Bivariable Analysis of EEG Signals - Quantitative EEG Analysis Methods and Clinical Applications - page 112

of the signals with respect to the other and assume that they will not be correlated
for large enough shifts [8].
Note that formally only the zero lag cross correlation can be considered to be a
symmetric descriptor. Indeed, the time delay in the definition of (4.1) introduces an
asymmetry that could, in principle, establish whether one of the signals leads or lags
the other in time. It should be mentioned, however, that a time delay between two
signals does not necessarily prove a certain driver-response causal relationship
between them. In fact, time delays could be caused by a third signal driving both
with a different delay or by internal delay loops of one of the signals [10].
Figure 4.2 shows the cross-correlation values for the three examples of Figure
4.1 as a function of the time delay . To visualize cross-correlation values with large
time delays, we used here a slight variant of (4.1) by introducing periodic boundary
conditions. The zero lag cross-correlation values are shown in Table 4.1. Here we
see that the tendency is in agreement with what we expect from the arguments of the
previous section; that is, Sync B > Sync C . However, the difference between

examples A and B is relatively small. In principle, one expects that for long enough
lags between the two signals the cross-correlation values should be close to zero.
However, fluctuations for large delays are still quite large.
Taking these fluctuations as an estimation of the error of the cross-correlation
values, one can infer that cross correlation cannot distinguish between the synchro-
nization levels of examples A and B. This is mainly due to the fact that cross correla-
tion is a linear measure and can poorly capture correlations between nonlinear
signals, as is the case for examples B and C with the presence of spikes. More
advanced nonlinear measures that are based on reconstruction of the signals in a
phase space could indeed clearly distinguish between these two cases [8].

Sync A >

4.2 Coherence Estimation

The coherence function gives an estimation of the linear correlation between two
signals as a function of the frequency. The main advantage over the cross-correla-
tion function described in the previous section is that coherence is sensitive to inter-
dependences that can be present in a limited frequency range. This is particularly
interesting in neuroscience to establish how coherence oscillations may interact in
different areas.
Let us first define the sample cross spectrum as the Fourier transform of the
cross-correlation function, or by using the Fourier convolution theorem, as

() ( ()( )()

*

C ω = Fx ω Fy ω (4.2)

xy

where ( Fx ) is the Fourier transform of x , <

N /2), and the asterisk indicates complex conjugation. The cross spectrum can be
estimated, for example, using the Welch method [11]. For this, the data is divided
into M epochs of equal length, and the spectrum of the signal is estimated as the
average spectrum of these M segments. The estimated cross spectrum C xy ()

are the discrete frequencies ( − N /2

ω is a

complex number, whose normalized amplitude