Multiple channels (multivariate) signals - Practical Biomedical Signal Analysis Using MATLAB - page 90

2002], is defined by the formula:

2 2

PLV n , m = cosφ n , m ( t ) + sinφ n , m ( t ) , (3.46)

where denotes averaging over time. From the above formula one can find how

the relative phase is distributed over the unit circle. If the two signals are phase
synchronized, the relative phase will occupy a small portion of the circle, which
means high phase coherence.
Other measures of phase synchronization (PS) are based on the calculation of the
distribution of phases [Pereda et al., 2005]. In the so-called stroboscopic approach
the phase of one of the oscillators is observed at those instants where that of the other
one attains a certain value.
The above methods require bandpass filtering of the signals. When the signals
are characterized by modulated natural frequency the relative phase distributions are
broad, which makes estimation of PS difficult. The problems connected with the
construction of histograms, mentioned in the context of finding generalized synchro-
nization, are also present in case of estimation of phase synchronization based on the
consideration of phase distributions.
PS does not allow determination of the direction of interaction between the chan-
nels of a process. Some attempts were made to judge the direction from the dynami-
cal model of the process under the assumption of weak dependence on the amplitude.

·

3.4.5 Testing the reliability of the estimators of directedness
Testing the null hypothesis about the lack or presence of causal relations between
time series is not straightforward, since analytical determination of the statistical
distributions of the estimators of directedness is difficult. The asymptotic distribution
of DTF under the null hypothesis of no information flow was derived by [Eichler,
2006]. He proposed a frequency dependent point-wise significance level that forms
an upper bound for the true unknown critical value of the asymptotic distribution of
DTF.
However in the absence of a general analytical approach the significance of the
directedness estimators is usually tested by means of surrogate data. Namely the
obtained results are compared with those computed for the time series with no de-
pendencies between channels.
The simplest approach is to randomly shuffle the samples of the data series, but
such procedure does not preserve the autocorrelation structures of the signals. The
use of such a white-noise version of data as a control condition turns the data into
a very unlikely realization of the physiological process. Therefore this procedure is
not recommended.
A better approach is to use the surrogates which preserve the spectral properties
of the time series and randomize only the phases of each signal. The procedure of
the generation of this type of data was described in (Sect. 1.6). The outcome of
this kind of test shows whether there is directedness among the set of signals. A