Interacting multiple model particle filter

Y. Boers and J.N. Driessen

Abstract: A new method for multiple model particle filtering for Markovian switching systems ispresented. This new method is a combination of the interacting multiple model (IMM) filter and a(regularised) particle filter. The mixing and interaction is similar to that in a conventional IMMfilter. However, in every mode a regularised particle filter is running. The regularised particle filterprobability density is a mixture of Gaussian probability densities. The proposed method is able todeal with nonlinearities and non-Gaussian noise. Furthermore, the new method keeps a fixednumber of particles in each mode, and therefore it does not suffer from the potential drawbacks ofexisting multiple model particle filters for Markovian switching systems.

1 Introduction

Estimating the (kinematic) state of non-cooperative targetsbased on sensor measurements often requires a multiplemodel formulation including a model for the mode switch-ing. Even for linear Gaussian models, the multiple modelfilter problem is characterised by a non-Gaussian a poster-iori probability density function, which at best can be viewedas a weighted sum of Gaussian probability densities.Although this allows one to run in parallel a bank of Kalmanfilters, each tuned to one specific mode sequence hypotheses,the number of hypotheses grows infinitely. One attempt toreduce the computational load is to apply hypothesis pruningand merging techniques; the IMM filter is a popular memberof this class of methods. Inevitably, this leads to approximatesolutions.

Since the models of the target dynamics and theobservations often exhibit nonlinearities, the filtering shouldtake into account, or at least be robust against, thesecharacteristics. Most multiple hypotheses filtering algor-ithms, mentioned above, rely on (extended) Kalman filters;it is well known that their performance is deteriorated bynonlinearities (see e.g., [1, 2]).

A filter that replaces the infinite enumeration by a suitableiterative (expectation maximisation) procedure has beenproposed in order to improve upon the approximate multiplehypotheses filtering techniques [3]. This is a batch techniquethat applies a form of a Kalman smoothing during everyiteration (M-step), and by applying a relinearisationprocedure this smoothing will improve upon the effectscaused by nonlinearities.

Recently, particle filters have been introduced. The firstworking particle filter has been reported in [1]. For a recentoverview of the field we refer to [4]. These filtersapproximate the a posteriori density on a stochastic gridin the state space. In the limiting case, i.e. an infinitely dense

grid, the true probability density is recovered. Particlefiltering methods can deal with nonlinearities in thedynamics and measurements. Furthermore, the assumptionthat the noises acting on the system are Gaussian can bedropped. Particle filter approaches for Markovian switchingsystems have also been proposed in [5–8]. These methodspropose augmentation of the state with the mode variableand straightforwardly apply a particle filter to thisaugmented state. A major drawback of these methods isthat there is no control over the number of particles in amode. In these methods, the number of particles in a specificmode is proportional to the mode probability, so if the modeprobability is very low, only a fraction of the total number ofparticles resides in that mode. This phenomenon is known tocause numerical problems and is also discussed in [9]. In [9],a multiple model particle filter is also used. There is,however, no interaction between the modes=classes, i.e. noMarkovian switching process is assumed and therefore theaforementioned problem does not occur. In this paper,however, we assume a non-trivial Markov switchingprocess for the mode transitions.

Here a new method for tracking a manoeuvring target,using a particle filter, is proposed. The basic idea is tocombine the interacting multiple model (IMM) approach[10, 11], with a particle filter approach. In the derivation ofthe standard IMM filter, a merging and filtering process aredefined. We adopt a regularised particle filter for thisfiltering step, and perform the merging step on theseprobability densities, represented by a Gaussian mixture.

One consequence of the discrete nature of the approxi-mation of the a posteriori density is that it cannot directly beapplied to an IMM framework as it is used in [10, 11].To obtain a good continuous approximation of the a post-eriori density, we use a regularised version of the bootstrapfilter as first reported in [12] for tracking targets in clutter.In this hybrid version of the bootstrap filter, the probabilitydensity function, that has been computed as a point massprobability density on a number of grid points in the statespace, is fitted to a continuous probability density functionthat is a sum of a prefixed number of Gaussian densityfunctions. Moreover, by using a hybrid type of sampling filteras an alternative for direct resampling, degeneracy in theeffective number of particles is avoided [12].

