Biomedical Engineering Reference
In-Depth Information
It is possible to prove that
z
G
.
z
;ƒ
z
/ with
z
D
E
.H
w
C / D H
ƒ
z
D
E
.
z
z
/
T
.
z
z
/
D HƒH
T
C R:
For the conditional probabilities, we find that
p
jƒ
z
j
p
.2/
n
exp
2
J
1
p
w
j
z
D
jƒjjRj
where J D .
w
w
/
T
ƒ
e
.
w
w
/ and ƒ
1
C H
T
R
1
Hand
w
MV
E
p
w
j
z
D ƒ
e
H
T
R
1
z
C ƒ
1
D
w
MAP
:
D ƒ
1
e
Moreover, we find that p
z
j
w
has average H
w
and ƒ
z
j
w
D R. If we maximize the
likelihood, we obtain
w
ML
D H
1
z
:
Again, it is possible to verify that the MV/MAP estimator is unbiased and the
ML estimator is obtained by the MAP, when ƒ
1
! 0.
Remark 6.1.
Contrarily to what previous examples may suggest, the coincidence of
MV and MAP is not true in general.
6.2.3
The Kalman Filter for Linear Problems
Kalman filter [
48
] is one of the most important algorithms of the twentieth century,
with an exceptional number of applications, ranging from robotics to mathematical
finance. It is concerned with the case of a dynamical system, when the variable to be
estimated is supposed to be the solution of a time-dependent linear system. Since for
the biomedical applications of interest here, dynamics is in general given by the time
discretization of a PDE (as we will see later on), here we consider a time-discrete
case, even though the time-continuous case can be investigated as well. The time
index will be denoted by k, and we represent the dynamics of interest (indexed by
k) of the system as
u
.k/
D A
k
1
u
.k
1/
C
b
.k
1/
(6.9)
where
b
.k
1/
is a
Gaussian white noise
in time representing the model error, i.e.,
b
.k/
G
.
0
; Q
k
/, and the errors are not correlated in time, i.e.,
E
b
.k/
b
.l/;T
D Q
k
ı
kl
:
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