Dynamic functional principal components
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Date
2012-11-07
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Abstract
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.