Using basis expansions for estimating functional PLS regression: Applications with chemometric data

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.