A Unified Principal Components Analysis for Stationary Functional Time Series

TL;DR

This paper introduces a unified approach called optimal functional filters for stationary functional time series, improving dimension reduction and forecasting accuracy over traditional FPCA methods.

Contribution

It develops a novel theoretical framework and estimation procedure for optimal functional filters, enhancing dimension reduction and prediction in functional time series analysis.

Findings

The proposed method outperforms existing FPCA approaches in simulations.

Theoretical properties of the new methodology are rigorously established.

Application to air pollution data demonstrates practical effectiveness.

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.