Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 584140, 25 pages
doi:10.1155/2012/584140
Research Article
Convergence Analysis for the SMC-MeMBer and
SMC-CBMeMBer Filters
Feng Lian,1 Chen Li,2 Chongzhao Han,1 and Hui Chen1
1
Ministry of Education Key Laboratory for Intelligent Networks and Network Security (MOE KLINNS),
School of Electronics and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, China
2
Software Engineering School, Xi’an Jiaotong University, Xi’an 710049, China
Correspondence should be addressed to Feng Lian, lianfeng1981@gmail.com
Received 27 February 2012; Revised 15 May 2012; Accepted 22 May 2012
Academic Editor: Qiang Ling
Copyright q 2012 Feng Lian et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The convergence for the sequential Monte Carlo SMC implementations of the multitarget multiBernoulli MeMBer filter and cardinality-balanced MeMBer CBMeMBer filters is studied here.
This paper proves that the SMC-MeMBer and SMC-CBMeMBer filters, respectively, converge to
the true MeMBer and CBMeMBer filters in the mean-square sense and the corresponding bounds
for the mean-square errors are given. The significance of this paper is in theory to present the
convergence results of the SMC-MeMBer and SMC-CBMeMBer filters and the conditions under
which the two filters satisfy mean-square convergence.
1. Introduction
Recently, the random finite-set- RFS- based multitarget tracking MTT approaches 1 have
attracted extensive attention. Although theoretically solid, the RFS-based approaches usually
involve intractable computations. By introducing the finite-set statistics FISSTs 2, Mahler
developed the probability hypothesis density PHD 3, and cardinalized PHD CPHD 4
filters, which have been shown to be a computationally tractable alternative to full multitarget
Bayes filters in the RFS framework. The sequential Monte Carlo SMC implementations for
the PHD and CPHD filters were devised by Zajic and Mahler 5, Sidenbladh 6, and Vo
et al. 7. Vo et al. 8, 9 devised the Gaussian mixture GM implementation for the PHD
and CPHD filters under the linear, Gaussian assumption on target dynamics, birth process,
and sensor model. However, the SMC-PHD and SMC-CPHD approaches require clustering
to extract state estimates from the particle population, which is expensive and unreliable
10, 11.
In 2007, Mahler proposed the multitarget multi-Bernoulli MeMBer 2 recursion,
which is an approximation to the full multitarget Bayes recursion using multi-Bernoulli
2
Journal of Applied Mathematics
RFSs under low clutter density scenarios. In 2009, Vo et al. showed that the MeMBer filter
overestimates the number of targets and proposed a cardinality-balanced MeMBer CBMeMBer filter 12 to reduce the cardinality bias. Then, the SMC and GM implementations for
the MeMBer and CBMeMBer filters were, respectively, proposed in nonlinear and linearGaussian dynamic and measurement models. The key advantage of this approach is that the
multi-Bernoulli representation allows reliable and inexpensive extraction of state estimates.
The Monte Carlo simulations given by Vo et al. showed that the SMC-CBMeMBer filter
outperforms the SMC-CPHD and hence SMC-PHD filter despite having smaller complexity
under certain range of signal settings.
Although the convergence results for the SMC-PHD and GM-PHD filters were
established by Clark and Bell 13 in 2006 and by Clark and Vo 14 in 2007, respectively, there
have been no results showing the asymptotic convergence for the SMC-MeMBer and SMCCBMeMBer filters. This paper demonstrates the mean-square convergence of the errors 15–
17 for the two filters. In other words, given simple sufficient conditions, the approximation
error of the multi-Bernoulli parameter set comprised of a set of weighted samples is proved
to converge to zero as the number of the samples tends to infinity at each stage of the two
algorithms. In addition, the corresponding bounds for the mean-square errors are obtained.
2. MeMBer and CBMeMBer Filters
A Bernoulli RFS Y i has probability 1 − r i of being empty, and probability r i 0 ≤ r i ≤ 1
of being a singleton whose only element is distributed according to a probability density pi .
The probability density of Y i is
π Y
i
i
1−r
i i
r p yi Y i ∅,
Y i {yi }.
2.1
A multi-Bernoulli RFS Y is a union of a fixed number of independent Bernoulli RFSs
i
described by the multi-Bernoulli
Y , i 1, . . . , M, that is, Y M
i1 Y . Y is thus completely
i i M
i
and the probability density
parameter set {r , p }i1 with the mean cardinality M
i1 r
2:
i
πY M j1
1−r
j
n r ij pij y
j
1≤i1 /
··· /
in ≤M j1
1 − r ij .
2.2
Throughout this paper, we abbreviate a probability density of the form 2.2 by π M
{r , pi }i1 .
By approximating the multitarget RFS as a multi-Bernoulli RFS at each time step,
Mahler proposed the MeMBer recursion, which propagated the multi-Bernoulli parameters
of the posterior multitarget density forward in time 2. The MeMBer filter is summarized as
follows.
i
Journal of Applied Mathematics
3
MeMBer Prediction
If at time k − 1, the posterior multitarget density is a multi-Bernoulli of the form πk−1 i
i
k−1
{rk−1 , pk−1 }M
i1 , then the predicted multitarget density is also a multi-Bernoulli and is given
by
πk|k−1 i
i
i
i
rP,k|k−1 , pP,k|k−1
Mk−1 MΓ,k
i
i
rΓ,k , pΓ,k
,
i1
2.3
i1
M
where {rΓ,k , pΓ,k }i1Γ,k are the parameters of the multi-Bernoulli RFS of births at time k:
i
i
i
rP,k|k−1 rk−1 pk−1 , pS,k ,
i
pP,k|k−1 xk i 1, . . . , Mk−1 ,
i
fk|k−1 xk | ·, pk−1 pS,k
,
i
pk−1 , pS,k
2.4
i 1, . . . , Mk−1 .
2.5
MeMBer Update
If at time k, the predicted multitarget density is a multi-Bernoulli of the form πk|k−1 M
i
i
{rk|k−1 , pk|k−1 }i1k|k−1 ; then the posterior multitarget density can be approximated by a multiBernoulli as follows:
πk ≈
where,
i
rL,k
i
i
rL,k , pL,k
Mk|k−1 i1
rU,k zk , pU,k ·; zk i
1 − pk|k−1 , pD,k
i
rk|k−1
,
i
i
1 − rk|k−1 pk|k−1 , pD,k
i
i
pL,k xk pk|k−1 xk rU,k zk κk zk Mk|k−1
i1
M
k|k−1
i
rα,k zk i1
i
i
rα,k zk i
rα,k zk ,
zk ∈ Zk ,
2.6
2.7
2.8
2.9
zk ∈ Zk ,
2.10
i 1, . . . , Mk|k−1 ,
2.11
i
rk|k−1 pk|k−1 xk ψk,zk xk ,
i
i
1 − rk|k−1 pk|k−1 , pD,k
i
pα,k xk ; zk , 1
,
i 1, . . . , Mk|k−1 ,
M
k|k−1
1
i
pα,k xk ; zk ,
pU,k xk ; zk M
k|k−1 i
rα,k zk i1
i1
i
pα,k xk ; zk zk ∈Zk
i 1, . . . , Mk|k−1 ,
1 − pD,k xk ,
i
1 − pk|k−1 , pD,k
1
i
i
rk|k−1 pk|k−1 , ψk,zk
,
i
i
1 − rk|k−1 pk|k−1 , pD,k
i 1, . . . , Mk|k−1 .
2.12
4
Journal of Applied Mathematics
By correcting the cardinality bias in the rU,k zk of the MeMBer update step, Vo et al.
proposed the CBMeMBer filter 12. The CBMeMBer recursions are the same as the MeMBer
recursions except the update of rU,k zk , which is revised as
∗
rU,k
zk κk zk 1
Mk|k−1
i1
M
k|k−1
i
rα,k zk i1
i
i
1 − rk|k−1 rα,k zk .
i
i
1 − rk|k−1 pk|k−1 , pD,k
2.13
Note that not 38 in 12 but 2.10 in our paper is used in the CBMeMBer update
step here. The reasons are 1 the 38 in 12 and the 2.10 in our paper are both the
i
approximations of 36 in 12 under the same assumption pk|k−1 , pD,k ≈ 1, but the latter
i
is more precise than former; 2 the 38 in 12 is unbounded at rk|k−1 1 while 2.10 in our
i
1.
paper is bounded at rk|k−1 1 as long as pD,k xk /
i
i
i
k
For the multi-Bernoulli representation πk {rk , pk }M
i1 , the probability rk indicates
i
how likely the ith hypothesized track is a true track, and the posterior density pk describes
k i
the distribution of the estimated current state of the track. Hence, M
i1 rk denotes the
multitarget number and the multitarget state estimate can be obtained by choosing the means
or modes from the posterior densities of the hypothesized tracks with existence probabilities
exceeding a given threshold.
