Biomedical Engineering Reference
In-Depth Information
Substituting Eqs. (
7.52
) and (
7.50
) into Eq. (
7.49
), and neglecting the constant
i
,
the residual coherence is expressed as
[
˃
T
˃
S
]
(ˆ)
ˆ
R
=
=
ʾ.
2
=
1
(7.53)
−
(ˆ)
2
−[
˃
T
˃
S
]
2
2
|
˃
T
|
|
˃
S
|
The above equation shows that the residual coherence is exactly equal to the corrected
imaginary coherence
ʾ
. That is, the corrected imaginary coherence is derived as the
coherence between the seed signal and the residual signal, which is obtained by
regressing out the seed signal from the target signal.
7.5.4 Phase Dependence of the Corrected
Imaginary Coherences
The coherence
ˆ
is complex valued, and it is expressed as the amplitude and phase,
such that
e
i
ʸ
,
ˆ
=|
ˆ
|
(7.54)
where
is omitted for simplicity.
When the coherence is caused by true brain interactions, we naturally interpret that
the amplitude
ʸ
is the phase of the coherence, and the notation
(
f
)
, which is equal to the magnitude coherence, represents the strength
of the brain interactions.
The imaginary coherence is expressed as
|
ˆ
|
(ˆ)
=|
ˆ
|
ʸ.
sin
(7.55)
The value of the imaginary coherence depends on the phase
. This is one weak point
of the imaginary coherence, because even if there is a strong brain interaction, the
imaginary coherence becomes very small when the phase
ʸ
ʸ
has a value close to one
of multiples of
.
Let us see how the corrected imaginary coherence
ˀ
ʾ
depends on the phase. Using
Eq. (
7.54
), we have
|
ˆ
|
sin
ʸ
ʾ
=
1
.
(7.56)
2
cos
2
−|
ˆ
|
ʸ
ʾ
The plots of
with respect to the phase are shown in Fig.
7.1
with four values of the
magnitude coherence
|
ˆ
|
. This figure shows that the corrected imaginary coherence
becomes asymptotically independent of the phasewith the limit of
1. However,
the corrected imaginary coherence has almost the same phase dependence as the
imaginary coherence, when the magnitude coherence
|
ˆ
|ₒ
|
ˆ
|
is less than 0.6. When
|
ˆ
|
becomes closer to 1, the corrected imaginary coherence
ʾ
becomes significantly
greater than the imaginary coherence
(ˆ)
.
