Digital Signal Processing Reference
In-Depth Information
(5.34), and after the integration we obtain
ð
f
(x(
nþ
2)
j
x(
nþ
1))
f
(x(
nþ
1)
j
y(1 :
n
))
d
x(
nþ
1)
f
(x(
nþ
2)
j
y(1 :
n
))
¼
ð
f
(x(
nþ
2)
j
x(
nþ
1))
X
M
w
(
m
)
(
n
)
d
(x(
nþ
1)
'
m¼
1
(
m
)
(
nþ
1))
d
x (
nþ
1)
x
¼
X
M
w
(
m
)
(
n
)
f
(x(
nþ
2)
j
x
(
m
)
(
nþ
1))
:
m¼
1
Further, we approximate this PDF by
f
(x(
nþ
2)
j
y(1 :
n
))
'
X
M
w
(
m
)
(
n
)
d
(x(
nþ
2)
x
(
m
)
(
nþ
2))
m¼
1
(
m
)
(
nþ
2) by drawing it from
f
(x(
nþ
2)
j
x
(
m
)
(
nþ
1)).
where as before we obtain x
Thus,
the approximation of
the predictive density
f
(x(
n þ
2)
j
y(1 :
n
))
is
obtained by
(
m
)
(
nþ
1) from
f
(x(
nþ
1)
j
x
(
m
)
(
n
))
1. drawing particles x
(
m
)
(
n þ
2) from
f
(x(
nþ
2)
j
x
(
m
)
(
nþ
1)); and
2. generating x
(
m
)
(
n þ
2) the weights
w
(
m
)
(
n
) and thereby
3. associating with the samples
x
(
m
)
(
nþ
2),
w
(
m
)
(
n
)}
m¼
1
.
forming the random measure {x
At this point, it is not difficult to generalize the procedure for obtaining the approxi-
mation of
f
(x(
nþ k
)
j
y(1 :
n
)) for any
k
.
0. We propagate the particle streams from
time instant
n
to time instant
n þ k
by using the transition PDFs
f
(x(
n þ i
)
j
x(
nþ i
2
1)),
i ¼
1, 2,
...
,
k
, and associating with the last set of generated particles x
(
m
)
(
n þ k
)
the weights
w
(
m
)
(
n
) to obtain the random measure {x
(
m
)
(
nþk
),
w
(
m
)
(
n
)}
m¼
1
.
5.9 SMOOTHING
In Section 5.2, we pointed out that sometimes we may prefer to estimate x(
n
) based on
data y(1 :
nþ k
), where
k
.
0. All the information about x(
n
) in that case is in the
smoothing PDF
f
(x(
n
)
j
y(
n þ k
)). With particle filtering, we approximate this density
with a random measure in a similar way as we approximated filtering PDFs.
When we deal with the problem of smoothing, we discriminate between two types
of smoothing. One of them is called fixed-lag smoothing and the interest there is in the
PDF
f
(x(
n
)
j
y(1 :
nþ k
)), where
k
is a fixed lag. The other is called fixed interval
smoothing, where the objective is to find the PDF
f
(x(
n
)
j
y(1 :
N
)) for any
n
where
0
n
,
N
.
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