3. SMC-MeMBer and SMC-CBMeMBer Filters
The SMC implementations of the MeMBer and CBMeMBer recursions are summarized as
follows.
SMC-MeMBer and SMC-CBMeMBer Predictions
i
i,L
Suppose that at time k − 1 the multi-Bernoulli posterior multitarget density πk−1 {rk−1 k−1 ,
i
i
i,L
i,L
k−1
pk−1 k−1 }M
is given and each pk−1 k−1 , i 1, . . . , Mk−1 , is comprised of a set of weighted
i1
i,j
i
i,j L
k−1
samples {ωk−1 , xk−1 }j1
:
i
i
i,L
pk−1 k−1 xk Lk−1
j1
i,j
k−1
i,j
i
3.1
i 1, . . . , Mk−1 .
ωk−1 δxi,j xk ,
i
Then, given proposal densities qk · | xk−1 , Zk and bk · | Zk , the predicted multii
i,L
i
i,L
i
i,Lk−1
rP,k|k−1
i
i,L
rk−1 k−1
i
i,Lk−1
pk−1 , pS,k ,
i
i,LΓ,k
k−1
k−1
k−1
, pP,k|k−1
}M
Bernoulli multitarget density πk|k−1 {rP,k|k−1
i1 ∪ {rΓ,k
computed as follows:
i 1, . . . , Mk−1 ,
i
i,LΓ,k
, pΓ,k
M
}i1Γ,k can be
3.2
Journal of Applied Mathematics
5
i
i
i,Lk−1
pP,k|k−1
xk Lk−1
j1
i,j
ω
P,k|k−1 δxi,j xk ,
i 1, . . . , Mk−1 ,
P,k|k−1
3.3
i
i
i,LΓ,k
pΓ,k
i
i,LΓ,k
where rΓ,k
i,j
i,j
LΓ,k
i,j
ω
Γ,k δxi,j xk ,
xk i 1, . . . , MΓ,k ,
Γ,k
j1
i,j
i,j
i 1, . . . , MΓ,k is given by birth model; xP,k|k−1 , ω
P,k|k−1 i 1, . . . , Mk−1 and
xΓ,k , ω
Γ,k i 1, . . . , MΓ,k are, respectively, given by
i,j
i
xP,k|k−1 ∼ qk
i,j
· | xk−1 , Zk ,
i,j
i
j 1, . . . , Lk−1 ,
i,j
ω
P,k|k−1 1
Li
k−1
i
xΓ,k ∼ bk · | Zk ,
i
j 1, . . . , LΓ,k ,
i,j
i,j
ωP,k|k−1 ,
ωP,k|k−1
i,j
i,j i,j
i,j
ωk−1 fk|k−1 xP,k|k−1 | xk−1 pS,k xk−1
,
i,j
i,j
i
qk xP,k|k−1 | xk−1 , Zk
j1
i,j
ωP,k|k−1
i,j
ω
Γ,k
1
Li
Γ,k
j1
i,j
i,j
ωΓ,k ,
i,j
ωΓ,k
ωΓ,k
3.4
3.5
i
j 1, . . . , Lk−1 ,
i,j pΓ,k xΓ,k
i,j
,
i
bk xΓ,k | Zk
i
j 1, . . . , LΓ,k .
3.6
SMC-MeMBer and SMC-CBMeMBer Updates
i
i,L
Suppose that at time k the predicted multi-Bernoulli multitarget density πk|k−1 {rk|k−1k|k−1 ,
i
i,L
i
i,L
M
pk|k−1k|k−1 }i1k|k−1 is given and each pk|k−1k|k−1 , i 1, . . . , Mk|k−1 , is comprised of a set of weighted
i,j
i
L
i,j
k|k−1
samples {ωk|k−1 , xk|k−1 }j1
:
i
Lk|k−1
i
i,Lk|k−1
pk|k−1 xk j1
i,j
3.7
i 1, . . . , Mk|k−1 .
ωk|k−1 δxi,j xk ,
k|k−1
Then, the multi-Bernoulli approximation of the SMC-MeMBer-updated multitarget
i
i,Lk|k−1
density πk ≈ {rL,k
i
i,Lk|k−1
, pL,k
i
i
i,Lk|k−1
updated multitarget density πk∗ ≈ {rL,k
can be computed as follows:
i
i,Lk|k−1
rL,k
i
L
M
L
k|k−1
k|k−1
}i1k|k−1 ∪ {rU,k
zk , pU,k
·; zk }zk ∈Zk and SMC-CBMeMBeri
i,Lk|k−1
, pL,k
M
i
∗,L
i
L
k|k−1
}i1k|k−1 ∪ {rU,kk|k−1 zk , pU,k
·; zk }zk ∈Zk
i
i,Lk|k−1
1 − pk|k−1 , pD,k
i
i,L
rk|k−1k|k−1
,
i
i
i,Lk|k−1
i,Lk|k−1
1 − rk|k−1
pk|k−1 , pD,k
i 1, . . . , Mk|k−1 ,
3.8
6
Journal of Applied Mathematics
i
i
i,Lk|k−1
pL,k
Lk|k−1
xk j1
i,j xk|k−1
1
−
p
D,k
i,j
ωk|k−1
δxi,j xk ,
i
k|k−1
i,Lk|k−1
1 − pk|k−1 , pD,k
i
i
∗,Lk|k−1
rU,k
M
k|k−1
1
L
k|k−1
rU,k
zk κk zk Mk|k−1
i1
zk κk zk ×
M
k|k−1
i1
i
rα,k
1
Mk|k−1
i1
Mk|k−1
i1
zk i
i1
i,Lk|k−1
rα,k
i
i,Lk|k−1
rα,k
zk ,
M
k|k−1
i
i,Lk|k−1
rα,k
zk i1
zk ∈ Zk ,
3.9
3.10
zk i
i
i,L
i,L
1 − rk|k−1k|k−1 rα,k k|k−1 zk ,
i
i
i,Lk|k−1
i,Lk|k−1
1 − rk|k−1
pk|k−1 , pD,k
1
L
k|k−1
pU,k
xk ; zk i
i,Lk|k−1
i 1, . . . , Mk|k−1 ,
i
i,Lk|k−1
pα,k
3.11
zk ∈ Zk ,
xk ; zk ,
zk ∈ Zk ,
3.12
where,
i
i,Lk|k−1
pα,k
i
i,j i
i,j i,Lk|k−1
ωk|k−1 rk|k−1 ψk,zk xk|k−1
xk ; zk δxi,j xk ,
i
i
k|k−1
i,Lk|k−1
i,Lk|k−1
j1
1 − rk|k−1
pk|k−1 , pD,k
Lk|k−1
i
i,Lk|k−1
rα,k
zk i 1, . . . , Mk|k−1 ,
3.13
i
i,L
pα,k k|k−1 xk ; zk , 1
i
i
i,L
i,L
rk|k−1k|k−1 pk|k−1k|k−1 , ψk,zk
,
i
i
i,L
i,L
1 − rk|k−1k|k−1 pk|k−1k|k−1 , pD,k
3.14
i 1, . . . , Mk|k−1 .
Resampling
To reduce the effect of degeneracy, we resample the particles for the multi-Bernoulli parameter set after the update step.
