Biomedical Engineering Reference
In-Depth Information
3.2.1.2 Estimation of Cumulants from EEGs
To estimate the third-order cumulant from an EEG data segment
s
of length
N
, sam-
pled with sampling period
Dt
, the following steps are performed:
1.
The data segment
s
is first divided into
R
smaller segments
s
i
, with
i
=
1, …,
N
.
2. Subtract the mean from each data segment
s
i
.
3.
If the data in each segment
i
is
s
i
(
n
) for
n
R
, each of length
M
such that
R
⋅
M
=
=
0, 1, …,
M
- 1, and with a
sampling period
Dt
such that
t
n
n·Dt,
an estimate of the third-order
cumulant per segment
s
i
is given by
1
v
(
)
()(
)(
)
=
∑
i
i
i
i
cll
,
=
snsn l sn l
+
+
(3.44)
312
1
2
M
nu
where
u
=
max(0,
−
l
1
,
−
l
2
);
v
=
min(
M
−
1,
M
−
1
−
l
1
,
M
−
1
−
l
2
);
l
1
·
Dt
= τ
1
,
= τ
2
. Higher-order cumulants can be estimated likewise [7].
4. Average the computed cumulants across the
R
segments:
and
l
2
·
Dt
1
R
(
)
=
∑
(
)
i
cll
,
=
cll
,
(3.45)
312
312
R
i
1
Thus,
c
3
(
l
1
,
l
2
) is the average of the estimated third-order cumulants per short
EEG segment
s
i
.
The preceding steps can be performed per EEG segment
i
over the available time
of recording to obtain the cumulants over time.
3.2.1.3 Frequency-Domain Higher-Order Statistics: Bispectrum and Bicoherence
Higher-order spectra (polyspectra) are defined by taking the multidimensional Fou-
rier transform of the higher order cumulants. Thus, the
r
th-order polyspectra are
defined as follows:
∞
∞
⎡
⎢
r
−
1
⎤
⎦
(
)
∑
∑
(
)
∑
S
ωω
,
,
,
ω
=
c
l
,
l
,
,
l
p
j
ω
l
(3.46)
⎥
r
1
2
r
−
1
r
1
2
r
−
1
i
i
l
=−∞
l
−
=−∞
i
=
1
1
r
1
Therefore, the
r
th-order cumulant must be absolutely summable for the
r
th-order spectra to exist.
Substituting
r
=
2 in (3.46), we get
∞
∑
()
() ( )(
)
S
ω
=
c
l
exp
−
j
ω
l
power spectrum
(3.47)
2
1
2
1
1
1
l
=−∞
1
Substituting
r
=
3 in (3.46), we instead get
∞
∑
∞
=
∑
(
)
(
) (
) (
)
S
ωω
,
=
c
l
,
l
exp
−
j
ω
l
−
j
ω
l
bispectrum
(3.48)
3
1
2
3
1
2
1
1
2
2
l
−∞
l
=−∞
1
2
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