G. KITAGAWA
3. MONTE CARLO FILTERING
In the
following
two
sections,
we show a Monte
Carlo based
algorithm
for recur-
sive evaluation of
the
predictor
and the
filter,
separately.
Two different
algorithms
for
smoothing
are discussed
in Section
4.
3.1 ONE STEP
AHEAD PREDICTION
In this
subsection,
we show that a set
of m
realizations
{pn)
.
p(m)
}
can
be
ob-
tained from {f)
..,
fm)
},
a
set
of
realizations
ofp(xn-
IYn-).
Let
{v),
..,Vm)}
be
independent
realizations
of the
system
noise
Vn.
Namely,
for
j
=
1,...,
m
fn-()1
'p(Xn-IY
1)I
V()
v
q(v).
(3.1)
Here
the
predictive
distribution
p(XnlYn-
)
can
be
expressed
by
p(Xn,Xn- ,Vn\Yn-_)dVndXn-
=
P(XnlXn-l,V n,Yn-l)p(VnlXn-l,Yn-I)p(Xn-llYn-l)dVndXn-
J/
p(XnlXn-l
,Vn)p(Vn)P(Xn-l
IYn-l)dVndxn-l
/
6(Xn -F(Xn-
,Vn))p(Vn)p(Xn-
IYn-!
)dvndXn-l, (3.2)
with
6(x)
being
the
delta
function.
Therefore,
if
we
define
p(i),
the
jth
particles,
by
P()
=
F(f,v
(3.3)
{p().
pm)
P
}
can be considered
as
independent
realizations
of the one
step
ahead
pre-
dictor
density,
p(Xn
Yn-l).
We note
that since
f)
and
v(j) are
independent
realization
from
p(xn- lYn,-)
and
p(vn),
respectively
the
prediction
formula
is valid even for
a
non-linear model.
3.2
FILTERING
Given the observation
yn
and the
particle,
pi),
compute
a(i),
the likelihood
of the
particle
pj)
based on the observation
yn.
That
is,
)
-p (ynPn )
=
r
(G (Yn,Pn
)))
OG
for
j
=
1,
...,
m.
Note that
G
is the inverse function
of
H,
and
r is
the
density
of the
observation
noise w.
Next,
we obtain
m
particles
{fn),...,
fn)
}
by
the
resampling
of
{pn
)..,
p)
},
with the
probabilities proportional
to
a
,...,
am,
respectively.
Namely,
define
f(j)
4
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