The main advantages of the new method that we proposehere are:

q IEE, 2003

IEE Proceedings online no. 20030741

doi: 10.1049/ip-rsn:20030741

The authors are with Thales Nederland, Zuidelijke Havenweg 40, 7554 RRHengelo, The Netherlands

Paper first received 10th February and in revised form 12th June 2003

IEE Proc.-Radar Sonar Navig., Vol. 150, No. 5, October 2003344

. the method is able to deal with nonlinearities and non-Gaussian noise in a mode; and. the method uses a fixed number of particles in each mode,independent of the mode probability.

2 System setup

Given the system

sðk þ 1Þ ¼ f ðsðkÞ; tðkÞ;mðkÞÞþ gðsðkÞ; tðkÞ;mðkÞÞwðk;mðkÞÞ; k 2 N ð1Þ

zðkÞ ¼ hðsðkÞ; tðkÞ;mðkÞÞ þ vðk;mðkÞÞ; k 2 N

where sðkÞ 2 RnðmðkÞÞ is the dynamical state of the system in

mode m(k) and mðkÞ 2 M � N is the modal state of thesystem. The process noise and the measurement noise arepossibly mode-dependent. Their densities are denotedby dwðk;mðkÞÞðwÞ and dvðk;mðkÞÞðvÞ: zðkÞ 2 R

pðmðkÞÞ are themeasurements in mode m(k).

Remark 1: Note that the dimension of the state space n(m(k))and the dimension of the measurement space p(m(k)) can bemode-dependent.

Furthermore the mode transition of the system ismodelled by a Markov chain with

Probfmðk þ 1Þ ¼ j jmðkÞ ¼ ig ¼ pij; 8i; j 2 M

The probability density of the initial state is known, sð0Þ �p0ðsÞ: Define the information up to and including time stepk as

ZðkÞ :¼ fzð1Þ; . . . ; zðkÞgThen the filtering problem that has to be solved is

Problem 1: Given a realisaion of Z(k) associated with (1)compute pðsðkÞjZðkÞÞ; i.e. the conditional probabilitydensity of the state s(k) given the set of measurements Z(k).

Remark 2: Note that indeed pðsðkÞjZðkÞÞ is a completesolution in the sense that all information on the state that isavailable is contained in this probability density function. Inthe case where (1) is linear, there is only one mode and boththe process and measurement noise are Gaussian, pðsðkÞjZðkÞÞ is also Gaussian and therefore it is fully characterisedby its mean and covariance. Furthermore, there exist closedand explicit update formulas for the mean and covarianceupdates, i.e. the well known Kalman filter. In the generalnonlinear and/or non-Gaussian case, however, to comple-tely solve the filtering problem, these first two centralmoments are not sufficient.

3 Algorithm

The algorithm that we propose here for systems having astructure like (1) is a combination of an IMM approach, toaccount for the switching, and a regularised particle filter, toaccount for nonlinearities in the dynamic models in thedifferent modes and possible non-Gaussian noise acting onthe system. Furthermore, the algorithm will process a fixednumber of particles in each mode.

Below we will roughly outline the new algorithm, i.e.algorithm 1. The algorithm has three stages, compare [10]and [11]:

. Interaction stage: In this stage, on the basis of the Markovmodel and the model likelihoods and the a posterioriprobability densities for the different modes on time stepðk � 1Þ, the initial densities pp

j0ðs0jðk � 1Þ jZðk � 1ÞÞ are

computed as Gaussian sum probability densities Note that inthe standard IMM filter only a mean and a covariance arecomputed.. Filtering stage: First a sample of size N is drawn in eachmode according to the probability density pp

