Generalized dynamic functional principal component analysis

In this paper, we explore dimension reduction for functional time series. We propose a generalized dynamic functional principal component analysis (GDFPCA) which does not rely on spectral density estimation and demonstrates strong empirical performance for both stationary and nonstationary functional time series. We define the generalized dynamic functional principal components (GDFPCs) as static factor time series in a functional dynamic factor model and obtain their multivariate representation from a truncation of the functional dynamic factor model. Estimation is based on a least-squares reconstruction criterion and implemented via a two-step procedure for the coefficient vectors of the loading curves under a basis expansion. We establish mean-square consistency of the reconstructed functional time series under weak stationarity. Simulation studies show that GDFPCA performs comparably to dynamic functional principal component analysis (DFPCA) for stationary data, while providing improved reconstruction accuracy in nonstationary settings, where both DFPCA and functional principal component analysis (FPCA) deteriorate. Applications to real datasets support the empirical advantages observed in the simulations.

💡 Research Summary

The paper introduces Generalized Dynamic Functional Principal Component Analysis (GDFPCA), a novel dimension‑reduction technique for functional time series that avoids the need for spectral density estimation. Traditional functional principal component analysis (FPCA) treats each observed curve as an independent functional datum, ignoring temporal dependence, while Dynamic FPCA (DFPCA) incorporates dependence by estimating spectral density operators and constructing frequency‑domain filters. Both approaches have limitations: FPCA discards serial correlation, and DFPCAs require accurate spectral estimation, which becomes problematic for non‑stationary or structurally changing series.

GDFPCA builds on the functional dynamic factor model (FDFM). The observed functional series (X_t(u)) is decomposed as \