Biomedical Engineering Reference
In-Depth Information
(ˆ
E
(
))
The imagina
ry and corrected
imaginary coherences are obtained as
f
and
(ˆ
E
(
))/
(
−
(ˆ
E
(
))
2
.
f
1
f
7.8 Statistical Thresholding of Coherence Images
In practical applications, we need to assess the statistical significance of the obtained
coherence images. The surrogate data method [
17
,
18
] has been used for this assess-
ment. In this method, the surrogate voxel spectra are created by multiplying random
phases with the original voxel spectra. The surrogate spectra for the seed and target
voxels,
˃
S
(
f
)
and
˃
T
(
f
)
, are expressed as
˃
S
=
˃
S
e
i
2
ˀʴ
S
=
˃
T
e
i
2
ˀʴ
T
and
˃
T
,
(7.107)
where
ʴ
S
and
ʴ
T
are uniform random numbers between 0 and 1, and the explicit
notation of
is again omitted for simplicity. Note that the surrogate spectra have
the same original power spectra but the phase relationship is destroyed bymultiplying
the random phases to the original spectra.
Any coherence-based metric, such as the magnitude/imaginary coherence or cor-
rected imaginary coherence, can be computed using the surrogate spectra,
(
f
)
˃
S
and
˃
T
. The metric computed using the surrogate spectra is denoted
. As an example,
the imaginary coherence is computed using the surrogate spectra. It is expressed as
3
ˉ
|
˃
T
˃
S
|
=
|
˃
T
˃
S
e
i
2
ˀʔʴ
|
ˉ
=
,
(7.108)
2
2
2
2
|
˃
T
|
|
˃
S
|
|
˃
T
|
|
˃
S
|
where
ʔʴ
=
ʴ
T
−
ʴ
S
. The generation of
ˉ
is repeated
B
times, and a total of
B
1
2
B
, are obtained. These
1
B
can form an
values of
ˉ
, denoted
ˉ
, ˉ
,...,ˉ
ˉ
,...,ˉ
empirical null distribution at each voxel.
We could derive a voxel-by-voxel statistical threshold using this empirical null
distribution. However, the statistical threshold derived in this manner does not take
the multiple comparisons into account and it generally leads to a situation in which
many false-positive voxels arise, i.e., many voxels that contain no brain interaction
are found to be interacting. To avoid this problem, the statistical significance is
determined using a procedure that takes multiple comparisons into account. For this
purpose, we can use the maximal statistics
4
[
19
,
20
].
To utilize maximum statistics, the values
ˉ
(
=
1
,...,
B
) are first standardized
and converted into pseudo-
t
values, such that
3
The absolute value of the imaginary coherence is usually computed, because its sign has no
meaning in expressing the connectivity.
4
We can apply another method such as the false discovery rate to this multiple comparison problem.