j0ðs0jðk � 1Þj

Zðk � 1ÞÞ: This sample is used as the basis for theprediction. The predicted sample is then weighted with thenew measurement. Now instead of resampling, which isdone in the standard sampling importance resampling filters,a Gaussian sum probability density is fitted to the particlecloud. This probability density is an approximation of thetrue a posteriori probability density of the state sjðkÞ on timek given all measurements up to time k, i.e. pp jðsjðkÞ jZðkÞÞ:In this stage also the model likelihoods are computed by thestandard IMM hypotheses calculations and then merged.Note that the mixture reduction, mentioned in the algorithm,is described in detail in [13]. Essentially, this mixturereduction reduces the number of Gaussian components inthe mixture, while preserving the mean and the overallcovariance. We emphasise that the mixture reduction is notcrucial when it comes to performance [14]. The number ofterms in the mixture should be high enough to capture allcharacteristics of the a posteriori probability density.To play safe one could take Nr ¼ N in the algorithm,i.e. completely leaving out the reduction. Furthermore,the choice of the scaling parameter �j is carried out similarlyto its choice in [14]. The value is given explicitly in thealgorithm.. Combination stage: In this stage, we obtain thea posteriori probability density of the state sðkÞ; ppðsðkÞjZðkÞÞ by combining the probability density functions of thedifferent modes taking into account the mode probabilities.This stage is quite similar to the standard IMM, where thesame is done only for the means and covariances of thestates in the different modes.

The algorithm solves problem 1 in a sense that itproduces an approximate solution to the probability densitypðsðkÞj ZðkÞÞ in the form of a Gaussian sum probabilitydensity.

For the sake of completeness we state here one cycleof the entire algorithm in detail.

Algorithm 1 (IMM particle filter):

. Interaction stageCompute:Mixing probabilities

mijjðk � 1jk � 1Þ ¼ 1

cj

pijmiðk � 1Þ

Normalising factors

cj ¼Xi2M

pijmiðk � 1Þ

A priori probability density in mode j

ppj0ðs0jðk � 1ÞjZðk � 1ÞÞ ¼

Xi2M

ppiðsiðk � 1jZðk � 1ÞÞ

mijjðk � 1jk � 1Þ

† Filtering stage8j 2 M draw N samples �ss

jl ðk � 1Þ according to

IEE Proc.-Radar Sonar Navig., Vol. 150, No. 5, October 2003 345

ppj0ðs0jðk � 1ÞjZðk � 1ÞÞ

and obtain:Predicted samples

ssljðkÞ ¼ f �ssl

jðk � 1; tðk � 1Þ; j� �

þ g �ssljðk � 1; tðk � 1Þ; j

� �wlðk � 1; jÞ

where wlðk � 1; jÞ are samples from dwðk�1; jÞðwÞ:Predicted output

zzljðkjk � 1Þ ¼ h ssl

jðkÞ; tðkÞ; j� �

Probability weight

�qqljðkÞ ¼ dvðk;jÞ zðkÞ � zzl

jðkjk � 1Þ� �

Normalising

~qqjðkÞ ¼XN

l¼1

�qqljðkÞ

Normalised probability masses

qlj ¼

�qqljðkÞ~qqjðkÞ

Mean of the state over the sample set

�ssjðkÞ ¼XN

l¼1

qljss

ljðkÞ

Covariance of the state over the sample set

PPjðkÞ ¼XN

l¼1

qlj ssl

jðkÞ � �ssjðkÞ� �

ssljðkÞ � �ssjðkÞ

� �t

From the conditional probability density function for thestate in mode j based on a mixture of N Gaussian densities

ppjNðsjðkÞjZðkÞÞ ¼

XN

l¼1

qljN ssl

jðkÞ; �jPPjðkÞ� �

where �j ¼ 0:5N�2=dj and dj is the dimension of the statespace.Obtain the probability density function for the state in modej after mixture reduction, i.e. based on a mixture of Nr � NGaussian densities.

pp jðsjðkÞjZðkÞÞ ¼XNr

l¼1

qr;lj N ssr;l

j ðkÞ; �jPPrj ðkÞ

� �

Mean of predicted output over the sample set

�hhjðkÞ ¼XN

l¼1

h ssljðkÞ; k; j

� �

Residual covariance over the sample set

SSjðkÞ ¼XN

l¼1

h ssljðkÞ; k; j

� �� �hhjðkÞ

� �h ssl

jðkÞ; k; j� �

� �hhjðkÞ� �t

Innovations

rljðkÞ ¼ zðkÞ � h ssl

jðkÞ; k; j� �

Probability density function for the innovations

pp jðrjðkÞjZðkÞÞ ¼XN

l¼1

qljNð0; SSjðkÞÞ ¼ Nð0; SSjðkÞÞ

Likelihoods

LljðkÞ ¼ N rl

jðkÞ; 0; SSjðkÞ� �

Mode probabilities

mjðkÞ ¼1

cLjðkÞcj

where

c ¼Xj2M

LjðkÞcj

† Combination stageObtain the a posteriori conditional probability densityfunction for the state