4. Convergence of the Mean-Square Errors for the SMC-MeMBer and
SMC-CBMeMBer Filters
To show the convergence results for the SMC-MeMBer and SMC-CBMeMBer filters, certain
conditions on the functions need to be met:
1 the transition
kernel fk|k−1 xk | xk−1 satisfies the Feller property 18, that is, for all
ϕ ∈ Cb Rd , ϕxk−1 φk|k−1 xk | xk−1 dxk−1 ∈ Cb Rd ;
Journal of Applied Mathematics
7
2 single-sensor/target likelihood density ψk,zk xk ∈ BRd ;
i
3 Qk are rational-valued random variables such that there exists p > 1, some constant
C, and α < p − 1 so that
N p
p
i
i
i Qk − Nωk q ≤ CN α q ,
E i1
with
N
i1
i
4.1
Qk N
for all vectors q q1 , . . . , qN ;
4 the importance sampling ratios are bounded, that is, there exists constants B1 and
i
i
i
B2 such that pΓ,k /bk ≤ B1 , i 1, . . . , MΓ,k , and fk|k−1 /qk ≤ B2 , i 1, . . . , Mk−1 ;
5 the resampling strategy is multinomial and hence unbiased 19.
First, the convergence of the mean-square errors for the initialization steps of the two
filters can easily be established by Lemma 0 in 13. Assuming that at time k 0, we can
i
sample exactly from the initial distribution p0 i 1, . . . , M0 . Then, for all ϕ ∈ BRd ,
2 i
c0
i,L0
i
≤ i , i 1, . . . , M0 ,
E r0
− r0
L0
2
i
2 d0
i,L
i
≤ ϕ i , i 1, . . . , M0
E
p0 0 , ϕ − p 0 , ϕ
L0
4.2
i
hold for some real numbers c0 > 0 and d0 > 0 which are independent of the number L0 of
the sampled particles at time k 0, i 1, . . . , M0 .
Also, the convergence of the mean-square errors for the resampling steps of the two
filters can easily be established by Assumption 5 and Lemma 5 in 19.
The main difficulty and greatest challenge is to prove the mean-square convergence
for the prediction steps and update steps of the two filters. They are, respectively, established
by Propositions 4.1 and 4.2.
Proposition 4.1. Suppose that, for all ϕ ∈ BRd ,
i
i,L
rk−1 k−1
2 ≤
ck−1
, i 1, . . . , Mk−1 ,
i
Lk−1
2
i
2 dk−1
i,Lk−1
i
≤ ϕ i , i 1, . . . , Mk−1 ,
E
pk−1 , ϕ − pk−1 , ϕ
Lk−1
E
−
i
rk−1
4.3
4.4
i
hold for some real numbers ck−1 > 0 and dk−1 > 0 which are independent of the number Lk−1 of the
resampled particles at time k − 1, i 1, . . . , Mk−1 .
8
Journal of Applied Mathematics
Then, after the prediction steps of the SMC-MeMBer and SMC-CBMeMBer filters at
time k:
2 i
cP,k|k−1
i,Lk−1
i
, i 1, . . . , Mk−1 ,
E rP,k|k−1 − rP,k|k−1
≤
4.5
i
Lk−1
2
i
2 dP,k|k−1
i,Lk−1
i
≤ ϕ
E
, i 1, . . . , Mk−1 ,
pP,k|k−1 , ϕ − pP,k|k−1 , ϕ
4.6
i
Lk−1
i
2
2 dΓ,k
i,LΓ,k
i
≤ ϕ i , i 1, . . . , MΓ,k ,
E
pΓ,k|k−1 , ϕ − pΓ,k|k−1 , ϕ
4.7
LΓ,k
hold for a constant dΓ,k > 0 and some real numbers cP,k|k−1 > 0 and dP,k|k−1 > 0 which are
i
independent of Lk−1 , i 1, . . . , Mk−1 . cP,k|k−1 and dP,k|k−1 are defined by A.8 and A.18,
respectively. The proof of Proposition 4.1 can be found in Appendix A.1.
Proposition 4.2. Suppose that, for all ϕ ∈ BRd ,
i
i,Lk|k−1
2 ≤
ck|k−1
, i 1, . . . , Mk|k−1 ,
i
Lk|k−1
i
2
2 dk|k−1
i,Lk|k−1
i
≤ ϕ i , i 1, . . . , Mk|k−1
E
pk|k−1 , ϕ − pk|k−1 , ϕ
Lk|k−1
E
rk|k−1
−
i
rk|k−1
4.8
4.9
i
hold for some real numbers ck|k−1 > 0 and dk|k−1 > 0 which are independent of the number Lk|k−1
of the predicted particles, i 1, . . . , Mk|k−1 . Then, after the update steps of the SMC-MeMBer and
SMC-CBMeMBer filters at time k:
2 i
i,Lk|k−1
cL,k
i
≤ i , i 1, . . . , Mk|k−1 ,
E rL,k
− rL,k
Lk|k−1
i
2
2 dL,k
i,Lk|k−1
i
≤ ϕ i , i 1, . . . , Mk|k−1 ,
, ϕ − pL,k , ϕ
E
pL,k
Lk|k−1
2 i
Lk|k−1
cU,k
≤ min , zk ∈ Zk ,
E rU,k zk − rU,k zk Lk|k−1
2 ∗
i
cU,k
∗,Lk|k−1
∗
≤ min , zk ∈ Zk ,
E rU,k zk − rU,k zk Lk|k−1
i
2 dU,k
2
Lk|k−1
≤ ϕ min , zk ∈ Zk
E
pU,k ·; zk , ϕ − pU,k ·; zk , ϕ
Lk|k−1
4.10
4.11
4.12
4.13
4.14
Journal of Applied Mathematics
9
∗
> 0, and dU,k > 0, which are independent
hold for some real numbers cL,k > 0, dL,k > 0, cU,k > 0, cU,k
i
∗
of Lk|k−1 . cL,k , dL,k , cU,k , cU,k
and dU,k are defined by A.29, A.35, A.47, A.55, and A.61,
1
M
k|k−1
∗
respectively. Lmin
minLk|k−1 , . . . , Lk|k−1
, min· denotes the minimum. In addition, cU,k , cU,k
,
k|k−1
and dU,k depend on the number of targets and decrease with the increase of the target number. From
i
∗,L
∗
A.47 and A.55, it can also be seen that cU,k
≥ cU,k . It indicates that rU,kk|k−1 zk may need more
i
L
k|k−1
particles than rU,k
zk to achieve the same mean-square error bound. The proof of the Proposition 4.2
can be found in Appendix A.2.
Propositions 4.1 and 4.2 show that the bounds for the mean-square error of the SMCMeMBer and SMC-CBMeMBer prediction steps and update steps at each stage depend on the
number of particles. The mean-square errors tend to zero as the number of particles tends to
infinity. The bounds for the mean-square errors of these quantities are inversely proportional
to the corresponding particle number.
Moreover, from the proofs of Propositions 4.1 and 4.2, it can be seen that
1 Assumptions 1, 3, and 4 ensure that 4.6 holds;
2 Assumption 4 ensures that 4.7 holds;
3 Assumption 2 ensures that 4.12, 4.13, and 4.14 hold;
4 Assumption 5 ensures the convergence of the mean-square errors for the
resampling steps of the two filters.
Assumptions 3, 4, and 5 are concerned with the SMC method. They can be satisfied as
long as the appropriate sampling strategies are chosen. Assumptions 1 and 2 are concerned
with the likelihood and target transition kernel. They may be too restrictive or unrealistic
for some practical applications. However, these convergence results give justification to the
SMC implementations of the MeMBer and CBMeMBer filters and show how the order of the
mean-square errors are reduced as the number of particles increases.
5. Simulations
Here, we briefly describe the application of the convergence results for the SMC-CBMeMBer
filter to the nonlinear MTT example presented in Example 1 of 12. The experiment settings
i
are the same as those of Example 1 except that the number of the particles Lk used for each
i
hypothesized track at time k. For convenience, we assume Lk L. Assumptions 1–5 are
satisfied in this example. So, the SMC-CBMeMBer filter converges to the ground truth in the
mean-square sense.
For the SMC-CBMeMBer filter, the estimates of the multitarget number and states,
which are derived from the particle multi-Bernoulli parameter set, are unbiased. Therefore,
via comparing the tracking performance of the algorithm in the various particle number L,
the convergence results for the SMC-CBMeMBer filter can be verified to a great extent.
