In many scientific experiments we need to face analysis with functional data, where the observations are sampled from random process, together with a potentially large number of non-functional covariates. The complex nature of functional data makes it difficult to directly apply existing methods to model selection and estimation. We propose and study a new class of penalized semiparametric functional linear regression to characterize the regression relation between a scalar response and multiple covariates, including both functional covariates and scalar covariates. The resulting method provides a unified and flexible framework to jointly model functional and non-functional predictors, identify important covariates, and improve efficiency and interpretability of the estimates. Featured with two types of regularization: the shrinkage on the effects of scalar covariates and the truncation on principal components of the functional predictor, the new approach is flexible and effective in dimension reduction. One key contribution of this paper is to study theoretical properties of the regularized semiparametric functional linear model. We establish oracle and consistency properties under mild conditions by allowing possibly diverging number of scalar covariates and simultaneously taking the infinite-dimensional functional predictor into account. We illustrate the new estimator with extensive simulation studies, and then apply it to an image data analysis.
Group for Research in Decision Analysis