On the use of Singular Spectrum Analysis

Abstract

Singular Spectrum Analysis (SSA) or Singular Value Decomposition (SVD) are often used to de-noise univariate time series or to study their spectral profile. Both techniques rely on the eigendecomposition of the correlation matrix estimated after embedding the signal into its delayed coordinates. In this work we show that the eigenvectors can be used to calculate the coefficients of a set of filters which form a filter bank. The properties of these filters are derived. In particular we show that their outputs can be grouped according to their frequency response. Furthermore, the frequency at the maximum of each frequency response and the corresponding eigenvalue can provide a power spectrum estimation of the time series. Two different applications illustrate how both characteristics can be applied to analyze wideband signals in order to achieve narrow-band signals or to infer their frequency occupation.

Circulant Singular Spectrum Analysis: A new automated procedure for signal extraction

“…introduce optimal thresholds for grouping eigenvectors linked to nearby frequencies in order to assign them to the same harmonic [21,36]; perform a spectral-based Fisher g test to asses certain principal components to the business cycle frequency [17]; considering eigenvectors as filters [23] group the outputs according to their frequency reponse [24]; and even apply cluster techniques for grouping the elementary components based on k-means [37] or hierarchical clustering [38]. Nevertheless, whatever procedure is used, the grouping of frequencies is made after the elementary components are extracted.…”

“…Those are the only eigenvectors that have information related to this frequency. As eigenvectors can be considered filters [23,24], these pair of eigenvectors extract elementary series linked to the same frequency without mixing harmonics of other frequencies. As a result, the two elementary series, when reconstructed in step 4, have spectral correlation close to 1 between them and close to zero with the remaining ones.…”

“…The common practice when applying SSA is to extract the Principal Components of the trajectory matrix and to identify afterwards the frequencies associated to the extracted components, by analysing their estimated periodogram [17,21,22] or response function [23,24] just to cite a few. Though there are fast computing algorithms for the eigenvalues and eigenvectors of Toeplitz matrices [25,26], the use of circulant matrices have a great advantage as their eigenvalues and eigenvectors have a closed form.…”

Noises investigations and image denoising in femtosecond stimulated Raman scattering microscopy

“…Therefore, the established practice we said before is not correct. Finally, the results obtained from statistical properties analysis show that a 1D technique based on singular spectrum analysis (SSA) [38,39] has been tailored for SRS imaging denoising. The results from SSA have been compared with two wellestablished and commonly applied algorithms: DWT and SVD, and a significant improvement of SNR, obtained using our adapted method, is demonstrated.…”