🤖 AI Summary
Traditional functional principal component analysis (FPCA) treats principal component estimates as deterministic quantities, neglecting their sampling variability and thereby yielding inaccurate uncertainty quantification. To address this, we propose Bayes-FPCA—a fast Bayesian FPCA framework that models principal components directly on the Stiefel manifold for the first time. It achieves efficient dimensionality reduction via orthogonal spline basis projection and constructs a uniform prior on the manifold using the polar decomposition. A stable MCMC sampling strategy is designed to respect the structural constraints inherent in FPCA. Evaluated on DASH4D continuous glucose monitoring data, Bayes-FPCA accurately characterizes postprandial glucose dynamics and substantially improves uncertainty estimation accuracy while accelerating computation by an order of magnitude. All code and simulation routines are publicly available.
📝 Abstract
Functional Principal Components Analysis (FPCA) is one of the most successful and widely used analytic tools for exploration and dimension reduction of functional data. Standard implementations of FPCA estimate the principal components from the data but ignore their sampling variability in subsequent inferences. To address this problem, we propose the Fast Bayesian Functional Principal Components Analysis (Fast BayesFPCA), that treats principal components as parameters on the Stiefel manifold. To ensure efficiency, stability, and scalability we introduce three innovations: (1) project all eigenfunctions onto an orthonormal spline basis, reducing modeling considerations to a smaller-dimensional Stiefel manifold; (2) induce a uniform prior on the Stiefel manifold of the principal component spline coefficients via the polar representation of a matrix with entries following independent standard Normal priors; and (3) constrain sampling using the assumed FPCA structure to improve stability. We demonstrate the application of Fast BayesFPCA to characterize the variability in mealtime glucose from the Dietary Approaches to Stop Hypertension for Diabetes Continuous Glucose Monitoring (DASH4D CGM) study. All relevant STAN code and simulation routines are available as supplementary material.