The standard deviation of the estimated cardinality distribution and the optimal
subpattern assignment OSPA multitarget miss-distance 20 of order p 2 with cut-off
c 100 m, which jointly captures differences in cardinality and individual elements between
two finite sets, are used to evaluate the performance of the method. Table 1 shows the timeaveraged standard deviation of the estimated cardinality distribution and the time-averaged
OSPA in various L via 200 MC simulation experiments.
10
Journal of Applied Mathematics
Table 1: Time-averaged standard deviation of the estimated cardinality distribution and time-averaged
OSPA m in various L.
Particle number L
Time-averaged standard deviation of the estimated
cardinality distribution from the SMC-CBMeMBer filter
OSPA m from the SMC-CBMeMBer filter
100
500
1000
1500
2000
2.69
2.03
1.48
1.23
1.04
65.2
51.9
42.3
34.8
27.7
Table 1 shows that both the standard deviation of the estimated cardinality distribution and OSPA decrease with the increase of the particle number L. This phenomenon can be
reasonably explained by the convergence results derived in this paper: first, the mean-square
error of the particle multi-Bernoulli parameter set decreases as the number of the particles
increases; then, the more precise estimates of the cardinality distribution and multitarget
states can be derived from the more precise particle multi-Bernoulli parameter set, which
eventually leads to the results presented in Table 1.
6. Conclusions and Future Work
This paper presents the mathematical proofs of the convergence for the SMC-MeMBer and
SMC-CBMeMBer filters and gives the bounds for the mean-square errors. In the linearGaussian condition, Vo et al. presented the analytic solutions to the MeMBer and CBMeMBer
recursions: GM-MeMBer and GM-CBMeMBer filters 12. The future work is focused on
studying the convergence results and error bounds for the two filters.
Appendix
A.
In deriving the proofs, we use the Minkowski inequality, which states that, for any two random variables X and Y in L2 ,
1/2
1/2
1/2
A.1
E X Y 2
≤ E X2
E X2
.
Using Minkowski’s inequality, we obtain that, for all ϕ ∈ BRd ,
E
i
r i,L
E
≤E
r
2 !1/2
i
pi,L , ϕ − r i pi , ϕ
i,Li
p
i,Li
2 !1/2
i
i
, ϕ − r i pi,L , ϕ r i pi,L , ϕ − r i pi , ϕ
2 !1/2
2 2 !1/2
i
i
i
pi,L , ϕ
r i,L − r i
r i E
pi,L , ϕ − pi , ϕ
2 !1/2
2 !1/2
i
i
≤ ϕE r i,L − r i
r i E
pi,L , ϕ − pi , ϕ
A.2
A.3
A.4
holds, i 1, . . . , M, for the multi-Bernoulli density π {r i , pi }M
i1 and its particle approxi
i
i
imation π L {r i,L , pi,L }M
.
i1
Journal of Applied Mathematics
11
A.1. Proof of Proposition 4.1
We first prove 4.5. From 2.4 and 3.2, we have
2 1/2
2 1/2
i
i
i
i,Lk−1
i,Lk−1
i,Lk−1
i
i
i
E rk−1
E rP,k|k−1 − rP,k|k−1
pk−1 , pS,k − rk−1 pk−1 , pS,k
A.5
by A.4
≤ pS,k E
i
i,L
rk−1 k−1
−
i
rk−1
2 1/2
i
rk−1 E
i
i,L
pk−1 k−1 , pS,k
−
i
pk−1 , pS,k
2 1/2
A.6
i
by 4.3, 4.4, and 0 ≤ rk−1 ≤ 1
"
√
ck−1 dk−1
≤ pS,k .
#
i
Lk−1
A.7
2
"
2 √
cP,k|k−1 pS,k ck−1 dk−1 .
A.8
So that 4.5 is proved with
Now turn to 4.6. From 2.5, we have
E
i
i,Lk−1
,ϕ
pP,k|k−1
−
2
1/2
i
pP,k|k−1 , ϕ
+⎞2 ⎤1/2
* fk|k−1 , pi pS,k
i
k−1
⎟⎥
⎢⎜ i,Lk−1
E⎣⎝ pP,k|k−1
,ϕ −
,ϕ ⎠ ⎦
i
pk−1 , pS,k
⎡⎛
A.9
adding and subtracting a new term
i
* fk|k−1 , pi,Lk−1 pS,k
+
k−1
⎢⎜ i,Li
k−1
⎜
E⎢
,ϕ
i
⎣⎝ pP,k|k−1 , ϕ −
i,Lk−1
pk−1 , pS,k
⎡⎛
*
i
i,L
fk|k−1 , pk−1 k−1 pS,k
i
i,L
pk−1 k−1 , pS,k
*
+
,ϕ
−
i
fk|k−1 , pk−1 pS,k
i
pk−1 , pS,k
⎞2 ⎤1/2
+
⎟⎥
⎥
,ϕ ⎟
⎠⎥
⎦
A.10
12
Journal of Applied Mathematics
using Minkowski’s inequality
⎡⎛
⎞2 ⎤1/2
i
* fk|k−1 , pi,Lk−1 pS,k
+
⎢⎜ i,Li
k−1
⎟ ⎥
⎥
⎢
k−1
≤ E⎢⎜
pP,k|k−1
,ϕ −
,ϕ ⎟
i
⎝
⎠ ⎥
⎦
⎣
i,Lk−1
pk−1 , pS,k
⎡⎛ ⎞2 ⎤ 1/2
i
i,Lk−1
i
,
p
,
ϕ
p
f
S,k
k|k−1
⎢⎜ k−1
pk−1 , pS,k fk|k−1 , ϕ ⎟ ⎥
⎢
⎟⎥
.
E⎢⎜
−
i
⎝
⎠⎥
i
⎦
⎣
i,Lk−1
pk−1 , pS,k
pk−1 , pS,k
A.11
By Assumption 3 and Lemma 1 in 13, we easily obtain that the first term in A.11
becomes
⎡⎛
⎞2 ⎤1/2
i
i,Lk−1
*
+
fk|k−1 , pk−1 pS,k
⎢⎜ i,Li ⎟ ⎥
⎢
k−1
⎟ ⎥
E⎢⎜
p
,
ϕ
−
,
ϕ
i
⎠ ⎥
⎦
⎣⎝ P,k|k−1
i,Lk−1
pk−1 , pS,k
⎛
⎞1/2
2
ϕ ⎜
fk|k−1 pS,k 2 ⎟
fk|k−1 pS,k ⎠
≤#
⎝
i
i
Lk−1 qk
A.12
i
since fk|k−1 /qk ≤ B2 by Assumption 4 and fk|k−1 pS,k ∈ Cb Rd by Assumption 1
ϕ 1/2
pS,k 2 B2 fk|k−1 pS,k 2
.
≤#
2
i
Lk−1
Adding and subtracting a new term in the second term of A.11, we have
⎡⎛ ⎤1/2
i
⎞2
i,Lk−1
i
⎢⎜ pk−1 , pS,k fk|k−1 , ϕ
pk−1 , pS,k fk|k−1 , ϕ ⎟ ⎥
⎢
⎟ ⎥
E⎢⎜
−
i
⎠ ⎥
i
⎣⎝
⎦
i,Lk−1
pk−1 , pS,k
pk−1 , pS,k
A.13
Journal of Applied Mathematics
⎡⎛ i
i
i,L
i,L
pk−1 k−1 , pS,k fk|k−1 , ϕ
pk−1 k−1 , pS,k fk|k−1 , ϕ
⎢⎜
⎜
E⎢
−
i
⎣⎝
i
i,Lk−1
p
,
p
S,k
pk−1 , pS,k
k−1
i,L
pk−1 k−1 , pS,k fk|k−1 , ϕ
i
i
pk−1 , pS,k
13
⎞2 ⎤1/2
i
pk−1 , pS,k fk|k−1 , ϕ ⎟ ⎥
⎟ ⎥
−
⎠ ⎥
i
⎦
pk−1 , pS,k
A.14
using Minkowski’s inequality
⎡⎛ ⎞2 ⎤1/2
i
i,Lk−1
,
p
,
ϕ
p
f
S,k
k|k−1
⎢⎜ k−1
i
⎟ ⎥
1
i,L
⎢
i
⎟ ⎥
≤ pk−1 , pS,k − pk−1 k−1 , pS,k
E⎢⎜
i
⎠ ⎥
⎝
i
⎦
⎣
i,Lk−1
pk−1 , pS,k
pk−1 , pS,k
1
i
pk−1 , pS,k
A.15
1/2
i
2
i,Lk−1
i
pk−1 , pS,k fk|k−1 , ϕ − pk−1 , pS,k fk|k−1 , ϕ
E
2 1/2
i,Li
fk|k−1 , ϕ i
k−1
≤2
pk−1 , pS,k − pk−1 , pS,k
E
i
pk−1 , pS,k
A.16
by 4.4
2
ϕ · pS,k 3
3 dk−1
≤2
4 i ,
inf pS,k
Lk−1
A.17
where inf· denotes the infimum.
