Principal component analysis of periodically correlated functional time series

TL;DR

This paper introduces a novel principal component analysis method for periodically correlated functional time series, leveraging operator-valued filters and Hilbert space theory, with practical implementation and demonstrated superiority over existing methods.

Contribution

It develops a new PCA framework for periodically correlated functional data, including filters, score processes, and inversion formulas, with theoretical properties and a computational R package.

Findings

Method outperforms existing tools on periodic functional data

Theoretical convergence and optimality properties established

Implemented in a custom R package for practical use

Abstract

Within the framework of functional data analysis, we develop principal component analysis for periodically correlated time series of functions. We define the components of the above analysis including periodic, operator-valued filters, score processes and the inversion formulas. We show that these objects are defined via convergent series under a simple condition requiring summability of the Hilbert-Schmidt norms of the filter coefficients, and that they poses optimality properties. We explain how the Hilbert space theory reduces to an approximate finite-dimensional setting which is implemented in a custom build R package. A data example and a simulation study show that the new methodology is superior to existing tools if the functional time series exhibit periodic characteristics.