Linear manifold modeling of multivariate functional data

Multivariate functional data are increasingly encountered in data analysis, while statistical models for such data are not well developed yet. Motivated by a case study where one aims to quantify the relationship between various longitudinally recorded behavior intensities for Drosophila flies, we propose a functional linear manifold model. This model reflects the functional dependency between the components of multivariate random processes and is defined through data-determined linear combinations of the multivariate component trajectories, which are characterized by a set of varying coefficient functions. The time-varying linear relationships that govern the components of multivariate random functions yield insights about the underlying processes and also lead to noisereduced representations of the multivariate component trajectories. The proposed functional linear manifold model is put to the task for an analysis of longitudinally observed behavioral patterns of flying, feeding, walking and resting over the lifespan of Drosophila flies and is also investigated in simulations.