Dynamic Functional Principal Component

In this paper, we address the problem of dimension reduction for time series of functional data $(X_t\colon t\in\mathbb{Z})$. Such {\it functional time series} frequently arise, e.g., when a continuous-time process is segmented into some smaller natural units, such as days. Then each~$X_t$ represents one intraday curve. We argue that functional principal component analysis (FPCA), though a key technique in the field and a benchmark for any competitor, does not provide an adequate dimension reduction in a time-series setting. FPCA indeed is a {\it static} procedure which ignores the essential information provided by the serial dependence structure of the functional data under study. Therefore, inspired by Brillinger's theory of {\it dynamic principal components}, we propose a {\it dynamic} version of FPCA, which is based on a frequency-domain approach. By means of a simulation study and an empirical illustration, we show the considerable improvement the dynamic approach entails when compared to the usual static procedure.