Finally, substituting A.12 and A.17 into A.11, 4.6 is proved with
dP,k|k−1
62
5#
"
2 2 2 2pS,k dk−1
.
pS,k B2 fk|k−1 pS,k infpS,k i
A.18
i
Now, turn to 4.7. By Lemma 0 in 13 and the boundedness of pΓ,k /bk ≤ B1 i 1, . . . , Mk−1 in Assumption 4, we get that 4.7 holds for a constant dΓ,k . This completes the
proof.
14
Journal of Applied Mathematics
A.2. Proof of Proposition 4.2
Now turn to 4.10. From 2.7 and 3.8, we have
E
i
rL,k
i
i,Lk|k−1
− rL,k
2 1/2
⎞2 ⎤1/2
i
i,Lk|k−1
1
−
p
,
p
D,k
i
⎟ ⎥
⎢⎜
1−
k|k−1
i,L
⎟ ⎥
⎢⎜ i
E⎢⎜rk|k−1
− rk|k−1k|k−1
⎟ ⎥
i
i
i
i
⎠ ⎦
⎣⎝
i,Lk|k−1
i,Lk|k−1
1 − rk|k−1 pk|k−1 , pD,k
1 − rk|k−1
pk|k−1 , pD,k
⎡⎛
i
pk|k−1 , pD,k
A.19
adding and subtracting a new term
⎡⎛
⎞2 ⎤1/2
i
i,Lk|k−1
i
1 − pk|k−1 , pD,k
⎢⎜
i
1 − pk|k−1 , pD,k
⎟ ⎥
i,L
⎢⎜
i
⎟ ⎥
⎢⎜
rk|k−1
− rk|k−1k|k−1
⎟ ⎥
i
i
i
i
⎢⎜
⎟ ⎥
1 − rk|k−1 pk|k−1 , pD,k
1 − rk|k−1 pk|k−1 , pD,k
⎢⎜
⎟ ⎥
⎢⎜
⎟ ⎥
i
i
A.20
E⎢⎜
i,Lk|k−1
i,Lk|k−1
⎟ ⎥
⎢⎜
⎟ ⎥
1 − pk|k−1 , pD,k
i
⎢⎜ i,Li 1 − pk|k−1 , pD,k
⎥
⎟ ⎥
i,L
⎢⎜r k|k−1
− rk|k−1k|k−1
⎢⎝ k|k−1
⎟
i
i
⎠ ⎥
i
i
i,Lk|k−1
i,Lk|k−1
⎣
⎦
1 − rk|k−1 pk|k−1 , pD,k
1 − rk|k−1
pk|k−1 , pD,k
using Minkowski’s inequality
2 1/2
i
i
i,Lk|k−1
i,Lk|k−1
i
i
E rk|k−1 1 − pk|k−1 , pD,k
− rk|k−1
1 − pk|k−1 , pD,k
≤
i
i
1 − rk|k−1 pk|k−1 , pD,k
⎞2 ⎤1/2
i
i
i,Lk|k−1
i,Lk|k−1
1 − pk|k−1 , pD,k
i
⎟ ⎥
⎢⎜ i,Li 1 − pk|k−1 , pD,k
i,L
⎟ ⎥
⎢⎜
E⎢⎜rk|k−1k|k−1
− rk|k−1k|k−1
⎟ ⎥ .
i
i
i
i
⎠ ⎦
⎣⎝
i,Lk|k−1
i,Lk|k−1
1 − rk|k−1 pk|k−1 , pD,k
1−r
,p
p
⎡⎛
k|k−1
k|k−1
D,k
A.21
The numerator of the first term in A.21 is
E
i
rk|k−1
1−
i
pk|k−1 , pD,k
i
i,Lk|k−1
− rk|k−1
2 1/2
i
i,Lk|k−1
1 − pk|k−1 , pD,k
i
i
i
2 1/2
i,Lk|k−1
i,Lk|k−1
i,Lk|k−1
i
i
i
E
rk|k−1
pk|k−1 , pD,k − rk|k−1 pk|k−1 , pD,k
rk|k−1 − rk|k−1
A.22
Journal of Applied Mathematics
15
using Minkowski’s inequality and then A.4
≤E
i
i,Lk|k−1
i
rk|k−1
2 1/2
− rk|k−1
pD,k E
i
rk|k−1
i
i,Lk|k−1
2 1/2
− rk|k−1
A.23
i
2 1/2
i,Lk|k−1
i
i
rk|k−1 E
pk|k−1 , pD,k − pk|k−1 , pD,k
by 4.8 and 4.9
√
#
i 1 pD,k ck|k−1 rk|k−1 pD,k dk|k−1
≤
.
#
i
Lk|k−1
A.24
The second term in A.21 is
⎞2 ⎤1/2
i
i
i,Lk|k−1
i,Lk|k−1
1 − pk|k−1 , pD,k
i
⎟ ⎥
⎢⎜ i,Li 1 − pk|k−1 , pD,k
i,L
⎟ ⎥
⎢⎜
E⎢⎜rk|k−1k|k−1
− rk|k−1k|k−1
⎟ ⎥
i
i
i
i
⎠ ⎦
⎣⎝
i,Lk|k−1
i,Lk|k−1
1 − rk|k−1 pk|k−1 , pD,k
1−r
,p
p
⎡⎛
k|k−1
k|k−1
D,k
⎞2
i
i,Lk|k−1
pk|k−1 , pD,k ⎟
⎢⎜ rk|k−1 − rk|k−1
⎟
⎢⎜
E⎢⎜
⎟
i
i
⎠
⎣⎝
i,Lk|k−1
i,Lk|k−1
1 − rk|k−1
pk|k−1 , pD,k
⎡⎛
i
i
i,Lk|k−1
⎛
i
⎜ rk|k−1
i,Lk|k−1
×⎜
⎝
i
pk|k−1 , pD,k
i
i,Lk|k−1
A.25
i
i,Lk|k−1
⎞2 ⎤1/2
− rk|k−1
pk|k−1 , pD,k ⎟ ⎥
⎟⎥
⎠⎦
i
i
1 − rk|k−1 pk|k−1 , pD,k
i
i,L
by 0 ≤ rk|k−1k|k−1 ≤ 1
2 1/2
i
i
i,Lk|k−1
i,Lk|k−1
i
i
E rk|k−1 pk|k−1 , pD,k − rk|k−1
pk|k−1 , pD,k
≤
1−
i
rk|k−1
i
pk|k−1 , pD,k
A.26
by A.4, 4.8 and 4.9
#
pD,k √ck|k−1 r i pD,k dk|k−1
k|k−1
.
≤ #
i
i
i
1 − rk|k−1 pk|k−1 , pD,k
Lk|k−1
A.27
16
Journal of Applied Mathematics
i
Substituting A.24 and A.27 into A.21, and then using 0 ≤ rk|k−1 ≤ 1, we get
E
i
rL,k
i
i,Lk|k−1
− rL,k
2 1/2
√
#
i 1 2pD,k ck|k−1 2rk|k−1 pD,k dk|k−1
≤
#
i
i
i
1 − rk|k−1 pk|k−1 , pD,k
Lk|k−1
A.28
√
#
1 2pD,k ck|k−1 2pD,k dk|k−1
≤
.
# i
1 − pD,k Lk|k−1
Finally, 4.10 is proved with
⎞2
#
pD,k √ck|k−1 2pD,k dk|k−1
1
2
⎟
⎜
⎝
⎠.
