Biomedical Engineering Reference
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were calculated between all signals pairwise. For the peak frequency the phases were
found and assuming the linear phase relations, the delays were calculated. The re-
sulting scheme of propagation presented in Figure 3.4 b) shows many false flows.
Similar results were obtained for bivariate AR model (Figure 3.4 c). This situation
is common to all bivariate methods; namely the propagation is found in each case
where the phase difference is present [Blinowska et al., 2004a]. Sometimes, for bi-
variate measures even the reverse propagation may be found [Kus et al., 2004]. The
results obtained by means of the multivariate method (DTF) show the correct scheme
of propagation (Figure 3.4 d).
Another problem is that, from correlation or coherence (also partial coherence), it
is not possible to detect reciprocal flows (interaction in both directions). Such pat-
terns of propagation may be found by means of estimators based on the causality
principle.
All non-linear methods described above are bivariate, so they suffer from the dis-
advantages pointed out above. There were some attempts to apply non-linear mea-
sures for number of channels bigger than two. Chen et al. [Chen et al., 2004] pro-
posed conditional extended Granger causality and applied it to 3 channel simulated
non-linear time series, but the advantages over the linear approach were not clearly
demonstrated and the influence of noise was not studied. The method requires deter-
mination of embedding dimension and neighborhood size, which are difficult to find
in an objective and optimal way. The same objections concern another methods aim-
ing to extend non-linear estimators for a number of channels higher than two. The
problems concerning state space reconstruction and noise sensitivity become even
more serious for a higher number of channels.
In the study devoted to comparison of linear and non-linear methods of cou-
pling [Netoff et al., 2006], non-linear estimators: mutual information, phase cor-
relation, continuity measure 2 were compared with correlation in case of non-linear
signals in the presence of noise. The authors found that any method that relies on an
accurate state space reconstruction will be inherently at a disadvantage over measures
which do not rely on such assumptions. Another finding was the high sensitivity to
noise of non-linear estimators. The authors conclude: “We have been as guilty as
any of our colleagues in being fascinated by the theory and methods of nonlinear
dynamics. Hence we have continually been surprised by robust capabilities of lin-
ear CC (correlation) to detect weak coupling in nonlinear systems, especially in the
presence of noise.”
We can conclude that non-linear methods are not recommended in most cases.
They might be used only when there is clear evidence that there is a good reason
to think that there is non-linear structure either in data themselves or in the interde-
pendence between them [Pereda et al., 2005]. We have to bear in mind that many
systems composed of highly non-linear components exhibit an overall linear type
of behavior; the examples may be some electronic devices, as well as brain signals
2 Measure similar to GS testing for continuity of mapping between neighboring points in one data set to
their corresponding points in the other data set.
 
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