Relationship between Singular Spectrum Analysis and Fourier analysis: Theory and application to the monitoring of volcanic activity

Singular Spectrum Analysis is a quite recent technique for the analysis of experimental time series, based on the singular value decomposition of certain Hankel matrices. However, the mathematical and physical interpretation of the singular values in this kind of application is not fully clarified. In this paper, using asymptotic properties of the eigenvalues of Toeplitz matrices, we show that SSA is related to Fourier analysis. Indeed, the singular values provide information that can be interpreted, and estimated efficiently, by means of the power spectrum of the time series. We apply our results to the continuous seismic signal recorded at Stromboli volcano in order to highlight precursors of a paroxysmal volcanic eruption.