ppðsðkÞjZðkÞÞ ¼Xj2M

pp jðsjðkÞjZðkÞÞmjðkÞ

Remark 3: Note that the output of the algorithm is thea posteriori probability density of the stage given allmeasurements, i.e. ppðsðkÞjZðkÞÞ: Based on this a posterioridensity an estimate of the state is easily obtained as

ssðkÞ :¼ EppðsðkÞjZðkÞÞðsðkÞÞ

The above estimate is referred to as the minimum varianceestimator. This state estimate can be used for outputpurposes.

In the following Section we will show that the proposedalgorithm is superior to the standard IMM algorithmreported in [10] and [11].

4 Target tracking example

We will apply the filter described by the above algorithm toa target tracking problem and compare its performance tothe performance of a standard ‘classical’ IMM filter, i.e. anIMM filter based on a bank of Kalman filters.

Consider the system

sðk þ 1Þ ¼ f ðsðkÞ; tðkÞ;mðkÞÞþ gðsðkÞ; tðkÞ;mðkÞÞwðk;mðkÞÞ; k 2 N ð2Þ

zðkÞ ¼ hðsðkÞ; tðkÞ;mðkÞÞ þ vðk;mðkÞÞ; k 2 N

where m 2 f1; 2; 3g; m ¼ 1 corresponds to the modestraight, m ¼ 2 corresponds to the mode circular manoeuvreor co-ordinated turn and m ¼ 3 corresponds to the verticalacceleration mode.

In the example T :¼ tðkÞ � tðk � 1Þ is constant. The stateevolutions within the different modes are given below.

f ðsðkÞ; tðkÞ; 1Þ ¼

1 0 0 T 0

0 1 0 0 T

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

0BBBBBB@

1CCCCCCA

sðkÞ

with state sðkÞ ¼ ½xðkÞ; yðkÞ; zðkÞ; vxðkÞ; vyðkÞ�t:IEE Proc.-Radar Sonar Navig., Vol. 150, No. 5, October 2003346

f ðsðkÞ; tðkÞ; 2Þ ¼1 0 0 sinðoðkÞTÞ=oðkÞ0 1 0 ð1 � cosðoðkÞTÞÞ=oðkÞ0 0 1 0

0 0 0 cosðoðkÞTÞ0 0 0 sinðoðkÞTÞ0 0 0 0

0 0 0 0

0BBBBBBBBBBB@

ðcosðoðkÞTÞ � 1Þ=oðkÞ 0 0

sinðoðkÞTÞ=oðkÞ 0 0

0 T 0

� sinðoðkÞTÞ 0 0

cosðoðkÞTÞ 0 0

0 1 0

0 0 1

1CCCCCCCCCCCA

sðkÞ

with state sðkÞ ¼ ½xðkÞ; yðkÞ; zðkÞ; vxðkÞ; vyðkÞ; vzðkÞ;oðkÞ�t:

f ðsðkÞ; tðkÞ; 3Þ ¼

1 0 0 T 0 0 0

0 1 0 0 T 0 0

0 0 1 0 0 T 12

T2

0 0 0 1 0 0 0

0 0 0 0 1 0 0

0 0 0 0 0 1 T

0 0 0 0 0 0 1

0BBBBBBBB@

1CCCCCCCCA

sðkÞ

with state sðkÞ ¼ ½xðkÞ; yðkÞ; zðkÞ; vxðkÞ; vyðkÞ; vzðkÞ; azðkÞ�t:The process noise input functions are given below

