Central Posterior Envelopes for Bayesian Functional Principal Component Analysis.

Area of Science:

• Statistics
• Functional Data Analysis
• Bayesian Inference

Background:

• Functional Principal Component Analysis (FPCA) is crucial for decomposing functional data but often relies on bootstrap methods.
• Bayesian Functional Principal Components Analysis (BFPCA) offers uncertainty quantification through posterior samples.
• Existing BFPCA methods lack robust visualization tools for summarizing posterior variations.

Purpose of the Study:

• To introduce Central Posterior Envelopes (CPEs) as a novel visualization tool for BFPCA.
• To enhance uncertainty quantification in functional data analysis using Bayesian methods.
• To apply CPEs in BFPCA for analyzing electroencephalography (EEG) power spectral densities (PSD).

Main Methods:

• Developed CPEs for BFPCA using functional depth measures (modified band depth and modified volume depth).
• Employed a latent factor model within a mixed effects modeling framework.
• Utilized modified multiplicative gamma process shrinkage priors on variance components.

Main Results:

• CPEs effectively summarize variation in posterior samples of mean functions and eigenfunctions in BFPCA.
• Simulations demonstrate the utility and performance of the proposed CPEs.
• Application to EEG PSD data revealed significant diagnostic group differences in children with autism spectrum disorder.

Conclusions:

• CPEs provide a powerful visualization and uncertainty quantification tool for BFPCA.
• The proposed Bayesian approach offers direct inference without bootstrap reliance.
• CPEs yield novel insights into neurophysiological differences in autism spectrum disorder using EEG data.