Biomedical Engineering Reference
In-Depth Information
from the previous step (leftmost circle and set of dots) is evolved under the model
resulting in propagation of the uncertainty, represented by the spread of the green
dots and size of the blue ovals. The Kalman filter algorithm yields a new analysis
that has reduced uncertainty. The lines connecting the dots from the background,
analysis, and observation represent the fact that the analysis is computed by finding
the optimal linear combination of the discrepancy between the background and
observation.
To derive the updated analysis u a n we seek a minimum variance, unbiased
estimator of the true state, u , that satisfies the recursive update equation
K n u b n +
K n y n .
u a n =
(17)
The matrices K and K can be thought of as telling us how much we should trust the
background and observations. Below all quantities are at timepoint n so we drop this
subscript. We first simplify Eq. ( 17 ) by eliminating the matrix K .Todosoconsider
the estimation error equations
u a =
u
+
u a ,
(18)
u b =
u
+
u b .
(19)
Substituting Eq. ( 17 )infor u a and solving for u a yield
K u b +
Ky o
u a =
u
.
(20)
Now substitute Eq. ( 19 )for u b and Eq. ( 12 )for y o
into Eq. ( 20 ). This gives, after
rearrangement,
K +
Ky o
K u b +
u a =[
I
]
u
+
K
ε .
(21)
Since we want the updated analysis to be unbiased, we force E
[
u a ]=
0. Taking the
expected value of both sides of Eq. ( 21 )weget
K +
K E
0
=[
KH
I
]
u
+
[
u b ]+
K E
[ ε ]
(22)
K +
=[
KH
I
]
u
,
(23)
where we have used the fact that
ε
is Gaussian with zero mean, and background is
unbiased. It follows that
K =
I
KH
.
(24)
Substituting this into Eq. ( 17 ) we get the Kalman filter update equation
y o
u a
=
u b
+
[
H u b
] .
K
(25)
In Eq. ( 25 ), K is called the Kalman Gain matrix. It represents the proper linear
combination of the discrepancy between the background and observations that
minimizes the variance of the updated analysis.
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