Singular Spectrum Analysis - Brief History

Brief History

The origins of SSA and, more generally, of subspace-based methods for signal processing, go back to the eighteenth century (Prony's method). A key development was the formulation of the spectral decomposition of the covariance operator of stochastic processes by Kari Karhunen and Michel Loève in the late 1940s (Loève, 1945; Karhunen, 1947).

Broomhead and King (1986a, b) and Fraedrich (1986) proposed to use SSA and multichannel SSA (M-SSA) in the context of nonlinear dynamics for the purpose of reconstructing the attractor of a system from measured time series. These authors provided an extension and a more robust application of the idea of reconstructing dynamics from a single time series based on the embedding theorem. Several other authors had already applied simple versions of M-SSA to meteorological and ecological data sets (Colebrook, 1978; Barnett and Hasselmann, 1979; Weare and Nasstrom, 1982).

Ghil, Vautard and their colleagues (Vautard and Ghil, 1989; Ghil and Vautard, 1991; Vautard et al., 1992; Ghil et al., 2002) noticed the analogy between the trajectory matrix of Broomhead and King, on the one hand, and the Karhunen–Loeve decomposition (Principal component analysis in the time domain), on the other. Thus, SSA can be used as a time-and-frequency domain method for time series analysis — independently from attractor reconstruction and including cases in which the latter may fail. The survey paper of Ghil et al. (2002) is the basis of the #Singular spectrum analysis (SSA) section of this article. A crucial result of the work of these authors is that SSA can robustly recover the "skeleton" of an attractor, including in the presence of noise. This skeleton is formed by the least unstable periodic orbits, which can be identified in the eigenvalue spectra of SSA and M-SSA. The identification and detailed description of these orbits can provide highly useful pointers to the underlying nonlinear dynamics.

The so-called ‘Caterpillar’ methodology is a version of SSA that was developed in the former Soviet Union, independently of the mainstream SSA work in the West. This methodology became known in the rest of the world more recently (Danilov and Zhigljavsky, Eds., 1997; Golyandina et al., 2001; Zhigljavsky, Ed., 2010; Golyandina and Zhigljavsky, 2013). ‘Caterpillar-SSA’ emphasizes the concept of separability, a concept that leads, for example, to specific recommendations concerning the choice of SSA parameters. This method is thoroughly described in #SSA as a model-free tool section of this article.