Research ArticleClustered Exact Daum-Huang Particle Flow Filter

Halit OumlrenbaG andMuharremMercimek

Control and Automation Engineering Yildiz Technical University Istanbul 34220 Turkey

Correspondence should be addressed to Halit Orenbas orenbasyildizedutr

Received 13 February 2019 Accepted 21 April 2019 Published 13 May 2019

Academic Editor Saeed Eftekhar Azam

Copyright copy 2019 Halit Orenbas andMuharremMercimekThis is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Unlike the conventional particle filters particle flow filters do not rely on proposal density and importance sampling they employflow of the particles through a methodology derived from the log-homotopy scheme and ensure successful migration of theparticles Amongst the efficient implementations of particle filters Exact Daum-Huang (EDH) filter pursues the calculation ofmigration parameters all together An improved version of it Localized Exact Daum-Huang (LEDH) filter calculates the migrationparameters separately In this study themain objective is to reduce the cost of calculation in LEDHfilters which is due to exhaustivecalculation of eachmigration parameterWe proposed the Clustered Exact Daum-Huang (CEDH) filterThemain impact of CEDHis the clustering of the particles considering the ones producing similar errors and then calculating the same migration parametersfor the particles within each clusterThrough clustering and handling the particles with high errors their engagement and influencecan be balanced and the system can greatly reduce the negative effects of such particles on the overall system We implement thefilter successfully for the scenario of high dimensional target tracking The results are compared to those obtained with EDH andLEDH filters to validate its efficiency

1 Introduction

For the analysis inference and comprehensive understand-ing of a dynamic system two models are essentially requiredThe first model is the system model which describes thechange of its states with respect to time and embodies theinformation to characterize the entire system The secondmodel is the measurement model which represents the noisemeasurements associated with the state vectors

The probabilistic state-space representations of thesemodels and the need for the information update upon havingnew measurements are preferably consistent with the prac-tical Bayesian approach [1] State-space approach providesgreat advantages in handling the multivariable nonlinearprocesses over conventional time series techniques [2]

Bayesian approach constructs a posterior probability den-sity function of a state based on available information for thepurpose of dynamic state estimation The use of a recursivefilter can be a good solution in that an estimation takes placewhen a new measurement is obtained The received data ispreferably processed sequentially rather than being processedas a whole in the recursive filter approach Therefore there is

no need to store the complete data and use it when estimationis required upon receiving an instant measurement Theprediction and update stages must be employed for stateestimation At the prediction stage the system model is usedto construct the pdf of the state for the next measurementtime The state is deteriorated due to unknown disturbanceshence during the prediction stage the pdf of the state ismostly deformed or distorted The update stage employs theBayes theorem to modify the prediction pdf Thus the latestmeasurements establish the comprehension of the target state[1]

Particle filters are the sequential Monte Carlo algorithmswhich can be applied to any state-space model [3] They arethe generalized form of the Kalman filters Particle repre-sentations of the probability densities are established withparticle filtersThe Bootstrap particle filter (BPF) updates theweight of each particle after drawing the particles from theprior distribution using the recent measurement likelihoods[4] If salient measurements are employed or the dimensionof the state is high then many particles from the priordistribution will be in regions with low likelihoodThis surely

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 8369565 8 pageshttpsdoiorg10115520198369565

2 Mathematical Problems in Engineering

results in assigning low weights to many particles in BPF[5] Hence Monte Carlo approximation for the posteriordistribution is established depending on too few particlesthe degeneracy due to weights leads to faulty posteriordistribution depictions [2 3]

The variance of the importanceweights can beminimizedusing the optimal proposal distribution approximation [6]Cutting-edge particle filters mimic the well-known strategyof the optimal proposal distribution approximation throughestablishing proposal distributions in efficiency [7] Auxiliaryparticle filters (APF) were put forward to sample the particlesmore appropriately considering the new measurements inhand [8]The variances ofMonte Carlo estimates are reducedwithRao-Blackwellised particle filters through isolating someof the states wisely [9]The unscented transformation aimingto employ an error-bounded deterministic sampling methodis adopted in the unscented particle filter [10]The approachesmentioned so far can be conveniently used under certaincircumstances within a certain range of problem setting [11]Particle filter use for the states in high dimensional spaceinvolved with the scenario of multitarget tracking still is achallenging task [12] Conventional particle filter methodscan be so naive to handle high dimensionality problem [13]There are promising methods providing good performancesuch as a well-knownmethod the equivalent weights particlefilter [14] It fights against the weight degeneracy but lacksthe statistical consistency of the conventional particle filtersThe separation of the state space through partitioning orsegmenting has been adopted in some studies [15 16] Thetechnique relies essentially on the factorization of the condi-tional posterior hence it has a narrow scope of applicationAn all-intentionally approach is the use of Markov ChainMonte Carlo (MCMC) method to aid the particle filters[17] The number of particles and consequently the cost ofcalculation grow exponentially when higher state dimensionsare dealt with [18]

An unconventional strategy was proposed by Daum andHuang as a nonlinear filtering approach [19ndash21] A homotopybetween the logarithms of unnormalized prior and posteriordistributions is presented at each step of Daum-Huang (DH)filters The homotopy depicts the particle flow which can beregarded as the solution to a partial differential equationTheparticle flow can be stated to be guaranteeing the gradualmigration of the particles to the regions with higher posterior[18]

Extraordinary performance is reported to be obtainedwhen DH filters are utilized in a number of nonlinearfiltering systems On the other hand the researchers alsoreport that the unqualified dependence of EDH filter on theextended or unscented Kalman filters (EKFUKF) which areexecuted in parallel yields performance drawbacks [22] Asolution to this is proposed in [18] through focusing on thecases that the system and the observation noise are bothin the Gaussian form the system map is linear but theobservations are in a highly nonlinear manner The modifiedversion is accordingly proposed and introduced as LEDHThemodification actually gives support in the linearization ofthe system and the calculation of each migration parameterof each particle individually rather than dealing with a

single calculation associating all particles It is not difficultto state that the cost of calculation is the weakness of thiseminent filter despite the fact that it has a remarkably goodperformance

Standard particle filters utilizing importanceweights havebeen reported to be struggling with exponentially increasingsample size which is essentially due to high dimension ofthe state space Therefore particle filtering inherits weightdegeneracy in case of having high dimensional filteringscenarios The problems arising from high dimensionalityhave been discussed in the literature regarding differentaspects

Recent studies on particle flow filters provide solutions tothe problems arising fromweight degeneracy Li et al presentnew filters which integrate deterministic particle flows intoa particle filter framework [23] The proposed theoreticalscheme can provide the adequacy of the particle filter Alsoit can sustain the efficiency of particle flow methods Li andCoates strive the computational burden of the particle flowparticle filter by incorporating clustering of the particles [24]In this study we deal solely with improving the efficiency ofparticle flowfilterwhich is inherently easier to implement andthus more suitable in practical scenarios The main impactof CEDH is the clustering of the particles considering theones producing similar errors and then calculating the samemigration parameters for the particles within each clusterThrough clustering and handling the particles with higherrors their engagement and influence can be balanced andthe system can greatly reduce the negative effects of suchparticles on the overall system

Surace et al [25] focus on the varying aspects of thecurse of dimensionality in continuous time filtering Theyinvestigate the use of optimal feedback control scheme thatdeals with importance weights Daum et al [26] extendedtheir studies through derivation of a new exact stochastic par-ticle flow filter using a theorem established by Gromov Theyconducted numerical experiments for a number of differenthigh dimensional problems In [27] the researchers com-bined the strengths of invertible particle flow and sequentialMarkov chain Monte Carlo (SMCMC) through constructinga composite Metropolis-Hastings (MH) kernel They alsoproposed a Gaussian mixture model- (GMM-) based particleflow algorithm to construct effective MH kernels

Ourmain objective is to reduce and balance the influenceof particles which produce high errors through clusteringin Particle Flow Filter This will provide reduced cost ofcalculationwhilemaintaining the performance of LEDHTheerror value for each particle is taken into account and k-means++ algorithm [28] is employed for clustering Such aclustering provides a significant benefit in the calculation ofmigration parameters of the particles which are designated tobe in the same cluster

Unlike the EDH filter which calculates of migrationparameters all together and LEDH filter which calculatesthe migration parameters separately the same migrationparameters are pursued for the particles in the same clusterThe number of the migration parameters accompanyingeach cluster can be reduced and this yields an ease on thecalculations of the LEDH

Mathematical Problems in Engineering 3

In this study for each target velocity position derivativeof the velocity and derivative of the position are dealt withthus we are involved with a tracking problem in the four-dimensional state space Four targets are tracked utilizing ourCEDH filter as well as EKF BPF EDH and LEDH filters andperformances are compared

The structure of this paper is as follows In Section 2theoretical base in terms of established techniques for particleflow filters is represented The implementation details of k-means++ algorithm are given and its use in CEDH filter isdescribed in Section 3 The application of CEDH filter ona descriptive scenario is detailed in Section 4 Performancecomparisons of the filters employed for multitarget trackingproblem in the high dimensional state space are given in thissection as well Section 5 puts concluding comments on theproposed filter discusses problems that arose in the studyand possible future work and lists practical pieces of adviceresulting from the scenario

2 Particle Flow Filter

Particle flow filter is the modified version of the conventionalparticle filters It essentially fights against the particle degen-eracy and ensures the fast convergence to the particles withthe highest posterior distribution in the next step throughemploying a logarithmic homotopy function The homotopyfunction depicts the transition between prior and posteriordistributions for the flow of particles The filter uses the Itostochastic Partial Differential Equation (PDE) which is usedto differentiate between the step parameter of the homotopyfunction and the state

We consider a discrete-time nonlinear filtering task withthe following models119909119896 = 119891119896 (119909119896minus1) + V119896 (1)119911 = 120574 (119909119896minus1) + 119899119896 (2)

where 119909119896 isin R119863 is a target state vector 119911119896 isin R119878 is a measure-ment vector V119896 isin R119863 is the process noise and 119899119896 isin R119878 is themeasurement noise 119891119896 R119863 997888rarr R119863 is a nonlinear map and120574119896 R119863 997888rarr R119878 is a nonlinear measurement map

Bayesian rule to define the unnormalized marginal pos-terior distribution is as follows119901 (119909119896 | 1199111119896) = 119901 (119911119896 | 119909119896) 119901 (119909119896 | 1199111119896minus1) (3)

Daum and Huang expressed the homotopy function in thisform 120601 (119909119896 120582) = log119892 (119909119896) + 120582 log ℎ (119909119896) (4)

where ℎ(119909119896) = 119901(119911119896 | 119909119896) 119892(119909119896) = 119901(119909119896 | 1199111119896minus1) and 120582 isthe real valued step parameter in the range of [0 1] andrepresents the intensity or amount of particle flow Thehomotopy function provides a continuous deformation fromlog119892(119909119896) (when 120582 = 0) to the logarithm of the unnormalizedposterior distribution log119901(119909119896 | 1199111119896) (when 120582 = 1)

In the original particle flow filter the flow is improved sothat the homotopy function remains constant as 120582 evolves

Partial differential equation use is required for this purposeand the following equation can be referred to solve andcalculate the flow of the particles120597120593120597119909 119889119909119889120582 + 120597120593120597120582 = 0 (5)

Daum and Huang derived the following expression utilizingthe Fokker-Planck equation and developed and generalizedthe filter through proposing EDH filter [21]120597120593120597119909120595 (119909 120582) + log (ℎ) = minus119879119903(120597120595120597119909) (6)

The solution of this equation leads to the exact flow of theprobability densityThe flow quantities of the particles can becalculated as given in [21]119889119909119889120582 = 119860 (120582) 119909 + 119887 (120582) (7)

where 119860 = minus12119875119867119879 (120582119867119875119867119879 + 119877)minus1119867 (8)

and 119887 = (119868 minus 2120582119860) [(119868 + 120582119860) 119875119867119879119877minus1119911 + 119860119909] (9)119875 and 119877 depict the covariance of the estimation error andmeasurement noise respectively 119867 is the measurementmatrix 119909 denotes the state variable that immigrates in eachcycle and 119889119909119889120582 represents the change of 119909 with respect to12058221 Implementation of Particle Flow Filter Algorithm 1includes the pseudocodes of the EDH filter implementationregarding the previously stated theoretical base Particlemigration task is realized following the steps on lines 7 to16 119873 is the number of particles and 119879 is the number oftime intervals The UKFEKF state and covariance matrixestimation are represented with119898 and 119875 respectively

The steps of UKFEKF estimation and UKFEKF updateare expressed on lines 6 and 17 respectivelyThemeasurementmatrix 119867119909 is calculated through linearizing the currentestimate 119909119896 and depicted on line 9 The same 119860 and 119887 valuesare used to update all particles

A simple but effective change stated in [18] is to replacethe lines of 7-19 of Algorithm 1 with the pseudocodes givenin Algorithm 2

There are two major changes in Algorithm 2 For eachparticle associated part of the measurement function islinearized and119860 119894 and 119887119894 values are calculated using119867119894matrixThus the calculation of the migration parameters for eachparticle is performed individually with varying values of 119860 119894and 119887119894 The other change is that the mean estimate from theUKFEKF is replaced with the state estimate from the Daum-Huang filter as stated on line 19 [18]

