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.

翻译：函数主成分分析（FPCA）是探索和降维函数数据最成功且应用最广泛的工具之一。FPCA的标准实现从数据中估计主成分，但在后续推断中忽略了其抽样变异性。为解决此问题，我们提出了快速贝叶斯函数主成分分析（Fast BayesFPCA），将主成分视为Stiefel流形上的参数。为确保效率、稳定性和可扩展性，我们引入了三项创新：(1) 将所有特征函数投影到正交样条基上，将建模问题简化为更低维的Stiefel流形；(2) 通过具有独立标准正态先验的矩阵极坐标表示，在Stiefel流形上诱导主成分样条系数的均匀先验；(3) 利用假定的FPCA结构约束抽样以提高稳定性。我们展示了Fast BayesFPCA在DASH4D CGM研究中表征餐后血糖变异性的应用。所有相关的STAN代码和模拟程序均作为补充材料提供。