Dynamic functional principal components

In this paper, we address the problem of dimension reduction for sequentially observed functional data (X_k : k ∈ Z). Such functional time series arise frequently, e.g., when a continuous time process is segmented into some smaller natural units, such as days. Then each Xk 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 is a static procedure which ignores valuable information in the serial dependence of the functional data. Therefore, inspired by Brillinger’s theory of dynamic principal components, we propose a 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 our method entails when compared to the usual (static) procedure. While the main part of the article outlines the ideas and the implementation of dynamic FPCA for functional Xk, we provide in the appendices a rigorous theory for general Hilbertian data