Biomedical Engineering Reference
In-Depth Information
problem is completely defined by specifying the prior distribution
p
(
x
0
). The posterior distribution
p
(
x
0:
t
|
u
1:
t
,
z
1:
t
) constitutes the complete solution to the sequential estimation problem, however many
times we are interested in one of the marginals, such as the
filtering density
p
(
x
t
|
u
1:
t
,
z
1:
t
), because
one does not need to keep track of all the model parameters, and still can answer many important
modeling questions.
According to (
5.6
), there are two models required to analyze and infer the state of a dynami-
cal system: the system model, which describes the evolution of the state with time, and the continu-
ous observation measurement model, which relates the noisy measurements to the state. There are
two stages to adapt the filtering density: prediction and update. The prediction stage uses the system
model
p
(
x
t
+ 1
|
x
t
) to propagate into the future the posterior probability density of the state given the
observation, called the Chapman-Kolmogorov equation
p
(
x
t
-1
|
u
1:
t
-1
,
z
1:
t
-1
) as follows
=
∫
p x u
(
|
,
z
)
p x
(
|
x
) (
p x
|
u
,
z
)
d
x
,
(5.7)
t
1
:
t
−
1
1
:
t
−
1
t
t
−
1
t
−
1
1
:
t
−
1
1
:
t
−
1
t
−
1
The update stage applies Bayes rule when new data (
u
t
,
z
t
) is observed
p z
(
|
x
,
u
) (
p x
|
x
,
z
)
t
t
t
t
1
:
t
-
1
1
:
t
-
1
(5.8)
p x
(
|
u
,
z
)
=
t
1
:
t
1
:
t
p u
(
|
u
,
z
-1
)
t
t
1
:
t
To evaluate (5.6) and (5.7), one needs to still compute the system model from the process noise
v
t
-1
as
=
∫
∫
p x
(
|
x
)
p x v
(
|
,
x
) (
p v
|
x
)
d
v
=
Ξ
(
x
v
x
) (
p v
)
d
v
(5.9)
-
-
t
t
-
1
t
t
-
1
t
-
1
t
-
1
t
-
1
t
-
1
t
t
-
1
t
-
1
t
-
1
t
-
1
where the notation Ξ(.) means that the computation is deterministic, and the conditional prob-
ability reduces to
p
(
v
t
-1
) because of the assumed independence of the noise. The likelihood density
function is determined by the measurement model
ˆ
(
∫
Ξ
(
|
,
)
=
(
-
,
)
-
n p n
d
n
) (
)
p z
x u
z
f u x
(5.10)
t
t
t
t
t
t
t
t
t
Although the evidence is estimated by
=
∫
p z z
(
|
,
u
)
p z
(
|
x u p x
,
) (
|
z
,
u
)
d
x
(5.11)
t
1
:
t
-
1
t
t
t
t
t
1
:
t
1
1
:
t
1
t
-
-
When one closely examines the procedure of computing these expressions, it is clear that the
difficulty is one of integrating PDFs. The most popular approach is to assume a Gaussian distri-
bution for the PDFs and proceed with closed form integration, which provides the Kalman filter
when the model is linear [
17
], and its variants (extended Kalman filter (EKF) and the unscented
