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Singular Spectrum Analysis (SSA)

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Digital Signal Processing and Spectral Analysis for Scientists

Part of the book series: Signals and Communication Technology ((SCT))

Abstract

This chapter is devoted to an approach of extracting periodic or quasi-periodic components from a random signal. Singular Spectrum Analysis (SSA) is not, in a strict sense, a simple spectral method, since it is aimed at representing the signal as a linear combination of elementary variability modes that are not necessarily harmonic components, but can exhibit amplitude and frequency modulations in time, and are data-adaptive, i.e., modeled on the data. It does not provide a stationary spectral estimate but can separate auto-coherent from random features. SSA is a non-parametric method, since it does not assume any specific model for the generation of the signal. It can also be viewed as a powerful de-noising technique; finally, it can be exploited as a tool for filling gaps in data records that is soundly based from a theoretical point of view. Examples the real-world applications of SSA are provided.

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Notes

  1. 1.

    Recall that in linear algebra, a Toeplitz matrix is a matrix in which each descending diagonal from left to right is constant, i.e., all elements in a diagonal are equal.

  2. 2.

    We must underline that SVD is different from the so-called eigen-decomposition used to perform diagonalization of a square matrix. Focussing on real matrices, matrix diagonalization is the process of taking a square matrix and converting it into a special type of matrix—a so-called diagonal matrix—that shares the same fundamental properties of the underlying matrix. Matrix diagonalization is equivalent to transforming the underlying system of equations into a special set of coordinate axes in which the matrix takes this diagonal form. Diagonalizing a matrix is also equivalent to finding the matrix’s eigenvalues, which turn out to be precisely the entries of the diagonalized matrix. Similarly, the eigenvectors make up the new set of axes corresponding to the diagonal matrix. The relationship between a diagonalized matrix, eigenvalues, and eigenvectors follows from an equation according to which a square matrix \(\varvec{R}\) can be decomposed into the form \(\varvec{R}= \varvec{A}\varvec{\varLambda }\varvec{A}^{-1}\), where \(\varvec{A}\) is a matrix composed of the eigenvectors of \(\varvec{R}\) arranged by columns, \(\varvec{\varLambda }\) is the diagonal matrix constructed from the corresponding eigenvalues, and \(\varvec{A}^{-1}\) is the matrix inverse of \(\varvec{A}\). It is obvious that if a square matrix \(\varvec{R}\) has a matrix of eigenvectors \(\varvec{A}\) that is not invertible, then \(\varvec{R}\) does not have an eigen-decomposition. However, in the case of the lag-covariance matrix, the eigenvectors are orthogonal to one another. The matrix \(\varvec{A}\) having the eigenvectors of \(\varvec{R}\) as its columns is thus an orthogonal matrix, so that its inverse is equal to its transpose: \(\varvec{A}^{-1}=\varvec{A}^T\). Then \(\varvec{R}\) can be written using a so-called SVD of the form \(\varvec{R}= \varvec{A}\varvec{\varLambda }\varvec{A}^{T}\).

    More generally, in linear algebra the SVD is a factorization of a real or complex matrix, which is not necessarily square—it can also be rectangular. Even if linear algebra is beyond the scope of the book, we may mention that some key differences between SVD and eigen-decomposition are the following:

    • the vectors forming the columns of the eigen-decomposition matrix \(\varvec{A}\) are not necessarily orthogonal. On the other hand, the vectors that in the SVD factorization play a similar role are orthonormal; therefore the corresponding matrices—two different matrices—involved in SVD are orthogonal;

    • these matrices involved in SVD are not necessarily the inverse of one another. They are usually not related to each other at all. In the eigen-decomposition, the matrices involved are inverses of each other, i.e., they are \(\varvec{A}\) and \(\varvec{A}^{-1}\);

    • in the SVD, the entries in the diagonal matrix \(\varvec{\varLambda }\) are all real and non-negative. In the eigen-decomposition, the entries of the corresponding matrix can be any complex number—negative, positive, imaginary, whatever;

    • the SVD always exists for any sort of rectangular or square matrix, whereas the eigen-decomposition only exists for square matrices, and even among square matrices sometimes it doesn’t exist.

  3. 3.

    The reference standard for the Oxygen isotopic composition in carbonates is the PDB standard, which is based on the CO\({_2}\) produced from Cretaceous belemnites of Pee Dee formation in South Carolina (Faure, G: Principles of isotope geology, Second Edition, John Wiley and Sons (1986)).

  4. 4.

    Siltation, in general, is the pollution of water by fine particulate terrestrial clastic material, with a particle size dominated by silt or clay. Here it refers to the increased accumulation of fine sediments on the river bottom.

  5. 5.

    Swift is a NASA mission with international participation. Within seconds of detecting a burst, Swift relays its location to ground stations, allowing both ground-based and space-based telescopes around the world the opportunity to observe the burst’s afterglow. Swift is part of NASA’s medium explorer (MIDEX) program and was launched into a low-Earth orbit on a Delta 7320 rocket on November 20, 2004.

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  1. Department of Physics, University of Turin, Turin, Italy

    Silvia Maria Alessio

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  1. Silvia Maria Alessio
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Alessio, S.M. (2016). Singular Spectrum Analysis (SSA). In: Digital Signal Processing and Spectral Analysis for Scientists. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-25468-5_12

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