Bivariable Analysis of EEG Signals - Quantitative EEG Analysis Methods and Clinical Applications - page 114

Table 4.1 Cross-Correlation, Coherence, and Phase
Synchronization Values for the Three Examples of Figure 4.1
Example c xy

xy

A 0.70 0.88 0.59

B 0.79 0.86 0.71

C 0.42 0.40 0.48

estimate each auto spectrum and cross spectrum, the coherence function of (4.3)
gives always a trivial value of 1.
Figure 4.3 shows the power spectra and coherence values for the three examples
of Figure 4.1. For the spectral estimates we used half-overlapping segments of 128
data points, tapered with a Hamming window in order to diminish border effects
[11]. In the case of example A, the spectrum resembles a power-law distribution
with the main activity concentrated between 1 and 10 Hz. This range of frequencies
had the largest coherence values. For examples B and C, a more localized spectral
distribution is seen, with a peak around 7 to 10 Hz and a harmonic around 15 Hz.
These peaks correspond to the frequency of the spikes of Figure 4.1.
It is already clear from the spectral distribution that there is a better matching
between the power spectra of the right and left channels of example B than for
example C. This is reflected in the larger coherence values of example B, with a sig-
nificant synchronization for this frequency range. In contrast, coherence values are
much lower for example C, seeming significant only for the low frequencies (below
6 Hz). In Table 4.1 the coherence values at a frequency of 9 Hz—the main frequency
of the spikes of examples B and C—are reported. As it was the case for the cross cor-
relation, note that the coherence function does not distinguish well between exam-
ples A and B. From Figure 4.3, there is mainly a difference for frequencies larger
than about 11 Hz, but this just reflects the lack of activity at this frequency range for
example A, whereas in example B it reflects the synchronization between the
high-frequency harmonics of the spikes. Even then, it is difficult to assess which fre-
quency should be taken to rank the overall synchronization of the three signals (but
some defenders of coherence may still argue that an overall synchronization value is
meaningless).

4.3 Mutual Information Analysis

The cross-correlation and coherence functions evaluate linear relationships between
two signals in the time and frequency domains, respectively. These measures are rel-
atively simple to compute and interpret but have the main disadvantage of being lin-
ear and, therefore, not sensitive to nonlinear interactions. In this section we describe
a measure that is sensitive to nonlinear interactions, but with the caveat that it is
usually more difficult to compute, especially for short datasets.
Suppose we have a discrete random variable X with M possible outcomes X 1 ,…,
X M , which can, for example, be obtained by partitioning of the X variables into M
bins. Each outcome has a probability p i ,i

= 1, …, M, with p i ≥ 0 and Σ p i = 1. A first

estimate of these probabilities is to consider p i = n i / N , where n i is the probability of