  The analysis of multivariate functional curves has the potential to yield
important scientific discoveries in domains such as healthcare, medicine,
economics and social sciences. However, it is common for real-world settings to
present longitudinal data that are both irregularly and sparsely observed,
which introduces important challenges for the current functional data
methodology. A Bayesian hierarchical framework for multivariate functional
principal component analysis is proposed, which accommodates the intricacies of
such irregular observation settings by flexibly pooling information across
subjects and correlated curves. The model represents common latent dynamics via
shared functional principal component scores, thereby effectively borrowing
strength across curves while circumventing the computationally challenging task
of estimating covariance matrices. These scores also provide a parsimonious
representation of the major modes of joint variation of the curves and
constitute interpretable scalar summaries that can be employed in follow-up
analyses. Estimation is carried out using variational inference, which combines
efficiency, modularity and approximate posterior density estimation, enabling
the joint analysis of large datasets with parameter uncertainty quantification.
Detailed simulations assess the effectiveness of the approach in sharing
information from sparse and irregularly sampled multivariate curves. The
methodology is also exploited to estimate the molecular disease courses of
individual patients with SARS-CoV-2 infection and characterise patient
heterogeneity in recovery outcomes; this study reveals key coordinated dynamics
across the immune, inflammatory and metabolic systems, which are associated
with survival and long-COVID symptoms up to one year post disease onset. The
approach is implemented in the R package bayesFPCA.
