:10.1214/11-AOS958.

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  • Aguilera, M., Escabiasa, M., Preda, C. and Saporta, G. (2010). Using basis expansions for estimating functional PLS regression: Applications with chemometric data. Chemom. Intell. Lab. 104 289–305.
  • Apanasovich, T. V. and Goldstein, E. (2008). On prediction error in functional linear regression. Statist. Probab. Lett. 78 1807–1810.
  • Baillo, A. (2009). A note on functional linear regression. J. Stat. Comput. Simul. 79 657–669.
  • Berg, C. and Szwarc, R. (2011). The smallest eigenvalue of Hankel matrices. Constr. Approx. 34 107–133.
  • Bro, R. and Eldén, L. (2009). PLS works. J. Chemom. 23 69–71.
  • Cai, T. T. and Hall, P. (2006). Prediction in functional linear regression. Ann. Statist. 34 2159–2179.
  • Cardot, H. and Sarda, P. (2008). Varying-coefficient functional linear regression models. Comm. Statist. Theory Methods 37 3186–3203.
  • Delaigle, A. and Hall, P. (2012). Achieving near-perfect classification for functional data. J. Roy. Statist. Soc. Ser. B 74 267–286.
  • Durand, J.-F. and Sabatier, R. (1997). Additive splines for partial least squares regression. J. Amer. Statist. Assoc. 92 1546–1554.
  • Escabias, M., Aguilera, A. M. and Valderrama, M. J. (2007). Functional PLS logit regression model. Comput. Statist. Data Anal. 51 4891–4902.
  • Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis. Springer, New York.
  • Frank, I. E. and Friedman, J. H. (1993). A statistical view of some chemometrics regression tools (with discussion). Technometrics 35 109–148.
  • Garthwaite, P. H. (1994). An interpretation of partial least squares. J. Amer. Statist. Assoc. 89 122–127.
  • Goutis, C. and Fearn, T. (1996). Partial least squares regression on smooth factors. J. Amer. Statist. Assoc. 91 627–632.
  • Hastie, T., Tibshirani, R. and Friedman, J. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd ed. Springer, New York.
  • Helland, I. S. (1990). Partial least squares regression and statistical models. Scand. J. Stat. 17 97–114.
  • Höskuldsson, A. (1988). PLS regression methods. J. Chemom. 2 211–228.
  • Hou, Q.-H., Lascoux, A. and Mu, Y.-P. (2005). Evaluation of some Hankel determinants. Adv. in Appl. Math. 34 845–852.
  • Krämer, N., Boulesteix, A. L. and Tutz, G. (2008). Penalized partial least squares with applications to B-spline transformations and functional data. Chemom. Intell. Lab. 94 60–69.
  • Krämer, N. and Sugiyama, M. (2011). The degrees of freedom of partial least squares regression. J. Amer. Statist. Assoc. 106 697–705.
  • Lange, K. (1999). Numerical Analysis for Statisticians. Springer, New York.
  • Lascoux, A. (1990). Inversion des matrices de Hankel. Linear Algebra Appl. 129 77–102.
  • Lorber, A., Wangen, L. E. and Kowalski, B. R. (1987). A theoretical foundation for the PLS algorithm. J. Chemom. 1 19–31.
  • Martens, H. and Naes, T. (1989). Multivariate Calibration. Wiley, New York.
  • Müller, H.-G. and Yao, F. (2010). Additive modelling of functional gradients. Biometrika 97 791–805.
  • Nguyen, D. V. and Rocke, D. M. (2004). On partial least squares dimension reduction for microarray-based classification: A simulation study. Comput. Statist. Data Anal. 46 407–425.
  • Phatak, A. and de Hoog, F. (2003). Exploiting the connection between PLS, Lanczos, and conjugate gradients: Alternative proofs of some properties of PLS. J. Chemom. 16 361–367.
  • Phatak, A., Reilly, P. M. and Penlidis, A. (2002). The asymptotic variance of the univariate PLS estimator. Linear Algebra Appl. 354 245–253.
  • Preda, C. and Saporta, G. (2005a). PLS regression on a stochastic process. Comput. Statist. Data Anal. 48 149–158.
  • Preda, C. and Saporta, G. (2005b). Clusterwise PLS regression on a stochastic process. Comput. Statist. Data Anal. 49 99–108.
  • Preda, C., Saporta, G. and Lévéder, C. (2007). PLS classification of functional data. Comput. Statist. 22 223–235.
  • Reiss, P. T. and Ogden, R. T. (2007). Functional principal component regression and functional partial least squares. J. Amer. Statist. Assoc. 102 984–996.
  • Wold, H. (1975). Soft modelling by latent variables: The non-linear iterative partial least squares (NIPALS) approach. In Perspectives in Probability and Statistics, Papers in Honour of M. S. Bartlett (J. Gani, ed.). Academic Press, London.
  • Wu, Y., Fan, J. and Müller, H.-G. (2010). Varying-coefficient functional linear regression. Bernoulli 16 730–758.
  • Yao, F. and Müller, H.-G. (2010). Functional quadratic regression. Biometrika 97 49–64.