18
Methods using Time
Structure
The model of independent component analysis (ICA) that we have considered so
far consists of mixing independent random variables, usually linearly. In many
applications, however, what is mixed is not random variables but time signals, or
time series. This is in contrast to the basic ICA model in which the samples of
x
have no particular order: We could shuffle them in any way we like, and this would
have no effect on the validity of the model, nor on the estimation methods we have
discussed. If the independent components (ICs) are time signals, the situation is quite
different.
In fact, if the ICs are time signals, they may contain much more structure than sim-
ple random variables. For example, the autocovariances (covariances over different
time lags) of the ICs are then well-defined statistics. One can then use such additional
statistics to improve the estimation of the model. This additional information can
actually make the estimation of the model possible in cases where the basic ICA
methods cannot estimate it, for example, if the ICs are gaussian but correlated over
time.
In this chapter, we consider the estimation of the ICA model when the ICs are
time signals,
s
i
(t)t =1 ::: T
,where
t
is the time index. In the previous chapters,
we denoted by
t
the sample index, but here
t

covariances between the values of the signal at different time points: cov
(x
i
(t)x
i
(t 
 ))
where

is some lag constant,
 =1 2 3:::
. If the data has time-dependencies,
the autocovariances are often different from zero.
In addition to the autocovariances of one signal, we also need covariances between
two signals: cov
(x
i
(t)x
j
(t   ))
where
i 6= j
. All these statistics for a given time
lag can be grouped together in the time-lagged covariance matrix
C
x

= E fx(t)x(t   )
T
g

go to zero, the lagged covariances are made zero as well:
E fy
i
(t)y
j
(t   )g =0
for all
i j 
(18.4)
The motivation for this is that for the ICs
s
i
(t)
, the lagged covariances are all zero due
to independence. Using these lagged covariances, we get enough extra information
to estimate the model, under certain conditions specified below. No higher-order
information is then needed.
SEPARATION BY AUTOCOVARIANCES
343
18.1.2 Using one time lag
In the simplest case, we can use just one time lag. Denote by

such a time lag, which
is very often taken equal to 1. A very simple algorithm can now be formulated to find
a matrix that cancels both the instantaneous covariances and the ones corresponding
to lag

.
Consider whitened data (see Chapter 6), denoted by
z

z

=
1
2
W
T
E fs(t)s(t   )
T
g + E fs(t   )s(t)
T
g]W = W
T

C
s

W
(18.8)
Due to the independence of the
s
i
(t)
, the time-lagged covariance matrix
C
s

=
E fs(t)s(t   )g
is diagonal; let us denote it by

The AMUSE algorithm
Thus we have a simple algorithm, called AMUSE [424],
for estimating the separating matrix
W
for whitened data:
1. Whiten the (zero-mean) data
x
to obtain
z(t)
.
2. Compute the eigenvalue decomposition of

C
z

=
1
2
C

+ C
T

]
,where
C

=
E fz(t)z(t   )g
is the time-lagged covariance matrix, for some lag

s
i
(t)
have
identical power spectra, that is, identical autocovariances, then no value of

makes
estimation possible.
18.1.3 Extension to several time lags
An extension of the AMUSE method that improves its performance is to consider
several time lags

instead of a single one. Then, it is enough that the covariances for
one of these time lags are different. Thus the choice of

is a somewhat less serious
problem.
In principle, using several time lags, we want to simultaneously diagonalize all the
corresponding lagged covariance matrices. It must be noted that the diagonalization
is not possible exactly, since the eigenvectors of the different covariance matrices
are unlikely to be identical, except in the theoretical case where the data is exactly
generated by the ICA model. So here we formulate functions that express the degree
of diagonalization obtained and find its maximum.
One simple way of measuring the diagonality of a matrix
M
is to use the operator
off
(M)=
X
i6=j

(W

C
z

W
T
)
(18.11)
Minimizing
J
1
under the constraint that
W
is orthogonal gives us the estimation
method. This minimization could be performed by (projected) gradient descent.
Another alternative is to adapt the existing methods for eigenvalue decomposition to
this simultaneous approximate diagonalization of several matrices. The algorithm
called SOBI (second-order blind identification) [43] is based on these principles, and
so is TDSEP [481].
z
SEPARATION BY AUTOCOVARIANCES
345
The criterion
J
1
can be simplified. For an orthogonal transformation,
W
,thesum
of the squares of the elements of

are the rows of
W
. Thus, minimizing
J
2
is equivalent to minimizing
J
1
.
An alternative method for measuring the diagonality can be obtained using the
approach in [240]. For any positive-definite matrix
M
,wehave
X
i
log m
ii
 log j det Mj
(18.13)
and the equality holds only for diagonal
M
. Thus, we could measure the nondiago-
nality of
M
by
F (M)=
X
i
log m
ii

W
T
)
(18.15)
Just as in maximum likelihood (ML) estimation,
W
decouples from the term involv-
ing the logarithm of the determinant. We obtain
J
3
(W)=
X
 2S
X
i
1
2
log(w
T
i

C
z

w
i
)  log j det Wj
1
2
log j det

This is in fact rather similar to the function
J
2
in (18.12). The only difference is
that the function
u
2
has been replaced by
1=2 log(u)
. What these functions have
1
This is because it equals trace
(WMW
T
(WMW
T
)
T
) =
trace
(WMM
T
W
T
) =
trace
(W
T
WMM
T