**Multi-channel SSA (or M-SSA)** is a natural extension of SSA to a multivariate time
series of vectors or maps, such as time-varying temperature or
pressure distributions over the globe.

Let {*X _{l,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, based on the
single-channel experience, one usually chooses L &le M.
Often M-SSA is applied to a few leading PCA components of the spatial
data, with *M* chosen large enough to extract detailed temporal and
spectral information from the multivariate time series.

Both SSA (*L*=1) and conventional PCA (*M*=1) analysis are special cases
of EEOF or M-SSA analysis. Both algorithms can be understood in practice in
terms of a ``window."

These windows are illustrated schematically in the following figure:

axes are spatial coordinate (or spatial PC label) *x*,
time *t*, and lag *s*; discrete values of these variables are labeled by* l*, *n*, and *j*, respectively.

Standard PCA slides a flat and narrow window, of length 1 and width *L*, over the data set's *N* fields, each of which contains *L* data
points. PCA thus identifies the spatial patterns, i.e.,
the EOFs, which account for a high proportion of the
variance in the *N* views of the data set thus obtained. Equivalently, PCA can
be described as sliding a long and narrow,
*N* x 1 window across the *L *input channels and identifying high-variance temporal patterns, i.e. the PCs, in
the corresponding *L* views.
In the above figure,
the former view of things corresponds to sliding
the
1 x *L* window parallel to the *x*-axis along the *t*-axis. In the latter view, one slides an *N* x 1 window that starts out by lying along the *t*-axis parallel
to the *x*-axis. This latter
case is also somewhat analogous to the trajectory-matrix version
of single-channel SSA, except
that in SSA we *reduce* the length of this long and narrow window to *M*, and
consider only one channel.

These two different window interpretations carry over into M-SSA. In
the first case one proceeds from spatial PCA to EEOFs. To do so,
we extend our 1 x *L* window by *M* lags to form an *M* x *L* window that lies in the horizontal (*x*,*s*)-plane.
By moving this window along the *t*-axis, we
search for spatio-temporal patterns, i.e. the
EEOFs, that maximize the variance in the * N'=N-M+1 *overlapping views of the time series thus obtained. The EEOFs
are the eigenvectors of the

The second conceptual route leads us from single-channel SSA
to M-SSA. To follow this route, we
reduce the length of the *N* x 1 window to
*N*' x 1.
This window still lies initially along the *t*-axis and we
search for temporal patterns, i.e. the *N*'-long PCs, that maximize the
variance in the *M* x *L* views of the time series. This is equivalent to SSA
of the univariate time series formed by stringing together all the channels of
the original multi-channel time series end-to-end, with the *complementary window* *N*' playing the role of *M* in SSA. This approach is a basis for computing reduced covraiance matrix in M-SSA.

The generalization of **Toeplitz method** of SSA to a multivariate time series
requires the constructiuon of a ``grand'' block-matrix
T_{X} that has form:

Each block T

where the normalizing factor depends on the range of summation:

Note that, unlike in the single-channel case, here T* _{l,l'}*
is
Toeplitz but not symmetric. Its
main diagonal contains the estimate of the lag-zero covariance of channels

An alternative approach to computing the lagged cross-covariances
is to form the multi-channel trajectory matrix
by first
augmenting each channel
of
*X *with *M* lagged copies of itself:

and then forming the full augmented

Thereafter, one computes the grand covariance matrix:

The blocks of C_{X}
are given by

with entries

where

Each block C* _{l,l'}*
contains,
like T

The trajectory-matrix method gradually slides, regardless
of *m*, the same *N*'-long windows
over the two channels.
To compute the first entry in the diagonal that
contains the lag-*m* covariances,
the two matching windows are situated so
that one starts at the first time point of the
trailing channel and at the (*m*+1)st point of
the leading channel.
Both windows are then slid forward by
one point in time to produce the second entry. This
results in slightly different values,
from entry to entry, until the last point of
the leading channel is covered and the
(*M*-*m*)th entry of the diagonal, which is the last one,
is calculated. This latter method thus
retains detailed information on the variation of
the lag-covariance estimates from one pair of segments of
the channels *l* and *l*' to another, while the Toeplitz
method produces a single, and smoother, global estimate for each lag *m*.

