Biomedical Engineering Reference
In-Depth Information
2000
1000
500
0
−
500
−
1000
0
5
10
15
20
25
30
Time (seconds)
s
(
2)
0
− τ
−
500
−
1000
0
2000
500
−
500
0
s
()
−
500
s
(
)
−
1000
− τ
−
1000
Figure 3.25
An EEG segment from a focal right temporal lobe cortical electrode, before and after
the onset of an epileptic seizure in the time domain and in the state space. (a) A 30-second epoch
s
(
t
)
of EEG (voltage in microvolts) of which 10 seconds are from prior to the onset of a seizure and 20 sec-
onds from during the seizure. (b) The three-dimensional state space representation of
s
(
t
)(
m
=3,
τ
=
20 ms).
mine the correlation integral (sum)
C
(
m
,
(radius in the state space
that corresponds to a multidimensional bin size) and for consecutive embedding
dimensions
m
. Another way to interpret
C
(
m
,
ε
) for a range of
ε
) in the state space is in terms of an
underlying multidimensional probability distribution. It is the self-similarity of this
distribution that
ε
and
d
quantify.
We define the correlation sum for a collection of points
s
i
=
ν
s
(
i
·
Dt
) in the vector
space to be the fraction of all possible pairs of points closer than a given distance
ε
,
using a particular norm
(e.g., the Euclidean or max) to measure this distance.
Thus, the basic formula for
C
(
m
,
||
·
||
ε
) is [64]
2
N
N
(
)
()
(
∑
=
∑
Cm
,
ε
=
Θ
ε
−−
s
s
(3.56)
)
i
j
NN
−
1
i
1
ji
=+
1
where
Θ
is the Heaviside step function,
Θ
(
s
)
=
0if
s
=
0 and
Θ
(
s
)
=
1 for
s
>
0. The
summation counts the pairs of points (
s
i
,
s
j
) whose distance is smaller than
ε
. In the
limit of an infinite amount of data (
N
8) and for small
ε
, we theoretically expect
C
D
to scale with
ε
exponentially, that is,
C
(
ε
)
≈ ε
and we can then define
D
and
ν
by
()
∂
ln
ε
∂ε
Cm
,
()
()
Dm
=
lim lim
and then
ν
=
lim
Dm
(3.57)
ln
ε0
→→∞
N
m
→∞
It is obvious that the limits of (3.57) cannot be satisfied in real data and approx-
imations have to be made. In finite data,
N
is limited by the size and stationarity of
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