1 − pD,k ⎛
cL,k
A.29
Now turn to 4.11. From 2.8 and 3.9, we have
E
i
pL,k , ϕ
2 1/2
i
i,Lk|k−1
− pL,k
,ϕ
⎡⎛
i
⎢⎜ pk|k−1
, ϕ 1 − pD,k
⎢⎜
E⎢⎜
−
⎣⎝ 1 − pi , p
k|k−1 D,k
⎞2 ⎤1/2
i
i,L
pk|k−1k|k−1 , ϕ 1 − pD,k ⎟ ⎥
⎟ ⎥
⎟ ⎥
i
⎠ ⎦
i,Lk|k−1
1 − pk|k−1 , pD,k
A.30
adding and subtracting a new term
⎡⎛
i
pk|k−1 , ϕ 1
⎢⎜
− pD,k
⎢⎜
E⎢⎜
⎣⎝ 1 − pi , p
k|k−1 D,k
i
i,L
pk|k−1k|k−1 , ϕ 1 − pD,k
−
i
1 − pk|k−1 , pD,k
⎞2 ⎤1/2
i
i
i,Lk|k−1
i,Lk|k−1
pk|k−1 , ϕ 1 − pD,k
pk|k−1 , ϕ 1 − pD,k ⎟ ⎥
⎟⎥
−
⎟⎥
i
i
⎠⎦
i,Lk|k−1
1 − pk|k−1 , pD,k
1 − pk|k−1 , pD,k
A.31
Journal of Applied Mathematics
17
using Minkowski’s inequality
E
i
pk|k−1 , ϕ 1
− pD,k
≤
1/2
i
2
i,Lk|k−1
− pk|k−1 , ϕ 1 − pD,k
i
1 − pk|k−1 , pD,k
⎡⎛ ⎢⎜
⎢⎜
E⎢⎜
⎣⎝
E
⎞2 ⎤1/2
⎞2 ⎛ i,Li
i
k|k−1
−
p
p
,
p
,
p
D,k ⎟ ⎥
⎟ ⎜ k|k−1 D,k
k|k−1
⎟ ⎥
⎟ ⎜
⎟ ⎥
⎟ ·⎜
i
⎠ ⎦
⎠ ⎝
1 − pk|k−1 , pD,k
pk|k−1 , ϕ 1 − pD,k
i
i,L
1 − pk|k−1k|k−1 , pD,k
i
i,Lk|k−1
i
pk|k−1 , ϕ 1
≤
− pD,k
1/2
i
2
i,Lk|k−1
− pk|k−1 , ϕ 1 − pD,k
i
1 − pk|k−1 , pD,k
A.33
2 1/2
i,Li
i
k|k−1
ϕE
pk|k−1 , pD,k − pk|k−1 , pD,k
A.32
i
1 − pk|k−1 , pD,k
by 4.9
#
ϕ dk|k−1
≤
# i .
1 − pD,k Lk|k−1
A.34
Finally, 4.11 is proved with
dk|k−1
2 .
1 − pD,k dL,k A.35
Now, turn to 4.12. From 2.9 and 3.10, we have
E
2 1/2
i
Lk|k−1
rU,k zk − rU,k zk ⎡⎛
⎢⎜
E⎢
⎣⎝
Mk|k−1
i1
κk zk i
i,Lk|k−1
rα,k
Mk|k−1
i1
Mk|k−1
zk i
i,Lk|k−1
rα,k
−
zk ⎞2 ⎤1/2
⎥
⎥
⎦
i
rα,k zk i1
⎟
⎠
Mk|k−1 i
κk zk i1 rα,k zk A.36
18
Journal of Applied Mathematics
adding and subtracting a new term
⎡⎛
Mk|k−1
i
i,Lk|k−1
Mk|k−1
Mk|k−1
zk i
i,L
rα,k k|k−1 zk i1
i1
⎢⎜
E⎣⎝
−
Mk|k−1 i
i
i,L
M
κk zk i1k|k−1 rα,k k|k−1 zk κk zk i1 rα,k zk rα,k
Mk|k−1
i
i,Lk|k−1
i
rα,k
zk rα,k zk i1
−
Mk|k−1 i
Mk|k−1 i
κk zk i1 rα,k zk κk zk i1 rα,k zk i1
⎞2 ⎤1/2
⎟ ⎥
⎠ ⎥
⎦
A.37
using Minkowski’s inequality
⎡⎛
⎞2 ⎤1/2
Mk|k−1 i,Li
Mk|k−1 i
k|k−1
i
rα,k zk − i1 rα,k
zk ⎟ ⎥
⎢⎜ Lk|k−1
i1
≤ E⎢
⎠⎥
Mk|k−1 i
⎣⎝rU,k zk ⎦
κk zk i1 rα,k zk A.38
⎡⎛
⎞2 ⎤1/2
Mk|k−1 i
Mk|k−1 i,Li
k|k−1
zk − i1 rα,k zk ⎟ ⎥
⎢⎜ i1 rα,k
E⎢
⎠⎥
M
⎣⎝
⎦
i
κk zk i1k|k−1 rα,k zk i
L
k|k−1
using κk zk ≥ 0, 0 ≤ rU,k
zk ≤ 1
2
Mk|k−1
i1
2 1/2
i
i,Lk|k−1
i
E rα,k
zk − rα,k zk ≤
Mk|k−1
i1
A.39
.
i
rα,k zk From 2.12 and 3.14, the expectation in the summation of A.39 is
E
i
i,Lk|k−1
rα,k
zk −
⎡⎛
i
rk|k−1
i
rα,k zk 2 1/2
i
pk|k−1 , ψk,zk
i
i,Lk|k−1
i
i,Lk|k−1
⎞2 ⎤1/2
pk|k−1 , ψk,zk
rk|k−1
⎟
⎢⎜
⎟
⎢⎜
E⎢⎜
−
⎟
i
i
i
⎠
⎣⎝ 1 − r i
i,Lk|k−1
i,Lk|k−1
p
,
p
D,k
k|k−1
k|k−1
pk|k−1 , pD,k
1 − rk|k−1
⎥
⎥
⎥
⎦
A.40
Journal of Applied Mathematics
19
adding and subtracting a new term
⎡⎛
i
i
i,Lk|k−1
i,Lk|k−1
i
i
,
ψ
r
p
k,zk
⎢⎜ k|k−1
rk|k−1 pk|k−1 , ψk,zk
k|k−1
⎢⎜
E⎢⎜
−
i
i
i
i
⎣⎝
i,L
i,L
i,L
i,L
1 − rk|k−1k|k−1 pk|k−1k|k−1 , pD,k
1 − rk|k−1k|k−1 pk|k−1k|k−1 , pD,k
⎞2 ⎤1/2
i
i
i
i
⎟
rk|k−1 pk|k−1 , ψk,zk
rk|k−1 pk|k−1 , ψk,zk
⎟
−
⎟
i
i
i
i
i,Lk|k−1
i,Lk|k−1
1 − rk|k−1 pk|k−1 , pD,k ⎠
1 − rk|k−1
pk|k−1 , pD,k
A.41
⎥
⎥
⎥
⎦
using Minkowski’s inequality
i
i
⎞2 ⎤1/2
i,Lk|k−1
i,Lk|k−1
i
i
p
,
ψ
,
ψ
p
−
r
r
k,zk
k,zk ⎟ ⎥
⎢⎜ k|k−1
k|k−1
k|k−1
k|k−1
⎟ ⎥
⎢⎜
≤ E⎢⎜
⎟ ⎥
i
i
⎠ ⎦
⎣⎝
i,Lk|k−1
i,Lk|k−1
pk|k−1 , pD,k
1 − rk|k−1
⎡⎛
i
rk|k−1
i
i
pk|k−1 , ψk,zk
i
1 − rk|k−1 pk|k−1 , pD,k
⎡⎛
i
i,Lk|k−1
⎢⎜ rk|k−1
⎢⎜
E⎢⎜
⎣⎝
i
⎞2 ⎤1/2
i,Lk|k−1
i
i
pk|k−1 , pD,k − rk|k−1 pk|k−1 , pD,k ⎟ ⎥
⎟⎥
⎟⎥
i
i
⎠⎦
i,Lk|k−1
i,Lk|k−1
1 − rk|k−1
pk|k−1 , pD,k
A.42
i
i,L
by 0 ≤ rk|k−1k|k−1 ≤ 1, Assumption 2 and A.4
ψk,z E
k
i
i,Lk|k−1
rk|k−1
−
i
rk|k−1
≤
i
i
rk|k−1 pk|k−1 , ψk,zk
i
i
1 − rk|k−1 pk|k−1 , pD,k
pD,k E
×
2 1/2
i
rk|k−1 E
i
2 1/2
i,Lk|k−1
i
pk|k−1 , ψk,zk − pk|k−1 , ψk,zk
1 − pD,k 2 1/2
i
i
2 1/2
i,Lk|k−1
i,Lk|k−1
i
i
i
rk|k−1 E
rk|k−1 − rk|k−1
pk|k−1 , pD,k − pk|k−1 , pD,k
1 − pD,k A.43
20
Journal of Applied Mathematics
i
by 4.8, 4.9, and 0 ≤ rk|k−1 ≤ 1
#
ψk,z √ck|k−1 dk|k−1
k
≤
.