Diagonalizing the *LM x LM*
matrix* C** _{X}* or

where

Selected RCs are obtained by convolving
the corresponding PCs with the EOFs. Thus, the *k*th RC at time *t* for channel *l* is given by:

The normalization factor *M*_{t} equals
*M*, except near the ends of the time series, and the sum of all the
RCs recovers the original time series, as it
does in the single-channel case.

The
(reduced) PCs are the eigenvectors of the
* N' x N' reduced
covariance matrix* with the elements:

This matrix, unlike the *LM* x *LM*
matrix C_{X}
is given by:

Whenever

As in the single-channel case, a test of statistical significance is
needed to avoid spatially smooth-looking but spurious oscillations
that might arise from M-SSA of finitely sampled noise processes. The
complementary *N*'-window approach allows the univariate SSA Monte Carlo test to be
extended to M-SSA in a straightforward manner, provided *LM* > *N*', so
that the reduced covariance matrix is completely determined and has full
rank. *N*' can always be chosen sufficiently
small, so that the complementary window *N*' used in
determines the spectral resolution.

The usefulness of the test depends in an essential way on the channels being uncorrelated at zero lag or very nearly so. When data is first processed by PCA, that is consists of leading PCs of spatial EOF analysis, the decorrelation condition holds exactly. When using time series from grid points that are sufficiently far from each other for decorrelation to be near-perfect, the test can still be useful.

In this test, the
data series together with a
large ensemble of red-noise surrogates are projected onto the
eigenmodes of the reduced covariance matrix of either the data or the
noise. The statistical significance of the projections is estimated
as in the single-channel test.
The noise surrogates are constructed to consist of univariate AR(1)
segments, one per channel, that match the data in
autocovariance at lag 0 and lag 1, channel-by-channel. The reason an essentially
univariate test can be applied is because the eigenmodes do not
depend on cross-channel lag-covariance, provided *N*' is interpreted
as the spectral window.

The EOFs rotation has been introduced orginally as modification of standard PCA.

Spatial orthogonality of EOFs and temporal orthogonality of PCs coming from PCA impose certain limits on physical interpretability. This is because physical processes are not independent, and therefore physical modes are expected in general to be non-orthogonal. To help overcome these difficulties and gain easy interpretation, a number of methods have been proposed.

Method based simply on rotating the EOF patterns, seems to be the most widely used because of its relative simplicity. The method attempts to rotate a fixed number of EOF patterns using typically an orthogonal rotation matrix subject to maximizing given simplicity criterion. The EOFs can be either unscaled or scaled by the square root of the associ- ated eigenvalues. The most well-known and used rotation algorithm is the **VARIMAX** criterion. Therefore **VARIMAX** attempts to simplify the structure of the EOFs patterns by pushing the EOF coefficients towards zero, or ±1.

Recently, Groth and Ghil (2011) have demonstrated that a classical M-SSA analysis suffers from a degeneracy problem, with eigenvectors not well separating between distinct oscillations when the corresponding eigenvalues are similar in size. This problem is a shortcoming of principal component analysis in general, not just of M-SSA in particular. In order to reduce mixture effects and to improve the physical interpretation, Groth and Ghil (2011) have proposed a subsequent varimax rotation of the ST-EOFs. To avoid a loss of spectral properties (Plaut and Vautard 1994), they have introduced a slight modification of the common varimax rotation that takes the spatio-temporal structure of ST-EOFs into account.

kSpectra implements** Varimax Rotation **both for **PCA** and **MSSA**. Benefits of **MSSA** with **Varimax Rotatio**n is demonstrated on Multivariate Small Signal example.

Allen, M.R., and A.W. Robertson, 1996: *Distinguishing modulated oscillations from coloured noise in multivariate datasets.
Climate Dynamics*, ** 12 **, 775-784.

Ghil, M., and
Vautard, R., 1991: *Interdecadal oscillations and the warming trend in
global temperature time series*, Nature, **350**,
324-327.

Groth A, Ghil M, 2011: *Multivariate singular spectrum analysis and the road to phase synchronization, *Phys Rev E 84(3 Pt 2), 036206.

Plaut, G., and R. Vautard, *Spells of low-frequency oscillations and weather regimes in the Northern Hemisphere*,J. Atmos. Sci., 51, 210 –236, 1994.

Preisendorfer, R. W., *Principal Component Analysis in Meteorology and Oceanography*, 425 pp., Elsevier Sci., New York, 1988.