Functional Principal Components Analysis (FPCA) is a widely used analytic tool for dimension reduction of functional data. Traditional implementations of FPCA estimate the principal components from the data, then treat these estimates as fixed in subsequent analyses. To account for the uncertainty of PC estimates, we propose FAST, a fully-Bayesian FPCA with three core components: (1) projection of eigenfunctions onto an orthonormal spline basis; (2) efficient sampling of the orthonormal spline coefficient matrix using a parameter expansion scheme based on polar decomposition; and (3) ordering eigenvalues during sampling. Extensive simulation studies show that FAST is very stable and performs better compared to existing methods. FAST is motivated by and applied to a study of 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）是一种广泛用于函数数据降维的分析工具。传统的FPCA实现从数据中估计主成分，然后在后续分析中将估计值视为固定值。为考虑主成分估计的不确定性，我们提出FAST方法，这是一种全贝叶斯FPCA框架，包含三个核心组成部分：（1）将特征函数投影到正交样条基上；（2）基于极分解的参数扩展方案实现正交样条系数矩阵的高效采样；（3）在采样过程中对特征值进行排序。大量仿真研究表明，FAST方法具有高度稳定性，且性能优于现有方法。FAST方法的提出源于并应用于"通过饮食方法控制高血压-糖尿病连续血糖监测"研究中餐后血糖变异性的分析。所有相关的STAN代码和仿真程序均已作为补充材料提供。