Probabilistic data association filter (PDAF)

The PDAF was first proposed by Bar-Shalom and Tse [20]. The algorithm assigns a probability, called the association probability, to every hypothesis associating a val- idated measurement to a target. The validated measurements refer to measurements that lie in the validation gate of a target at the current time. A validation gate centred around the predicted measurement of the target set up to select the set of validated measurements is (z(k) z(kjk^ 1)) T S 1 (k)(z(k) z(kjk 1)) (6.2)

whereS(k)is the covariance of the innovation anddetermines the size of the gate. The set of validated measurements at timekis

Z(k)=z i (k); i=1;:::;m k (6.3) wherez i

(k)is theith measurement in the validation region at timek. This is also known as the ‘all neighbour’ modified filter.

The standard PDAF equations are as follows [195]: State prediction ^ x(kjk 1)=F^x (k 1jk 1) (6.4) Measurement prediction ^ z(kjk 1)=Hx(kjk^ 1) (6.5) Innovation ofith measurement i (k)=z i (k) z(kjk^ 1) (6.6) Covariance prediction P(kjk 1)=FP(k 1jk 1)F T +GQG T (6.7) Innovation covariance and Kalman gain

S(k)=HP(kjk 1)H T +R (6.8) K(k)=P(kjk 1)H T S(k) 1 (6.9) Updated covariance if target originated measurements were known

P o

(kjk)=P(kjk 1) K(k)S(k)K(k) T

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Overall covariance update

(k) = mk X i=1 i (k) i (k) (6.11) P(kjk) = P o (kjk)+K(k)[ o (k)S(k)+ mk X i=1 [ i (k) i (k) i (k) T ] (k)(k) T ]K T (k) (6.12) wherem

kis the number of validated returns at

kth instant. Updated state estimate

^

x(kjk) = x (kjk^ 1)+K(k)(k) (6.13)

The PDAF association probabilities are [205, 50]

i (k)= p i (k) P m(k ) i=0 p i (k) where p i (k)= 8 < : (1 P d P g ) if i=0 P d (2) M=2 jS(k )j 1=2 exp[ 1 2 r i (k) 2 ] if [(k)]=1; i6=0 0 otherwise and= m k V(k ) ,V(k)= M=2 (M=2+1) M jS(k)j 1=2 ,[205] (k)=

1 if the return belongs to the validation gate of the target 0 otherwise

Mis the dimension of the state vector andis the clutter density.P

dis the proba- bility of detecting the correct return andP

gis the probability of validating a detected return.

A weighted average of the state, estimated under all the hypotheses associating different measurements (returns) to a particular target serves as the PDAF estimate of the state of that target. The associating of different measurements to a particular target serves as the PDAF estimate of the state of that target. Hence, the association probabilities are used as weights.

The disadvantages of the PDAF are:

Miss-tracking. Since the PDAF ignores the interference from other targets, it may, sometimes, result in miss-tracking of closely-spaced targets [82]. Hence, may perform poorly while tracking is crossing targets or when the targets are close to each other.

Suboptimal Bayesian approach. The PDAF is a suboptimal Bayesian approach to the problem of tracking when the source of measurement data is uncertain, due to clutter and missed detections.

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Single target only. The PDAF is designed for the association of a single target in a clutter. The number of false alarms is usually modelled by a Poisson density and false alarms are assumed to be distributed uniformly in space. When there are multiple targets in a surveillance area, the PDAF does not perform well, because the false alarm model is no longer valid. It is due to the persistent returns generated by interfering targets.

Need to provide separate track initiation and deletion algorithms. PDAF as- sumes that a track has been established and, therefore, track initiation and deletion algorithms have to be provided separately.

Mainly good for tracking non-manoeuvring targets in cluttered environments. If the target undertakes a manoeuvre, it is highly likely that the PDAF will lose the target track [153].

Various modifications to combine and improve the PDAF with the state estima- tion are suggested. Most of these algorithms are discussed in Bar-Shalom and Li [18]. Here, we will just state the algorithms, their assumptions and advantages.

Multiple model probabilistic data association filter (MMPDAF). This is a com- bination of the non-switching multiple model and the PDAF. The result is an adaptive estimator that can adjust itself to the ‘true model’ of the target while in a cluttered environment. The assumptions of the MMPDAF are:

– The system obeys one out ofNmodels.

– The models do not switch in time.

– The models differ only in the process and/or measurement noise levels.

The MMPDAF can track a manoeuvring target in clutter with a number of PDAFs, based on different models, that are running in parallel [18].

IMM combined with PDAF (IMMPDAF). The IMMPDAF can be used to

– Initiate tracks.

– Carry out track maintenance for manoeuvring targets. – Terminate tracks.

This is accomplished with the following set of models:

– Model 1: Undetectable target (no target).

– Model 2: True target moving with nearly constant velocity. – Model 3: Motion with large acceleration increments. – Model 4: Nearly constant acceleration motion.

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This approach assumes that the modelling assumptions are correct.

In [194], it is claimed that the IMMPDAF is incapable of performing track detection when the signal-to-noise ratio (SNR) is low [194]. Hence, this tech- nique may not recover from a track loss situation.

One improvement in a IMMPDAF is to use the amplitude information. Hence, the technique is known as the IMMPDAFAI. IMMPDAFAI uses the statistical information of the detector output for target returns and clutter to improve track maintenance for low SNR targets in a dense cluttered environment [138]. The amplitude likelihood ratio

i = p 1 (a i (k)) p o (a i (k)) (6.14) is used to modify the standard PDAF association probabilities to include am- plitude discrimination. p

o

(a) andp 1

(a)are probability density functions of the amplitude if it is due to noise only and if it is originated from a target, respectively. The amplitude likelihood ratio term affects the association prob- abilities by favouring a high amplitude measurement.

Directional probability data association (DPDA) [50]. The DPDA incorpo- rates the directional information and the association probabilities which are estimated using both the Mahalanobis distance and the track direction. Exponentially weighted probabilistic data association filter (EWPDA) [153].

In [153], it is claimed that the algorithm can track a manoeuvring target in a cluttered environment.