4 Mathematical Problems in Engineering

1 Initialization Draw 1199091198940119873119894=1 from the prior 119901(1199090)2 Set 1199090 and1198980 as the mean 1198750 as the covariance matrix3 for 119896 = 1 to 119879 do4 Propagate particles 119909119894119896minus1 = 119891119896(119909119894119896minus1) + V1198965 Calculate the mean value 1199091198966 Apply UKFEKF prediction (119898119896minus1|119896minus1 119875119896minus1|119896minus1) 997888rarr (119898119896|119896minus1 119875119896|119896minus1)7 for 119895 = 1 to119873120582 do8 Set 120582 = 119895Δ1205829 Calculate119867119909 by linearizing 120574119896() at 11990911989610 Calculate 119860 and 119887 from (8) and (9) using 119875119896|119896minus1 119909 and11986711990911 for 119894 = 1 to119873 do12 Evaluate 119889119909119894119896119889120582 for each particle from (7)13 Migrate particles 119909119894119896 = 119909119894119896 + Δ120582(119889119909119894119896119889120582)14 endfor15 Re-evaluate 119909119896 using the updated particles 11990911989411989616 endfor17 Apply UKFEKF update (119898119896|119896minus1 119875119896|119896minus1) 997888rarr (119898119896|119896 119875119896|119896)18 Estimate 119909119896 from the particles 119909119894119896 using 119875119896|11989619 Redraw particles 119909119894119896 sim 119873(119909119896 119875119896|119896) (Optional)20 endfor

Algorithm 1 Exact Flow Daum-Huang Filter

7 for 119895 = 1 to119873120582 do8 Set 120582 = 119895Δ1205829 for 119894 = 1 to119873 do10 Calculate119867119909119894 by linearizing 120574119896() at 11990911989411 Calculate 119860119894 and 119887119894 from (8) and (9) using 119875119896|119896minus1 119909 and119867119909119894 12 Evaluate 119889119909119894119896119889120582 for each particle from (7)13 Migrate particles 119909119894119896 = 119909119894119896 + Δ120582(119889119909119894119896119889120582)14 endfor15 Re-evaluate 119909119896 using the updated particles 11990911989411989616 endfor17 Apply UKFEKF update (119898119896|119896minus1 119875119896|119896minus1) 997888rarr (119898119896|119896 119875119896|119896)18 Estimate 119909119896 from the particles 119909119894119896 using 119875119896|11989619 Set119898119896|119896 = 11990911989620 Redraw particles 119909119894119896 sim 119873(119909119896 119875119896|119896) (Optional)

Algorithm 2 Local Exact Flow Daum-Huang Filter

3 Clustered Particle Flow Filter

EDH filter calculates migration parameters for all of theparticles LEDH filter calculates the migration parametersindividually for each particle CEDHfilter employs clusteringof the particles considering the ones producing similarerrors The calculation cost is significantly reduced throughcalculating common migration parameters for the particleswithin each cluster

In this study k-means++ algorithm [28] is adoptedto fulfill the clustering demand of CEDH filter The well-known k-means class methods aim at minimizing the meansquare distance between points in the same set [29] Arthurand Vassilvitskii proposed an algorithm that is 119874(log 119896)-competitive It strengthens the clustering scheme with arandomized seeding technique and improves both the speedand the accuracy in clustering

31 Implementation of Clustered Particle Flow Filter The costof calculation is anticipated to be high in LEDH filter sinceit pursues the calculation of the flow for each particle in eachpseudo-time interval The main objective of the CEDH filteris that it reduces the cost of calculation through implementingthe steps of a clustering methodology Thus it reduces theinfluence of particles having a high error margin on theoverall system and therefore improves the performance of theLEDH filter The pseudocodes for the implementation of theCEDH filter are specified in Algorithm 3119873119888 is the number of clusters Algorithm 3 like the LEDHfilter changes the steps on lines 7 to 19 of the Algorithm 1Clustering step takes place over the error margin of the par-ticles as expressed on line 12 The migration parameters arecalculated up to the number of clusters through employingthe steps depicted on lines 13 to 17 Therefore in CEDH

Mathematical Problems in Engineering 5

7 for 119895 = 1 to119873120582 do8 Set 120582 = 119895Δ1205829 for 119894 = 1 to119873 do10 Calculate119867119909 by linearizing 120574119896() at 11990911989411 endfor12 Cluster particles over the margin of error13 for 119888 = 1 to119873119888 do14 Calculate 119860119888 and 119887119888 from (6) and (7) using 119875119896|119896minus1 119909 and119867119909119888 15 Evaluate 119889119909119888119896119889120582 for each particle from (5)16 Migrate particles 119909119888119896 = 119909119888119896 + Δ120582(119889119909119888119896119889120582)17 endfor18 Eliminate the largest cluster with the biggest error19 Re-evaluate 119909119896 using the updated particles 11990911989411989620 endfor21 Apply UKFEKF update (119898119896|119896minus1 119875119896|119896minus1) 997888rarr (119898119896|119896 119875119896|119896)22 Estimate 119909119896 from the particles 119909119894119896 using 119875119896|11989623 Redraw particles 119909119894119896 sim 119873(119909119896 119875119896|119896) (Optional)

Algorithm 3 Clustered Exact Flow Daum-Huang Filter

filter the particles are not associated with a single parametercalculation Also themigration parameters are not calculatedfor each individual particle

CEDH filters fight to reduce the cost of calculationin LEDH filters which typically occurs due to exhaustivecalculation of each migration parameter In addition whilereducing the cost of calculation that the particles with higherror margin can be systematically clustered together leadsto significant performance improvement The assessmentof the performance and calculation speed as the outcomesof CEDH filter implementation for the scenario of multi-dimensional target tracking will be presented in the nextsection

4 Simulation and Results

The particle filters are applied to a multitarget tracking prob-lem which is adapted from [30] A wireless sensor networkmodel of 25 acoustic amplitude sensor nodes located at theintersections of the grids on a 40119898 119909 40119898 rectangular regionis usedThemodel is shown in Figure 1There are four targetsmoving along two axes independentlyTheoverall state vectorcontains four states for these four targetsTherefore it is in the16-dimensional state space

The independentmovementmodel of four targets (119875 = 4)moving at a constant speed can be expressed as119909(119901)

119896= 119865119909(119901)119896minus1

+ 119882119901V(119901)119896 (10)

where 119909119901119896= [119909119901119896 119910119901119896 119901119896 119910119901119896] consists of the x-y position and

x-y velocity components of the corresponding target

119865 = [[[[[[1 0 1 00 1 0 10 0 1 00 0 0 1

]]]]]] (11)

is the transition matrix V(119901)119896

sim 119873(0 1205902V119881) is the process noise1205902V is set to 001 and the covariance of the process noise forthe filters is set as

119882119901 = (05 0 02 00 05 0 021 0 1 00 0 0 1 ) (12)

All targets emit sounds with amplitude 120595 and each sensorrecords the sum of amplitudes Thus measurement functionfor the 119904-th sensor located at 119877119904 is additive119911119904 (119909119896) = 119875sum

119901=1

1205951003817100381710038171003817100381710038171003817(119909(119901)119896 119910(119901)119896

)119879 minus 119877210038171003817100381710038171003817100381710038172 + 1198890 (13)

where 1198890 = 01 and 120595 = 10 in our simulations The measure-ments are perturbed by Gaussian noise There are 119873119904 = 25sensors located at grid intersections as shown in Figure 1Thefour targets are initializedwith states [8 8 001minus001] [9 34minus0001 0003] [34 32 002 minus 01] and [35 12 0001 minus 0001]respectively

Multitarget tracking scenario associated with the wirelesssensor network model was simulated employing the filters ofEKF BPF EDH LEDH and CEDH for different measure-ment sets Each run starts with a different initial distributionFigure 1 shows the true routes of the targets and the estimatedroutes obtained with CEDH The simulation consists of 40time steps and each algorithm is run separately 100 timesThe optimal mass transfer (OMAT) metric is calculated bytaking the mean of each time interval The average OMATerror calculated for EKF BPF EDH LEDH and CEDH isgiven in Figure 2

As shown in Figure 2 the EKF filter has the highesterror margin It is followed by the BPF EDH and LEDHrespectively CEDH filters with different cluster numbers arethe best performing filters at all time intervals

6 Mathematical Problems in Engineering

5 10 15 20 25 30 35 400X (m)

Target 1 (true)Target 2 (true)Target 3 (true)Target 4 (true)

Target 1 (est)Target 2 (est)Target 3 (est)Target 4 (est)

0

5

10

15

20

25

30

35

40

Y (m

)

Figure 1 One example of true and estimated trajectories using CEDH

30 32 341

15

2x2

5 10 15 20 25 30 35 400time step

BPFEKFEDHLEDH

CEDH (10)CEDH (30)CEDH (50)

0

1

2

3

4

5

6

7

8

9

aver

age O

MAT

erro

r (m

)

Figure 2 Average OMAT errors of the filters

Mathematical Problems in Engineering 7

Table 1 Detailed information about the filters

Algorithm EKF BPF EDH LEDH CEDH CEDH CEDHParticle num NA 10000 500 500 500 500 500120582 NA NA 30 30 30 30 30Cluster num NA NA NA NA 10 30 50Avg err (m) 447 379 324 260 150 141 137Exec time (s) 003 157 048 4473 2305 2479 2842

0 10 20 30 40 50 60 70 80 90 100trial

0

5

10

15

20

25

30

35

40

45

50

exec

utio

n tim

e (s)

2 4 6 8 10 12

0

2

4 x2

BPFEKFEDHLEDH

CEDH (10)CEDH (30)CEDH (50)

Figure 3 Execution times of the filters

Detailed information about the filters is given in Table 1The computer used for the simulations isMacBookAir (Early2015) which has a 16GHz Intel Core i5 processor 4GB of1600MHz DDR3 memory and an Intel HD Graphics 60001536 MB graphics card

As shown in Figure 3 the CEDH algorithm ensuredsignificant improvement in terms of calculation cost Itcorresponds to approximately half of the execution timeof the LEDH algorithm Increasing the number of clustersimproves performance however this improvement bringsextra cost of calculation to the filter

5 Conclusions

We introduced Clustered Exact Flow Daum-Huang ParticleFilter in this study While improving the performance of

the LEDH filter CEDH also reduces the cost of calculationthrough utilizing a clustering strategy The consideration ofclustering of the particles with high error margins ensuresthat the influence of the particles on the overall system canbe reduced In CEDHfilter the particles are not accompaniedwith the calculation of same parameter as it occurs in EDHfilter CEDHfilter succeeds in reducing the cost of calculationin LEDH filter arising due to comprehensive calculation ofeachmigration parameterThe calculation cost is significantlyreduced by calculating commonmigration parameters for theparticles with similar error values The given multidimen-sional target tracking problem demonstrates the success ofthe CEDH filter In the future work the effect of clustering onthe systemwill be examined and the focuswill be on the selec-tion of the appropriate number of clusters and elimination ofthe appropriate cluster with the help of machine learning

8 Mathematical Problems in Engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] M S Arulampalam S Maskell N Gordon and T Clapp ldquoAtutorial on particle filters for online nonlinearnon-GaussianBayesian trackingrdquo IEEE Transactions on Signal Processing vol50 no 2 pp 174ndash188 2002

[2] M West and J Harrison Bayesian Forecasting and DynamicModels Springer Series in Statistics New York NY USA 2ndedition 1997

[3] G Welch and G Bishop ldquoAn introduction to the Kalmanfilterrdquo Technical report UNC-CH Computer Science TechnicalReport 95041 1995

[4] N J Gordon D J Salmond and A F M Smith ldquoNovel ap-proach to nonlinearnon-gaussian Bayesian state estimationrdquoIEE proceedings F vol 140 no 2 pp 107ndash113 1993

[5] P Bunch and S Godsill ldquoApproximations of the optimalimportance density using Gaussian particle flow importancesamplingrdquo Journal of the American Statistical Association vol111 no 514 pp 748ndash762 2016

[6] A Doucet S Godsill and C Andrieu ldquoOn sequential MonteCarlo sampling methods for Bayesian filteringrdquo Statistics andComputing vol 10 no 3 pp 197ndash208 2000

[7] J Cornebise E Moulines and J Olsson ldquoAdaptive methods forsequential importance sampling with application to state spacemodelsrdquo Statistics and Computing vol 18 no 4 pp 461ndash4802008

[8] M K Pitt and N Shephard ldquoFiltering via simulation auxiliaryparticle filtersrdquo Journal of the American Statistical Associationvol 94 no 446 pp 590ndash599 1999

[9] A Doucet N de Freitas N Gordon and S J Russell ldquoRao-Blackwellised particle filtering for dynamic Bayesian networksrdquoin Proceedings of the Uncertainty in Artificial Intelligence (UAI)pp 176ndash183 Springer San Francisco Calif USA 2000

[10] R van der Merwe A Doucet N De Freitas and E Wan ldquoTheunscented particle filterrdquo in Proceedings of the Neural Info ProcSys (NIPS) pp 584ndash590 Denver Co USA

[11] A Beskos D Crisan andA Jasra ldquoOn the stability of sequentialMonte Carlo methods in high dimensionsrdquo e Annals ofApplied Probability vol 24 no 4 pp 1396ndash1445 2014

[12] D Crisan and B Rozovskiie Oxford Handbook of NonlinearFiltering Oxford University Press 2011

[13] P Bui Quang C Musso and F Le Gland ldquoAn insight into theissue of dimensionality in particle filteringrdquo in Proceedings ofthe 2010 13th International Conference on Information Fusion(FUSION 2010) pp 1ndash8 Edinburgh July 2010

[14] M Ades and P J van Leeuwen ldquoThe equivalent-weights particlefilter in a high-dimensional systemrdquo Quarterly Journal of theRoyalMeteorological Society vol 141 no 687 pp 484ndash503 2015

