Central Posterior Envelopes for Bayesian Longitudinal Functional Principal Component Analysis

Longitudinally observed functional data are commonly encountered in biomedical studies. Under the weak separability assumption of the high dimensional covariance, the recently proposed Bayesian longitudinal functional principal component analysis (B-LFPCA) achieves the decomposition of the multidimensional signal into highly interpretable lower dimensional summaries, including eigenfunctions that capture directions of variation in the data along the longitudinal and functional dimensions. B-LFPCA provides uncertainty quantification of the estimated functional decomposition components through simultaneous parametric credible bands formed using the posterior sample. However, these traditional summaries are inherently based on point-wise summaries of the estimated functional components and do not take into account the functional nature of the estimated quantities. We introduce central posterior envelopes (CPEs) for uncertainty quantification of the low-dimensional B-LFPCA decomposition components based on functional depth ordering of the posterior estimates. The proposed CPEs are fully data-driven visualization tools, displaying the most-central regions of the posterior sample at specified $\alpha$ -level percentile contours. Modified band depth and modified volume depth are utilized to order posterior sample of functional decomposition components, including the mean function and the marginal longitudinal and functional eigenfunctions. The proposed CPEs are applied to analyze the longitudinally observed Event Related Potentials (ERPs) recorded during an implicit learning paradigm, leading to novel insights on longitudinal learning trends across a group of autistic kids and their neurotypical peers. Finally, effectiveness of the proposed CPEs is demonstrated through extensive simulations that explore different scenarios of increased variability in the longitudinal functional data.