Biomedical Engineering Reference
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the data, whereas
is limited from below by the finite accuracy of the data, noise,
and the inevitable lack of near neighbors in the state space at small length scales. In
addition, for large
m
and for a finite
D
to exist, we theoretically expect that
D
would
converge to
ε
2
d
1). Also, the previous estimator
of correlation dimension is biased toward small values when the pairs included in
the correlation sum are statistically dependent simply because of oversampling of
the continuous signal in the time domain and/or inclusion of common components
in successive state vectors [e.g.,
s
(
t
-
ν
for large values of
m
(e.g., for
m
>
τ
) is a common component in the vectors
s
(
t
)
and
s
(
t
)]. Then, it is highly probable that the embedded vectors
s
(
t
) at successive
times
t
are nearby in the state space. In the process of estimation of the correlation
dimension, the presence of such temporal correlations may lead to serious underesti-
mation of
− τ
. A solution to this problem is to exclude such pairs of points in (3.56).
Thus, the lower limit in the second sum in (3.56) is changed, taking in consideration
a correlation time
t
min
=
ν
n
min
⋅
Dt
(Theiler's correction) [72] as follows:
2
N
N
(
)
()
(
∑
∑
Cm
,
ε
=
Θ
ε
−−
s
s
(3.58)
)(
)
i
j
Nn
−
Nn
−
−
1
i
=
1
ji n
=+
min
min
min
Note that
t
min
is not necessarily equal to the average correlation time [i.e., the
time lag at which the autocorrelation function of
s
(
t
) has decayed to 1/
e
of its value
at lag zero]. It has rather to do with the time spanned by a state vector's components,
that is, with (
m
-1)
τ
.
Application: Estimation of Correlation Integrals and Dimensions from EEGs
A reliable estimation of the correlation dimension
requires a large number of data
points [73, 74]. However, due to the nonstationarity of EEGs, a maximum length
T
for the EEG segment under analysis (typically on the order of 10 seconds), which
also depends on the patient's state and could be derived by measure(s) of
nonstationarity, has to be considered in the estimation of
ν
ν
[74]. A scaling region of
ln
C
(
m
,
ε
) versus ln
ε
for the estimation of
D
(
m
) is considered true if it occurs for
ε <<
σ
is the standard deviation either of the one-dimensional data, or the size of
the attractor in the
m
-dimensional state space. If the thus estimated
D
(
m
) versus
m
reaches a plateau with increasing
m
, the value of the plateau is a rough estimate of
, where
σ
.
We show the application of the correlation dimension for the estimation of the
complexity of the EEG attractor during an epileptic seizure. The procedure for esti-
mating the correlation dimension of an EEG segment described earlier is applied to a
10-second EEG segment recorded from a focal electrode within a seizure. The
TISEAN software package [73] was used to produce the results shown in Figure
3.26.
Figure 3.26(a) shows the ln
C
(
m
,
ν
ε
) versus ln
ε
for
m
=
2upto
m
=
20. The raw
EEG data were normalized to
±
1 before the estimation of
C
, and therefore 0
< ε <
1.
Figure 3.26(b) shows the local slopes
D
(
m
,
ε
) versus ln
ε
, estimated in local regions of
ε
, for
m
=
2upto
m
=
20 in step 2. It is relatively clear that, as
ε
increases from zero,
the first plateau of
D
(
m
,
ε
) with
m
is observed in the
ε
range of
−
3.0
=
ln
ε
<
−
2.0 (i.e.,
0.14) for
m
larger than 10. The fact that the plateau is not readily discern-
ible reflects the influence of the limited number of data points and of possible super-
imposed noise to the data. Nevertheless, the value
0.05
= ε <
ν
of the formed plateau is in the
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