[15] P M Djuric T Lu and M F Bugallo ldquoMultiple particlefilteringrdquo in Proc Intl Conf Acoustics Speech and Signal Proc(ICASSP) vol 3 pp 1181ndash1184 2007

[16] A Beskos D Crisan A Jasra K Kamatani and Y Zhou ldquoAstable particle filter for a class of high-dimensional state-spacemodelsrdquo Advances in Applied Probability vol 49 no 1 pp 24ndash48 2017

[17] F Septier S K Pang A Carmi and S Godsill ldquoOn MCMC-based particle methods for Bayesian filtering Application tomultitarget trackingrdquo in Proceedings of the 3rd IEEE Interna-tional Workshop on Computational Advances in Multi-SensorAdaptive Processing (CAMSAP 2009) pp 360ndash363 ArubaDutch Antilles December 2009

[18] T Ding and M J Coates ldquoImplementation of the Daum-Huang exact- flow particle filterrdquo in Proceedings of the 2012 IEEEStatistical Signal Processing Workshop SSP pp 257ndash260 USAAugust 2012

[19] F Daum and J Huang ldquoNonlinear filters with log-homotopyrdquoin Proc SPIE Conf Signal and Data Processing of Small TargetsSan Diego CA USA 2007

[20] F Daum I Kadar and J Huang ldquoNonlinear filters with particleflow induced by log-homotopyrdquo in Proceedings of the SPIEDefense Security and Sensing p 733603 Orlando FloridaUSA

[21] F Daum I Kadar J Huang and A Noushin ldquoExact particleflow for nonlinear filtersrdquo in Proceedings of the SPIE DefenseSecurity and Sensing p 769704 Orlando Florida

[22] F Daum J Huang and A Noushin ldquoCoulombs law particleflow for nonlinear filtersrdquo in Proc SPIE Conf Signal and DataProc San Diego CA USA 2011

[23] Y Li L Zhao and M Coates ldquoParticle flow for particlefilteringrdquo in Proceedings of the IEEE Interna- tional Conferenceon Acoustics Speech and Signal Processing (ICASSP) pp 20ndash25Shanghai China March 2016

[24] Y Li and M Coates ldquoFast Particle Flow Particle Filters viaClusteringrdquo in Proceedings of the 19th Inter- national Conferenceon Information Fusion Heidelberg Germany 2016

[25] S C Surace A Kutschireiter and J-P Pfister ldquoHow to avoid thecurse of dimen- sionality scalability of particle filters with andwithout importance weightsrdquo httpsarxivorgabs170307879

[26] F Daum J Huang and A Noushin ldquoNewTheory and Numer-ical Results for Gromovrsquos Method for Stochastic Particle FlowFiltersrdquo in Proceedings of the 21st International Conference onInformation Fusion FUSION 2018 pp 108ndash115 UK July 2018

[27] Y Li S Pal and M Coates ldquoInvertible Particle Flow-basedSequential MCMC with ex- tension to Gaussian Mixture noisemodelsrdquo IEEE Transactions on Signal Processing 2019

[28] D Arthur and S Vassilvitskii ldquok-means++ The advantages ofcareful seedingrdquo in Proceedings of the ACM-SIAM Symposiumon Discrete Algorithms pp 1027ndash1035 2007

[29] S P Lloyd ldquoLeast squares quantization in PCMrdquo IEEE Transac-tions on Information eory vol 28 no 2 pp 129ndash137 1982

[30] O Hlinka O Sluciak F Hlawatsch P M Djuric and MRupp ldquoDistributed Gaussian particle filtering using likelihoodconsensusrdquo in Proceedings of the ICASSP 2011 - 2011 IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP) pp 3756ndash3759 Prague Czech Republic May 2011

Mathematical Problems in Engineering 3

In this study for each target velocity position derivativeof the velocity and derivative of the position are dealt withthus we are involved with a tracking problem in the four-dimensional state space Four targets are tracked utilizing ourCEDH filter as well as EKF BPF EDH and LEDH filters andperformances are compared

The structure of this paper is as follows In Section 2theoretical base in terms of established techniques for particleflow filters is represented The implementation details of k-means++ algorithm are given and its use in CEDH filter isdescribed in Section 3 The application of CEDH filter ona descriptive scenario is detailed in Section 4 Performancecomparisons of the filters employed for multitarget trackingproblem in the high dimensional state space are given in thissection as well Section 5 puts concluding comments on theproposed filter discusses problems that arose in the studyand possible future work and lists practical pieces of adviceresulting from the scenario

2 Particle Flow Filter

Particle flow filter is the modified version of the conventionalparticle filters It essentially fights against the particle degen-eracy and ensures the fast convergence to the particles withthe highest posterior distribution in the next step throughemploying a logarithmic homotopy function The homotopyfunction depicts the transition between prior and posteriordistributions for the flow of particles The filter uses the Itostochastic Partial Differential Equation (PDE) which is usedto differentiate between the step parameter of the homotopyfunction and the state

We consider a discrete-time nonlinear filtering task withthe following models119909119896 = 119891119896 (119909119896minus1) + V119896 (1)119911 = 120574 (119909119896minus1) + 119899119896 (2)

where 119909119896 isin R119863 is a target state vector 119911119896 isin R119878 is a measure-ment vector V119896 isin R119863 is the process noise and 119899119896 isin R119878 is themeasurement noise 119891119896 R119863 997888rarr R119863 is a nonlinear map and120574119896 R119863 997888rarr R119878 is a nonlinear measurement map

Bayesian rule to define the unnormalized marginal pos-terior distribution is as follows119901 (119909119896 | 1199111119896) = 119901 (119911119896 | 119909119896) 119901 (119909119896 | 1199111119896minus1) (3)

Daum and Huang expressed the homotopy function in thisform 120601 (119909119896 120582) = log119892 (119909119896) + 120582 log ℎ (119909119896) (4)

where ℎ(119909119896) = 119901(119911119896 | 119909119896) 119892(119909119896) = 119901(119909119896 | 1199111119896minus1) and 120582 isthe real valued step parameter in the range of [0 1] andrepresents the intensity or amount of particle flow Thehomotopy function provides a continuous deformation fromlog119892(119909119896) (when 120582 = 0) to the logarithm of the unnormalizedposterior distribution log119901(119909119896 | 1199111119896) (when 120582 = 1)

In the original particle flow filter the flow is improved sothat the homotopy function remains constant as 120582 evolves

Partial differential equation use is required for this purposeand the following equation can be referred to solve andcalculate the flow of the particles120597120593120597119909 119889119909119889120582 + 120597120593120597120582 = 0 (5)

Daum and Huang derived the following expression utilizingthe Fokker-Planck equation and developed and generalizedthe filter through proposing EDH filter [21]120597120593120597119909120595 (119909 120582) + log (ℎ) = minus119879119903(120597120595120597119909) (6)

The solution of this equation leads to the exact flow of theprobability densityThe flow quantities of the particles can becalculated as given in [21]119889119909119889120582 = 119860 (120582) 119909 + 119887 (120582) (7)

where 119860 = minus12119875119867119879 (120582119867119875119867119879 + 119877)minus1119867 (8)

and 119887 = (119868 minus 2120582119860) [(119868 + 120582119860) 119875119867119879119877minus1119911 + 119860119909] (9)119875 and 119877 depict the covariance of the estimation error andmeasurement noise respectively 119867 is the measurementmatrix 119909 denotes the state variable that immigrates in eachcycle and 119889119909119889120582 represents the change of 119909 with respect to12058221 Implementation of Particle Flow Filter Algorithm 1includes the pseudocodes of the EDH filter implementationregarding the previously stated theoretical base Particlemigration task is realized following the steps on lines 7 to16 119873 is the number of particles and 119879 is the number oftime intervals The UKFEKF state and covariance matrixestimation are represented with119898 and 119875 respectively

The steps of UKFEKF estimation and UKFEKF updateare expressed on lines 6 and 17 respectivelyThemeasurementmatrix 119867119909 is calculated through linearizing the currentestimate 119909119896 and depicted on line 9 The same 119860 and 119887 valuesare used to update all particles

A simple but effective change stated in [18] is to replacethe lines of 7-19 of Algorithm 1 with the pseudocodes givenin Algorithm 2

There are two major changes in Algorithm 2 For eachparticle associated part of the measurement function islinearized and119860 119894 and 119887119894 values are calculated using119867119894matrixThus the calculation of the migration parameters for eachparticle is performed individually with varying values of 119860 119894and 119887119894 The other change is that the mean estimate from theUKFEKF is replaced with the state estimate from the Daum-Huang filter as stated on line 19 [18]

4 Mathematical Problems in Engineering

1 Initialization Draw 1199091198940119873119894=1 from the prior 119901(1199090)2 Set 1199090 and1198980 as the mean 1198750 as the covariance matrix3 for 119896 = 1 to 119879 do4 Propagate particles 119909119894119896minus1 = 119891119896(119909119894119896minus1) + V1198965 Calculate the mean value 1199091198966 Apply UKFEKF prediction (119898119896minus1|119896minus1 119875119896minus1|119896minus1) 997888rarr (119898119896|119896minus1 119875119896|119896minus1)7 for 119895 = 1 to119873120582 do8 Set 120582 = 119895Δ1205829 Calculate119867119909 by linearizing 120574119896() at 11990911989610 Calculate 119860 and 119887 from (8) and (9) using 119875119896|119896minus1 119909 and11986711990911 for 119894 = 1 to119873 do12 Evaluate 119889119909119894119896119889120582 for each particle from (7)13 Migrate particles 119909119894119896 = 119909119894119896 + Δ120582(119889119909119894119896119889120582)14 endfor15 Re-evaluate 119909119896 using the updated particles 11990911989411989616 endfor17 Apply UKFEKF update (119898119896|119896minus1 119875119896|119896minus1) 997888rarr (119898119896|119896 119875119896|119896)18 Estimate 119909119896 from the particles 119909119894119896 using 119875119896|11989619 Redraw particles 119909119894119896 sim 119873(119909119896 119875119896|119896) (Optional)20 endfor

Algorithm 1 Exact Flow Daum-Huang Filter

7 for 119895 = 1 to119873120582 do8 Set 120582 = 119895Δ1205829 for 119894 = 1 to119873 do10 Calculate119867119909119894 by linearizing 120574119896() at 11990911989411 Calculate 119860119894 and 119887119894 from (8) and (9) using 119875119896|119896minus1 119909 and119867119909119894 12 Evaluate 119889119909119894119896119889120582 for each particle from (7)13 Migrate particles 119909119894119896 = 119909119894119896 + Δ120582(119889119909119894119896119889120582)14 endfor15 Re-evaluate 119909119896 using the updated particles 11990911989411989616 endfor17 Apply UKFEKF update (119898119896|119896minus1 119875119896|119896minus1) 997888rarr (119898119896|119896 119875119896|119896)18 Estimate 119909119896 from the particles 119909119894119896 using 119875119896|11989619 Set119898119896|119896 = 11990911989620 Redraw particles 119909119894119896 sim 119873(119909119896 119875119896|119896) (Optional)

Algorithm 2 Local Exact Flow Daum-Huang Filter

3 Clustered Particle Flow Filter

EDH filter calculates migration parameters for all of theparticles LEDH filter calculates the migration parametersindividually for each particle CEDHfilter employs clusteringof the particles considering the ones producing similarerrors The calculation cost is significantly reduced throughcalculating common migration parameters for the particleswithin each cluster

In this study k-means++ algorithm [28] is adoptedto fulfill the clustering demand of CEDH filter The well-known k-means class methods aim at minimizing the meansquare distance between points in the same set [29] Arthurand Vassilvitskii proposed an algorithm that is 119874(log 119896)-competitive It strengthens the clustering scheme with arandomized seeding technique and improves both the speedand the accuracy in clustering

31 Implementation of Clustered Particle Flow Filter The costof calculation is anticipated to be high in LEDH filter sinceit pursues the calculation of the flow for each particle in eachpseudo-time interval The main objective of the CEDH filteris that it reduces the cost of calculation through implementingthe steps of a clustering methodology Thus it reduces theinfluence of particles having a high error margin on theoverall system and therefore improves the performance of theLEDH filter The pseudocodes for the implementation of theCEDH filter are specified in Algorithm 3119873119888 is the number of clusters Algorithm 3 like the LEDHfilter changes the steps on lines 7 to 19 of the Algorithm 1Clustering step takes place over the error margin of the par-ticles as expressed on line 12 The migration parameters arecalculated up to the number of clusters through employingthe steps depicted on lines 13 to 17 Therefore in CEDH

Mathematical Problems in Engineering 5

7 for 119895 = 1 to119873120582 do8 Set 120582 = 119895Δ1205829 for 119894 = 1 to119873 do10 Calculate119867119909 by linearizing 120574119896() at 11990911989411 endfor12 Cluster particles over the margin of error13 for 119888 = 1 to119873119888 do14 Calculate 119860119888 and 119887119888 from (6) and (7) using 119875119896|119896minus1 119909 and119867119909119888 15 Evaluate 119889119909119888119896119889120582 for each particle from (5)16 Migrate particles 119909119888119896 = 119909119888119896 + Δ120582(119889119909119888119896119889120582)17 endfor18 Eliminate the largest cluster with the biggest error19 Re-evaluate 119909119896 using the updated particles 11990911989411989620 endfor21 Apply UKFEKF update (119898119896|119896minus1 119875119896|119896minus1) 997888rarr (119898119896|119896 119875119896|119896)22 Estimate 119909119896 from the particles 119909119894119896 using 119875119896|11989623 Redraw particles 119909119894119896 sim 119873(119909119896 119875119896|119896) (Optional)

