Efficient Bayesian functional principal component analysis of irregularly-observed multivariate curves.

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 conducted using variational inference, ensuring that accurate posterior approximation and robust uncertainty quantification are achieved. The algorithm also introduces a novel variational message passing fragment for multivariate functional principal component Gaussian likelihood that enables modularity and reuse across models. 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 long-COVID symptoms up to one year post disease onset. The approach is implemented in the R package bayesFPCA.