Biomedical Engineering Reference
In-Depth Information
Usually, the importance density
q
(
x
i
k
|
x
i
k
-1
,
z
k
) is often chosen as the prior density
p
(
x
i
k
|
x
i
k
-1
) for
convenience. The importance density method requires the generation of new samples from
p
(
x
i
k
|
x
i
k
-1
).
The new sample can be generated by the system model with a process noise sample
v
i
k
-1
generated
according to the PDF of
v
k
-1
. For the importance density that we choose the posterior filtered density
p
(
x
k
|
z
1:
k
) can be approximated as
N
∑
δ
i
i
p x
(
|
z
)
≈
w x
(
x
)
-
k
1
k
k
k
k
(5.27)
i
=
1
where
i
i
i
w
∝
w p z
(
|
x
)
(5.28)
-
1
k
k
k
k
It can be shown that when the number of the samples is very large, (
5.27
) approaches the true
posterior density
p
(
x
k
|
z
1:
k
).
The particle filter based on important sampling displays a phenomenon called degeneracy
[
36
]. All but a few particles have negligible weight after several iterations, which implies a large
computational effort to update the particles with a minute contribution in the estimation of the
posterior density and results in a loss of diversity in the particle pool. When a significant degen-
eracy appears,
resampling
is implemented to reduce degeneracy [
36
]. The basic idea of resampling
is to eliminate the particles with small weight and to concentrate on particles with large weights.
Resampling is applied at every time index, so that the samples are i.i.d. from the discrete uniform
density with
w
i
k
-1
= 1/
N
. The weight then changes proportionally given by
i
k
i
k
w
∝
p
(
z
|
x
)
(5.29)
k
The weights given by (
5.29
) are normalized every time before resampling. The assump-
tions behind resampling are weak. The state dynamics
and measurement func-
f
(
x
,
v
)
k
k
−
k
1 −
1
tion
xh
are required to be known. The realization of the posterior density of the state
x
k
given the measurement
z
k
is sampled from the process noise distribution
v
k
and the prior. Finally,
the likelihood function
p
(
z
k
|
x
k
) is also required to be available to generate new weights. The new
sample
(
,
n
)
k
k
k
i
k
i
k
−
is generated by setting
with a process noise sample
x
~
p
(
x
|
x
)
x
=
f
(
x
,
v
)
k
1
k
k
k
−
1 −
k
1
i
k
k
v
.
The particle filter can be used as a statistical learning and probabilistic inference technique to
infer the hand position of a subject from multielectrode recording of neural activity in motor cortex
[
26
].
, where
is the PDF of
v
~
p
(
v
)
p
(
v
)
−
1
v
k
−
1
v
k
−
−
The hand movement (position, velocity, and acceleration) can be modeled as the system state
x
k
.
The firing rate of the neurons can be modeled as the observation (measurement)
z
k
. A non-Gaussian
model of “tuning” that characterizes the response of the cell firing
z
k
conditioned on hand velocity
x
k