2 # i
Lk|k−1
1 − pD,k A.44
i
From 2.12, 0 ≤ rk|k−1 ≤ 1 and Assumption 2, the denominator of A.39 is
M
k|k−1
i1
i
rα,k zk M
k|k−1
i1
i
i
M
k|k−1
rk|k−1 pk|k−1 , ψk,zk
i
i
rk|k−1 pk|k−1 , ψk,zk
≥
i
i
1 − rk|k−1 pk|k−1 , pD,k
i1
≥ inf ψk,zk
M
k|k−1
i1
A.45
i
rk|k−1 inf ψk,zk nk|k−1 ,
M
i
where nk|k−1 i1k|k−1 rk|k−1 is the number of the predicted targets at time k.
Substituting A.44 and A.43 into A.39, we get
E
2 1/2
i
Lk|k−1
rU,k zk − rU,k zk ⎛
#
√
k|k−1
2ψk,zk ck|k−1 dk|k−1 M
1
⎜
≤ ⎝#
2 i
inf ψk,zk nk|k−1 i1
1 − pD,k
L
k|k−1
#
√
2ψk,zk ck|k−1 dk|k−1 Mk|k−1
≤
2 # min ,
inf ψk,z nk|k−1 1 − pD,k L
M
⎟
⎠
A.46
k|k−1
k
1
⎞
k|k−1
where Lmin
minLk|k−1 , . . . , Lk|k−1
, min· denotes the minimum.
k|k−1
Finally, 4.12 is proved with
cU,k
⎞2
⎛ #
√
2ψk,zk ck|k−1 dk|k−1 Mk|k−1
⎟
⎜
⎝
⎠.
2
inf ψk,zk nk|k−1 1 − pD,k A.47
i
Now, turn to 4.13. First, from 2.12, 0 ≤ rk|k−1 ≤ 1 and Assumption 2, we have
i
rα,k zk i
i
rk|k−1 pk|k−1 , ψk,zk
ψk,z k
.
≤
i
i
1 − pD,k 1 − rk|k−1 pk|k−1 , pD,k
A.48
Journal of Applied Mathematics
E
21
Then, from 2.13, 3.11, and A.39, we get
2 1/2
i
∗,L
∗
zk rU,kk|k−1 zk − rU,k
⎛
≤
⎞2 ⎤1/2 ⎞
⎜
zk 1 − rk|k−1
⎟⎥ ⎟
⎢⎜ rα,k
⎜
⎟⎥ ⎟
⎢⎜
⎜ ⎡⎛
⎤
⎢⎜
⎞2 1/2
⎟⎥ ⎟
i
⎜
i
i
⎜
⎟⎥ ⎟
⎢
i,Lk|k−1
⎢⎜ 1 − r i,Lk|k−1 pi,Lk|k−1 , pD,k ⎟ ⎥ ⎟
Mk|k−1 ⎜
zk ⎠ ⎥
⎜ ⎢⎝ rα,k
⎜
⎟⎥ ⎟
⎢
k|k−1
k|k−1
⎜E ⎣
⎦ E⎢⎜
⎟⎥ ⎟
i1
i
⎜
⎟⎥ ⎟
⎢⎜
i
i
−rα,k zk ⎜
⎜
⎟⎥ ⎟
⎢
r
z
1
−
r
k
⎜
α,k
k|k−1
⎟⎥ ⎟
⎢⎜
⎜
⎝
⎠⎦ ⎟
–
⎣
⎝
⎠
i
i
1 − rk|k−1 pk|k−1 , pD,k
⎡⎛
Mk|k−1
i1
i
i,Lk|k−1
i
i,Lk|k−1
i
rα,k zk A.49
adding and subtracting a new term in the second expectation in the summation
⎛ 2 1/2
⎞
i
i,Lk|k−1
≤
i
⎜ E rα,k
zk − rα,k zk ⎜
⎜
⎜
⎡⎛
i
i
⎜
i,Lk|k−1
i,Lk|k−1
⎜
i
i
zk 1 − rk|k−1
⎜
⎢⎜ rα,k
1
−
r
r
z
k
α,k
k|k−1
⎢⎜
Mk|k−1 ⎜
⎜
⎢⎜
−
i
i
i
i
⎜
⎢⎜
i,L
i,L
i,L
i,L
i1
⎜
⎢⎜ 1 − r k|k−1 p k|k−1 , pD,k 1 − r k|k−1 p k|k−1 , pD,k
⎜
⎢⎜
k|k−1
k|k−1
k|k−1
k|k−1
⎜ E⎢⎜
⎜
⎢⎜
i
i
i
i
⎜
⎢⎜
1
−
r
1
−
r
r
r
z
z
k
k
α,k
α,k
k|k−1
k|k−1
⎜
⎢⎜
⎜
⎢⎜ −
i
i
⎝
i
i
⎣⎝
i,Lk|k−1
i,Lk|k−1
1 − rk|k−1
pk|k−1 , pD,k 1 − rk|k−1 pk|k−1 , pD,k
Mk|k−1
i1
⎞2 ⎤1/2
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
i
rα,k zk A.50
It holds that using Minkowski’s inequality for the second term in the summation
⎛ ⎞
2 1/2
i
i,Lk|k−1
i
⎜E r
⎟
zk − rα,k zk α,k
⎜
⎟
⎜
⎟
⎜ ⎡⎛
⎟
⎞
⎤
1/2
2
⎜
⎟
i
i
i
⎜
⎟
i,Lk|k−1
i,Lk|k−1
i,Lk|k−1
i
i
i
⎜ ⎢⎜ r
⎟
⎟
⎥
zk − rα,k zk rα,k zk rk|k−1 − rα,k
zk rk|k−1 ⎟ ⎥
⎜ ⎢⎜ α,k
⎟
⎜
⎟
E
⎟
⎜
⎥
⎢
Mk|k−1 ⎜ ⎣⎝
i
i
⎟
⎠⎦
i,Lk|k−1
i,Lk|k−1
⎜
⎟
i1
pk|k−1 , pD,k
1 − rk|k−1
⎜
⎟
⎜
⎟
⎜
⎟
⎞
⎤
⎡⎛
1/2
2
i
i
⎜
⎟
i,Lk|k−1
i,Lk|k−1
i
i
⎜
i
i
pk|k−1 , pD,k −rk|k−1 pk|k−1 , pD,k ⎟ ⎥ ⎟
⎜ rα,k zk 1 − rk|k−1
⎟
⎢⎜rk|k−1
⎜
⎟⎥ ⎟
⎢⎜
E⎢⎜
⎜
⎟
⎟
⎥
i
i
⎝ 1−r i pi , p
⎠⎦ ⎠
⎣⎝
i,L
i,L
k|k−1
k|k−1 D,k
1 − rk|k−1k|k−1 pk|k−1k|k−1 , pD,k
≤
Mk|k−1 i
rα,k zk i1
A.51
22
Journal of Applied Mathematics
i
i,L
by 0 ≤ rk|k−1k|k−1 ≤ 1, A.4, and Minkowski’s inequality
⎛
2 1/2
i
i,L
⎜ 2− pD,k E r k|k−1 zk −r i zk α,k
α,k
⎜
⎜
⎜ ⎡⎛
⎞2 ⎤1/2
i
⎜
i,Lk|k−1
i
i
i
⎜
z
r
−r
z
r
r
⎜ ⎢⎜ α,k k k|k−1 α,k k k|k−1
⎟⎥
⎜ E⎣⎝
⎠⎦
i
i
i
i,Lk|k−1
i,Lk|k−1
i,Lk|k−1
Mk|k−1 ⎜
i
⎜
r
z
r
−r
z
r
k
α,k k k|k−1
⎜
i1
⎛ α,k
k|k−1
⎜
2 1/2
i
⎜
i,Lk|k−1
i
⎜
p E r
i
i
−rk|k−1
⎜
k|k−1
rα,k zk 1−rk|k−1 ⎜
⎜ D,k
⎜
⎜
⎜
⎜
⎜
i
i
i
2 1/2
i,Lk|k−1
⎝ 1−rk|k−1 pk|k−1 , pD,k ⎝
i
i
rk|k−1 E
pk|k−1 , pD,k − pk|k−1 , pD,k
≤
Mk|k−1 i
rα,k zk 1 − pD,k i1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎞⎟
⎟
⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠⎟
⎠
A.52
i
i,L
using 0 ≤ rk|k−1k|k−1 ≤ 1 and Minkowski’s inequality again for the second term in the
summation
⎛
2 1/2
2 1/2⎞
i
i
i,Lk|k−1
i,Lk|k−1
i
i
i
⎜ 3− pD,k E rα,k
⎟
zk −rα,k zk rα,k zk E rk|k−1 −rk|k−1
⎜
⎟
⎜
⎟
⎞
⎛
⎜
⎟
1/2
2
i
⎜
⎟
Mk|k−1 ⎜
⎜
i,Lk|k−1
⎟ ⎟
i
i
i
