Functional Principal Component Regression and Functional Partial Least-squares Regression: An Overview and a Comparative Study

Functional data analysis is a field of growing importance in Statistics. In particular, the functional linear model with scalar response is surely the model that has attracted more attention in both theoretical and applied research. Two of the most important methodologies used to estimate the parameters of the functional linear model with scalar response are functional principal component regression and functional partial least-squares regression. We provide an overview of estimation methods based on these methodologies and discuss their advantages and disadvantages. We emphasise that the role played by the functional principal components and by the functional partial least-squares components that are used in estimation appears to be very important to estimate the functional slope of the model. A functional version of the best subset selection strategy usual in multiple linear regression is also analysed. Finally, we present an extensive comparative simulation study to compare the performance of all the considered methodologies that may help practitioners in the use of the functional linear model with scalar response.