A Riemann Manifold Model Framework for Longitudinal Changes in Physical Activity Patterns

Physical activity (PA) is significantly associated with many health outcomes. The wide usage of wearable accelerometer-based activity trackers in recent years has provided a unique opportunity for in-depth research on PA and its relations with health outcomes and interventions. Past analysis of activity tracker data relies heavily on aggregating minute-level PA records into day-level summary statistics, in which important information of diurnal PA patterns is lost. In this paper we propose a novel functional data analysis approach based on theory of Riemann manifolds for modeling PA records and longitudinal changes in PA temporal patterns. We model smoothed minute-level PA of a day as one-dimensional Riemann manifolds and longitudinal changes in PA in different visits as deformations between manifolds. A key quantity of the deformations are the initial momenta, which are vector fields representing directions and magnitudes to drag each point on a baseline PA curve towards a follow-up visit curve. The variability in changes of PA among a cohort of subjects is characterized via variability in the deformation momenta. Functional principal components analysis is further adopted to model deformation momenta and PC scores are used as a proxy in modeling the relation between changes in PA and health outcomes and/or interventions. We conduct comprehensive analyses on data from two clinical trials: Reach for Health (RfH) and Metabolism, Exercise and Nutrition at UCSD (MENU), focusing on the effect of interventions on longitudinal changes in PA patterns and how different patterns of changes in PA influence weight loss, respectively. Important modes of variation in PA are identified to be significantly associated with lifestyle interventions/health outcomes.