Singular Spectrum Analysis

The Singular Spectrum Analysis (SSA) techniqueis a powerful technique of time series analysisincorporating the elements of classical time series analysis,multivariate statistics, multivariate geometry, dynamical systemsand signal processing.

The birth of SSA is usually associated withthe publication of papers by Broomhead (e.g. Broomhead and King,1986) while the ideas of SSA were independently developed in Russia(St. Petersburg, Moscow) and in several groups in the UK and USA. Atpresent, the papers dealing with the methodological aspects and theapplications of SSA number several hundred (see, for example,Vautard et al., 1992; Ghil and Taricco, 1997; Allen and Smith, 1986;Danilov, 1997; Yiou et al., 2000 and references therein). A thoroughdescription of the theoretical and practical foundations of the SSAtechnique (with several examples) can be found in Danilov andZhigljavsky (1997) and Golyandina et al. (2001). An elementaryintroduction to the subject can be found in Elsner and Tsonis(1996).

The SSA technique is a nonparametric spectral estimation method based on embedding a time series $X(t)$: $\{\; extit\; t=1,N\}$ in a vector space of dimension $M$. SSA proceeds by diagonalizing the $M\; imes\; M$ lag-covariance matrix $\{\; extbf\; C\}\_X$ of $X(t)$ to obtain spectral information on the time series, assumed to be stationary in the weak sense. The matrix $\{\; extbf\; C\}\_X$ can be estimated directly from the data as a Toeplitz matrix with constant diagonals (Vautard and Ghil, 1989), i.e., its entries $c\_\{ij\}$ depend only on the lag $|i-j|$:
$c\_\{ij\}\; =\; frac\{1\}\{N-|i-j\; sum\_\{t=1\}^\{N-|i-j\; X(t)\; X(t+|i-j|).$

An alternative way to compute $\{\; extbf\; C\}\_X$, is by using the $N\text{'}\; imes\; M$ ``trajectory matrix" $\{\; extbf\; D\}$ that is formed by $M$ lag-shifted copies of $\{it\; X(t)\}$, which are $N\text{'}\; =N-M+1$ long; then

$\{\; extbf\; C\}\_X\; =\; frac\{1\}\{N\text{'}\}\; \{\; extbf\; D\}^\{\; m\; t\}\; \{\; extbf\; D\}.$

The $M$ eigenvectors $\{\; extbf\; E\}\_k$ of the lag-covariance matrix $\{\; extbf\; C\}\_\; X$ are called temporal empirical orthogonal functions (EOFs). The eigenvalues $lambda\_k$ of $\{\; extbf\; C\}\_\{X\}$ account for the partial variance in thedirection $\{\; extbf\; E\}\_k$ and the sum of the eigenvalues, i.e., the trace of$\{\; extbf\; C\}\_\{X\}$, gives the total variance of the original time series$X(t)$. The name of the method derives from the singular values $lambda^\{1/2\}\_k$ of $\{\; extbf\; C\}\_\{X\}$.

Decomposition and reconstruction

Projecting the time series onto each EOF yields the correspondingtemporal principal components (PCs) $\{\; extbf\; A\}\_k$:
$A\_k(t)\; =\; sum\_\{j=1\}^\{M\}\; X(t+j-1)\; E\_k(j).$

An oscillatory mode is characterized by a pair ofnearly equal SSA eigenvalues and associated PCs that are in approximate phase quadrature (Ghil et al., 2002). Such a pair can represent efficiently a nonlinear, anharmonic oscillation. 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 $\{\; extit\; sines\}$ and $\{\; extit\; cosines\}$ used in the Fourier Transform.

The window width $M$ determines the longest periodicity captured by SSA. Signal-to-noise separation can be obtained by merely inspecting the slope break in a "scree diagram" of eigenvalues $lambda\_k$ or singular values $lambda^\{1/2\}\_k$ vs. $k$. The point $k^*\; =\; S$ at which this break occurs should not be confused with a ``dimension" $D$ of the underlying deterministic dynamics (Vautard and Ghil, 1989).

