Information Technology Reference
In-Depth Information
analysis is valid when
l
=
1; for
l
>
1, tuples from original
Y
, each of length
l
,
should be shuffled instead.
The shuffling was based on generation of white Gaussian noise and reordering of
the original data samples of the driver data series according to the order indicated
by the generated noise values (i.e., random permutation of all indices 1
N
and
reordering of the
Y
time series accordingly). Transfer entropy TE
s
values of the
shuffled datasets were calculated at the optimal radius
r
∗
from the original data.
If the TE values obtained from the original time series (TE
o
) were greater than
T
th standard deviations from the mean of the TE
s
values, the null hypothesis was
rejected at the
,...,
α
=
0
.
01 level (The value of
T
th depends on the desired level of
confidence 1-
and the number of the shuffled data segments generated, i.e., the
degrees of freedom of the test). Similar surrogate methods have been employed to
assess uncertainty in other empirical distributions [4, 21, 16].
α
15.2.4 Detecting Causality Using Transfer Entropy
Since it is difficult to expect a truly unidirectional flow of information in real-world
data (where flow is typically bidirectional), we have defined the causality measure
net transfer entropy (NTE) that quantifies the driving of
X
by
Y
as
NTE
(
Y
→
X
)=
TE
(
Y
→
X
)
−
TE
(
X
→
Y
)
.
(15.6)
Positive values of NTE(
Y
X
) denote that
Y
drives (causes)
X
, while negative
values denote the reverse case. Values of NTE close to 0 may imply either equal
bidirectional flow or no flow of information (then, the values of TE will help decide
between these two plausible scenarios). Since NTE is based on the difference be-
tween the TEs per direction, we expect this metric to generally be less biased than
TE in the detection of the driver. In the next section, we test the ability of TE and
NTE to detect direction and causality in coupled nonlinear systems and also test
their performance against measurement (observation) noise.
→
15.3 Simulation Example
In this section, we show the application of the method of TE to nonlinear data gen-
erated from two coupled, nonidentical,
R ossler
-type oscillators
i
and
j
[7], each
governed by the following general differential equations:
2
j
=
1
,
i
=
j
=
−
ω
−
+
∑
−
ε
,
x
i
i
y
i
z
i
ε
ji
x
j
ii
x
i
y
i
=
ω
i
x
i
+
α
i
y
i
,
z
i
=
β
i
x
i
+
(15.7)
z
i
(
x
i
−
γ
i
)
,
