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Central Posterior Envelopes for Bayesian Functional Principal Component Analysis
Volume 21, Issue 4 (2023), pp. 715–734
Joanna Boland   Donatello Telesca   Catherine Sugar     All authors (8)

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https://doi.org/10.6339/23-JDS1085
Pub. online: 19 January 2023      Type: Statistical Data Science      Open accessOpen Access

Received
5 July 2022
Accepted
13 January 2023
Published
19 January 2023

Abstract

Bayesian methods provide direct uncertainty quantification 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 hierarchical 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 from resting state electroencephalography where they lead to novel insights on diagnostic group differences among children diagnosed with autism spectrum disorder and their typically developing peers across age.

Supplementary material

 Supplementary Material
Supplementary material online includes: derived posterior distributions for model estimation; algorithm for alignment of posterior eigenfunction estimates; details on data generation for simulation studies and additional simulation results; pre-processing of the EEG data featured in Section 5; and plots illustrating band depth, and estimates from simulation studies and data analysis. The R code for the proposed methodology is made publicly available on the Github page https://github.com/dsenturk/FDpostSumms_BFPCA, along with a tutorial for step-by-step implementation using simulated data.

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Copyright
2023 The Author(s). Published by the School of Statistics and the Center for Applied Statistics, Renmin University of China.
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Open access article under the CC BY license.

Keywords
electroencephalography functional data analysis modified band depth modified volume depth uncertainty quantification

Funding
This research was supported by National Institute of Mental Health [R01 MH122428 (DS, DT, CS, SJ)].

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