A Unified Principal Components Analysis for Stationary Functional Time Series

Abstract: Functional time series (FTS) data have become increasingly available in real-world applications. Research on such data typically focuses on two objectives: curve reconstruction and forecasting, both of which require efficient dimension reduction. While functional principal component analysis (FPCA) serves as a standard tool, existing methods often fail to achieve simultaneous parsimony and optimality in dimension reduction, thereby restricting their practical implementation. To address this limitation, we propose a novel notion termed optimal functional filters, which unifies and enhances conventional FPCA methodologies. Specifically, we establish connections among diverse FPCA approaches through a dependence-adaptive representer for stationary FTS. Building on this theoretical foundation, we develop an estimation procedure for optimal functional filters that enables both dimension reduction and prediction within a Bayesian modeling framework. Theoretical properties are established for the proposed methodology, and comprehensive simulation studies validate its superiority over competing approaches. We further illustrate our method through an application to reconstructing and forecasting daily air pollutant concentration trajectories.