In modern experiments, functional and nonfunctional data are often encountered simultaneously when observations are sampled from random processes and high-dimensional scalar covariates. It is difficult to apply existing methods for model selection and estimation. We propose a new class of partially functional linear models to characterize the regression between a scalar response and covariates of both functional and scalar types. The new approach provides a unified and flexible framework that simultaneously takes into account multiple functional and ultrahigh-dimensional scalar predictors, enables us to identify important features, and offers improved interpretability of the estimators. The underlying processes of the functional predictors are considered to be infinite-dimensional, and one of our contributions is to characterize the effects of regularization on the resulting estimators. We establish the consistency and oracle properties of the proposed method under mild conditions, demonstrate its performance with simulation studies, and illustrate its application using air pollution data.

We explore the functional linear regression by focusing on the large-scale scenario that the scalar response is associated with potentially an ultra-large number of functional predictors, leading to a more challenging model framework than the classical case. The emphasis is to establish rigorous procedures for testing general hypothesis on an arbitrary subset of regression coefficient functions. Specifically, we exploit the techniques developed for post-regularization inference, and propose a new test for the large-scale functional linear regression based on a decorrelated score function that separates the primary and nuisance parameters in functional spaces. Likewise, we also devise the corresponding decorrelated Wald and likelihood ratio tests and establish the exact equivalence among these three tests for the model under consideration. The proposed test is shown uniformly convergent to the prescribed significance, and its finite sample performance is illustrated via simulation studies and a dataset arising from the Human Connectome Project for identifying brain regions associated with emotional tasks.

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We investigate a partially functional linear model by focusing on the heterogenous error scenario that the scalar response is associated with an ultra-large number of both functional predictors and scalar covariates. Moreover, the model does not require the standard condition on eigenvalue decay for functional predictors, leading to a more challenging and general framework. The target is to establish rigorous inferential procedure for hypothesis testing on an arbitrary subset of both regression functions and scalar coefficients. To be specific, we devise a confidence region for post-regularization inference via a pseudo score function that is not decorrelated due to heterogenous errors. The proposed test does not require the estimation consistency of functional part and is shown uniformly convergent to the prescribed significance, and its finite sample performance is illustrated via simulation studies and an application to a motivating functional magnetic resonance imaging brain image data.