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improvements and modifications for the estimation of TE, is a robust measure for
detecting the direction and the rate of the net information flow in coupled nonlinear
systems even under severe noise conditions (e.g., SNR
=
3dB).
15.4 Discussion and Conclusion
In this study, we suggested and implemented improvements for the estimation of
transfer entropy (TE), a measure of the direction and the level of information flow
between coupled subsystems, built upon it to introduce a new measure of informa-
tion flow, and showed their application to a simulation example. The two innovations
we introduced in the TE estimation were: (a) the distance in the state space at which
the required probabilities should be estimated and (b) the use of surrogate data to
evaluate the statistical significance of the estimated TE values. The new estimator
for TE was shown to be consistent and reliable when applied to complex signals gen-
erated by systems in their chaotic regime. A more practical estimator of TE, that av-
erages the values of TE produced in an intermediate range of distances
r
in the state
space, was shown to be robust to additive noise up to
S
/
=
3 dB, and could reliably
and significantly detect the direction of information flow for a wide range of cou-
pling strengths, even for coupling strengths close to 0. Our analysis in this chapter
dealt with only pairwise (bivariate) interactions between subsystems and as such,
it does not detect both direct and indirect interactions among multiple subsystems
at the time resolution of the sampling period of the data involved. A multivariate
extension of TE to detect information flow between more than two subsystems is
straightforward. Such an extension could also be proven useful in distinguishing be-
tween direct and indirect interactions [6,5], and thus further enhance TE's capability
to detect causal interactions from experimental data.
A new measure of causality, namely net transfer of entropy [NTE(
i
N
j
)], was
then introduced for a system
i
driving a system
j
(see Equation (15.6)). NTE of the
system
i
measures the outgoing net flow of information from the driving
i
to the
driven
j
system, that is, it takes into consideration both incoming TE to and outgo-
ing TE from the driving system
i
. Our simulation example herein also showed the
importance of NTE for the identification of the driving system in a pair of coupled
systems for a range of coupling strengths and noise levels. We believe that our ap-
proach to estimating information flow between coupled systems can have several
potential applications to coupled complex systems in diverse scientific fields, from
medicine and biology, to physics and engineering.
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Acknowledgments
This work was supported in part by NSF (Grant ECS-0601740) and the Sci-
ence Foundation of Arizona (Competitive Advantage Award CAA 0281-08).
