J.Q. Smith and Antonio Santos
simplified,
2 2 7 2 ∖ ..2 2 ..2 2 __ ..2 X
/ I *∖ α αt ( 1 I αt ʌ , μt,k , μt,k μt,k ʌ ∕qo∖
g(y⅛) = exp (j-2- (41 + yy) + . . ( )
After sampling k from a distribution proportional to (32), the particle αt-1,k is
chosen, and the density, assuming the role of prior density, assumes a Gaussian form
with mean μt,k = φαt- 1 ,k and variance σiη. This is combined with a Gaussian density
with mean αt2 and variance σ2 =2. The approximating density thus becomes:
g (αtlαt- 1 ,k, α2) = Я μkk,σt,k}
(33)
where
2
μt,k
2 μt,k + σηα2
2 + σ2η
(34)
and
2
σt,k
(35)
After the particles have been sampled, they must be resampled in order to take into
account the target density. They are resampled using the second stage weights
log wj =
πt,j =
αtj
2
wj
y2 + α,j
2β2 exp (αt,j )
22
t2 )
(36)
m,
j=1 wj
j=1,...,m
(37)
Following the resampling stage, an approximation of the target posterior distribution
of the states at t is available, which will be used as a prior distribution to update
the states at t + 1.
To summarize, the particles at t - 1 propagated to update the distribution of
the states at t are chosen randomly according to the weights defined in (32). These
weights are influenced by the information contained in yt . By conditioning on each
particle chosen through the first stage weights, new particles are sampled. As these
come from an approximating distribution, a second step is necessary. The particles
are resampled using the weights defined in (36)-(37). Our modification, outlined
above, makes this second order APF straightforward and quick to implement.
G.E.M.F - F.E.U.C.
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