  Many modern data sets require inference methods that can estimate the shared
and individual-specific components of variability in collections of matrices
that change over time. Promising methods have been developed to analyze these
types of data in static cases, but very few approaches are available for
dynamic settings. To address this gap, we consider novel models and inference
methods for pairs of matrices in which the columns correspond to multivariate
observations at different time points. In order to characterize common and
individual features, we propose a Bayesian dynamic factor modeling framework
called Time Aligned Common and Individual Factor Analysis (TACIFA) that
includes uncertainty in time alignment through an unknown warping function. We
provide theoretical support for the proposed model, showing identifiability and
posterior concentration. The structure enables efficient computation through a
Hamiltonian Monte Carlo (HMC) algorithm. We show excellent performance in
simulations, and illustrate the method through application to a social
synchrony experiment.
