Biomedical Engineering Reference
In-Depth Information
In the linearized formulation
H
and
M
are related by the following
additive noise
relation
U
C D
M
)
Z
H
C D
M
:
(6.57)
Here, the
observation operator
between
H
and
M
(see the example in Sect.
6.2.2
for Gaussian vectors) is actually the inverse of a (discrete) differential operator;
Z D DS
1
R
in
M
in
has been introduced after (
6.52
) and it describes the deterministic
relation between the velocity and the normal stress. The random variable accounts
for the measurement noise. We make the assumption of mutual independence of
U
and .Since
H
and
U
are related by a linear relation this implies the independence
of
H
and . As a consequence, the p.d.f. of is independent of any realization of
H
and the likelihood function, p
M
j
H
, can be expressed as
p
M
j
H
.M/ D p
M
j
H
. C ZH/D p
.M ZH/:
(6.58)
Next, we consider the realization M D
d
(the vector of available velocity measures
introduced previously), we have
p
H
j
M
.H/ / p
M
j
H
.
d
/p
H
.H/ D p
.
d
ZH/p
H
.H/:
(6.59)
Now we make the assumption that all variables are Gaussian and we define the a
priori distribution and the noise distribution as follows:
p
H
D g
H
/ exp
2
.H H
0
/
T
ƒ
H
.H H
0
/
;
1
p
D g
/ exp
2
.
0
/
T
ƒ
1
.
0
/
(6.60)
1
I
where H
0
and
0
are the expectation values and ƒ
H
and ƒ
are the correlation
matrices for
H
and , respectively. The analysis of Sect.
6.2.2
shows that the
posterior distribution p
H
j
M
is a Gaussian distribution itself with covariance and
mean given by
ƒ
H
j
M
D .ƒ
H
C Z
T
ƒ
1
Z/
1
;
E
(6.61)
.
H
/ D ƒ
H
j
M
.Z
T
ƒ
1
.
d
0
/ C ƒ
H
H
0
/:
We recall that the mean value of the posterior distribution is the value that maximizes
p
H
j
M
, and then, by definition, it is the MAP estimator of
H
,say
H
MAP
.Onthe
other hand, the value that maximizes the likelihood function, with respect to H,
corresponds to the ML estimator for
H
and has the following expression
H
ML
D .Z
T
ƒ
1
Z/
1
.Z
T
ƒ
1
.
d
0
//:
(6.62)
In treating the nonlinearity we consider an iterative approach similar to the
deterministic one described in the previous section; in fact, also in this case, we
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