Information Technology Reference
In-Depth Information
ε
21
at a value of
r
∗
chosen in the
a
black
and
blue lines
respectively) estimated over
linear region of the ln
C
n
(
vs. ln
r
of the original data, using MI-suggested
k
values
(
k
values decrease from 16 to 14 as
r
)
1. (The corresponding
values for
k
through the use of AF changed from 14 to 12 with the increase of
ε
21
.) From this figure, it is clear that the TE
o
(2
ε
21
increases) and with
l
=
→
1) is significantly greater than
μ
(
TE
s
)+
2
.
68
×
σ
(
TE
s
)
almost over the entire range of coupling, where
μ
(
TE
s
)
is
the mean of TE
s
over 50 surrogate values and 2
.
68
×
σ
(
TE
s
)
is the error bar on the
distribution of TE
s
(49 degrees of freedom) at the
α
=
0
.
005 level. [For very small
values of coupling (
ε
21
<
0
.
02), detection of the direction of information flow is not
possible (
p
1,
proportional to the increase of coupling in that direction, and no significant change
in the direction 1
>
0
.
05).] Also, TE
o
shows a progressive increase in the direction 2
→
→
2. In Fig. 15.2b, the TE
o
(
ε
21
)
and the mean and 99.5% error bars
on the distribution of TE
s
(
ε
21
)
are illustrated for a pair of arbitrary chosen values
for
k
and
l
(e.g.,
k
=
l
=
5). Neither a statistically significant preferential direction of
information flow (1
1) nor a statistically significant progressive increase
in TE
o
values with the increase of coupling
→
2or2
→
ε
21
were observed, due to erroneous
selection of
k
and
l
for the estimation of TE.
In Fig. 15.2c, we present the same quantities as in Fig. 15.2a, but they now are
estimated as averages of TEs over an intermediate range of values of
r
[
<
<
σ
/5
ln
r
/5] (that is, not at
r
∗
). We also observe that the TE values in Fig. 15.2c are larger
than the ones in Fig. 15.2a with
r
2
σ
r
∗
and that it is possible from Fig. 15.2c to detect
the correct direction of flow and its significant changes with the strength of coupling.
This result is very important for the estimation of TE in practical applications, where
an optimal
r
∗
is difficult to obtain. In Fig. 15.2d, we show the values of the measure
of causality NTE and its statistical significance for the detection of direction and
strength of coupling in the two coupled oscillator system over a range of
=
ε
21
.NTE
was also estimated as an average of NTEs over intermediate values of
r
[
σ
/5
<
ln
r
/5]. From the statistically significant values of NTE, it is clear that oscillator
2 drives oscillator 1 and the degree of driving increases proportional to the increase
in their coupling.
<
2
σ
15.3.2 Robustness to Noise
In order to assess the practical usefulness of this methodology for the detection
of causality in noise-corrupted data, Gaussian noise with variance corresponding
to a 10 or 3 dB signal-to-noise ratio (SNR) was added independently to the
X
se-
ries of the original data from each of the two coupled
R ossler
systems. The noisy
data were then processed in the same way as the noise-free data, including testing
against the null hypothesis that an obtained value of TE at each coupling value
ε
21
is not statistically significant. The corresponding TE
o
and TE
s
values are illustrated
in Fig. 15.2e, f for each of the two SNR values, respectively. It is noteworthy that
only at extremely low values of coupling (
0.02) NTE cannot detect the direc-
tion of information flow. Thus, it appears that the NTE, along with the suggested
ε
21
<
