Bayesian Functional Factor Analysis and Bayesian Functional Principal Component Analysis

This research explores the application of Functional Data Analysis (FDA) to uncover time-varying patterns in complex dynamic systems. FDA treats data as smooth, continuous functions rather than discrete observations, enabling the analysis of temporal dependencies and underlying dynamic structures. Building on this framework, we apply Bayesian Principal Component Analysis (BPCA) and Bayesian Factor Analysis (BFA) to enhance the study of time-dependent data. BPCA is extended to functional data to address challenges in dimensionality reduction for high-dimensional and noisy datasets, providing robust uncertainty quantification and improved interpretability of temporal principal components. Complementing this, BFA is applied at ten distinct time points, using smoothing splines to visualize dynamic factor loadings and scores, capturing the evolution of latent structures over time. Together, these Bayesian approaches advance our understanding of temporal correlations in dynamic systems, offering a flexible and detailed methodology for applications such as gene regulatory networks and other evolving processes.