Algorithm 3 Clustered Exact Flow Daum-Huang Filter

filter the particles are not associated with a single parametercalculation Also themigration parameters are not calculatedfor each individual particle

CEDH filters fight to reduce the cost of calculationin LEDH filters which typically occurs due to exhaustivecalculation of each migration parameter In addition whilereducing the cost of calculation that the particles with higherror margin can be systematically clustered together leadsto significant performance improvement The assessmentof the performance and calculation speed as the outcomesof CEDH filter implementation for the scenario of multi-dimensional target tracking will be presented in the nextsection

4 Simulation and Results

The particle filters are applied to a multitarget tracking prob-lem which is adapted from [30] A wireless sensor networkmodel of 25 acoustic amplitude sensor nodes located at theintersections of the grids on a 40119898 119909 40119898 rectangular regionis usedThemodel is shown in Figure 1There are four targetsmoving along two axes independentlyTheoverall state vectorcontains four states for these four targetsTherefore it is in the16-dimensional state space

The independentmovementmodel of four targets (119875 = 4)moving at a constant speed can be expressed as119909(119901)

119896= 119865119909(119901)119896minus1

+ 119882119901V(119901)119896 (10)

where 119909119901119896= [119909119901119896 119910119901119896 119901119896 119910119901119896] consists of the x-y position and

x-y velocity components of the corresponding target

119865 = [[[[[[1 0 1 00 1 0 10 0 1 00 0 0 1

]]]]]] (11)

is the transition matrix V(119901)119896

sim 119873(0 1205902V119881) is the process noise1205902V is set to 001 and the covariance of the process noise forthe filters is set as

119882119901 = (05 0 02 00 05 0 021 0 1 00 0 0 1 ) (12)

All targets emit sounds with amplitude 120595 and each sensorrecords the sum of amplitudes Thus measurement functionfor the 119904-th sensor located at 119877119904 is additive119911119904 (119909119896) = 119875sum

119901=1

1205951003817100381710038171003817100381710038171003817(119909(119901)119896 119910(119901)119896

)119879 minus 119877210038171003817100381710038171003817100381710038172 + 1198890 (13)

where 1198890 = 01 and 120595 = 10 in our simulations The measure-ments are perturbed by Gaussian noise There are 119873119904 = 25sensors located at grid intersections as shown in Figure 1Thefour targets are initializedwith states [8 8 001minus001] [9 34minus0001 0003] [34 32 002 minus 01] and [35 12 0001 minus 0001]respectively

Multitarget tracking scenario associated with the wirelesssensor network model was simulated employing the filters ofEKF BPF EDH LEDH and CEDH for different measure-ment sets Each run starts with a different initial distributionFigure 1 shows the true routes of the targets and the estimatedroutes obtained with CEDH The simulation consists of 40time steps and each algorithm is run separately 100 timesThe optimal mass transfer (OMAT) metric is calculated bytaking the mean of each time interval The average OMATerror calculated for EKF BPF EDH LEDH and CEDH isgiven in Figure 2

As shown in Figure 2 the EKF filter has the highesterror margin It is followed by the BPF EDH and LEDHrespectively CEDH filters with different cluster numbers arethe best performing filters at all time intervals

6 Mathematical Problems in Engineering

5 10 15 20 25 30 35 400X (m)

Target 1 (true)Target 2 (true)Target 3 (true)Target 4 (true)

Target 1 (est)Target 2 (est)Target 3 (est)Target 4 (est)

0

5

10

15

20

25

30

35

40

Y (m

)

Figure 1 One example of true and estimated trajectories using CEDH

30 32 341

15

2x2

5 10 15 20 25 30 35 400time step

BPFEKFEDHLEDH

CEDH (10)CEDH (30)CEDH (50)

0

1

2

3

4

5

6

7

8

9

aver

age O

MAT

erro

r (m

)

Figure 2 Average OMAT errors of the filters

Mathematical Problems in Engineering 7

Table 1 Detailed information about the filters

Algorithm EKF BPF EDH LEDH CEDH CEDH CEDHParticle num NA 10000 500 500 500 500 500120582 NA NA 30 30 30 30 30Cluster num NA NA NA NA 10 30 50Avg err (m) 447 379 324 260 150 141 137Exec time (s) 003 157 048 4473 2305 2479 2842

0 10 20 30 40 50 60 70 80 90 100trial

0

5

10

15

20

25

30

35

40

45

50

exec

utio

n tim

e (s)

2 4 6 8 10 12

0

2

4 x2

BPFEKFEDHLEDH

CEDH (10)CEDH (30)CEDH (50)

Figure 3 Execution times of the filters

Detailed information about the filters is given in Table 1The computer used for the simulations isMacBookAir (Early2015) which has a 16GHz Intel Core i5 processor 4GB of1600MHz DDR3 memory and an Intel HD Graphics 60001536 MB graphics card

As shown in Figure 3 the CEDH algorithm ensuredsignificant improvement in terms of calculation cost Itcorresponds to approximately half of the execution timeof the LEDH algorithm Increasing the number of clustersimproves performance however this improvement bringsextra cost of calculation to the filter

5 Conclusions

We introduced Clustered Exact Flow Daum-Huang ParticleFilter in this study While improving the performance of

the LEDH filter CEDH also reduces the cost of calculationthrough utilizing a clustering strategy The consideration ofclustering of the particles with high error margins ensuresthat the influence of the particles on the overall system canbe reduced In CEDHfilter the particles are not accompaniedwith the calculation of same parameter as it occurs in EDHfilter CEDHfilter succeeds in reducing the cost of calculationin LEDH filter arising due to comprehensive calculation ofeachmigration parameterThe calculation cost is significantlyreduced by calculating commonmigration parameters for theparticles with similar error values The given multidimen-sional target tracking problem demonstrates the success ofthe CEDH filter In the future work the effect of clustering onthe systemwill be examined and the focuswill be on the selec-tion of the appropriate number of clusters and elimination ofthe appropriate cluster with the help of machine learning

8 Mathematical Problems in Engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] M S Arulampalam S Maskell N Gordon and T Clapp ldquoAtutorial on particle filters for online nonlinearnon-GaussianBayesian trackingrdquo IEEE Transactions on Signal Processing vol50 no 2 pp 174ndash188 2002

[2] M West and J Harrison Bayesian Forecasting and DynamicModels Springer Series in Statistics New York NY USA 2ndedition 1997

[3] G Welch and G Bishop ldquoAn introduction to the Kalmanfilterrdquo Technical report UNC-CH Computer Science TechnicalReport 95041 1995

[4] N J Gordon D J Salmond and A F M Smith ldquoNovel ap-proach to nonlinearnon-gaussian Bayesian state estimationrdquoIEE proceedings F vol 140 no 2 pp 107ndash113 1993

[5] P Bunch and S Godsill ldquoApproximations of the optimalimportance density using Gaussian particle flow importancesamplingrdquo Journal of the American Statistical Association vol111 no 514 pp 748ndash762 2016

[6] A Doucet S Godsill and C Andrieu ldquoOn sequential MonteCarlo sampling methods for Bayesian filteringrdquo Statistics andComputing vol 10 no 3 pp 197ndash208 2000

[7] J Cornebise E Moulines and J Olsson ldquoAdaptive methods forsequential importance sampling with application to state spacemodelsrdquo Statistics and Computing vol 18 no 4 pp 461ndash4802008

[8] M K Pitt and N Shephard ldquoFiltering via simulation auxiliaryparticle filtersrdquo Journal of the American Statistical Associationvol 94 no 446 pp 590ndash599 1999

[9] A Doucet N de Freitas N Gordon and S J Russell ldquoRao-Blackwellised particle filtering for dynamic Bayesian networksrdquoin Proceedings of the Uncertainty in Artificial Intelligence (UAI)pp 176ndash183 Springer San Francisco Calif USA 2000

[10] R van der Merwe A Doucet N De Freitas and E Wan ldquoTheunscented particle filterrdquo in Proceedings of the Neural Info ProcSys (NIPS) pp 584ndash590 Denver Co USA

[11] A Beskos D Crisan andA Jasra ldquoOn the stability of sequentialMonte Carlo methods in high dimensionsrdquo e Annals ofApplied Probability vol 24 no 4 pp 1396ndash1445 2014

[12] D Crisan and B Rozovskiie Oxford Handbook of NonlinearFiltering Oxford University Press 2011

[13] P Bui Quang C Musso and F Le Gland ldquoAn insight into theissue of dimensionality in particle filteringrdquo in Proceedings ofthe 2010 13th International Conference on Information Fusion(FUSION 2010) pp 1ndash8 Edinburgh July 2010

[14] M Ades and P J van Leeuwen ldquoThe equivalent-weights particlefilter in a high-dimensional systemrdquo Quarterly Journal of theRoyalMeteorological Society vol 141 no 687 pp 484ndash503 2015

[15] P M Djuric T Lu and M F Bugallo ldquoMultiple particlefilteringrdquo in Proc Intl Conf Acoustics Speech and Signal Proc(ICASSP) vol 3 pp 1181ndash1184 2007

[16] A Beskos D Crisan A Jasra K Kamatani and Y Zhou ldquoAstable particle filter for a class of high-dimensional state-spacemodelsrdquo Advances in Applied Probability vol 49 no 1 pp 24ndash48 2017

[17] F Septier S K Pang A Carmi and S Godsill ldquoOn MCMC-based particle methods for Bayesian filtering Application tomultitarget trackingrdquo in Proceedings of the 3rd IEEE Interna-tional Workshop on Computational Advances in Multi-SensorAdaptive Processing (CAMSAP 2009) pp 360ndash363 ArubaDutch Antilles December 2009

[18] T Ding and M J Coates ldquoImplementation of the Daum-Huang exact- flow particle filterrdquo in Proceedings of the 2012 IEEEStatistical Signal Processing Workshop SSP pp 257ndash260 USAAugust 2012

[19] F Daum and J Huang ldquoNonlinear filters with log-homotopyrdquoin Proc SPIE Conf Signal and Data Processing of Small TargetsSan Diego CA USA 2007

[20] F Daum I Kadar and J Huang ldquoNonlinear filters with particleflow induced by log-homotopyrdquo in Proceedings of the SPIEDefense Security and Sensing p 733603 Orlando FloridaUSA

[21] F Daum I Kadar J Huang and A Noushin ldquoExact particleflow for nonlinear filtersrdquo in Proceedings of the SPIE DefenseSecurity and Sensing p 769704 Orlando Florida

[22] F Daum J Huang and A Noushin ldquoCoulombs law particleflow for nonlinear filtersrdquo in Proc SPIE Conf Signal and DataProc San Diego CA USA 2011

[23] Y Li L Zhao and M Coates ldquoParticle flow for particlefilteringrdquo in Proceedings of the IEEE Interna- tional Conferenceon Acoustics Speech and Signal Processing (ICASSP) pp 20ndash25Shanghai China March 2016

[24] Y Li and M Coates ldquoFast Particle Flow Particle Filters viaClusteringrdquo in Proceedings of the 19th Inter- national Conferenceon Information Fusion Heidelberg Germany 2016

[25] S C Surace A Kutschireiter and J-P Pfister ldquoHow to avoid thecurse of dimen- sionality scalability of particle filters with andwithout importance weightsrdquo httpsarxivorgabs170307879

[26] F Daum J Huang and A Noushin ldquoNewTheory and Numer-ical Results for Gromovrsquos Method for Stochastic Particle FlowFiltersrdquo in Proceedings of the 21st International Conference onInformation Fusion FUSION 2018 pp 108ndash115 UK July 2018

[27] Y Li S Pal and M Coates ldquoInvertible Particle Flow-basedSequential MCMC with ex- tension to Gaussian Mixture noisemodelsrdquo IEEE Transactions on Signal Processing 2019

[28] D Arthur and S Vassilvitskii ldquok-means++ The advantages ofcareful seedingrdquo in Proceedings of the ACM-SIAM Symposiumon Discrete Algorithms pp 1027ndash1035 2007

[29] S P Lloyd ldquoLeast squares quantization in PCMrdquo IEEE Transac-tions on Information eory vol 28 no 2 pp 129ndash137 1982

[30] O Hlinka O Sluciak F Hlawatsch P M Djuric and MRupp ldquoDistributed Gaussian particle filtering using likelihoodconsensusrdquo in Proceedings of the ICASSP 2011 - 2011 IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP) pp 3756ndash3759 Prague Czech Republic May 2011

4 Mathematical Problems in Engineering

1 Initialization Draw 1199091198940119873119894=1 from the prior 119901(1199090)2 Set 1199090 and1198980 as the mean 1198750 as the covariance matrix3 for 119896 = 1 to 119879 do4 Propagate particles 119909119894119896minus1 = 119891119896(119909119894119896minus1) + V1198965 Calculate the mean value 1199091198966 Apply UKFEKF prediction (119898119896minus1|119896minus1 119875119896minus1|119896minus1) 997888rarr (119898119896|119896minus1 119875119896|119896minus1)7 for 119895 = 1 to119873120582 do8 Set 120582 = 119895Δ1205829 Calculate119867119909 by linearizing 120574119896() at 11990911989610 Calculate 119860 and 119887 from (8) and (9) using 119875119896|119896minus1 119909 and11986711990911 for 119894 = 1 to119873 do12 Evaluate 119889119909119894119896119889120582 for each particle from (7)13 Migrate particles 119909119894119896 = 119909119894119896 + Δ120582(119889119909119894119896119889120582)14 endfor15 Re-evaluate 119909119896 using the updated particles 11990911989411989616 endfor17 Apply UKFEKF update (119898119896|119896minus1 119875119896|119896minus1) 997888rarr (119898119896|119896 119875119896|119896)18 Estimate 119909119896 from the particles 119909119894119896 using 119875119896|11989619 Redraw particles 119909119894119896 sim 119873(119909119896 119875119896|119896) (Optional)20 endfor

