Filtrated Grouping in Multiple Functional Regression

Abstract: In this article, we develop a novel covariate grouping framework in the context of multiple functional regression, in which a scalar response is associated with multiple functional covariates. We apply this approach to examine the relationship between chronological age and gait angular kinematics in a cohort of healthy individuals. This application is motivated by the need to understand and communicate the risk of chronic joint disease associated with aging by studying how age influences gait patterns. A key challenge stems from the significant interdependence among various joints, which provides important insights into how movement coordination evolves with aging. This limitation drives the primary objective of this work: to develop an efficient methodology to unravel both the association between chronological age and joint kinematics, and the coordination across different joints. To achieve this goal, we develop a forest-structured covariate grouping framework in which different functional covariates are aggregated hierarchically based on the level of coefficient homogeneity. This approach allows for the analysis of both common and idiosyncratic effects of covariates in a nuanced, multi-resolution manner. The identification of the forest structure is entirely data-driven and requires no prior knowledge, providing valuable insights into the interdependence among covariates. Compared to existing methods, the proposed regression framework demonstrates superior predictive power and offers more insightful interpretability. In addition, the proposed framework is broadly applicable and can be readily extended to analyze other types of multivariate functional data.