gðsðkÞ; tðkÞ; 1Þ ¼

12

asT2 0 0

0 12

asT2 0

0 0 12

asT2

asT 0 0

0 asT 0

0BBBB@

1CCCCA

gðsðkÞ; tðkÞ; 2Þ ¼ RðsðkÞÞC; where

RðsðkÞÞ ¼

ca �sa 0 0 0 0 0

sa ca 0 0 0 0 0

0 0 1 0 0 0 0

0 0 0 ca �sa 0 0

0 0 0 sa ca 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2

x þ v2y þ v2

z

q

0BBBBBBBBB@

1CCCCCCCCCA

where ca ¼ cosðatan2ðvy=vxÞÞ and sa ¼ sinðatan2ðvy=vxÞÞ

C ¼

12

alongT2 0 0

0 12

alatT2 0

0 0 12

avertT2

alongT 0 0

0 alatT 0

0 0 avertT

0 alat 0

0BBBBBBBB@

1CCCCCCCCA

gðsðkÞ; tðkÞ; 3Þ ¼

12

aazT2 0 0

0 12

aazT2 0

0 0 12

aazT2

aazT 0 0

0 aazT 0

0 0 aazT

0 0 aaz

0BBBBBBBB@

1CCCCCCCCA

The measurement function h is mode- and time-independentand is given by

h1ðsÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2 þ z2

qh2ðsÞ ¼ atan2ðy=xÞ

h3ðsÞ ¼ atan z=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

q� �

h4ðsÞ ¼x·vx þ y·vy þ z·vzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2 þ y2 þ z2p

The mode transition matrix ½ pij� is given by

0:96 0:02 0:02

0:02 0:96 0:02

0:02 0:02 0:96

0@

1A

Remark 4: Note that in all modes the measurements arenonlinear, whereas in the co-ordinated turn mode thedynamics are also nonlinear.

12 13 14 15 16 17 18 190

50

100

150

200

250

300

time, s

posi

tion

erro

r, m

12 13 14 15 16

a

b

17 18 190

50

100

150

200

250

300

time, s

posi

tion

erro

r, m

Fig. 1 Position error

a Standard IMM filterb IMM hybrid bootstrap filter

IEE Proc.-Radar Sonar Navig., Vol. 150, No. 5, October 2003 347

We measure the range, the bearing, the elevation and therange rate or Doppler speed. The measurement noise like theprocess noise is white and Gaussian.

We compare the standard IMM algorithm, see [10] and[11], to algorithm 1 for a typical scenario.

Remark 5: We would like to stress the fact that, although inthis particular example we have assumed white Gaussiannoise as process noise, algorithm 1 is able to deal with anyprocess noise as long as samples can be drawn from theprocess noise distribution. One can argue that in mode 3 it ismore realistic to assume uniform process noise, i.e. becausethe third model should be able to handle dynamics notcovered by the straight or the co-ordinated turn model. Thismeans that extreme manoeuvres should be covered by thethird model, and therefore the probability on an extrememanoeuvre (maximal acceleration) should not be lower thanthe probability for non-manoeuvring (zero acceleration).This is realised by a uniform distributed process noisemodel for the third mode.

. Manoeuvrability is as ¼ 0:5ms�2 in the straight model.For the circular manoeuvre model it is along ¼ avert ¼15ms�2 longitudinal and vertical and alat ¼ 20ms�2 lateral.In the vertical acceleration mode all acceleration parametersare set to aaz ¼ 20ms�2

. Update time is T ¼ 0:5 s:

. Measurement noise standard deviations are15 m for therange measurement, 2 mrad for the bearing and elevationand 2ms�1 for the Doppler speed.. We run the IMM particle filter with N ¼ 1000 samples ineach mode. Furthermore, we impose that after mixturereduction we obtain a Gaussian sum probability densityfunction which consists of 15 terms in each mode.

In Figs. 1–4, data corresponding to the standard IMMfilter solution and to the IMM hybrid bootstrap filter solutionare shown.

The scenario is such that the first five time steps, i.e.t ¼ 12 s until t ¼ 14 s correspond to a straight trajectory.The next time steps until t ¼ 18 s are a hybrid between thecircular manoeuvre and the vertical acceleration model, andthe last three time steps are a hybrid between straighttrajectory and circular manoeuvre.

We have done 1000 runs of the same trajectory withdifferent realisations of measurements noise. In 254 out ofthose 1000 realisations the standard IMM filter did notdiverge and performed almost equally well as the IMMhybrid bootstrap filter. In 746 out of those 1000 realisationsthe standard IMM filter diverges. We will now discuss arealisation that is representative for those cases in which thestandard IMM diverges. We observe that the standard IMMfilter diverges after t ¼ 14 s: We can see this in Figs. 1 and 2.

