High-dimensional inference for functional regression with an application to the Alzheimer's disease magnetoencephalography study.

Alzheimer's disease (AD) is a progressive, chronic neurodegenerative disorder affecting millions worldwide. A new clinical magnetoencephalography (MEG) study was conducted to identify neural activity biomarkers and key brain regions in AD. Traditional methods for analyzing MEG data, which typically extract features from power spectral density, suffer from information loss. Furthermore, functional regression with variable selection tends to produce non-robust results, making it less ideal for drawing reliable scientific conclusions. To address these challenges, we propose a high-dimensional hypothesis testing (HDHT) framework for functional covariates and introduce a rigorous inference process to support scientific conclusions. We establish the theoretical properties of the HDHT framework and validate its performance through simulation studies. Applying the HDHT framework to the AD MEG data, we identify 19 important regions associated with cognitive functions that align with established AD pathophysiology. These findings suggest that the non-invasive MEG can be a potential low-risk and low-toxicity modality for monitoring neurodegenerative progression.