Penalized versions of functional PLS regression

• PLS is considered to estimate the functional linear model with scalar response.
• Two penalized estimation approaches are proposed to improve the estimation.
• Discrete and continuous penalties can be used with basis expansions of the curves.
• K-fold cross-validation is used for model selection.
• A simulation study and an application with chemometric functional data is developed.

Least squares estimation of the functional linear regression model with scalar response is an ill-posed problem due to the infinite dimension of the functional predictor. Dimension reduction approaches as principal component regression or partial least squares regression are proposed and widely used in applications. In both cases the interpretation of the model could be difficult because of the roughness of the coefficient regression function. In this paper, two penalized estimations of this model based on modifying the partial least squares criterion with roughness penalties for the weight functions are proposed. One introduces the penalty in the definition of the norm in the functional space, and the other one in the cross-covariance operator. A simulation study and several applications on real data show the efficiency of the penalized approaches with respect to the non-penalized ones.