Algorithm 1 Exact Flow Daum-Huang Filter

7 for 119895 = 1 to119873120582 do8 Set 120582 = 119895Δ1205829 for 119894 = 1 to119873 do10 Calculate119867119909119894 by linearizing 120574119896() at 11990911989411 Calculate 119860119894 and 119887119894 from (8) and (9) using 119875119896|119896minus1 119909 and119867119909119894 12 Evaluate 119889119909119894119896119889120582 for each particle from (7)13 Migrate particles 119909119894119896 = 119909119894119896 + Δ120582(119889119909119894119896119889120582)14 endfor15 Re-evaluate 119909119896 using the updated particles 11990911989411989616 endfor17 Apply UKFEKF update (119898119896|119896minus1 119875119896|119896minus1) 997888rarr (119898119896|119896 119875119896|119896)18 Estimate 119909119896 from the particles 119909119894119896 using 119875119896|11989619 Set119898119896|119896 = 11990911989620 Redraw particles 119909119894119896 sim 119873(119909119896 119875119896|119896) (Optional)

Algorithm 2 Local Exact Flow Daum-Huang Filter

3 Clustered Particle Flow Filter

EDH filter calculates migration parameters for all of theparticles LEDH filter calculates the migration parametersindividually for each particle CEDHfilter employs clusteringof the particles considering the ones producing similarerrors The calculation cost is significantly reduced throughcalculating common migration parameters for the particleswithin each cluster

In this study k-means++ algorithm [28] is adoptedto fulfill the clustering demand of CEDH filter The well-known k-means class methods aim at minimizing the meansquare distance between points in the same set [29] Arthurand Vassilvitskii proposed an algorithm that is 119874(log 119896)-competitive It strengthens the clustering scheme with arandomized seeding technique and improves both the speedand the accuracy in clustering

31 Implementation of Clustered Particle Flow Filter The costof calculation is anticipated to be high in LEDH filter sinceit pursues the calculation of the flow for each particle in eachpseudo-time interval The main objective of the CEDH filteris that it reduces the cost of calculation through implementingthe steps of a clustering methodology Thus it reduces theinfluence of particles having a high error margin on theoverall system and therefore improves the performance of theLEDH filter The pseudocodes for the implementation of theCEDH filter are specified in Algorithm 3119873119888 is the number of clusters Algorithm 3 like the LEDHfilter changes the steps on lines 7 to 19 of the Algorithm 1Clustering step takes place over the error margin of the par-ticles as expressed on line 12 The migration parameters arecalculated up to the number of clusters through employingthe steps depicted on lines 13 to 17 Therefore in CEDH

Mathematical Problems in Engineering 5

7 for 119895 = 1 to119873120582 do8 Set 120582 = 119895Δ1205829 for 119894 = 1 to119873 do10 Calculate119867119909 by linearizing 120574119896() at 11990911989411 endfor12 Cluster particles over the margin of error13 for 119888 = 1 to119873119888 do14 Calculate 119860119888 and 119887119888 from (6) and (7) using 119875119896|119896minus1 119909 and119867119909119888 15 Evaluate 119889119909119888119896119889120582 for each particle from (5)16 Migrate particles 119909119888119896 = 119909119888119896 + Δ120582(119889119909119888119896119889120582)17 endfor18 Eliminate the largest cluster with the biggest error19 Re-evaluate 119909119896 using the updated particles 11990911989411989620 endfor21 Apply UKFEKF update (119898119896|119896minus1 119875119896|119896minus1) 997888rarr (119898119896|119896 119875119896|119896)22 Estimate 119909119896 from the particles 119909119894119896 using 119875119896|11989623 Redraw particles 119909119894119896 sim 119873(119909119896 119875119896|119896) (Optional)

Algorithm 3 Clustered Exact Flow Daum-Huang Filter

filter the particles are not associated with a single parametercalculation Also themigration parameters are not calculatedfor each individual particle

CEDH filters fight to reduce the cost of calculationin LEDH filters which typically occurs due to exhaustivecalculation of each migration parameter In addition whilereducing the cost of calculation that the particles with higherror margin can be systematically clustered together leadsto significant performance improvement The assessmentof the performance and calculation speed as the outcomesof CEDH filter implementation for the scenario of multi-dimensional target tracking will be presented in the nextsection

4 Simulation and Results

The particle filters are applied to a multitarget tracking prob-lem which is adapted from [30] A wireless sensor networkmodel of 25 acoustic amplitude sensor nodes located at theintersections of the grids on a 40119898 119909 40119898 rectangular regionis usedThemodel is shown in Figure 1There are four targetsmoving along two axes independentlyTheoverall state vectorcontains four states for these four targetsTherefore it is in the16-dimensional state space

The independentmovementmodel of four targets (119875 = 4)moving at a constant speed can be expressed as119909(119901)

119896= 119865119909(119901)119896minus1

+ 119882119901V(119901)119896 (10)

where 119909119901119896= [119909119901119896 119910119901119896 119901119896 119910119901119896] consists of the x-y position and

x-y velocity components of the corresponding target

119865 = [[[[[[1 0 1 00 1 0 10 0 1 00 0 0 1

]]]]]] (11)

is the transition matrix V(119901)119896

sim 119873(0 1205902V119881) is the process noise1205902V is set to 001 and the covariance of the process noise forthe filters is set as

119882119901 = (05 0 02 00 05 0 021 0 1 00 0 0 1 ) (12)

All targets emit sounds with amplitude 120595 and each sensorrecords the sum of amplitudes Thus measurement functionfor the 119904-th sensor located at 119877119904 is additive119911119904 (119909119896) = 119875sum

119901=1

1205951003817100381710038171003817100381710038171003817(119909(119901)119896 119910(119901)119896

)119879 minus 119877210038171003817100381710038171003817100381710038172 + 1198890 (13)

where 1198890 = 01 and 120595 = 10 in our simulations The measure-ments are perturbed by Gaussian noise There are 119873119904 = 25sensors located at grid intersections as shown in Figure 1Thefour targets are initializedwith states [8 8 001minus001] [9 34minus0001 0003] [34 32 002 minus 01] and [35 12 0001 minus 0001]respectively

Multitarget tracking scenario associated with the wirelesssensor network model was simulated employing the filters ofEKF BPF EDH LEDH and CEDH for different measure-ment sets Each run starts with a different initial distributionFigure 1 shows the true routes of the targets and the estimatedroutes obtained with CEDH The simulation consists of 40time steps and each algorithm is run separately 100 timesThe optimal mass transfer (OMAT) metric is calculated bytaking the mean of each time interval The average OMATerror calculated for EKF BPF EDH LEDH and CEDH isgiven in Figure 2

As shown in Figure 2 the EKF filter has the highesterror margin It is followed by the BPF EDH and LEDHrespectively CEDH filters with different cluster numbers arethe best performing filters at all time intervals

6 Mathematical Problems in Engineering

5 10 15 20 25 30 35 400X (m)

Target 1 (true)Target 2 (true)Target 3 (true)Target 4 (true)

Target 1 (est)Target 2 (est)Target 3 (est)Target 4 (est)

0

5

10

15

20

25

30

35

40

Y (m

)

Figure 1 One example of true and estimated trajectories using CEDH

30 32 341

15

2x2

5 10 15 20 25 30 35 400time step

BPFEKFEDHLEDH

CEDH (10)CEDH (30)CEDH (50)

0

1

2

3

4

5

6

7

8

9

aver

age O

MAT

erro

r (m

)

Figure 2 Average OMAT errors of the filters

Mathematical Problems in Engineering 7

Table 1 Detailed information about the filters

Algorithm EKF BPF EDH LEDH CEDH CEDH CEDHParticle num NA 10000 500 500 500 500 500120582 NA NA 30 30 30 30 30Cluster num NA NA NA NA 10 30 50Avg err (m) 447 379 324 260 150 141 137Exec time (s) 003 157 048 4473 2305 2479 2842

0 10 20 30 40 50 60 70 80 90 100trial

0

5

10

15

20

25

30

35

40

45

50

exec

utio

n tim

e (s)

2 4 6 8 10 12

0

2

4 x2

BPFEKFEDHLEDH

CEDH (10)CEDH (30)CEDH (50)

Figure 3 Execution times of the filters

Detailed information about the filters is given in Table 1The computer used for the simulations isMacBookAir (Early2015) which has a 16GHz Intel Core i5 processor 4GB of1600MHz DDR3 memory and an Intel HD Graphics 60001536 MB graphics card

As shown in Figure 3 the CEDH algorithm ensuredsignificant improvement in terms of calculation cost Itcorresponds to approximately half of the execution timeof the LEDH algorithm Increasing the number of clustersimproves performance however this improvement bringsextra cost of calculation to the filter

5 Conclusions

We introduced Clustered Exact Flow Daum-Huang ParticleFilter in this study While improving the performance of

the LEDH filter CEDH also reduces the cost of calculationthrough utilizing a clustering strategy The consideration ofclustering of the particles with high error margins ensuresthat the influence of the particles on the overall system canbe reduced In CEDHfilter the particles are not accompaniedwith the calculation of same parameter as it occurs in EDHfilter CEDHfilter succeeds in reducing the cost of calculationin LEDH filter arising due to comprehensive calculation ofeachmigration parameterThe calculation cost is significantlyreduced by calculating commonmigration parameters for theparticles with similar error values The given multidimen-sional target tracking problem demonstrates the success ofthe CEDH filter In the future work the effect of clustering onthe systemwill be examined and the focuswill be on the selec-tion of the appropriate number of clusters and elimination ofthe appropriate cluster with the help of machine learning

8 Mathematical Problems in Engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] M S Arulampalam S Maskell N Gordon and T Clapp ldquoAtutorial on particle filters for online nonlinearnon-GaussianBayesian trackingrdquo IEEE Transactions on Signal Processing vol50 no 2 pp 174ndash188 2002

[2] M West and J Harrison Bayesian Forecasting and DynamicModels Springer Series in Statistics New York NY USA 2ndedition 1997

[3] G Welch and G Bishop ldquoAn introduction to the Kalmanfilterrdquo Technical report UNC-CH Computer Science TechnicalReport 95041 1995

[4] N J Gordon D J Salmond and A F M Smith ldquoNovel ap-proach to nonlinearnon-gaussian Bayesian state estimationrdquoIEE proceedings F vol 140 no 2 pp 107ndash113 1993

[5] P Bunch and S Godsill ldquoApproximations of the optimalimportance density using Gaussian particle flow importancesamplingrdquo Journal of the American Statistical Association vol111 no 514 pp 748ndash762 2016

[6] A Doucet S Godsill and C Andrieu ldquoOn sequential MonteCarlo sampling methods for Bayesian filteringrdquo Statistics andComputing vol 10 no 3 pp 197ndash208 2000

[7] J Cornebise E Moulines and J Olsson ldquoAdaptive methods forsequential importance sampling with application to state spacemodelsrdquo Statistics and Computing vol 18 no 4 pp 461ndash4802008

[8] M K Pitt and N Shephard ldquoFiltering via simulation auxiliaryparticle filtersrdquo Journal of the American Statistical Associationvol 94 no 446 pp 590ndash599 1999

[9] A Doucet N de Freitas N Gordon and S J Russell ldquoRao-Blackwellised particle filtering for dynamic Bayesian networksrdquoin Proceedings of the Uncertainty in Artificial Intelligence (UAI)pp 176ndash183 Springer San Francisco Calif USA 2000

[10] R van der Merwe A Doucet N De Freitas and E Wan ldquoTheunscented particle filterrdquo in Proceedings of the Neural Info ProcSys (NIPS) pp 584ndash590 Denver Co USA

[11] A Beskos D Crisan andA Jasra ldquoOn the stability of sequentialMonte Carlo methods in high dimensionsrdquo e Annals ofApplied Probability vol 24 no 4 pp 1396ndash1445 2014

[12] D Crisan and B Rozovskiie Oxford Handbook of NonlinearFiltering Oxford University Press 2011

[13] P Bui Quang C Musso and F Le Gland ldquoAn insight into theissue of dimensionality in particle filteringrdquo in Proceedings ofthe 2010 13th International Conference on Information Fusion(FUSION 2010) pp 1ndash8 Edinburgh July 2010

[14] M Ades and P J van Leeuwen ldquoThe equivalent-weights particlefilter in a high-dimensional systemrdquo Quarterly Journal of theRoyalMeteorological Society vol 141 no 687 pp 484ndash503 2015

[15] P M Djuric T Lu and M F Bugallo ldquoMultiple particlefilteringrdquo in Proc Intl Conf Acoustics Speech and Signal Proc(ICASSP) vol 3 pp 1181ndash1184 2007

[16] A Beskos D Crisan A Jasra K Kamatani and Y Zhou ldquoAstable particle filter for a class of high-dimensional state-spacemodelsrdquo Advances in Applied Probability vol 49 no 1 pp 24ndash48 2017

[17] F Septier S K Pang A Carmi and S Godsill ldquoOn MCMC-based particle methods for Bayesian filtering Application tomultitarget trackingrdquo in Proceedings of the 3rd IEEE Interna-tional Workshop on Computational Advances in Multi-SensorAdaptive Processing (CAMSAP 2009) pp 360ndash363 ArubaDutch Antilles December 2009

