Multiple channels (multivariate) signals - Practical Biomedical Signal Analysis Using MATLAB - page 91

similar approach is used to test for non-linearity of the time series. The problem of
construction of bivariate surrogate data was considered in [Andrzejak et al., 2003a].
In case of multichannel data the test indicates presence or lack of dependencies, but it
doesn't indicate if the character of dependence is linear or non-linear. The generation
of surrogate data for multichannel series destroying only the non-linear part of the
interdependence is a problem which is not solved at the moment in a satisfactory
way.
In case of estimators operating in time-frequency space such as SDTF, the problem
of reliability testing is more difficult, since all realizations of the process are used
to compute the estimator. In this case the bootstrap method [Zoubir and Boashash,
1988] is applied. It relies on construction of the new artificial time series consisting
of the same number of realizations as the original data. The realizations are created
by drawing randomly (with repetitions) trials from the original set. Let us assume
that the original ensemble of trials consists of M realizations:

r 1 r 2 r 3 r 4 r 5 r k r k + 1 r M [ r ]=[ , , , , ,..., , ,..., ] (3.47)

New ensemble of trials may be:

r 3 r 1 r 3 r 7 r 5 r 5 r M

[ ]=[ , ,..., , ,..., ,..., , ]

r boot (3.48)

Repeating such procedure for all k channels L times (in practice 100 or more times)
we obtain L ensembles of realizations and we can calculate the average SDT F and
its variance as:

L
i = 1 ( SDT F − SDT F )

2

var ( SDT F )= (3.49)

( L − 1 )

The method may be used for evaluation of the errors of any estimator based on en-
semble averaging.

3.5 Comparison of the multichannel estimators of coupling be-
tween time series
Among approaches to quantify the coupling between multivariate time series, we
can distinguish between linear and non-linear methods as well as approaches based
on bivariate and multivariate estimators. For a long time the standard methods of
establishing relations between signals have been cross-correlation and coherence.
However, in case of these bivariate measures we don't know if the two channels are
really coupled or if there is another channel that drives them.
This point may be elucidated by simulation illustrating the situation where the ac-
tivity is measured in different distances from the source ( Figure 3.4) . The signals in
channels 2-5 were constructed by delaying the signal from channel one (the source)
by 1, 2, 3, 4 samples, respectively, and adding the noise in each step. The coherences