Biomedical Engineering Reference
In-Depth Information
In this chapter, the whole time series data
y
1
,
y
2
,...,
y
K
is collectively denoted
y
,
and the whole time series data
x
1
,
x
2
,...,
x
K
are collectively denoted
x
. We assume
that the noise
ʵ
is Gaussian and is identically and independently distributed across
time, i.e.,
,
ʛ
−
1
ʵ
∼
N(
ʵ
|
0
),
(B.13)
where we omit the notation of the time index
k
from
is a diagonal
precision matrix of which the
j
th diagonal entry is equal to the noise precision for the
j
th observation data. Then, using Eqs. (
B.13
) and (
C.3
), the conditional probability
p
ʵ
.InEq.(
B.13
),
ʛ
(
y
k
|
x
k
)
is obtained as
Hx
k
,
ʛ
−
1
p
(
y
k
|
x
k
)
=
N(
y
k
|
).
(B.14)
The conditional probability of the whole time series of
y
k
given the whole time series
of
x
k
is given by
p
(
y
|
x
)
=
p
(
y
1
,...,
y
K
|
x
1
,...,
x
K
)
K
K
Hx
k
,
ʛ
−
1
=
p
(
y
k
|
x
k
)
=
1
N(
y
k
|
).
(B.15)
k
=
1
k
=
The prior distribution of
x
k
is assumed to be Gaussian and independent across
time:
,
ʦ
−
1
p
(
x
k
)
=
N(
x
k
|
0
).
(B.16)
The prior distribution for the whole time series of
x
k
is expressed as
K
K
,
ʦ
−
1
p
(
x
)
=
p
(
x
1
,...,
x
K
)
=
p
(
x
k
)
=
1
N(
x
k
|
0
).
(B.17)
k
=
1
k
=
In this case, the posterior probability is independent across time, and given by
K
p
(
x
|
y
)
=
p
(
x
1
,...,
x
K
|
y
1
,...,
y
K
)
=
p
(
x
k
|
y
k
).
(B.18)
k
=
1
(
x
k
|
y
k
)
The posterior probability
p
can be derived by substituting Eqs. (
B.16
) and
(
B.14
) into Bayes' rule:
p
(
x
k
|
y
k
)
∝
p
(
y
k
|
x
k
)
p
(
x
k
).
(B.19)
is performed in the following manner. Since we
know that the posterior distribution is also Gaussian, the posterior distribution is
assumed to be
Actual computation of
p
(
x
k
|
y
k
)
x
k
,
ʓ
−
1
p
(
x
k
|
y
k
)
=
N(
x
k
|¯
),
(B.20)
