Abstract
The partial least squares approach has been particularly successful in spectrometric prediction in chemometrics. By treating the spectral data as realizations of a stochastic process, the functional partial least squares can be applied. Motivated by the spectral data collected from oriented strand board furnish, we propose a sparse version of the functional partial least squares regression. The proposed method aims at achieving locally sparse (i.e., zero on certain sub-regions) estimates for the functional partial least squares bases, and more importantly, the locally sparse estimate for the slope function. The new approach applies a functional regularization technique to each iteration step of the functional partial least squares and implements a computational method that identifies nonzero sub-regions on which the slope function is estimated. We illustrate the proposed method with simulation studies and two applications on the oriented strand board furnish data and the particulate matter emissions data.
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References
Asencio, M., G. Hooker, and H. O. Gao (2014). Functional convolution models. Statistical Modelling 14(4), 315–335
Boulesteix, A.-L. and K. Strimmer (2006). Partial least squares: a versatile tool for the analysis of high-dimensional genomic data. Briefings in Bioinformatics 8(1), 32–44
Cardot, H., F. Ferraty, and P. Sarda (2003). Spline estimators for the functional linear model. Statistica Sinica 13, 571–591
Chun, H., Keleş, S.: Sparse partial least squares regression for simultaneous dimension reduction and variable selection. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 72(1), 3–25 (2010)
Clark, N., Gautam, M., Wayne, W., Lyons, D., Thompson, G., Zielinska, B.: Heavy-duty vehicle chassis dynamometer testing for emissions inventory, air quality modeling, source apportionment and air toxics emissions inventory: E55/59 all phases. Technical report, Coordinating Research Council, Alpharetta (2007)
Cook, R. D. and L. Forzani (2019). Partial least squares prediction in high-dimensional regression. The Annals of Statistics 47(2), 884–908
de Boor, C.: A practical Guide to Splines. Springer-Verlag, New York (2001)
Delaigle, A., Hall, P.: Achieving near perfect classification for functional data. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 74(2), 267–286 (2012)
Delaigle, A. and P. Hall (2012b). Methodology and theory for partial least squares applied to functional data. The Annals of Statistics 40(1), 322–352
Escabias, M., A. M. Aguilera, and M. J. Valderrama (2007). Functional PLS logit regression model. Computational Statistics and Data Analysis 51(10), 4891–4902
Fan, J. and R. Li (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96(456), 1348–1360
Frank, L. E. and J. H. Friedman (1993). A statistical view of some chemometrics regression tools. Technometrics 35(2), 109–135
Garthwaite, P. H. (1994). An interpretation of partial least squares. Journal of the American Statistical Association 89(425), 122–127
Guan, T., Z. Lin, and J. Cao (2020). Estimating truncated functional linear models with a nested group bridge approach. Journal of Computational and Graphical Statistics 29(3), 620–628
Hall, P., Hooker, G.: Truncated linear models for functional data. Journal of Royal Statistical Society, Series B (Statistical Methodology) 78(3), 637–653 (2016)
Helland, I. S. (1990). Partial least squares regression and statistical models. Scandinavian Journal of Statistics 17(2), 97–114
James, G. M., J. Wang, and J. Zhu (2009). Functional linear regression that’s interpretable. The Annals of Statistics 37(5A), 2083–2108
Krämer, N. and M. Sugiyama (2011). The degrees of freedom of partial least squares regression. Journal of the American Statistical Association 106(494), 697–705
Lin, Z., J. Cao, L. Wang, and H. Wang (2017). Locally sparse estimator for functional linear regression models. Journal of Computational and Graphical Statistics 26(2), 306–318
Lin, Z., L. Wang, and J. Cao (2016). Interpretable functional principal component analysis. Biometrics 72, 846–854
Martens, H., Næs, T.: Multivariate Calibration. John Wiley & Sons, New York (1992)
Marx, B.D., Eilers, P.H.C.: Generalized linear regression on sampled signals and curves: A P-spline approach. Technometrics 41(1), 1–13 (1999)
Nie, Y., Cao, J.: Sparse functional principal component analysis in a new regression framework. Computational Statistics and Data Analysis 152, 107016 (2020)
Nie, Y., L. Wang, B. Liu, and J. Cao (2018). Supervised functional principal component analysis. Statistics and Computing 28(3), 713–723
Preda, C. and G. Saporta (2005). PLS regression on a stochastic process. Computational Statistics and Data Analysis 48, 149–158
Preda, C., Saporta, G., Lévéder, C.: PLS classification of functional data. Comput. Statistics 22(2), 223–235 (2007)
Reiss, P. T. and R. T. Ogden (2007). Functional principal component regression and functional partial least squares. Journal of the American Statistical Association 102(479), 984–996
Sang, P., L. Wang, and J. Cao (2017). Parametric functional principal component analysis. Biometrics 73, 802–810
Schwartz, W. R., Kembhavi, A., Harwood, D., Davis, L. S.: Human detection using partial least squares analysis. In: 2009 IEEE 12th International Conference on Computer Vision, pp. 24–31 (2009)
Shi, H., J. Dong, L. Wang, and J. Cao (2021). Functional principal component analysis for longitudinal data with informative dropout. Statistics in Medicine 40, 712–724
Wold, H.: Soft modelling by latent variables: The non-linear iterative partial least squares (NIPALS) approach. J. Appl. Probab. 12(S1), 117–142 (1975)
Zhou, J., N.-Y. Wang, and N. Wang (2013). Functional linear model with zero-value coefficient function at sub-regions. Statistica Sinica 23, 25–50
Acknowledgements
The authors would like to thank the Editor, the Associate Editor, and two reviewers for their valuable comments, which are very helpful to improve this work. This work was supported by a discovery grant from the Natural Sciences and Engineering Research Council of Canada (NSERC) to J. Cao.
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Guan, T., Lin, Z., Groves, K. et al. Sparse functional partial least squares regression with a locally sparse slope function. Stat Comput 32, 30 (2022). https://doi.org/10.1007/s11222-021-10066-y
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DOI: https://doi.org/10.1007/s11222-021-10066-y