Unified Principal Components Analysis of Irregularly Observed Functional Time Series

Irregularly observed functional time series (FTS) are increasingly available in many real-world applications. To analyze FTS, it is crucial to account for both serial dependencies and the irregularly observed nature of functional data. However, existing methods for FTS often rely on specific model assumptions in capturing serial dependencies, or cannot handle the irregular observational scheme of functional data. To solve these issues, one can perform dimension reduction on FTS via functional principal component analysis (FPCA) or dynamic FPCA. Nonetheless, these methods may either be not theoretically optimal or too redundant to represent serially dependent functional data. In this article, we introduce a novel dimension reduction method for FTS based on dynamic FPCA. Through a new concept called optimal functional filters, we unify the theories of FPCA and dynamic FPCA, providing a parsimonious and optimal representation for FTS adapting to its serial dependence structure. This framework is referred to as principal analysis via dependency-adaptivity (PADA). Under a hierarchical Bayesian model, we establish an estimation procedure for dimension reduction via PADA. Our method can be used for both sparsely and densely observed FTS, and is capable of predicting future functional data. We investigate the theoretical properties of PADA and demonstrate its effectiveness through extensive simulation studies. Finally, we illustrate our method via dimension reduction and prediction of daily PM2.5 data.