A Monte-Carlo test (Allen and Robertson, 1996) can be applied to ascertain the statistical significance of the oscillatory pairs detected by SSA. The entire time series or parts of it that correspond to trends, oscillatory modes or noise can be reconstructed by using linear combinations of the PCs and EOFs, which provide the reconstructed components (RCs) $\{\; extbf\; R\}\_k$:
$R\_\{\; k\}(t)\; =\; frac\{1\}\{M\_t\}\; sum\_\{kin\; \{\; extit\; K\; sum\_\{j=\{L\_t^\{U\_t\}A\_k(t-j+1)E\_k(j);$
here $extit\; K$ is the set of EOFs on which the reconstruction is based. The values of the normalization factor $M\_t$, as well as of the lower and upper bound of summation $L\_t$ and $U\_t$, differ between the central part of the time series and the vicinity of its endpoints (Ghil et al., 2002).

Multivariate extension

Multi-channel SSA (or M-SSA) is a natural extension of SSA to an $L$-channel time series of vectors or maps with $N$ data points $\{X\_\{l\}(t):\; l=1,dots,\; L;\; t=1,dots,\; N\}$. In the meteorological literature, extended EOF (EEOF) analysis is often assumed to be synonymous with M-SSA. The two methods are both extensions of classical principal component analysis (PCA) but they differ in emphasis: EEOF analysis typically utilizes a number $L$ of spatial channels much greater than the number $M$ of temporal lags, thus limiting the temporal and spectral information. In M-SSA, on the other hand, one usually chooses $L\; leq\; M$. Often M-SSA is applied to a few leading PCs of the spatial data, with $M$ chosen large enough to extract detailed temporal and spectral information from the multivariate time series (Ghil et al., 2002).

Spatio-temporal gap filling

The gap-filling version of SSA can be used to analyze data sets that are unevenly sampled or contain missing data (Kondrashov and Ghil, 2006). For a univariate time series, the SSA gap filling procedure utilizes temporal correlations to fill in the missing points. 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 $\{\; extbf\; C\}\_X$ and its EOFs $\{\; extbf\; E\}\_k$; and (ii) cross-validation is used to optimize the window width $M$ and the number of leading SSA modes to fill the gaps with the iteratively estimated ``signal," while the noise is discarded.

References

* Allen, M.R., and A.W. Robertson: Distinguishing modulated oscillations from coloured noise in multivariate datasets, Clim. Dyn., 12, 775--784, 1996.
*Ghil, M., R. M. Allen, M. D. Dettinger, K. Ide, D. Kondrashov, et al.: Advanced spectral methods for climatic time series, Rev. Geophys. 40(1), 3.1--3.41, doi: 10.1029/2000RG000092, 2002.
*Kondrashov, D., and M. Ghil: [http://www.nonlin-processes-geophys.net/13/151/2006/npg-13-151-2006.html Spatio-temporal filling of missing points in geophysical data sets] , Nonlin. Processes Geophys., 13, 151--159, 2006.
*Vautard, R., and M. Ghil: Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series, Physica D, 35, 395--424, 1989.
* Danilov, D. and Zhigljavsky, A. (Eds.). (1997): [http://www.gistatgroup.com/cat/books.html Principal Components of Time Series: the Caterpillar method] , University of St. Petersburg Press. (In Russian.)
* Golyandina, N., Nekrutkin, V. and Zhigljavsky, A. (2001): [http://www.amazon.com/Analysis-Time-Structure-Related-Techniques/dp/1584881941/ref=tag_tdp_sv_edpp_i Analysis of Time Series Structure: SSA and related techniques] . Chapman and Hall/CRC.

* Elsner, J.B. and Tsonis, A.A. (1996). Singular Spectral Analysis. A New Tool in Time Series nalysis. Plenum Press.

ee also

*Multitaper method
*Short-time Fourier transform
*Spectral density estimation

External links

* [http://www.atmos.ucla.edu/tcd/ssa Singular Spectrum Analysis--Multi-taper Method (SSA-MTM) Toolkit] freeware from UCLA.
*kSpectra Toolkit for Mac OS X from [http://www.spectraworks.com SpectraWorks.]
* [http://www.math.uni-bremen.de/~theodore/ssawiki/ Yet another SSAwiki page.]