Biomedical Engineering Reference
In-Depth Information
In the above equation
A
ij
(
f
)
is an element of
A
(
f
)
—a Fourier transform of MVAR
H
denotes the Hermitian
transpose. The PDC from
j
to
i
represents the relative coupling strength of the in-
teraction of a given source (signal
j
), with regard to some signal
i
as compared to
all of
j
's connections to other signals. From the normalization condition it follows
that PDC takes values from the interval
P
ij
model coefficients
a
j
(
f
)
is
j
-th column of
A
(
f
)
,andthe
(
.
)
. PDC shows only direct flows
between channels. Although it is a function operating in the frequency domain, the
dependence of
A
∈
[
0
,
1
]
(
)
on the frequency has not a direct correspondence to the power
spectrum. Figure 3.3 shows PDC for the simulation scheme presented in Figure 3.2
c).
f
FIGURE 3.3:
The PDC results for the simulation shown in
Figure 3.2
c). The
same convention of presentation as in
Figs. 3.1
and 3.2.
The definition of PDC was re-examined by the authors, who considered the fact
that PDC dependence on a signal's dynamic ranges, as modified by gains, obscures
the PDC ability to correctly pinpoint the direction of information flow [Baccal´aetal.,
2006]. They proposed the so-called generalized partial directed coherence (GPDC)
defined by the formula:
1
σ
i
A
ij
(
f
)
GPDC
ij
(
f
)=
∑
k
=
1
(3.38)
1
σ
k
A
ki
(
A
ki
(
f
)
f
)
Other problems inherent to PDC were pointed out by [Schelter et al., 2009]. The
most important one was that PDC measures the strength of influence relative to a
given signal source. It means that, in particular, adding further variables that are in-
fluenced by the source variable decreases the PDC, although the relationship between
source and target processes remains unchanged. To counteract this unfortunate prop-
erty in [Schelter et al., 2009], another kind of normalization of PDC was proposed.
This problem will be considered further in
Sect. 3.5.
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