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From the above formulation, it is clear that probabilities of a vector in the state
space at the n th time step are compared with ones of vectors in the state space at the
(
th time step, and, therefore, the units of TE are in bits/time step, where time
step in simulation studies is the algorithm's (e.g., Runge-Kutta) iteration step (or
a multiple of it if one downsamples the generated raw data before the calculation
of TE). In real life applications (like in electroencephalographic (EEG) data), the
time step corresponds to the sampling period of the sampled (digital) data. In this
sense, the units of TE denote that TE actually estimates the rate of the flow of infor-
mation. The multidimensional joint probabilities in Equation (15.4) are estimated
through the generalized correlation integrals C n (
n
+
1
)
r
)
in the state space of embedding
dimension p
=
k
+
l
+
1 [14] as
x ( k )
y ( l )
P r (
x n + 1
,
,
)=
n
n
x n + 1
x m + 1
x ( k )
x ( k )
1
N
N
1
(15.5)
Θ
r
n
m
m
=
0
y ( l )
y ( l m
n
=
C n + 1
(
r
) ,
where
Θ (
x
>
0
)=
1;
Θ (
x
=
0
)=
0,
|·|
is the maximum distance norm, and the
subscript
is included in C to signify the dependence of C on the time index
n (note that averaging over n is performed in the estimation of TE, using Equa-
tion (15.5) into Equation (15.3)). In the rest of the chapter we use the notation C n (
(
n
+
1
)
r
)
or C n + 1 (
interchangeably. Equation (15.5) is in fact a simple form of a kernel den-
sity estimator, where the kernel is the Heaviside function
r
)
. It has been shown that
this approach may present some practical advantages over the box-counting meth-
ods for estimating probabilities in a higher dimensional space. We also found that
the use of a more elaborate kernel (e.g., a Gaussian or one which takes into account
the local density of the states in the state space) than the Heaviside function does
not necessarily improve the ability of the measure to detect direction and strength
of coupling. Distance metrics other than the maximum norm, such as the Euclidean
norm, may also be considered, however, at the cost of increased computation time.
In order to avoid a bias in the estimation of the multidimensional probabilities, tem-
porally correlated pairs of points are excluded from the computation of C n (
Θ
r
)
by
means of the Theiler correction and a window of
(
p
1
)
l
=
k points in dura-
tion [23].
The estimation of joint probabilities between two different time series requires
concurrent calculation of distances in both state spaces (see Equation (15.4)). There-
fore, in the computation of C n (
, the use of a common value of radius r in both state
spaces is desirable. In order to establish a common radius r in the state space of X
and Y , the data are first normalized to zero mean (
r
)
μ =
σ =
1).
In previous publications [18, 19], using simulation examples (unidirectional as well
as bidirectional coupling in two and three coupled oscillator model configurations),
we have found that the TE values obtained for only a certain range of r accurately
detect the direction and strength of coupling. In general, when any of the joint prob-
abilities ( C n (
0) and unit variance (
r
)
) in log scale is plotted against the corresponding radius r in log scale,
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