264
Chapter 14: Patterns in Time: SSA and MSSA
constant equal to 1, and there is no such arbitrariness in the choice of the
dot product as mentioned in Chapter 13.
The k-th EOF p" is the k-th eigenvector of the Toeplitz matrix (sorted
into decreasing order of the eigenvalues). It is is a lag sequence of length m* .
Thus, the time patterns associated to these EOFs are called time-EOFs or
T-EOFs. The SSA expansion in terms of the T-EOFs writes
m·
(Xt+~, Xt+u., ... Xt+m·~) = L>~,,(t)P"
(14.5)
"=1
where again the projection coefficients a,,(t), that we shall call the timeprincipal components (T-PCs), are obtained by calculating the dot product
of the state vector with the T-EOFs,
m·
al;(t) = L Xt+i~P~
(14.6)
i=1
Under this form, the T-PCs can be interpreted as moving averages of the
original signal, the averages being weighted by the coordinates of the TEOFs. T-PCs are therefore filtered versions of the time series. This has
important spectral consequences (see Section 14.3.4).
14.3.2 Space-Time EOFs
MSSA is designed to analyse multichannel time series. The original signal to
be analysed consists of a random vector Xt of dimension L. In general L is
sm aller than m*, the total number of variables. Embedding this series into
the delay-coordinate space is obtained in the same way as for SSA. The (big)
new state vector Y t writes
Yt = (Xt+~,1, Xt+~,2" .Xt+~t,L;
Xt+2~,1" .Xt+2~IL;Xt+m·~,l" .Xt+m·~,L)
(14.7)
This state vector contains both space and time information. It consists, as in
EEOF analysis, of sequences of consecutive maps. The big covariance matrix
contains the covariance between each pair of coordinates of X at lags sm aller
than m*. The EOFs are now sequences of patterns and are called ST-EOFs.
The MSSA expansion takes a form similar to (14.4) and (14.5).
From a dynamical point of view, the embedding theorem also applies with
multichannel time series provided that m* L ~ 2m + 1. In the same way as
SSA does, the "macroscopic approximation" consists in the approximation
of the dynamics by a linear system having oscillatory components.
14.3.3 Oscillatory Pairs
Let us first take the following example: consider a progressive wave in the
finite interval -y ~ y ~ y
Chapter 14: Patterns in Time: SSA and MSSA
constant equal to 1, and there is no such arbitrariness in the choice of the
dot product as mentioned in Chapter 13.
The k-th EOF p" is the k-th eigenvector of the Toeplitz matrix (sorted
into decreasing order of the eigenvalues). It is is a lag sequence of length m* .
Thus, the time patterns associated to these EOFs are called time-EOFs or
T-EOFs. The SSA expansion in terms of the T-EOFs writes
m·
(Xt+~, Xt+u., ... Xt+m·~) = L>~,,(t)P"
(14.5)
"=1
where again the projection coefficients a,,(t), that we shall call the timeprincipal components (T-PCs), are obtained by calculating the dot product
of the state vector with the T-EOFs,
m·
al;(t) = L Xt+i~P~
(14.6)
i=1
Under this form, the T-PCs can be interpreted as moving averages of the
original signal, the averages being weighted by the coordinates of the TEOFs. T-PCs are therefore filtered versions of the time series. This has
important spectral consequences (see Section 14.3.4).
14.3.2 Space-Time EOFs
MSSA is designed to analyse multichannel time series. The original signal to
be analysed consists of a random vector Xt of dimension L. In general L is
sm aller than m*, the total number of variables. Embedding this series into
the delay-coordinate space is obtained in the same way as for SSA. The (big)
new state vector Y t writes
Yt = (Xt+~,1, Xt+~,2" .Xt+~t,L;
Xt+2~,1" .Xt+2~IL;Xt+m·~,l" .Xt+m·~,L)
(14.7)
This state vector contains both space and time information. It consists, as in
EEOF analysis, of sequences of consecutive maps. The big covariance matrix
contains the covariance between each pair of coordinates of X at lags sm aller
than m*. The EOFs are now sequences of patterns and are called ST-EOFs.
The MSSA expansion takes a form similar to (14.4) and (14.5).
From a dynamical point of view, the embedding theorem also applies with
multichannel time series provided that m* L ~ 2m + 1. In the same way as
SSA does, the "macroscopic approximation" consists in the approximation
of the dynamics by a linear system having oscillatory components.
14.3.3 Oscillatory Pairs
Let us first take the following example: consider a progressive wave in the
finite interval -y ~ y ~ y