Biomedical Engineering Reference
In-Depth Information
the estimation problem based on the VBFA algorithm, Eq. (
5.135
) is rewritten as
s
k
u
k
y
k
=
Ls
k
+
Au
k
+
ʵ
=[
L
,
A
]
+
ʵ
=
A
s
z
k
+
ʵ
,
(5.136)
s
k
,
u
k
]
T
, and
A
s
where
z
k
. This equation indicates that the problem
of estimating
s
k
can be expressed by the factor analysis model using the augmented
factor vector
z
k
and the augmented mixing matrix
A
s
. Equation (
5.136
) is, in princi-
ple, the same as Eq. (
5.107
). Therefore, the algorithm developed here is very similar
to the PFA algorithm described in Sect.
5.4
.
=[
=[
L
,
A
]
5.5.2 Probability Model
We assume the prior distribution for
s
k
to be zero-mean Gaussian, such that
,
ʦ
−
1
p
(
s
k
)
=
N(
s
k
|
0
),
(5.137)
where
3 (non-diagonal) precision matrix of this prior distribution. The
prior distribution of the factor vector
u
k
is assumed to be:
ʦ
is a 3
×
(
u
k
)
=
N(
u
k
|
,
).
p
0
I
(5.138)
Then, the prior distribution for
z
k
is derived as
,
ʦ
−
1
p
(
z
k
)
=
p
(
s
k
)
p
(
u
k
)
=
N(
s
k
|
0
)N(
u
k
|
0
,
I
)
exp
s
k
exp
2
u
k
u
k
1
/
2
1
/
2
2
1
2
s
k
ʦ
I
2
1
=
−
−
ˀ
ˀ
exp
z
k
1
/
2
2
1
,
ʦ
−
1
2
z
k
ʦ
=
−
=
N(
z
k
|
0
),
(5.139)
ˀ
where
ʦ
0
0
I
ʦ
=
.
Equation (
5.139
) indicates that the prior distribution of
z
k
is the mean-zero Gaussian
with the precision matrix of
ʦ
. The sensor noise is also assumed to be zero-mean
Gaussian with a diagonal precision matrix
ʛ
. Hence, we have
Au
k
,
ʛ
−
1
A
s
z
k
,
ʛ
−
1
p
(
y
k
|
s
k
,
u
k
)
=
N(
y
k
|
Ls
k
+
)
=
N(
y
k
|
).
(5.140)
We assume the same prior distribution for the mixing matrix
A
,asshowninEq.(
5.42
).
