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and
t
is the temporal interval between two samples. Eq. (10.5) can then be
rewritten as:
−
1
X
(
f
)
=
Λ
(
f
)
E
(
f
)
=
H
(
f
)
E
(
f
)
(10.7)
H
(
f
) is the transfer matrix of the system, whose element
H
ij
represents the
connection between the
j-th
input and the
i-th
output of the system.
10.2.1.1
. Directed transfer function
The Directed Transfer Function, representing the causal influence of the cortical
waveform estimated in the
j-th
ROI on that estimated in the
i-th
ROI is defined in
terms of elements of the transfer matrix
H
, is:
( )
( )
2
2
(10.8)
θ
f
=
H
f
ij
ij
In order to compare the results obtained for cortical waveforms with different
power spectra, normalization can be performed by dividing each estimated DTF
by the squared sums of all elements of the relevant row, thus obtaining the so-
called normalized DTF:
2
( )
H
f
( )
ij
γ
f
=
(10.9)
ij
N
∑
=
( )
2
H
f
im
m
1
γ
2
ij
(f) expresses the ratio of influence of the cortical waveform estimated in
the
j-th
ROI on the cortical waveform estimated in the
i-th
ROI, with respect
to the influence of all the estimated cortical waveforms. Normalized DTF values
are in the interval [0 1], and the normalization condition:
N
( )
∑
=
γ
2
f
=
1
(10.10)
in
n
1
is applied.
10.2.1.2.
Partial directed coherence
In order to distinguish between direct and cascade flows, another estimator
describing the direct causal relations between signals, the Partial Directed
Coherence (PDC), was proposed in 2001. Like DTF, it is defined in terms of
MVAR coefficients transformed to the frequency domain. The definition of
Partial Directed Coherence (PDC) is:
