In the context of functional data analysis, functional linear regression serves as a fundamental tool to handle the relationship between a scalar response and a functional covariate. With the aid of Karhunen-Loève expansion of a stochastic process, a functional linear model can be written as an infinite linear combination of functional principal component scores. A reduced form is fitted in practice for dimension reduction; it is essentially converted to a multiple linear regression model. Though the functional linear model is easy to implement and interpret in applications, it may suffer from an inadequate fit due to this specific linear representation. Additionally, effects of scalar predictors which may be predictive of the scalar response are neglected in the functional linear model. Prediction accuracy can be enhanced greatly by incorporating effects of these scalar predictors.
We propose a functional semiparametric additive model, which models the effect of a functional covariate nonparametrically and models several scalar covariates in a linear form. Our model is applied to ADHD study, in which the response is ADHD index and rs-fMRI serves as the functional covariate.