Biomedical Engineering Reference
In-Depth Information
ʱ
In the equation above, we use the update equation for
in Eq. (
5.79
). Therefore,
ignoring constant terms, the free energy is finally expressed as
K
K
K
2
log
|
ʛ
|
1
2
1
2
M
2
log
|
ʱ
|
y
k
ʛ
u
k
ʓ
¯
F
=
|
ʓ
|
−
y
k
+
1
¯
u
k
+
|
ʨ
|
.
(5.100)
k
=
1
k
=
Since the free energy is limited (i.e., upper bounded) by the likelihood log
p
(
y
|
ʸ
)
(where
collectively expresses the hyperparameters), increasing the free energy
increases the likelihood.
ʸ
5.3.4 Summary of the VBFA Algorithm
Let us summarize the VBFA algorithm. The algorithm is based on the factor analysis
model in Eq. (
5.2
). The algorithm estimates the factor activity
u
k
, the mixing matrix
A
, and the sensor-noise precision matrix
. In the estimation, an overspecified value
is set for the model order
L
, and appropriate initial values are set for
ʛ
¯
A
,
ʨ
, and
ʛ
.
In the E-step,
ʓ
and
u
k
(
k
¯
=
1
,...,
K
) are updated according to Eqs. (
5.58
) and
¯
A
, and
(
5.59
). In the M-step, values of
ʨ
are updated using Eqs. (
5.68
), and (
5.70
).
are updated using Eqs. (
5.79
) and (
5.86
). Since
we cannot compute the marginal likelihood, the free energy in Eq. (
5.100
)—which
is the lower limit of the marginal likelihood—is used for monitoring the progress of
the VBEM iteration. That is, if the increase of
The hyperparameters,
ʱ
and
ʛ
in Eq. (
5.100
) becomes very small
with respect to the iteration count, the VBEM iteration can be terminated.
Using the VBFA algorithm, the estimate of the signal component in the sensor
data,
F
y
k
, is given by
y
k
]=
¯
A
=
E
)
[
Au
]=
E
A
[
A
]
E
u
[
u
u
k
.
¯
(5.101)
(
A
,
u
u
k
and
¯
A
obtained when the VBEM
iteration is terminated. The sample covariance matrix can be computed using only
the signal component, and such a covariance matrix is derived as
In the equation above, we use the values of
¯
(
T
E
u
E
A
Au
k
u
k
A
T
K
K
1
K
1
K
R
=
E
Au
k
)(
Au
k
)
=
(
u
,
A
)
k
=
1
k
=
1
E
A
A
E
u
u
k
u
k
A
T
E
A
AR
uu
A
T
K
1
K
1
K
=
=
,
(5.102)
k
=
1
where
R
uu
is defined in Eq. (
5.15
). We can show
E
A
AR
uu
A
T
+
ʛ
−
1
tr
R
uu
ʨ
−
1
=
¯
AR
uu
¯
A
T
,
(5.103)
