Singular Spectrum Analysis - AbsoluteAstronomy.com

Singular spectrum analysis (SSA) combines elements of classical time series

Time series

In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...

analysis, multivariate statistics

Multivariate statistics

Multivariate statistics is a form of statistics encompassing the simultaneous observation and analysis of more than one statistical variable. The application of multivariate statistics is multivariate analysis...

, multivariate geometry, dynamical systems and signal processing

Signal processing

Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...

. Its roots lie in the classical Karhunen (1946)–Loève (1945, 1978) spectral decomposition of time series

Time series

and random fields and in the Mañé (1981)–Takens (1981) embedding theorem

Takens' theorem

In mathematics, a delay embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of a dynamical system...

.

In practice, SSA is a nonparametric spectral estimation method based on embedding a time series

Time series

in a vector space of dimension

. SSA proceeds by diagonalizing the

lag-covariance matrix

to obtain spectral information on the time series, assumed to be stationary

Stationary process

In the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...

in the weak sense. The matrix

can be estimated directly from the data as a Toeplitz matrix with constant diagonals (Vautard and Ghil, 1989), i.e., its entries

depend only on the lag

An alternative way to compute

, is by using the

``trajectory matrix"

that is formed by

lag-shifted copies of

, which are

long; then

The

eigenvectors

of the lag-covariance matrix

are called temporal empirical orthogonal functions (EOFs)

Empirical orthogonal functions

In statistics and signal processing, the method of empirical orthogonal function analysis is a decomposition of a signal or data set in terms of orthogonal basis functions which are determined from the data. It is the same as performing a principal components analysis on the data, except that the...

. The eigenvalues

account for the partial variance in the
direction

and the sum of the eigenvalues, i.e., the trace of

, gives the total variance of the original time series

. The name of the method derives from the singular values

Decomposition and reconstruction

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). 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

and

used in the Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

.

The window width

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

or singular values

vs.

. The point

at which this break occurs should not be confused with a ``dimension"

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)

here

is the set of EOFs on which the reconstruction is based. The values of the normalization factor

, as well as of the lower and upper bound of summation

and

, 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

-channel time series of vectors or maps with

data points

. 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)

Principal components analysis

Principal component analysis is a mathematical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of uncorrelated variables called principal components. The number of principal components is less than or equal to...

but they differ in emphasis: EEOF analysis typically utilizes a number

of spatial channels much greater than the number

of temporal lags, thus limiting the temporal and spectral information. In M-SSA, on the other hand, one usually chooses

. Often M-SSA is applied to a few leading PCs of the spatial data, with

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

and its EOFs

; and (ii) cross-validation is used to optimize the window width

and the number of leading SSA modes to fill the gaps with the iteratively estimated "signal," while the noise is discarded.

Brief history

Broomhead and King (1986a, b) proposed to use SSA and M-SSA in the context of nonlinear dynamics for the purpose of reconstructing the attractor

Attractor

An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...

of a system from measured time series. These authors provided an extension and a more robust application of the Mañé (1981)-Takens (1981) idea of reconstructing dynamics from a single time series.

Ghil, Vautard and associates (Vautard and Ghil, 1989; Ghil and Vautard, 1991; Vautard et al., 1992) noticed the analogy between the trajectory matrix of Broomhead and King, on the one hand, and Karhunen (1946)-Loève (1945) 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

Time series

analysis — independently from attractor

Attractor

An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...

reconstruction and including cases in which the latter may fail.

At present, the papers dealing with the methodological aspects and the applications of SSA number in the hundreds. Introductions to and reviews of the literature are provided by Elsner and Tsonis (1996), Danilov and Zhigljavsky (1997), Golyandina et al. (2001), and Ghil et al. (2002).

External links

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Decomposition and reconstruction

Multivariate extension

Spatio-temporal gap filling

Brief history

See also

External links