4.3 Existing smc Algorithms
4.3.1 Variable-Rate Particle Filter
In this section, we describe filtering for
pdp
s viasequential Monte Carlo(
smc
) methods. All three algorithms presented in this section may be viewed as special cases of the genericsir
algorithm from Section 2.3.1. Hence, we always use the same symbolsX1Wn,n,n,Pn andGn to referto the ‘states’, normalised and unnormalised extended target measures, proposal kernels and unnormalised incremental weights even though the particular form of these quantities may change between the next three subsections. The actual (marginal) target measure,Qn 2M.zEn/, and its
normalising constant,z
n, are defined as in Subsection 4.2.1.
The first particle filter for
pdp
s, termed variable-rate particle filter(
vrpf
), was proposed by Godsill and Vermaak (2004). Thevrpf
can be viewed as an application of the genericsmc
algorithm to a slightly reparametrised model described in the following.Let 0D t0 < t1; < t2 < : : : be a sequence of strictly increasing (non-
random) times, wheretp, forp > 1, represents the time of thepth
smc
step. Moreover, let.p;k; p;k/denote thekth jump time in the interval .tp 1; tpand its associated jump size. Letkp 0 be the total number of
XnWD.kn; n;1Wkn; n;1Wkn/, forn >1, takes values in (a subset of) EnWD 1 [ kD0 .fkg T.tn 1;tn;k Φ k /;
X1WD.k1; 1;1Wk1; 1;0Wk1/takes values in (a subset of) E1WD
1 [
kD0
.fkg T.0;t1;k ΦkC1/:
Let.n/ WDsupfm2Nn 1jkm>0gbe the index of the last interval
of the form.tp 1; tpbefore.tn 1; tnin which the
pdp
has had a jump,with the convention that if.n/ D 1, then we set.n/;k.n/ D0D0
and.n/;k.n/ D0. The target distribution is then given by
n WD n=z n onE1WnWD
pnD1Ep, where n.dx1Wn/ WDS.tn; .n/;k.n//q 0.d0/g.y.0;tnj.0;tn/ Y p2 zDn 1T.tp 1;tp ;kp.p;1Wkp/ q.dp;1j.p/;k.p/; p;1; .p/;k.p// f.dp;1j.p/;k.p// kp Y jD2 q.dp;jjp;j 1; p;j; p;j 1/f.dp;jjp;j 1/:Here,Dzn WD fp 2Nn jkp >0gis the collection of indices of intervals
of the form.tp 1; tpthat contain at least one jump. Fort 2.tn 1; tn, the
pdp
is then defined byt WD
(
F.t; n;j; n;j/; ifkn >0 andt 2Œn;j; n;jC1/, F.t; .n/;k.n/; .n/;k.n//; otherwise,
with the convention thatn;knC1Dtn. The extended distribution
n then
4.3 Existing
smc
Algorithms At Stepn, the algorithm generates a particle Xn from the proposalkernel
Pn.dxnjx1Wn 1/WDPn;1.dknjx1Wn 1/
Pn;2.dn;1Wkndn;1Wknjkn; x1Wn 1/:
In the above equation, the kernels on the right hand side are selected in such a way that the usual absolute-continuity conditions are satisfied. At Step 1, the kernelP1;2also samples a value for0.
The unnormalised incremental weight at Stepnis then given by the
following expressions. IfknD0, then with some abuse of the notation
for Radon–Nikodým derivatives,
Gn.x1Wn/D S.tn; .n/;k.n// S.t n 1; .n/;k.n// g .y .tn 1;tnj.n/;k.n/; .n/;k.n// Pn;1.knjx1Wn 1/ :
Ifkn1, then again with some abuse of notation, Gn.x1Wn/ D S .t n; n;kn/ S.t n 1; .n/;k.n// g.y.tn 1;n;1/j.n/;k.n/; .n/;k.n// P n.xnjx1Wn 1/ g.y.n;kn;tnjn;kn; n;kn/ q.n;1j.n/;k.n/; n;1; .n/;k.n//f .n;1j.n/;k.n// kn Y jD2 g.y.n;j 1;n;j/jn;j 1; n;j 1/ q.n;jjn;j 1; n;j; n;j 1/f.n;jjn;j 1/:
As shown in Whiteley et al. (2011), the
vrpf
can suffer from severe sample impoverishment. This is because at Stepn, jumps are only pro-posed in the interval.tn 1; tnand only based on information available
jumps in.tn 1; tn, as they usually are in
pdp
s, then at later steps, thisinformation can only be incorporated by reweighting particle paths. This can increase the variance of the particle weights which in turn aggravates the sample-impoverishment problem outlined in Subsection 2.4.1.
The