Central Posterior Envelopes for Bayesian Functional Principal Component Analysis.

Bayesian methods provide direct inference in functional data analysis applications without reliance on bootstrap techniques. A major tool in functional data applications is the functional principal component analysis which decomposes the data around a common mean function and identifies leading directions of variation. Bayesian functional principal components analysis (BFPCA) provides uncertainty quantification on the estimated functional model components via the posterior samples obtained. We propose central posterior envelopes (CPEs) for BFPCA based on functional depth as a descriptive visualization tool to summarize variation in the posterior samples of the estimated functional model components, contributing to uncertainty quantification in BFPCA. The proposed BFPCA relies on a latent factor model and targets model parameters within a mixed effects modeling framework using modified multiplicative gamma process shrinkage priors on the variance components. Functional depth provides a center-outward order to a sample of functions. We utilize modified band depth and modified volume depth for ordering of a sample of functions and surfaces, respectively, to derive at CPEs of the mean and eigenfunctions within the BFPCA framework. The proposed CPEs are showcased in extensive simulations. Finally, the proposed CPEs are applied to the analysis of a sample of power spectral densities (PSD) from resting state electroencephalography (EEG) where they lead to novel insights on diagnostic group differences among children diagnosed with autism spectrum disorder and their typically developing peers across age.