Information Technology Reference
In-Depth Information
Fig. 15.1: (
a
) Unidirectionally coupled oscillators with
05. (
b
) Mutual in-
formation MI vs.
k
(the first minimum of the mutual information between the
X
variables of the oscillators is denoted by a downward arrow at
k
ε
21
=
0
.
=
16). (
c
)ln
C
vs.
=
=
ln
r
with
k
1, where
C
and
r
denote average joint probability and radius,
respectively (
dotted line
for direction of flow 1
16,
l
→
2;
solid line
for direction of flow
2
1). (
d
) Autocorrelation function (AF) vs.
k
(the delay at which the AF decreases
to 1
→
/
e
of its maximum value is denoted by a
downward arrow
at
k
=
14).
of the driving system is sufficient to produce a considerable change in the dynamics
of the driven system within one time step (and hence only immediate interactions
between
X
and
Y
are assumed to be detected in the analysis herein). When larger
values for
l
were employed (i.e., a delayed influence of
Y
on
X
), detection of infor-
mation flow from
Y
to
X
was also possible. These results are not presented in this
chapter.
15.2.2.3 Selection of Radius
r
The multi-dimensional transitional probabilities involved in the definition of transfer
entropy (Equation (15.3)) are calculated by joint probabilities using the conditional
probability formula
P
(
A
|
B
)=
P
(
A
,
B
)
/
P
(
B
)
. One can then reformulate the transfer
entropy as
TE
(
Y
→
X
)=
log
2
P
(
x
n
+
1
,
x
(
k
)
y
(
l
)
n
x
(
k
)
n
(15.4)
x
(
k
)
y
(
l
)
,
)
P
(
N
n
1
P
(
x
n
+
1
,
,
)
n
.
∑
n
n
=
P
(
x
n
+
1
,
x
(
k
)
)
P
(
x
(
k
)
n
,
y
(
l
)
n
)
n