[18] T Ding and M J Coates ldquoImplementation of the Daum-Huang exact- flow particle filterrdquo in Proceedings of the 2012 IEEEStatistical Signal Processing Workshop SSP pp 257ndash260 USAAugust 2012

[19] F Daum and J Huang ldquoNonlinear filters with log-homotopyrdquoin Proc SPIE Conf Signal and Data Processing of Small TargetsSan Diego CA USA 2007

[20] F Daum I Kadar and J Huang ldquoNonlinear filters with particleflow induced by log-homotopyrdquo in Proceedings of the SPIEDefense Security and Sensing p 733603 Orlando FloridaUSA

[21] F Daum I Kadar J Huang and A Noushin ldquoExact particleflow for nonlinear filtersrdquo in Proceedings of the SPIE DefenseSecurity and Sensing p 769704 Orlando Florida

[22] F Daum J Huang and A Noushin ldquoCoulombs law particleflow for nonlinear filtersrdquo in Proc SPIE Conf Signal and DataProc San Diego CA USA 2011

[23] Y Li L Zhao and M Coates ldquoParticle flow for particlefilteringrdquo in Proceedings of the IEEE Interna- tional Conferenceon Acoustics Speech and Signal Processing (ICASSP) pp 20ndash25Shanghai China March 2016

[24] Y Li and M Coates ldquoFast Particle Flow Particle Filters viaClusteringrdquo in Proceedings of the 19th Inter- national Conferenceon Information Fusion Heidelberg Germany 2016

[25] S C Surace A Kutschireiter and J-P Pfister ldquoHow to avoid thecurse of dimen- sionality scalability of particle filters with andwithout importance weightsrdquo httpsarxivorgabs170307879

[26] F Daum J Huang and A Noushin ldquoNewTheory and Numer-ical Results for Gromovrsquos Method for Stochastic Particle FlowFiltersrdquo in Proceedings of the 21st International Conference onInformation Fusion FUSION 2018 pp 108ndash115 UK July 2018

[27] Y Li S Pal and M Coates ldquoInvertible Particle Flow-basedSequential MCMC with ex- tension to Gaussian Mixture noisemodelsrdquo IEEE Transactions on Signal Processing 2019

[28] D Arthur and S Vassilvitskii ldquok-means++ The advantages ofcareful seedingrdquo in Proceedings of the ACM-SIAM Symposiumon Discrete Algorithms pp 1027ndash1035 2007

[29] S P Lloyd ldquoLeast squares quantization in PCMrdquo IEEE Transac-tions on Information eory vol 28 no 2 pp 129ndash137 1982

[30] O Hlinka O Sluciak F Hlawatsch P M Djuric and MRupp ldquoDistributed Gaussian particle filtering using likelihoodconsensusrdquo in Proceedings of the ICASSP 2011 - 2011 IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP) pp 3756ndash3759 Prague Czech Republic May 2011

Mathematical Problems in Engineering 5

7 for 119895 = 1 to119873120582 do8 Set 120582 = 119895Δ1205829 for 119894 = 1 to119873 do10 Calculate119867119909 by linearizing 120574119896() at 11990911989411 endfor12 Cluster particles over the margin of error13 for 119888 = 1 to119873119888 do14 Calculate 119860119888 and 119887119888 from (6) and (7) using 119875119896|119896minus1 119909 and119867119909119888 15 Evaluate 119889119909119888119896119889120582 for each particle from (5)16 Migrate particles 119909119888119896 = 119909119888119896 + Δ120582(119889119909119888119896119889120582)17 endfor18 Eliminate the largest cluster with the biggest error19 Re-evaluate 119909119896 using the updated particles 11990911989411989620 endfor21 Apply UKFEKF update (119898119896|119896minus1 119875119896|119896minus1) 997888rarr (119898119896|119896 119875119896|119896)22 Estimate 119909119896 from the particles 119909119894119896 using 119875119896|11989623 Redraw particles 119909119894119896 sim 119873(119909119896 119875119896|119896) (Optional)

Algorithm 3 Clustered Exact Flow Daum-Huang Filter

filter the particles are not associated with a single parametercalculation Also themigration parameters are not calculatedfor each individual particle

CEDH filters fight to reduce the cost of calculationin LEDH filters which typically occurs due to exhaustivecalculation of each migration parameter In addition whilereducing the cost of calculation that the particles with higherror margin can be systematically clustered together leadsto significant performance improvement The assessmentof the performance and calculation speed as the outcomesof CEDH filter implementation for the scenario of multi-dimensional target tracking will be presented in the nextsection

4 Simulation and Results

The particle filters are applied to a multitarget tracking prob-lem which is adapted from [30] A wireless sensor networkmodel of 25 acoustic amplitude sensor nodes located at theintersections of the grids on a 40119898 119909 40119898 rectangular regionis usedThemodel is shown in Figure 1There are four targetsmoving along two axes independentlyTheoverall state vectorcontains four states for these four targetsTherefore it is in the16-dimensional state space

The independentmovementmodel of four targets (119875 = 4)moving at a constant speed can be expressed as119909(119901)

119896= 119865119909(119901)119896minus1

+ 119882119901V(119901)119896 (10)

where 119909119901119896= [119909119901119896 119910119901119896 119901119896 119910119901119896] consists of the x-y position and

x-y velocity components of the corresponding target

119865 = [[[[[[1 0 1 00 1 0 10 0 1 00 0 0 1

]]]]]] (11)

is the transition matrix V(119901)119896

sim 119873(0 1205902V119881) is the process noise1205902V is set to 001 and the covariance of the process noise forthe filters is set as

119882119901 = (05 0 02 00 05 0 021 0 1 00 0 0 1 ) (12)

All targets emit sounds with amplitude 120595 and each sensorrecords the sum of amplitudes Thus measurement functionfor the 119904-th sensor located at 119877119904 is additive119911119904 (119909119896) = 119875sum

119901=1

1205951003817100381710038171003817100381710038171003817(119909(119901)119896 119910(119901)119896

)119879 minus 119877210038171003817100381710038171003817100381710038172 + 1198890 (13)

where 1198890 = 01 and 120595 = 10 in our simulations The measure-ments are perturbed by Gaussian noise There are 119873119904 = 25sensors located at grid intersections as shown in Figure 1Thefour targets are initializedwith states [8 8 001minus001] [9 34minus0001 0003] [34 32 002 minus 01] and [35 12 0001 minus 0001]respectively

Multitarget tracking scenario associated with the wirelesssensor network model was simulated employing the filters ofEKF BPF EDH LEDH and CEDH for different measure-ment sets Each run starts with a different initial distributionFigure 1 shows the true routes of the targets and the estimatedroutes obtained with CEDH The simulation consists of 40time steps and each algorithm is run separately 100 timesThe optimal mass transfer (OMAT) metric is calculated bytaking the mean of each time interval The average OMATerror calculated for EKF BPF EDH LEDH and CEDH isgiven in Figure 2

As shown in Figure 2 the EKF filter has the highesterror margin It is followed by the BPF EDH and LEDHrespectively CEDH filters with different cluster numbers arethe best performing filters at all time intervals

6 Mathematical Problems in Engineering

5 10 15 20 25 30 35 400X (m)

Target 1 (true)Target 2 (true)Target 3 (true)Target 4 (true)

Target 1 (est)Target 2 (est)Target 3 (est)Target 4 (est)

0

5

10

15

20

25

30

35

40

Y (m

)

Figure 1 One example of true and estimated trajectories using CEDH

30 32 341

15

2x2

5 10 15 20 25 30 35 400time step

BPFEKFEDHLEDH

CEDH (10)CEDH (30)CEDH (50)

0

1

2

3

4

5

6

7

8

9

aver

age O

MAT

erro

r (m

)

Figure 2 Average OMAT errors of the filters

Mathematical Problems in Engineering 7

Table 1 Detailed information about the filters

Algorithm EKF BPF EDH LEDH CEDH CEDH CEDHParticle num NA 10000 500 500 500 500 500120582 NA NA 30 30 30 30 30Cluster num NA NA NA NA 10 30 50Avg err (m) 447 379 324 260 150 141 137Exec time (s) 003 157 048 4473 2305 2479 2842

0 10 20 30 40 50 60 70 80 90 100trial

0

5

10

15

20

25

30

35

40

45

50

exec

utio

n tim

e (s)

2 4 6 8 10 12

0

2

4 x2

BPFEKFEDHLEDH

CEDH (10)CEDH (30)CEDH (50)

Figure 3 Execution times of the filters

Detailed information about the filters is given in Table 1The computer used for the simulations isMacBookAir (Early2015) which has a 16GHz Intel Core i5 processor 4GB of1600MHz DDR3 memory and an Intel HD Graphics 60001536 MB graphics card

As shown in Figure 3 the CEDH algorithm ensuredsignificant improvement in terms of calculation cost Itcorresponds to approximately half of the execution timeof the LEDH algorithm Increasing the number of clustersimproves performance however this improvement bringsextra cost of calculation to the filter

5 Conclusions

We introduced Clustered Exact Flow Daum-Huang ParticleFilter in this study While improving the performance of

the LEDH filter CEDH also reduces the cost of calculationthrough utilizing a clustering strategy The consideration ofclustering of the particles with high error margins ensuresthat the influence of the particles on the overall system canbe reduced In CEDHfilter the particles are not accompaniedwith the calculation of same parameter as it occurs in EDHfilter CEDHfilter succeeds in reducing the cost of calculationin LEDH filter arising due to comprehensive calculation ofeachmigration parameterThe calculation cost is significantlyreduced by calculating commonmigration parameters for theparticles with similar error values The given multidimen-sional target tracking problem demonstrates the success ofthe CEDH filter In the future work the effect of clustering onthe systemwill be examined and the focuswill be on the selec-tion of the appropriate number of clusters and elimination ofthe appropriate cluster with the help of machine learning

8 Mathematical Problems in Engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] M S Arulampalam S Maskell N Gordon and T Clapp ldquoAtutorial on particle filters for online nonlinearnon-GaussianBayesian trackingrdquo IEEE Transactions on Signal Processing vol50 no 2 pp 174ndash188 2002

[2] M West and J Harrison Bayesian Forecasting and DynamicModels Springer Series in Statistics New York NY USA 2ndedition 1997

[3] G Welch and G Bishop ldquoAn introduction to the Kalmanfilterrdquo Technical report UNC-CH Computer Science TechnicalReport 95041 1995

[4] N J Gordon D J Salmond and A F M Smith ldquoNovel ap-proach to nonlinearnon-gaussian Bayesian state estimationrdquoIEE proceedings F vol 140 no 2 pp 107ndash113 1993

[5] P Bunch and S Godsill ldquoApproximations of the optimalimportance density using Gaussian particle flow importancesamplingrdquo Journal of the American Statistical Association vol111 no 514 pp 748ndash762 2016

[6] A Doucet S Godsill and C Andrieu ldquoOn sequential MonteCarlo sampling methods for Bayesian filteringrdquo Statistics andComputing vol 10 no 3 pp 197ndash208 2000

[7] J Cornebise E Moulines and J Olsson ldquoAdaptive methods forsequential importance sampling with application to state spacemodelsrdquo Statistics and Computing vol 18 no 4 pp 461ndash4802008

[8] M K Pitt and N Shephard ldquoFiltering via simulation auxiliaryparticle filtersrdquo Journal of the American Statistical Associationvol 94 no 446 pp 590ndash599 1999

[9] A Doucet N de Freitas N Gordon and S J Russell ldquoRao-Blackwellised particle filtering for dynamic Bayesian networksrdquoin Proceedings of the Uncertainty in Artificial Intelligence (UAI)pp 176ndash183 Springer San Francisco Calif USA 2000

[10] R van der Merwe A Doucet N De Freitas and E Wan ldquoTheunscented particle filterrdquo in Proceedings of the Neural Info ProcSys (NIPS) pp 584ndash590 Denver Co USA

[11] A Beskos D Crisan andA Jasra ldquoOn the stability of sequentialMonte Carlo methods in high dimensionsrdquo e Annals ofApplied Probability vol 24 no 4 pp 1396ndash1445 2014

[12] D Crisan and B Rozovskiie Oxford Handbook of NonlinearFiltering Oxford University Press 2011

[13] P Bui Quang C Musso and F Le Gland ldquoAn insight into theissue of dimensionality in particle filteringrdquo in Proceedings ofthe 2010 13th International Conference on Information Fusion(FUSION 2010) pp 1ndash8 Edinburgh July 2010

[14] M Ades and P J van Leeuwen ldquoThe equivalent-weights particlefilter in a high-dimensional systemrdquo Quarterly Journal of theRoyalMeteorological Society vol 141 no 687 pp 484ndash503 2015

[15] P M Djuric T Lu and M F Bugallo ldquoMultiple particlefilteringrdquo in Proc Intl Conf Acoustics Speech and Signal Proc(ICASSP) vol 3 pp 1181ndash1184 2007

[16] A Beskos D Crisan A Jasra K Kamatani and Y Zhou ldquoAstable particle filter for a class of high-dimensional state-spacemodelsrdquo Advances in Applied Probability vol 49 no 1 pp 24ndash48 2017

[17] F Septier S K Pang A Carmi and S Godsill ldquoOn MCMC-based particle methods for Bayesian filtering Application tomultitarget trackingrdquo in Proceedings of the 3rd IEEE Interna-tional Workshop on Computational Advances in Multi-SensorAdaptive Processing (CAMSAP 2009) pp 360ndash363 ArubaDutch Antilles December 2009

[18] T Ding and M J Coates ldquoImplementation of the Daum-Huang exact- flow particle filterrdquo in Proceedings of the 2012 IEEEStatistical Signal Processing Workshop SSP pp 257ndash260 USAAugust 2012

