A Riemann Manifold Model Framework for Longitudinal Changes in Physical Activity Patterns
Jingjing Zou, Tuo Lin, Chongzhi Di, John Bellettiere, Marta M. Jankowska, Sheri J. Hartman, Dorothy D. Sears, Andrea Z. LaCroix, Cheryl L. Rock, Loki Natarajan
身体活动（PA）与许多健康结果显著相关。近年来，基于可穿戴加速度计的活动跟踪器的广泛使用为深入研究PA及其与健康结果和干预措施的关系提供了独特的机会。过去对活动跟踪器数据的分析在很大程度上依赖于将分钟级PA记录汇总为日级汇总统计，其中PA时间/昼夜模式的重要信息丢失。在本文中，我们提出了一种新的功能数据分析方法的基础上黎曼流形建模PA及其纵向变化。我们模型平滑分钟级PA的一天一维黎曼流形和纵向变化PA在不同的访问之间的变形流形。通过变形的可变性来表征一组受试者中PA变化的可变性。进一步采用功能主成分分析对变形进行建模，并将PC分数用作PA变化与健康结果和/或干预之间关系的建模代理。我们对两项临床试验的数据进行了全面分析：UCSD的Reach for Health（RfH）和Metabolism，Exercise and Nutrition（MENU），重点关注干预措施对PA模式纵向变化的影响以及PA的不同变化模式如何影响减肥。所提出的方法揭示了独特的变化模式，包括整体增强PA，增强早晨PA，以及每个研究队列特定的活动时间变化。研究结果为PA和健康的纵向变化研究带来了新的见解，并有可能促进设计有效的健康干预措施和指南。
英文摘要：
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 PA temporal/diurnal patterns is lost. In this paper we propose a novel functional data analysis approach based on Riemann manifolds for modeling PA and its longitudinal changes. 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. The variability in changes of PA among a cohort of subjects is characterized via variability in the deformation. Functional principal component analysis is further adopted to model the deformations 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 modes of changes in PA influence weight loss, respectively. The proposed approach reveals unique modes of changes including overall enhanced PA, boosted morning PA, and shifts of active hours specific to each study cohort. The results bring new insights into the study of longitudinal changes in PA and health and have the potential to facilitate designing of effective health interventions and guidelines.
Principal Component Analysis for Functional Data on Riemannian Manifolds and Spheres
Functional data analysis on nonlinear manifolds has drawn recent interest. Sphere-valued functional data, which are encountered for example as movement trajectories on the surface of the earth, are an important special case. We consider an intrinsic principal component analysis for smooth Riemannian manifold -valued functional data and study its asymptotic properties. Riemannian functional principal component analysis (RFPCA) is carried out by first mapping the manifold-valued data through Riemannian logarithm maps to tangent spaces around the time-varying Fréchet mean function, and then perfo…
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