Principal Component Analysis using Frequency Components of Multivariate Time Series

TL;DR

This paper introduces a spectral domain method for dimension reduction in multivariate stationary time series, enabling the segmentation into independent subseries based on spectral coherence, with proven asymptotic properties and practical applications.

Contribution

It develops a novel spectral domain approach for segmenting multivariate time series into independent subseries using eigendecomposition of spectral matrices.

Findings

Method accurately segments series in simulations.

Consistent cross-spectrum testing validates segmentation.

Application to wind data demonstrates practical utility.

Abstract

Dimension reduction techniques for multivariate time series decompose the observed series into a few useful independent/orthogonal univariate components. We develop a spectral domain method for multivariate second-order stationary time series that linearly transforms the observed series into several groups of lower-dimensional multivariate subseries. These multivariate subseries have non-zero spectral coherence among components within a group but have zero spectral coherence among components across groups. The observed series is expressed as a sum of frequency components whose variances are proportional to the spectral matrices at the respective frequencies. The demixing matrix is then estimated using an eigendecomposition on the sum of the variance matrices of these frequency components and its asymptotic properties are derived. Finally, a consistent test on the cross-spectrum of pairs of components is used to find the desired segmentation into the lower-dimensional subseries. The numerical performance of the proposed method is illustrated through simulation examples and an application to modeling and forecasting wind data is presented.