Title: Generalized Heterogeneous Functional Model with Applications to Mobile Health Data

Abstract: Physical activity plays a pivotal role in human health, and it has been suggested that there is a strong relationship between physical activity and various diseases such as mental disorder and Parkinson's disease (PD). However, the underlying mechanism of this relationship is still unclear. One of the primary challenges of depicting this relationship is the inherent heterogeneity among people. To fill this gap,  we propose a  generalized heterogeneous functional method (GHFM) within the generalized functional data analysis framework, which can estimate coefficient functions and subgroup information simultaneously and accommodates generalized outcomes. Unlike traditional homogeneous methods, proposed approach distinguishes the relationship between physical activity and diseases within different subgroups, providing a more comprehensive and systematic depiction. Additionally, we propose a pre-clustering method to improve computational efficiency for large samples.  Simulation studies demonstrate the superior performance of our method in various settings compared to existing approaches. In applications, we investigate the influence of physical activity on the risk of mental disorder measured by neuroticism scores and risk of PD in UK-Biobank dataset.  In a dataset of 79,246 subjects for neuroticism scores, our method identifies four distinct subgroups and estimates their respective coefficient functions. Similarly, in a dataset of 80,692 subjects for PD, our method identifies three distinct subgroups and estimates their coefficient functions. We present scientific interpretation for each subgroup, and these findings  could contribute to identifying disease risks in mobile health applications in the future.