Information Technology Reference
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linear and nonlinear systems. The estimation of these interactions, especially when
the systems' structure is unknown, holds promise for the understanding of the
mechanisms of their interactions and for a subsequent design and implementation
of appropriate schemes to control their behavior. Traditionally, cross-correlation
and coherence measures have been the mainstay of assessing statistical interde-
pendence among coupled systems. These measures, however, do not provide reli-
able information about directional interdependence, i.e., if one system drives the
other.
To study the directional aspect of interactions, many other approaches have been
employed [24, 22, 11, 18, 19]. One of these approaches is based on the improvement
of the prediction of a series' future values by incorporating information from an-
other time series. Such an approach was originally proposed by Wiener [24] and
later formalized by Granger in the context of linear regression models of stochas-
tic processes. Granger causality was initially formulated for linear models, and it
was then extended to nonlinear systems by (a) applying to local linear models in
reduced neighborhoods, estimating the resulting statistical quantity and then aver-
aging it over the entire dataset [20] or (b) considering an error reduction that is
triggered by added variables in global nonlinear models [2].
Despite the relative success of the above approaches in detecting the direction
of interactions, they essentially are model-based (parametric) methods (linear or
nonlinear), i.e., these approaches either make assumptions about the structure of the
interacting systems or the nature of their interactions, and as such they may suffer
from the shortcomings of modeling systems/signals of unknown structure. For a
detailed review of parametric and nonparametric (linear and nonlinear) measures of
causality, we refer the reader to [9, 15]. To overcome this problem, an information
theoretic approach that identifies the direction of information flow and quantifies the
strength of coupling between complex systems/signals has recently been suggested
[22]. This method was based on the study of transitional probabilities of the states
of systems under consideration. The resulted measure was termed transfer entropy
(TE).
We have shown [18, 19] that the direct application of the method as proposed
in [22] may not always give the expected results. We show that tuning of certain
parameters involved in the TE estimation plays a critical role in detecting the correct
direction of the information flow between time series. We propose a methodology
to also test the significance of the TE values using surrogate data analysis and we
demonstrate its robustness to measurement noise. We then employ the improved TE
method to define a new measure, the net transfer entropy (NTE). Results from the
application of the improved
TE
and
NTE
show that these measures are robust in
detecting the direction and strength of coupling under noisy conditions.
The organization of the rest of this chapter is as follows. The measure of
TE and the estimation problems we identified, as well as the improvements and
practical adjustments that we introduced, are described in Section 15.2. In Sec-
tion 15.3, results from the application of this method to a system of coupled
R ossler
oscillators are shown. These results are discussed and conclusions are drawn in
Section 15.4.
