Using basis expansions for estimating functional PLS regression. Applications with chemometric data
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Functional dataPLS regressionBasis expansion methodsB-splines
Ana M. Aguilera, Manuel Escabias, Cristian Preda, Gilbert Saporta, Using basis expansions for estimating functional PLS regression: Applications with chemometric data, Chemometrics and Intelligent Laboratory Systems, Volume 104, Issue 2, 2010, Pages 289-305, ISSN 0169-7439, https://doi.org/10.1016/j.chemolab.2010.09.007
SponsorshipProject P06-FQM-01470 from Consejería de Innovación, Ciencia y Empresa. Junta de Andalucía, Spain; Project MTM2007-63793 from Dirección General de Investigación, Ministerio de Educación y Ciencia, Spain
There are many chemometric applications, such as spectroscopy, where the objective is to explain a scalar response from a functional variable (the spectrum) whose observations are functions of wavelengths rather than vectors. In this paper, PLS regression is considered for estimating the linear model when the predictor is a functional random variable. Due to the infinite dimension of the space to which the predictor observations belong, they are usually approximated by curves/functions within a finite dimensional space spanned by a basis of functions. We show that PLS regression with a functional predictor is equivalent to finite multivariate PLS regression using expansion basis coefficients as the predictor, in the sense that, at each step of the PLS iteration, the same prediction is obtained. In addition, from the linear model estimated using the basis coefficients, we derive the expression of the PLS estimate of the regression coefficient function from the model with a functional predictor. The results provided by this functional PLS approach are compared with those given by functional PCR and discrete PLS and PCR using different sets of simulated and spectrometric data.