Biomedical Engineering Reference
In-Depth Information
represent the required posterior density function by a set of random samples, which are called par-
ticles, with associated weights, and to estimate the posterior density of the state given measurement
based on these samples and weights. This Monte Carlo characterization becomes an equivalent
representation to the functional description of the posterior PDF of the states when the number of
the samples is large enough, and the particle filter solution will also approach the optimal Bayesian
estimation.
i
i
k
N
i
i
N
i
{
x
k
w
,
}
{
x
}
Let
=
denote a random measure of the posterior PDF
P
(
x
0:
k
|
z
1:
k
), where
=
0
1
0
k
1
N
å
=
1
i
k
N
i
w
i
{
w
}
1
. Then, the
is a set of
N
states up to time
k
with associated normalized weights
=
,
1
i
=
posterior density at time
k
can be represented by a discrete weighted approximation,
N
∑
δ
i
i
p x
(
|
z
)
w x
(
x
)
≈
-
.
(5.24)
0
:
k
1
:
k
k
0
:
k
0
:
k
i
=
1
The weights are chosen according to the principle of Importance Sampling, which generates
samples easily from a proposal Importance Density defined as
q
(
x
0:
k
|
z
1:
k
) [
34, 35
],
i
p x
(
|
z
z
)
)
i
0
:
k
1
:
k
w
∝
k
i
q x
(
|
0
:
k
1
:
k
i
i
i
i
p z
(
|
x
,
z
)
p x
(
|
x
,
z
)
p
(
x
|
z
)
-
-
-
k
0
:
k
1
:
k
1
k
0
:
k
-
1
1
:
k
1
0
:
k
1
-
1
:
k
1
=
i
p z
(
|
z
) (
q x
|
z
)
k
1
:
k
-
1
0
:
k
1
:
k
i
i
i
i
p
(
z
|
x
)
p x
(
|
x
) (
p
x
|
z
)
k
k
k
k
-
1
0
:
k
-
1
1
:
k
-
1
=
i
p z
(
|
z
1
) (
q
x
|
z
)
-
k
1
:
k
0
:
k
1
:
k
i
i
i
i
p z
(
|
x
) (
p x
|
x
) (
p x
|
z
)
-
k
k
k
k
1
0
:
k
-
1
1
:
k
-
1
∝
i
q x
(
|
z
)
.
(5.25)
0
:
k
1
:
k
If the importance density is only dependent on
x
k
-1
and
z
k
, we can discard the path
x
i
0:
k
-1
and the
observation history
z
1:
k
to simply modify the weight by
i
i
i
i
p z
(
|
x
) (
p
x
|
x
)
p x
(
|
z
)
i
k
k
k
k
-
1
0
:
k
-
1
1
:
k
-
1
w
∝
k
i
q x
(
|
z
)
0
:
k
1
:
k
i
i
i
i
p
(
z
|
x
)
p x
(
|
x
) (
p x
|
z
)
k
k
k
k
-
1
0
:
k
-
1
1
:
k
-
1
∝
i
i
i
q x
(
|
x
,
z
)
q
(
x
|
z
)
k
0
:
k
-
1
1
:
k
0
:
k
-
1
1
:
k
-
1
i
i
i
p z
k
(
|
x
) (
p x
|
x
)
i
k
k
k
-
1
=
w
k
-
1
i
i
q x
(
|
x
,
z
)
.
(5.26)
k
k
-
1
k
