Dynamic Functional Principal Component

TL;DR

This paper introduces a dynamic functional principal component analysis method that incorporates serial dependence in functional time series, offering improved dimension reduction over traditional static FPCA.

Contribution

It develops a novel frequency-domain based dynamic FPCA that captures serial dependence, enhancing dimension reduction for functional time series.

Findings

Dynamic FPCA outperforms static FPCA in simulations.

The method effectively captures serial dependence.

Empirical results demonstrate significant improvement.

Abstract

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.