Digital Signal Processing Reference
In-Depth Information
Furthermore, if the weights from the previous time step were all equal (because of
resampling), the previous update equation becomes even simpler
w
(
m
)
(
n
)
¼ f
( y(
n
)
j
x
(
m
)
(
n
))
(
m
)
(
n
) is only proportional to the likelihood of that
that is, the weight of the particle x
particle.
Next, we explain in more detail the steps of the SIR method. In the first step we
draw candidate particles x
(
m
)
c
(
n
) according to
(
m
)
c
(
m
)
(
n
1)),
m ¼
1, 2,
...M:
x
(
n
)
f
(x(
n
)
j
x
In the second step, we compute the non-normalized weights of these particles by
w
(
m
)
(
n
)
¼ f
( y(
n
)
j
x
(
m
)
c
(
n
)),
m ¼
1, 2,
... M
and then normalize them so that they sum up to one, that is, we use
w
(
m
)
(
n
)
¼
w
(
m
)
(
n
)
P
j¼
1
w
(
j
)
(
n
)
:
(
n
),
w
(
m
)
(
n
)}
m¼
1
,
(
m
)
c
With the completion of this step, we have the randommeasure {x
which should be used for computations of desired estimates.
In the third and final step we perform resampling of the particles x
(
m
c
(
n
) according
to the multinomial probability mass function defined by the weights
w
(
m
)
(
n
). Namely,
we draw indices
k
m
, for
m ¼
1, 2,
...
,
M
, where
Pr
(
k
m
¼ i
)
¼ w
(
i
)
(
n
). Once the
indexes are drawn, we set
(
m
)
(
n
)
¼
x
(
k
m
)
c
x
(
n
)
:
(
m
)
(
n
) are set to
w
(
m
)
(
n
)
¼
1
/
M
. The whole SIR
procedure is summarized in Table 5.3.
If resampling was not implemented as a third step, we have
The weights of the particles
x
(
m
)
(
n
)
¼
x
(
m
)
c
x
(
n
)
w
(
m
)
(
n
)
¼ w
(
m
)
(
n
1)
f
( y(
n
)
j
x
(
m
)
c
(
n
))
for
m ¼
1, 2,
...M
, and then we normalize the weights. After normalization, we would
typically perform a test to decide if resampling is needed. If it was necessary, we would
implement it as described; if not, we would proceed with the next time step. Recall that
with resampling some particles are removed and the ones that are preserved may be
replicated. We reiterate that, in general, resampling does not have to be implemented
at every time step, but here we adopt to do so. If resampling is not performed at the end
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