Spectral structure of singular spectrum decomposition for time series

TL;DR

This paper explores the spectral structure of Singular Spectrum Analysis (SSA) for time series, providing insights into filter-based decomposition and guiding the selection of window length for better spectral analysis.

Contribution

It introduces a spectral interpretation of SSA using eigenvector-based filters, aiding in understanding and choosing the window length parameter.

Findings

Spectral decomposition of SSA can be understood through eigenvector-derived filters.

The spectral structure depends on the window length, influencing analysis results.

Guidelines for selecting window length based on spectral insights are proposed.

Abstract

Singular spectrum analysis (SSA) is a nonparametric and adaptive spectral decomposition of a time series. The singular value decomposition of the trajectory matrix and the anti-diagonal averaging leads to a time-series decomposition. In this algorithm, a single free parameter, window length $K$, is involved which is the FIR filter length for the time series. There are no generally accepted criterion for the proper choice of the window length $K$. Moreover, the proper window length depends on the specific problem which we are interested in. Thus, it is important to monitor the spectral structure of the SSA decomposition and its window length dependence in detail for the practical application. In this paper, based on the filtering interpretation of SSA, it is shown that the decomposition of the power spectrum for the original time series is possible with the filters constructed from the eigenvectors of the lagged-covariance matrix. With this, we can obtain insights into the spectral structure of the SSA decomposition and it helps us for the proper choice of the window length in the practical application of SSA.