This problem is a shortcoming of principal component analysis in general, not just of M-SSA in particular. Groth and Ghil (2011) have demonstrated that a classical M-SSA analysis suffers from a degeneracy problem, namely the EOFs do not separate well between distinct oscillations when the corresponding eigenvalues are similar in size. In practice, SSA is a nonparametric spectral estimation method based on embedding a time series. This method is thoroughly described in § SSA as a model-free tool of this article. ‘Caterpillar-SSA’ emphasizes the concept of separability, a concept that leads, for example, to specific recommendations concerning the choice of SSA parameters. 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 Golyandina et al., 2018). 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. The identification and detailed description of these orbits can provide highly useful pointers to the underlying nonlinear dynamics. This skeleton is formed by the least unstable periodic orbits, which can be identified in the eigenvalue spectra of SSA and M-SSA. 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. (2002) is the basis of the § Methodology section of this article.
#Kspectra for windows series
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. 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. 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). 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. A key development was the formulation of the fspectral decomposition of the covariance operator of stochastic processes by Kari Karhunen and Michel Loève in the late 1940s (Loève, 1945 Karhunen, 1947).īroomhead 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. The origins of SSA and, more generally, of subspace-based methods for signal processing, go back to the eighteenth century ( Prony's method). 4 Relation between SSA and other methods.
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