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.40

50

100

150

200

250

300

350

x position, m (x104)

y po

sitio

n, m

0.7 0.8 0.9 1.0 1.1 1.2

measurementtrue positionfiltered position

1.3

a

b

1.40

50

100

150

200

250

300

350

x position, m (x104)

y po

sitio

n, m

Fig. 2 Horizontal target position

a Standard IMM filterb IMM hybrid bootstrap filter

12 13 14 15 16

a

b

17 18 19–20

–10

0

10

20

30

40

time, s

z po

sitio

n, m

12 13 14 15 16 17 18 19–20

–10

0

10

20

30

40

time, s

z po

sitio

n, m

measurementtrue positionfiltered position

Fig. 3 Target altitude

a Standard IMM filterb IMM hybrid bootstrap filter

IEE Proc.-Radar Sonar Navig., Vol. 150, No. 5, October 2003348

The reason for this phenomena is that in the standard IMMfilter the angular speed is wrongly estimated. This is due tothe nonlinearity in the dynamics in the circular manoeuvremodel. Within the circular manoeuvre model, the trueprobability density of the angular speed given themeasurements is bimodal. The filter chooses, withinthe circular manoeuvre model, the wrong solution for theangular speed. Due to the measurement noise realisation,this results in a bad estimate of the angular speed at t ¼ 14 s:The standard IMM filter does not recover from thismisinterpretation and is also not robust against it; itdiverges. The IMM regularised particle filter, however,can deal with this situation and we see that the filter also

performs well after t ¼ 14 s: Also, if we look at Fig. 4, wesee that the mode probabilities of the IMM particle filter arein accordance with the trajectory, i.e. straight until timet ¼ 14 s and a hybrid between circular and verticalacceleration from there until t ¼ 18 s; from where it ishybrid between straight trajectory and circular manoeuvre.We stress the fact that the mode probabilities for thestandard IMM cannot be trusted after t ¼ 14 s: This isbecause the filter has diverged.

5 Conclusions

In this paper we have presented a new method for multiplemodel filtering. This method is based on a combination ofthe standard IMM filter and a regularised particle filter. Thenew method has been shown to work well. By means of anexample we have shown that in a difficult target trackingsituation, i.e. nonlinear dynamics, nonlinear measurementsand different types of possible manoeuvres, the standardIMM filter performed poorly whereas the newly proposedfilter performed well.

6 References

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6 McGinnity, S., and Irwin, G.W.: ‘Manoevering target tracking using amultiple-model bootstrap filter’ in Doucet, A., de Freitas, N. andGordon, N. (Eds.): ‘Sequential Monte Carlo methods in practice’(Springer, New York, 2001), pp. 247–271

7 Musso, C., Oudjane, N., and Le Gland, F.: ‘Improving regularizedparticle filters’ in Doucet, A., de Freitas, N. and Gordon, N. (Eds.):‘Sequential Monte Carlo methods in practice’ (Springer, New York,2001), pp. 247–271

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11 Mazor, E., Averbuch, A., Bar-Shalom, Y., and Dayan, J.: ‘Interactingmultiple model methods in target tracking: a survey’, IEEE Trans.Aerosp. Electron. Syst., 1998, 34, pp. 103–123

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13 Salmond, D.J.: ‘Mixture reduction algorithms for target tracking inclutter’. Proc. SPIE Conf. on Signal and data processing of smalltargets, 1990

14 Salmond, D.J., Fisher, D., and Gordon, N.J.: ‘Tracking in the presenceof intermittent spurious objects and clutter’, Proc. SPIE–Int. Soc. Opt.Eng., 1998, 3373, pp. 460–474

12 13 14 15 16 17 18 190

0.2

0.4

0.6

0.8

1.0

time, s

prob

abili

ties

12 13 14 15 16

a

b

17 18 190

0.2

0.4

0.6

0.8

1.0

time, s

straightcircularvertical

prob

abili

ties

Fig. 4 Mode probabilities

a Standard IMM filterb IMM hybrid bootstrap filter

IEE Proc.-Radar Sonar Navig., Vol. 150, No. 5, October 2003 349