⎟
⎜
rk|k−1 −rk|k−1
⎜ r zk 1 − r
i1
⎟ ⎟
⎜ pD,k E
⎟ ⎟
⎜
α,k
k|k−1
⎜
⎟ ⎟
⎜
⎜
⎟
⎜
1/2
⎟ ⎟
⎜
⎜
i
i
2
i
⎟ ⎠
⎜
i,Lk|k−1
⎝ 1 − rk|k−1 pk|k−1 , pD,k ⎜ i
⎟
i
⎠
⎝ r
E
pk|k−1 , pD,k − pk|k−1 , pD,k
k|k−1
≤
Mk|k−1 i
1 − pD,k rα,k zk i1
A.53
i
by 0 ≤ rk|k−1 ≤ 1, 4.8, 4.9 A.44, A.45, and Assumption 2
⎛
#
⎞
i
ψk,z √ck|k−1 dk|k−1 M
pD,k k|k−1
1
4
−
r
k
k|k−1
⎜
⎟
≤
#
⎝
⎠
3 i
inf ψk,zk nk|k−1 i1
1 − pD,k
L
k|k−1
#
ψk,z √ck|k−1 dk|k−1 4Mk|k−1 − nk|k−1 Mk|k−1 pD,k k
.
≤
#
3 1 − pD,k inf ψk,zk nk|k−1 Lmin
k|k−1
A.54
Journal of Applied Mathematics
23
Finally, 4.13 is proved with
⎞
⎛
#
6 2
ψk,z √ck|k−1 dk|k−1 5 k
4 − pD,k Mk|k−1
⎟
⎜
⎝ − pD,k ⎠ .
3
n
k|k−1
infψk,zk 1 − pD,k
∗
cU,k
A.55
Now turn to 4.14. From 2.10 and 3.12, we get
E
i
2 1/2
Lk|k−1
pU,k ·; zk , ϕ − pU,k ·; zk , ϕ
⎡⎛ ⎢⎜
⎢⎜
E⎢⎜
⎣⎝
Mk|k−1
i
i,Lk|k−1
pα,k
i1
Mk|k−1
i1
xk ; zk , ϕ
i
i,Lk|k−1
rα,k
M
k|k−1
−
zk i1
i
pα,k xk ; zk , ϕ
Mk|k−1
i1
i
rα,k zk ⎞2 ⎤1/2
⎟
⎟
⎟
⎠
⎥
⎥
⎥
⎦
A.56
adding and subtracting a new term
⎡⎛ Mk|k−1 i,Li
Mk|k−1 i,Li
k|k−1
k|k−1
p
p
z
ϕ
z
ϕ
·;
,
·;
,
k
k
⎢⎜
i1
i1
α,k
α,k
⎢⎜
E⎢⎜
−
i
M
k|k−1 i
⎣⎝
Mk|k−1 i,Lk|k−1
rα,k zk i1
r
z
k
i1
α,k
Mk|k−1
i1
i
i,Lk|k−1
pα,k
Mk|k−1
i1
·; zk , ϕ
⎞ ⎤1/2
2
i
k|k−1
pα,k ·; zk , ϕ ⎟ ⎥
i1
⎟⎥
⎟⎥
Mk|k−1 i
⎠⎦
r
z
k
i1
α,k
M
−
i
rα,k zk A.57
using Minkowski’s inequality
⎡⎛
⎞⎞2 ⎤1/2
⎛
Mk|k−1 i
*Mk|k−1
+ Mk|k−1 i,Li
k|k−1
i
zk − i1 rα,k zk ⎟⎟ ⎥
⎢⎜ i,Lk|k−1
⎜ i1 rα,k
≤ E⎢
pα,k
·; zk , ϕ ⎝
⎠⎠ ⎥
i
⎣⎝
⎦
i,Lk|k−1
M
M
i
k|k−1
k|k−1
i1
rα,k zk i1 rα,k
zk i1
E
Mk|k−1
i1
i
i,Lk|k−1
pα,k
·; zk −
Mk|k−1
i1
Mk|k−1
i1
2 1/2
i
pα,k ·; zk , ϕ
A.58
i
rα,k zk by 2.12 and 3.14
2ϕ
≤ M
k|k−1
i1
i
rα,k zk ⎡5
62 ⎤1/2
M
M
k|k−1
k|k−1
i
i,Lk|k−1
i
E⎣
r
zk −
r zk ⎦
i1
α,k
i1
α,k
A.59
24
Journal of Applied Mathematics
by A.39 and A.47
ϕ
cU,k .
≤#
Lmin
k|k−1
A.60
Finally, 4.14 is proved with
dU,k cU,k
⎞2
⎛ #
√
2ψk,zk ck|k−1 dk|k−1 Mk|k−1.
⎟
⎜
⎝
⎠.
2
infψk,zk nk|k−1 1 − pD,k A.61
This completes the proof.
Nomenclature
xk :
zk :
nk :
mk :
k
:
Xk {xi,k }ni1
mk
Zk {zi,k }i1 :
fk|k−1 xk | xk−1 :
pS,k xk−1 :
pD,k xk :
κk zk :
ψk,zk xk fk zk | xk :
δx ·:
Rd :
Cb Rd :
BRd :
πY i :
πY :
π {r i , pi }M
i1 :
State vector of a single target at time k
Single measurement vector at time k
Number of existing targets at time k
Number of measurements collected at time k
Finite set of multitarget state-vectors at time k
Finite set of measurements collected at time k
Single-target Markov transition density at time k
Probability of target survival at time k
Probability of detection at time k
Intensity of Poisson clutter process at time k
Single-sensor/target likelihood density at time k
Dirac delta function centered at x
d-dimensional real space
Set of continuous bounded functions on Rd
Set of bounded Borel measurable functions on Rd
Probability density of a Bernoulli random finite set RFS Y i
i
Probability density of multi-Bernoulli RFS Y M
i1 Y
i i M
Abbreviation of πY . {r , p }i1 is the multi-Bernoulli parameter set
i
i,Li
i,Li M
i,Li
,p
}i1 : Particle approximation of π {r i , pi }M
, pi,L π {r
i1 . r
denotes that r i , pi is comprised of the number Li of the
particles
· :
Supremum norm. ϕ sup| ϕ |, sup· denotes the supremum
·, ·:
Inner product. If the measure in ·, · is continuous, it defines the
integral inner product; if the measure in ·, · is discrete, it defines
the summation inner product.
Acknowledgments
This research work was supported by the Natural Science Foundation of China 61004087,
61104051, 61005026, China Postdoctoral Science Foundation 20100481338, and Fundamental Research Funds for the Central University.
Journal of Applied Mathematics
25
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