Biomedical Engineering Reference
In-Depth Information
To determine
K
we first find a convenient form for the analysis covariance matrix
P
a
, then minimize variance which is defined by the cost function
J
(
K
)=
tr
(
P
a
)
,
(26)
where
tr
stands for the trace of a matrix. Below we assume that the observation and
background error are uncorrelated; i.e., E
u
b
]=
T
Then since
u
a
and
u
a
differ only by the state vector,
u
, the analysis covariance matrix is given by
[
ε
E
[
u
b
ε
]=
0
.
u
a
u
a
]
P
a
=
E
[
(27)
E
u
b
u
b
(
T
K
T
T
=
(
I
−
KH
)
I
−
KH
)
+
ε
(28)
u
b
(
T
K
T
T
+
K
ε
I
−
KH
)
ε
.
(29)
Using standard properties of expected value and the assumption of uncorrelated
errors we see that Eq. (
29
) simplifies to
u
b
u
b
](
T
T
K
T
P
a
=(
I
−
KH
)
E
[
I
−
KH
)
+
K
E
[
εε
]
(30)
T
KRK
T
=(
I
−
KH
)
P
b
(
I
−
KH
)
+
.
(31)
HP
b
H
T
Next let
N
=
+
R
. Expanding Eq. (
31
) we deduce that
KHP
b
H
T
K
T
P
b
H
T
K
T
KRK
T
P
a
=
P
b
−
KHP
b
+
−
+
(32)
P
b
H
T
K
T
HP
b
H
T
K
T
=
P
b
−
KHP
b
−
+
K
(
+
R
)
(33)
P
b
H
T
K
T
KNK
T
=
P
b
−
KHP
b
−
+
.
(34)
To derive the minimizer we differentiate Eq. (
26
) with respect to
K
and apply the
identities
∂
∂
ABA
T
A
(
tr
[
]) =
2
AB
,
(35)
∂
∂
]) =
∂
∂
B
T
A
(
tr
[
AB
A
(
tr
[
BA
]) =
.
(36)
This yields
∂
J
2
P
b
H
T
K
=
−
+
2
KN
.
(37)
∂
Setting the derivative to zero and solving for
K
we find the candidate for
K
is
P
b
H
T
N
−
1
K
=
.
(38)
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