Information Technology Reference
In-Depth Information
all the time series. The advantages of MVAR modeling of multichannel EEG
signals in order to compute efficient connectivity estimates have recently been
stressed. Kus
et al
. demonstrated the superiority of MVAR multichannel
modeling with respect to the pair-wise autoregressive approach. Another popular
estimator, the Partial Directed Coherence (PDC), based on MVAR coefficients
transformed into the frequency domain was recently proposed, as a factorization
of the Partial Coherence. The PDC is of particular interest because of its ability
to distinguish direct and indirect causality flows in the estimated connectivity
pattern. If another “true” flow exists from region
x2
to region
x3
, the PDC
estimator does not add an “erroneous” causality flow between the signal recorded
from region
x1
to region
x3
. This property is particularly interesting in its
application to brain signals, where the interpretation of a direct connection
between two cortical regions is straightforward.
10.2.1.
MultiVariate AutoRegressive models
The approach based on multivariate autoregressive models (MVAR) can
simultaneously model a whole set of signals. Let
X
be a set of estimated cortical
time series:
X
=
[
x
(
t
),
x
(
t
),...,
x
(
t
)]
(10.3)
1
2
N
where
t
refers to time and
N
is the number of cortical areas considered. Given an
MVAR process which is an adequate description of the data set
X
:
p
(
)
( )
=
Λ
(
k
)
X
t
−
k
=
E
t
(10.4)
k
0
where
X
(
t
) is the data vector in time;
E
(
t
)
=
[
e
1
(
t
),…,
e
N
] is a vector of
multivariate zero-mean uncorrelated white noise processes; Λ(1), Λ(2), …,Λ(
p
)
are the
N × N
matrices of model coefficients (Λ(0) =
I
);
and
p
is the model order.
The
p
order is chosen by means of the Akaike Information Criteria (AIC) for
MVAR processes. In order to investigate the spectral properties of the examined
process, the Eq. (10.4) is transformed into the frequency domain:
Λ
(
f
)
X
(
f
)
=
E
(
f
)
(10.5)
where:
p
( )
−
j
2
π
f
∆
tk
=
Λ
f
=
Λ
(
k
)
e
(10.6)
k
0