[19] F Daum and J Huang ldquoNonlinear filters with log-homotopyrdquoin Proc SPIE Conf Signal and Data Processing of Small TargetsSan Diego CA USA 2007

[20] F Daum I Kadar and J Huang ldquoNonlinear filters with particleflow induced by log-homotopyrdquo in Proceedings of the SPIEDefense Security and Sensing p 733603 Orlando FloridaUSA

[21] F Daum I Kadar J Huang and A Noushin ldquoExact particleflow for nonlinear filtersrdquo in Proceedings of the SPIE DefenseSecurity and Sensing p 769704 Orlando Florida

[22] F Daum J Huang and A Noushin ldquoCoulombs law particleflow for nonlinear filtersrdquo in Proc SPIE Conf Signal and DataProc San Diego CA USA 2011

[23] Y Li L Zhao and M Coates ldquoParticle flow for particlefilteringrdquo in Proceedings of the IEEE Interna- tional Conferenceon Acoustics Speech and Signal Processing (ICASSP) pp 20ndash25Shanghai China March 2016

[24] Y Li and M Coates ldquoFast Particle Flow Particle Filters viaClusteringrdquo in Proceedings of the 19th Inter- national Conferenceon Information Fusion Heidelberg Germany 2016

[25] S C Surace A Kutschireiter and J-P Pfister ldquoHow to avoid thecurse of dimen- sionality scalability of particle filters with andwithout importance weightsrdquo httpsarxivorgabs170307879

[26] F Daum J Huang and A Noushin ldquoNewTheory and Numer-ical Results for Gromovrsquos Method for Stochastic Particle FlowFiltersrdquo in Proceedings of the 21st International Conference onInformation Fusion FUSION 2018 pp 108ndash115 UK July 2018

[27] Y Li S Pal and M Coates ldquoInvertible Particle Flow-basedSequential MCMC with ex- tension to Gaussian Mixture noisemodelsrdquo IEEE Transactions on Signal Processing 2019

[28] D Arthur and S Vassilvitskii ldquok-means++ The advantages ofcareful seedingrdquo in Proceedings of the ACM-SIAM Symposiumon Discrete Algorithms pp 1027ndash1035 2007

[29] S P Lloyd ldquoLeast squares quantization in PCMrdquo IEEE Transac-tions on Information eory vol 28 no 2 pp 129ndash137 1982

[30] O Hlinka O Sluciak F Hlawatsch P M Djuric and MRupp ldquoDistributed Gaussian particle filtering using likelihoodconsensusrdquo in Proceedings of the ICASSP 2011 - 2011 IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP) pp 3756ndash3759 Prague Czech Republic May 2011

6 Mathematical Problems in Engineering

5 10 15 20 25 30 35 400X (m)

Target 1 (true)Target 2 (true)Target 3 (true)Target 4 (true)

Target 1 (est)Target 2 (est)Target 3 (est)Target 4 (est)

0

5

10

15

20

25

30

35

40

Y (m

)

Figure 1 One example of true and estimated trajectories using CEDH

30 32 341

15

2x2

5 10 15 20 25 30 35 400time step

BPFEKFEDHLEDH

CEDH (10)CEDH (30)CEDH (50)

0

1

2

3

4

5

6

7

8

9

aver

age O

MAT

erro

r (m

)

Figure 2 Average OMAT errors of the filters

Mathematical Problems in Engineering 7

Table 1 Detailed information about the filters

Algorithm EKF BPF EDH LEDH CEDH CEDH CEDHParticle num NA 10000 500 500 500 500 500120582 NA NA 30 30 30 30 30Cluster num NA NA NA NA 10 30 50Avg err (m) 447 379 324 260 150 141 137Exec time (s) 003 157 048 4473 2305 2479 2842

0 10 20 30 40 50 60 70 80 90 100trial

0

5

10

15

20

25

30

35

40

45

50

exec

utio

n tim

e (s)

2 4 6 8 10 12

0

2

4 x2

BPFEKFEDHLEDH

CEDH (10)CEDH (30)CEDH (50)

Figure 3 Execution times of the filters

Detailed information about the filters is given in Table 1The computer used for the simulations isMacBookAir (Early2015) which has a 16GHz Intel Core i5 processor 4GB of1600MHz DDR3 memory and an Intel HD Graphics 60001536 MB graphics card

As shown in Figure 3 the CEDH algorithm ensuredsignificant improvement in terms of calculation cost Itcorresponds to approximately half of the execution timeof the LEDH algorithm Increasing the number of clustersimproves performance however this improvement bringsextra cost of calculation to the filter

5 Conclusions

We introduced Clustered Exact Flow Daum-Huang ParticleFilter in this study While improving the performance of

the LEDH filter CEDH also reduces the cost of calculationthrough utilizing a clustering strategy The consideration ofclustering of the particles with high error margins ensuresthat the influence of the particles on the overall system canbe reduced In CEDHfilter the particles are not accompaniedwith the calculation of same parameter as it occurs in EDHfilter CEDHfilter succeeds in reducing the cost of calculationin LEDH filter arising due to comprehensive calculation ofeachmigration parameterThe calculation cost is significantlyreduced by calculating commonmigration parameters for theparticles with similar error values The given multidimen-sional target tracking problem demonstrates the success ofthe CEDH filter In the future work the effect of clustering onthe systemwill be examined and the focuswill be on the selec-tion of the appropriate number of clusters and elimination ofthe appropriate cluster with the help of machine learning

8 Mathematical Problems in Engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] M S Arulampalam S Maskell N Gordon and T Clapp ldquoAtutorial on particle filters for online nonlinearnon-GaussianBayesian trackingrdquo IEEE Transactions on Signal Processing vol50 no 2 pp 174ndash188 2002

[2] M West and J Harrison Bayesian Forecasting and DynamicModels Springer Series in Statistics New York NY USA 2ndedition 1997

[3] G Welch and G Bishop ldquoAn introduction to the Kalmanfilterrdquo Technical report UNC-CH Computer Science TechnicalReport 95041 1995

[4] N J Gordon D J Salmond and A F M Smith ldquoNovel ap-proach to nonlinearnon-gaussian Bayesian state estimationrdquoIEE proceedings F vol 140 no 2 pp 107ndash113 1993

[5] P Bunch and S Godsill ldquoApproximations of the optimalimportance density using Gaussian particle flow importancesamplingrdquo Journal of the American Statistical Association vol111 no 514 pp 748ndash762 2016

[6] A Doucet S Godsill and C Andrieu ldquoOn sequential MonteCarlo sampling methods for Bayesian filteringrdquo Statistics andComputing vol 10 no 3 pp 197ndash208 2000

[7] J Cornebise E Moulines and J Olsson ldquoAdaptive methods forsequential importance sampling with application to state spacemodelsrdquo Statistics and Computing vol 18 no 4 pp 461ndash4802008

[8] M K Pitt and N Shephard ldquoFiltering via simulation auxiliaryparticle filtersrdquo Journal of the American Statistical Associationvol 94 no 446 pp 590ndash599 1999

[9] A Doucet N de Freitas N Gordon and S J Russell ldquoRao-Blackwellised particle filtering for dynamic Bayesian networksrdquoin Proceedings of the Uncertainty in Artificial Intelligence (UAI)pp 176ndash183 Springer San Francisco Calif USA 2000

[10] R van der Merwe A Doucet N De Freitas and E Wan ldquoTheunscented particle filterrdquo in Proceedings of the Neural Info ProcSys (NIPS) pp 584ndash590 Denver Co USA

[11] A Beskos D Crisan andA Jasra ldquoOn the stability of sequentialMonte Carlo methods in high dimensionsrdquo e Annals ofApplied Probability vol 24 no 4 pp 1396ndash1445 2014

[12] D Crisan and B Rozovskiie Oxford Handbook of NonlinearFiltering Oxford University Press 2011

[13] P Bui Quang C Musso and F Le Gland ldquoAn insight into theissue of dimensionality in particle filteringrdquo in Proceedings ofthe 2010 13th International Conference on Information Fusion(FUSION 2010) pp 1ndash8 Edinburgh July 2010

[14] M Ades and P J van Leeuwen ldquoThe equivalent-weights particlefilter in a high-dimensional systemrdquo Quarterly Journal of theRoyalMeteorological Society vol 141 no 687 pp 484ndash503 2015

[15] P M Djuric T Lu and M F Bugallo ldquoMultiple particlefilteringrdquo in Proc Intl Conf Acoustics Speech and Signal Proc(ICASSP) vol 3 pp 1181ndash1184 2007

[16] A Beskos D Crisan A Jasra K Kamatani and Y Zhou ldquoAstable particle filter for a class of high-dimensional state-spacemodelsrdquo Advances in Applied Probability vol 49 no 1 pp 24ndash48 2017

[17] F Septier S K Pang A Carmi and S Godsill ldquoOn MCMC-based particle methods for Bayesian filtering Application tomultitarget trackingrdquo in Proceedings of the 3rd IEEE Interna-tional Workshop on Computational Advances in Multi-SensorAdaptive Processing (CAMSAP 2009) pp 360ndash363 ArubaDutch Antilles December 2009

[18] T Ding and M J Coates ldquoImplementation of the Daum-Huang exact- flow particle filterrdquo in Proceedings of the 2012 IEEEStatistical Signal Processing Workshop SSP pp 257ndash260 USAAugust 2012

[19] F Daum and J Huang ldquoNonlinear filters with log-homotopyrdquoin Proc SPIE Conf Signal and Data Processing of Small TargetsSan Diego CA USA 2007

[20] F Daum I Kadar and J Huang ldquoNonlinear filters with particleflow induced by log-homotopyrdquo in Proceedings of the SPIEDefense Security and Sensing p 733603 Orlando FloridaUSA

[21] F Daum I Kadar J Huang and A Noushin ldquoExact particleflow for nonlinear filtersrdquo in Proceedings of the SPIE DefenseSecurity and Sensing p 769704 Orlando Florida

[22] F Daum J Huang and A Noushin ldquoCoulombs law particleflow for nonlinear filtersrdquo in Proc SPIE Conf Signal and DataProc San Diego CA USA 2011

[23] Y Li L Zhao and M Coates ldquoParticle flow for particlefilteringrdquo in Proceedings of the IEEE Interna- tional Conferenceon Acoustics Speech and Signal Processing (ICASSP) pp 20ndash25Shanghai China March 2016

[24] Y Li and M Coates ldquoFast Particle Flow Particle Filters viaClusteringrdquo in Proceedings of the 19th Inter- national Conferenceon Information Fusion Heidelberg Germany 2016

[25] S C Surace A Kutschireiter and J-P Pfister ldquoHow to avoid thecurse of dimen- sionality scalability of particle filters with andwithout importance weightsrdquo httpsarxivorgabs170307879

[26] F Daum J Huang and A Noushin ldquoNewTheory and Numer-ical Results for Gromovrsquos Method for Stochastic Particle FlowFiltersrdquo in Proceedings of the 21st International Conference onInformation Fusion FUSION 2018 pp 108ndash115 UK July 2018

[27] Y Li S Pal and M Coates ldquoInvertible Particle Flow-basedSequential MCMC with ex- tension to Gaussian Mixture noisemodelsrdquo IEEE Transactions on Signal Processing 2019

[28] D Arthur and S Vassilvitskii ldquok-means++ The advantages ofcareful seedingrdquo in Proceedings of the ACM-SIAM Symposiumon Discrete Algorithms pp 1027ndash1035 2007

[29] S P Lloyd ldquoLeast squares quantization in PCMrdquo IEEE Transac-tions on Information eory vol 28 no 2 pp 129ndash137 1982

[30] O Hlinka O Sluciak F Hlawatsch P M Djuric and MRupp ldquoDistributed Gaussian particle filtering using likelihoodconsensusrdquo in Proceedings of the ICASSP 2011 - 2011 IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP) pp 3756ndash3759 Prague Czech Republic May 2011

Mathematical Problems in Engineering 7

Table 1 Detailed information about the filters

Algorithm EKF BPF EDH LEDH CEDH CEDH CEDHParticle num NA 10000 500 500 500 500 500120582 NA NA 30 30 30 30 30Cluster num NA NA NA NA 10 30 50Avg err (m) 447 379 324 260 150 141 137Exec time (s) 003 157 048 4473 2305 2479 2842

0 10 20 30 40 50 60 70 80 90 100trial

0

5

10

15

20

25

30

35

40

45

50

exec

utio

n tim

e (s)

2 4 6 8 10 12

0

2

4 x2

BPFEKFEDHLEDH

CEDH (10)CEDH (30)CEDH (50)

Figure 3 Execution times of the filters

Detailed information about the filters is given in Table 1The computer used for the simulations isMacBookAir (Early2015) which has a 16GHz Intel Core i5 processor 4GB of1600MHz DDR3 memory and an Intel HD Graphics 60001536 MB graphics card

As shown in Figure 3 the CEDH algorithm ensuredsignificant improvement in terms of calculation cost Itcorresponds to approximately half of the execution timeof the LEDH algorithm Increasing the number of clustersimproves performance however this improvement bringsextra cost of calculation to the filter

5 Conclusions

We introduced Clustered Exact Flow Daum-Huang ParticleFilter in this study While improving the performance of

the LEDH filter CEDH also reduces the cost of calculationthrough utilizing a clustering strategy The consideration ofclustering of the particles with high error margins ensuresthat the influence of the particles on the overall system canbe reduced In CEDHfilter the particles are not accompaniedwith the calculation of same parameter as it occurs in EDHfilter CEDHfilter succeeds in reducing the cost of calculationin LEDH filter arising due to comprehensive calculation ofeachmigration parameterThe calculation cost is significantlyreduced by calculating commonmigration parameters for theparticles with similar error values The given multidimen-sional target tracking problem demonstrates the success ofthe CEDH filter In the future work the effect of clustering onthe systemwill be examined and the focuswill be on the selec-tion of the appropriate number of clusters and elimination ofthe appropriate cluster with the help of machine learning

