P. XIE ET AL.
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[7]. Still it is based on static probability distribution thus
didn’t accounting for system dynamics. Transfer entropy
was proposed by Thomas Schreiber which is on the basis
of transition (dynamic) probabilities computation [8].
Many researches have found it a modelfree measure of
effective connectivity for the neurosciences [9]. Goure
vitch used transfer entropy to detect the information flow
between auditory c ortical neurons [10]. The main advan
tage of this measure is that it is nonlinear and dynamic;
furthermore it has directional sense to define information
transfer.
The author has proposed the concept of information
transfer index (ITI) based on joint complexity entropy for
studies on mechanical fault diagnosis [11]. Taking ad
vantages of transition (dynamic) probabilities in describ
ing dynamic information interaction process, modifica
tion is made to the original definition of ITI. In this paper
the modified ITI based on transition probabilities has
been introduced and used to describe the information
transfer of different coupling models and experimental
data.
2. Method
2.1. Calculation of Information Transfer Index
Let X and Y be two signals recorded from two associated
systems, the original ITI is defined as:
()( )()
( )
+ ,
[0,1]
c cc
xy
c
HX HYHXY
ITI HX
→
= ∈
(1)
where
,
are the complexity entropy of
signal X and Y,
is their joint complexity
entropy. This calculation can describe the amount of in
formation in X that shared by Y. Though it isn’t ac
counting for system dynamics and it cannot discriminate
against common history and input signals. Taking ad
vantages of transition probabilities, the new definition is
based on the concept that if the future of a signal Y is
better predicted with the observation of the past and
present of a signal X, then it is believed that there is in
formation transmitted from X to Y. To quantify the in
fluence of X on the system Y, the modified ITI is calcu
lated as:
() ()
( )
+
, [0,1]

n mn
tuttu tt
xy n
tu t
HyyHyx y
ITI Hy y
++
→
−
= ∈
(2)
where
is the entropy of the process Y con
ditional on its past. The ITI indicates the directed infor
mation interactions by measuring the uncertainty reduc
tion via conditional entropy. It quantifies how much the
past of a process X influence the transition probabilities
of another process Y. We are interested in the deviation
from the following generalized Markov condition.
() ()
 ,
n mn
tuttu tt
HyyHyx y
++
=
(3)
where
,
. When
the transition probabilities or dynamics of Y are inde
pendent of the past of X, (3) is fully satisfied, and we
infer an absence of directed interaction from X to Y. Oth
er way there is information flow from X to Y. ITI natu
rally incorporates direc tional and dynamical information,
because it is inherently asymmetric and based on transi
tion probabilities.
Sensible causality hypotheses are formulated in terms
of the underlying systems rather than on the signals being
actually measured. To overcome this problem recon
structing the full state space of a dynamical system from
the observed signals is needed. In this work, we use de
laycoordinates to create a set of vectors in a higher di
mensional space according to (4) to map our scalar time
series into trajectories in a state space of high dimension.
() () ()()
( )
( )
,,2,...,1
d
xxt xtxtxtd
t
ττ τ
=− −−−
(4)
2.2. Parameter Selection
This procedure depends on two parameters, the dimen
sion d and the delay τ of the embedding. The two para
meters considerably affect the outcome of the ITI esti
mates. For instance, a low value of d can not sufficiently
unfold the state space of a system. On the other hand, a
too large dimensionality may lower the estimation accu
racy and significantly enlarges the computing time. A
popular option is to take the delay embedding τ as the
autocorrelation decay time of the signal. To determine
the embedding dimension, the Cao criterion offers an
algorithm based on false neig hb ors com putation [12].
3. Simulation and EEG Experiment
3.1. Simulation Data
To test the ability of ITI to detect the direction of infor
mation flow and identify the relationship between two
time series. We used four different models, i.e. indepen
dent, linear, quadratic and threshold models.
1) The first test case we used two independent time se
ries X and Y generated by t he followin g process e s.
(5)
(6 )
where the coefficients
and
are drawn from a
normalized Gaussian distribution,
and
are inde
pendent Gaussian noise of unit variance.
2) The second test case consisted in simulating a li
near causal interaction between the two systems. We