Singular spectrum analysis
Its roots lie in the classical Karhunen (1946)–Loève (1945, 1978) spectral decomposition of time series and random fields and in the Mañé (1981)–Takens (1981) embedding theorem.
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).
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.
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.
In practice, SSA is a nonparametric spectral estimation method based on embedding a time series
Projecting the time series onto each EOF yields the corresponding temporal principal components (PCs)
: An oscillatory mode is characterized by a pair of nearly equal SSA eigenvalues and associated PCs that are in approximate phase quadrature (Ghil et al., 2002).
This is due to the fact that a single pair of data-adaptive SSA eigenmodes often will capture better the basic periodicity of an oscillatory mode than methods with fixed basis functions, such as the sines and cosines used in the Fourier transform.
chosen large enough to extract detailed temporal and spectral information from the multivariate time series (Ghil et al., 2002).
However, Groth and Ghil (2015) have demonstrated possible negative effects of this variance compression on the detection rate of weak signals when the number
This practice can further affect negatively the judicious reconstruction of the spatio-temporal patterns of such weak signals, and Groth et al. (2016) recommend retaining a maximum number of PCs, i.e.,
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 order to reduce mixture effects and to improve the physical interpretation, Groth and Ghil (2011) have proposed a subsequent VARIMAX rotation of the spatio-temporal EOFs (ST-EOFs) of the M-SSA.
To avoid a loss of spectral properties (Plaut and Vautard 1994), they have introduced a slight modification of the common VARIMAX rotation that does take the spatio-temporal structure of ST-EOFs into account.
Alternatively, a closed matrix formulation of the algorithm for the simultaneous rotation of the EOFs by iterative SVD decompositions has been proposed (Portes and Aguirre, 2016).
Two trajectory matrices can be organized as either vertical (VMSSA) or horizontal (HMSSA) as was recently introduced in Hassani and Mahmoudvand (2013), and it was shown that these constructions lead to better forecasts.
Singular spectrum analysis (SSA) and the maximum entropy method (MEM) have been combined to predict a variety of phenomena in meteorology, oceanography and climate dynamics (Ghil et al., 2002, and references therein).
Experience shows that this approach works best when the partial variance associated with the pairs of RCs that capture these modes is large (Ghil and Jiang, 1998).
In fact, the optimal order p obtained for the individual RCs is considerably lower than the one given by the standard Akaike information criterion (AIC) or similar ones.
For a multivariate data set, gap filling by M-SSA takes advantage of both spatial and temporal correlations.
In either case: (i) estimates of missing data points are produced iteratively, and are then used to compute a self-consistent lag-covariance matrix
and the number of leading SSA modes to fill the gaps with the iteratively estimated "signal," while the noise is discarded.
The areas where SSA can be applied are very broad: climatology, marine science, geophysics, engineering, image processing, medicine, econometrics among them.
System of series can be forecasted analogously to SSA recurrent and vector algorithms (Golyandina and Stepanov, 2005).
Business cycles plays a key role in macroeconomics, and are interest for a variety of players in the economy, including central banks, policy-makers, and financial intermediaries.
It is shown that SSA in such series produces a special kind of filter, whose form and spectral properties are derived, and that forecasting the single reconstructed component reduces to a moving average.
The application of SSA in this bivariate framework produces a smoothed series of the common root component.
Schoellhamer (2001) shows that the straightforward idea to formally calculate approximate inner products omitting unknown terms is workable for long stationary time series.
The method have proved to be useful in different engineering problems (e.g. Mohammad and Nishida (2011) in robotics), and has been extended to the multivariate case with corresponding analysis of detection delay and false positive rate.
