Biomedical Engineering Reference
In-Depth Information
(
−
)
since
Pq
holds, the estimation problem is not an ill-posed problem.
The EM algorithm for the Gaussian model is described in Sect. B.5 in the Appendix.
According to Eqs. (B.24) and (B.25), the E-step update equations are given by
K
P
G
T
ʓ
=
ʛ
G
+
ʦ
(8.117)
=
ʓ
−
1
G
T
x
¯
ʛ
y
(8.118)
The parameters,
ʦ
and
ʛ
, are estimated in the M step of the EM algorithm. The
update equation for
ʦ
is obtained from Eq. (B.42) with setting
K
=
1 in that equation,
resulting in
diag
+
ʓ
−
1
ʦ
−
1
x
T
=
x
¯
¯
,
(8.119)
where diag
indicates a diagonal matrix whose diagonal elements are equal to
those of a matrix in the parentheses. According to Eq. (B.45), the noise precision
matrix
[·]
ʛ
is given by
diag
ʓ
−
1
G
T
ʛ
−
1
2
=
y
−
G
x
¯
+
G
.
(8.120)
The estimate of the MVAR coefficients is obtained as
, after the EM iteration
is terminated. This algorithm is similar to, but considerably simpler than the one
proposed in [
15
].
x
¯
8.8 Numerical Examples
8.8.1 Experiments Using Bivariate Causal Time Series
Numerical experiments were performed to illustrate the properties of the causality
measures described in this chapter. The source-space causality analysis was applied
in which source time series are first estimated from simulated MEG recordings,
and the MVAR coefficients are then computed using the time series at selected
voxels. Here, we use a sensor alignment of the 275 whole-head MEG sensor array
from Omega
TM
(VMS Medtech, Coquitlam, Canada) neuromagnetometer. Three
sources are assumed to exist on the vertical single plane of
x
0 cm, as in the
numerical experiments in the previous chapters. The source-sensor configuration and
the coordinate system are depicted in Fig.
8.1
. As shown in this figure, we assume
three sources and the time series of these three sources are denoted
s
1
(
=
t
)
,
s
2
(
t
)
,
and
s
3
(
. The first experiments assume a causal relationship between bivariate time
series, and an information flow exists from
s
1
(
t
)
t
)
to
s
2
(
t
)
. The time series
s
1
(
t
)
and
s
2
(
t
)
are generated by using the MVAR process [
11
]