8 Mathematical Problems in Engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] M S Arulampalam S Maskell N Gordon and T Clapp ldquoAtutorial on particle filters for online nonlinearnon-GaussianBayesian trackingrdquo IEEE Transactions on Signal Processing vol50 no 2 pp 174ndash188 2002

[2] M West and J Harrison Bayesian Forecasting and DynamicModels Springer Series in Statistics New York NY USA 2ndedition 1997

[3] G Welch and G Bishop ldquoAn introduction to the Kalmanfilterrdquo Technical report UNC-CH Computer Science TechnicalReport 95041 1995

[4] N J Gordon D J Salmond and A F M Smith ldquoNovel ap-proach to nonlinearnon-gaussian Bayesian state estimationrdquoIEE proceedings F vol 140 no 2 pp 107ndash113 1993

[5] P Bunch and S Godsill ldquoApproximations of the optimalimportance density using Gaussian particle flow importancesamplingrdquo Journal of the American Statistical Association vol111 no 514 pp 748ndash762 2016

[6] A Doucet S Godsill and C Andrieu ldquoOn sequential MonteCarlo sampling methods for Bayesian filteringrdquo Statistics andComputing vol 10 no 3 pp 197ndash208 2000

[7] J Cornebise E Moulines and J Olsson ldquoAdaptive methods forsequential importance sampling with application to state spacemodelsrdquo Statistics and Computing vol 18 no 4 pp 461ndash4802008

[8] M K Pitt and N Shephard ldquoFiltering via simulation auxiliaryparticle filtersrdquo Journal of the American Statistical Associationvol 94 no 446 pp 590ndash599 1999

[9] A Doucet N de Freitas N Gordon and S J Russell ldquoRao-Blackwellised particle filtering for dynamic Bayesian networksrdquoin Proceedings of the Uncertainty in Artificial Intelligence (UAI)pp 176ndash183 Springer San Francisco Calif USA 2000

[10] R van der Merwe A Doucet N De Freitas and E Wan ldquoTheunscented particle filterrdquo in Proceedings of the Neural Info ProcSys (NIPS) pp 584ndash590 Denver Co USA

[11] A Beskos D Crisan andA Jasra ldquoOn the stability of sequentialMonte Carlo methods in high dimensionsrdquo e Annals ofApplied Probability vol 24 no 4 pp 1396ndash1445 2014

[12] D Crisan and B Rozovskiie Oxford Handbook of NonlinearFiltering Oxford University Press 2011

[13] P Bui Quang C Musso and F Le Gland ldquoAn insight into theissue of dimensionality in particle filteringrdquo in Proceedings ofthe 2010 13th International Conference on Information Fusion(FUSION 2010) pp 1ndash8 Edinburgh July 2010

[14] M Ades and P J van Leeuwen ldquoThe equivalent-weights particlefilter in a high-dimensional systemrdquo Quarterly Journal of theRoyalMeteorological Society vol 141 no 687 pp 484ndash503 2015

[15] P M Djuric T Lu and M F Bugallo ldquoMultiple particlefilteringrdquo in Proc Intl Conf Acoustics Speech and Signal Proc(ICASSP) vol 3 pp 1181ndash1184 2007

[16] A Beskos D Crisan A Jasra K Kamatani and Y Zhou ldquoAstable particle filter for a class of high-dimensional state-spacemodelsrdquo Advances in Applied Probability vol 49 no 1 pp 24ndash48 2017

[17] F Septier S K Pang A Carmi and S Godsill ldquoOn MCMC-based particle methods for Bayesian filtering Application tomultitarget trackingrdquo in Proceedings of the 3rd IEEE Interna-tional Workshop on Computational Advances in Multi-SensorAdaptive Processing (CAMSAP 2009) pp 360ndash363 ArubaDutch Antilles December 2009

[18] T Ding and M J Coates ldquoImplementation of the Daum-Huang exact- flow particle filterrdquo in Proceedings of the 2012 IEEEStatistical Signal Processing Workshop SSP pp 257ndash260 USAAugust 2012

[19] F Daum and J Huang ldquoNonlinear filters with log-homotopyrdquoin Proc SPIE Conf Signal and Data Processing of Small TargetsSan Diego CA USA 2007

[20] F Daum I Kadar and J Huang ldquoNonlinear filters with particleflow induced by log-homotopyrdquo in Proceedings of the SPIEDefense Security and Sensing p 733603 Orlando FloridaUSA

[21] F Daum I Kadar J Huang and A Noushin ldquoExact particleflow for nonlinear filtersrdquo in Proceedings of the SPIE DefenseSecurity and Sensing p 769704 Orlando Florida

[22] F Daum J Huang and A Noushin ldquoCoulombs law particleflow for nonlinear filtersrdquo in Proc SPIE Conf Signal and DataProc San Diego CA USA 2011

[23] Y Li L Zhao and M Coates ldquoParticle flow for particlefilteringrdquo in Proceedings of the IEEE Interna- tional Conferenceon Acoustics Speech and Signal Processing (ICASSP) pp 20ndash25Shanghai China March 2016

[24] Y Li and M Coates ldquoFast Particle Flow Particle Filters viaClusteringrdquo in Proceedings of the 19th Inter- national Conferenceon Information Fusion Heidelberg Germany 2016

[25] S C Surace A Kutschireiter and J-P Pfister ldquoHow to avoid thecurse of dimen- sionality scalability of particle filters with andwithout importance weightsrdquo httpsarxivorgabs170307879

[26] F Daum J Huang and A Noushin ldquoNewTheory and Numer-ical Results for Gromovrsquos Method for Stochastic Particle FlowFiltersrdquo in Proceedings of the 21st International Conference onInformation Fusion FUSION 2018 pp 108ndash115 UK July 2018

[27] Y Li S Pal and M Coates ldquoInvertible Particle Flow-basedSequential MCMC with ex- tension to Gaussian Mixture noisemodelsrdquo IEEE Transactions on Signal Processing 2019

[28] D Arthur and S Vassilvitskii ldquok-means++ The advantages ofcareful seedingrdquo in Proceedings of the ACM-SIAM Symposiumon Discrete Algorithms pp 1027ndash1035 2007

[29] S P Lloyd ldquoLeast squares quantization in PCMrdquo IEEE Transac-tions on Information eory vol 28 no 2 pp 129ndash137 1982

[30] O Hlinka O Sluciak F Hlawatsch P M Djuric and MRupp ldquoDistributed Gaussian particle filtering using likelihoodconsensusrdquo in Proceedings of the ICASSP 2011 - 2011 IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP) pp 3756ndash3759 Prague Czech Republic May 2011

8 Mathematical Problems in Engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] M S Arulampalam S Maskell N Gordon and T Clapp ldquoAtutorial on particle filters for online nonlinearnon-GaussianBayesian trackingrdquo IEEE Transactions on Signal Processing vol50 no 2 pp 174ndash188 2002

[2] M West and J Harrison Bayesian Forecasting and DynamicModels Springer Series in Statistics New York NY USA 2ndedition 1997

[3] G Welch and G Bishop ldquoAn introduction to the Kalmanfilterrdquo Technical report UNC-CH Computer Science TechnicalReport 95041 1995

[4] N J Gordon D J Salmond and A F M Smith ldquoNovel ap-proach to nonlinearnon-gaussian Bayesian state estimationrdquoIEE proceedings F vol 140 no 2 pp 107ndash113 1993

[5] P Bunch and S Godsill ldquoApproximations of the optimalimportance density using Gaussian particle flow importancesamplingrdquo Journal of the American Statistical Association vol111 no 514 pp 748ndash762 2016

[6] A Doucet S Godsill and C Andrieu ldquoOn sequential MonteCarlo sampling methods for Bayesian filteringrdquo Statistics andComputing vol 10 no 3 pp 197ndash208 2000

[7] J Cornebise E Moulines and J Olsson ldquoAdaptive methods forsequential importance sampling with application to state spacemodelsrdquo Statistics and Computing vol 18 no 4 pp 461ndash4802008

[8] M K Pitt and N Shephard ldquoFiltering via simulation auxiliaryparticle filtersrdquo Journal of the American Statistical Associationvol 94 no 446 pp 590ndash599 1999

[9] A Doucet N de Freitas N Gordon and S J Russell ldquoRao-Blackwellised particle filtering for dynamic Bayesian networksrdquoin Proceedings of the Uncertainty in Artificial Intelligence (UAI)pp 176ndash183 Springer San Francisco Calif USA 2000

[10] R van der Merwe A Doucet N De Freitas and E Wan ldquoTheunscented particle filterrdquo in Proceedings of the Neural Info ProcSys (NIPS) pp 584ndash590 Denver Co USA

[11] A Beskos D Crisan andA Jasra ldquoOn the stability of sequentialMonte Carlo methods in high dimensionsrdquo e Annals ofApplied Probability vol 24 no 4 pp 1396ndash1445 2014

[12] D Crisan and B Rozovskiie Oxford Handbook of NonlinearFiltering Oxford University Press 2011

[13] P Bui Quang C Musso and F Le Gland ldquoAn insight into theissue of dimensionality in particle filteringrdquo in Proceedings ofthe 2010 13th International Conference on Information Fusion(FUSION 2010) pp 1ndash8 Edinburgh July 2010

[14] M Ades and P J van Leeuwen ldquoThe equivalent-weights particlefilter in a high-dimensional systemrdquo Quarterly Journal of theRoyalMeteorological Society vol 141 no 687 pp 484ndash503 2015

[15] P M Djuric T Lu and M F Bugallo ldquoMultiple particlefilteringrdquo in Proc Intl Conf Acoustics Speech and Signal Proc(ICASSP) vol 3 pp 1181ndash1184 2007

[16] A Beskos D Crisan A Jasra K Kamatani and Y Zhou ldquoAstable particle filter for a class of high-dimensional state-spacemodelsrdquo Advances in Applied Probability vol 49 no 1 pp 24ndash48 2017

[17] F Septier S K Pang A Carmi and S Godsill ldquoOn MCMC-based particle methods for Bayesian filtering Application tomultitarget trackingrdquo in Proceedings of the 3rd IEEE Interna-tional Workshop on Computational Advances in Multi-SensorAdaptive Processing (CAMSAP 2009) pp 360ndash363 ArubaDutch Antilles December 2009

[18] T Ding and M J Coates ldquoImplementation of the Daum-Huang exact- flow particle filterrdquo in Proceedings of the 2012 IEEEStatistical Signal Processing Workshop SSP pp 257ndash260 USAAugust 2012

[19] F Daum and J Huang ldquoNonlinear filters with log-homotopyrdquoin Proc SPIE Conf Signal and Data Processing of Small TargetsSan Diego CA USA 2007

[20] F Daum I Kadar and J Huang ldquoNonlinear filters with particleflow induced by log-homotopyrdquo in Proceedings of the SPIEDefense Security and Sensing p 733603 Orlando FloridaUSA

[21] F Daum I Kadar J Huang and A Noushin ldquoExact particleflow for nonlinear filtersrdquo in Proceedings of the SPIE DefenseSecurity and Sensing p 769704 Orlando Florida

[22] F Daum J Huang and A Noushin ldquoCoulombs law particleflow for nonlinear filtersrdquo in Proc SPIE Conf Signal and DataProc San Diego CA USA 2011

[23] Y Li L Zhao and M Coates ldquoParticle flow for particlefilteringrdquo in Proceedings of the IEEE Interna- tional Conferenceon Acoustics Speech and Signal Processing (ICASSP) pp 20ndash25Shanghai China March 2016

[24] Y Li and M Coates ldquoFast Particle Flow Particle Filters viaClusteringrdquo in Proceedings of the 19th Inter- national Conferenceon Information Fusion Heidelberg Germany 2016

[25] S C Surace A Kutschireiter and J-P Pfister ldquoHow to avoid thecurse of dimen- sionality scalability of particle filters with andwithout importance weightsrdquo httpsarxivorgabs170307879

[26] F Daum J Huang and A Noushin ldquoNewTheory and Numer-ical Results for Gromovrsquos Method for Stochastic Particle FlowFiltersrdquo in Proceedings of the 21st International Conference onInformation Fusion FUSION 2018 pp 108ndash115 UK July 2018

[27] Y Li S Pal and M Coates ldquoInvertible Particle Flow-basedSequential MCMC with ex- tension to Gaussian Mixture noisemodelsrdquo IEEE Transactions on Signal Processing 2019

[28] D Arthur and S Vassilvitskii ldquok-means++ The advantages ofcareful seedingrdquo in Proceedings of the ACM-SIAM Symposiumon Discrete Algorithms pp 1027ndash1035 2007

[29] S P Lloyd ldquoLeast squares quantization in PCMrdquo IEEE Transac-tions on Information eory vol 28 no 2 pp 129ndash137 1982

[30] O Hlinka O Sluciak F Hlawatsch P M Djuric and MRupp ldquoDistributed Gaussian particle filtering using likelihoodconsensusrdquo in Proceedings of the ICASSP 2011 - 2011 IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP) pp 3756ndash3759 Prague Czech Republic May 2011

Hindawiwwwhindawicom Volume 2018

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Mathematical Problems in Engineering

Applied MathematicsJournal of

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Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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