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defined as
p
XY
ij
ln
p
XY
H
(
X,Y
)=
−
ij
.
(14.2)
i,j
where
p
XY
ij
which is the joint probability of
X
=
X
i
and
Y
=
Y
j
.
The cross
information between
X
and
Y
,
CMI
(
X,Y
), is then given by
CMI
(
X,Y
)=
H
(
Y
)
−
H
(
X
|
Y
)=
H
(
X
)
−
H
(
Y
|
X
)
(14.3)
=
H
(
X
)+
H
(
Y
)
−
H
(
X,Y
)
(14.4)
=
f
XY
(
x,y
)log
2
f
XY
(
x,y
)
f
X
(
x
)
f
Y
(
y
)
dxdy.
(14.5)
The cross mutual information is nonnegative. If these two random variables
X,Y
are independent,
f
XY
(
x,y
)=
f
X
(
x
)
f
Y
(
y
),then
CMI
(
X,Y
)=0,which
implies that there is no relationship or correlation between
X
and
Y
. The proba-
bilities are estimated using the histogram based box counting method. The random
variables representing the observed number of pairs of point measurements in his-
togram cell (
i,j
),row
i
and column
j
,are respectively
k
ij
,k
i.
and
k
.j
. Here, we
assume the probability of a pair of point measurements outside the area covered
by histogram is negligible, therefore
i,j
P
ij
=1[9, 10, 19].
Fig. 14.2.
A 10 sec. EEG epoch for
RTD
2,
RTD
4 and
RTD
6
Figure 14.2 illustrates an example of 10-second epochs recorded from the right
mesial temporal depth (R(T)D) region. Figure 14.3 displays the scatter plots for
pair-wise EEG signals shown in Fig. 14.2. Figure 14.4 shows the CMI values mea-
sured from EEG electrode pairs from Fig. 14.2. From the scatter plot, it is clear
that EEG recordings between
R
(
T
)
D
2 and
R
(
T
)
D
4 have weak linear correla-
tion which have also yielded lower CMI values in Fig. 14.4. The stronger linear
relationship is discovered between
R
(
T
)
D
4 and
R
(
T
)
D
6 and this linear correla-
tion pattern has resulted in higher CMI values in Fig. 14.4. Before measuring the
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