The proposal compared with a Kalman filter

Here, the ability of the proposed soft and soft systematic resamplers to accu-rately represent the posterior and to retain more information is demonstrated.

For this testing, a 2D linear Gaussian model is used so that comparison can be made with the Kalman filter [20] which is known to give the optimal pos-terior. The target state is characterised by the position and velocity of the target as x_k= (x, v_x)^T. The process model is

x_k =

"

1 T 0 1

#

x_k−1+ q_k−1 (5.29)

with T = 1s and the process noise q_k−1 ∼ N (0, Q) where

Q =





0.95 0.2 0.2 0.75



 The observation model

zk= xk+ rk (5.30)

generates a clean though noisy point target measurement. The term “clean measurement” implies there are no false alarms, clutter detections, missed observations and/or out-of-sequence (OOS) observations³. The sensor noise is rk ∼ N (0, R) with R = σ²I having the variance σ² = 0.5 along each dimension.

Accuracy of posterior representation

Fig. 5.13 shows the KS statistic of various filters for varying numbers of particles. It can be observed that the PF operating with the systematic re-sampler exhibits the best consensus with the theoretically optimal posterior representation obtained using the Kalman filter. It can also be observed that

3A OOS observation is a target detection at time k but received at a later time sample k + τ .

the performance of the soft resampler is slightly lower than that of the sys-tematic resampler because of continually discarding the low weight particles.

The figure also indicates that with increasing β the soft systematic resampler exhibits better agreement with the theoretically optimal posterior distribu-tion. The soft redistribution exhibits similar performance to that of the soft renormalisation and hence is not shown.

10² 10³

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Number of particles (I)

KS statistic

Soft systematic: (α, β)= (0.95, 1) Soft systematic: (α, β)= (0.95, 4) Soft−renormalised

Systematic Multinomial

Figure 5.13: KS statistic versus the number of the particles I. The results are averaged over 50 instances, each having 50 time steps.

Fig. 5.14 shows the cdfs of the standard PFs (in the position dimension) operating on soft, soft systematic and systematic resamplers. The figure corresponds to a single time sample when the target is making a sharp ma-noeuvre. The proposed resamplers are compared with the systematic re-sampler because the latter has been found to exhibit the best performance

in terms of accurately representing the posterior in Fig. 5.13. In Fig. 5.14, the cdf in blue is obtained from (5.20) and the cdf in red is obtained from (5.21). It can be observed from the figure that while the soft resampler in Fig. 5.14(d) does not properly treat the tails of the cdf because the method discards the low weight particles all the time, the soft systematic resampler in Fig. 5.14(b) overcomes this problem by stochastically resampling the par-ticles located near the tails of the cdf. For increasing β, the performance of

−50 0 5

Figure 5.14: cdfs for PFs operated using soft, soft systematic and systematic resamplers.

These cdfs correspond to the position of a target. The legend of Fig. 5.14(a) applies to Fig. 5.14(b,c,d) also.

the proposed soft systematic resampler will approach that of the systematic resampler. Fig. 5.14(c) indicates that the systematic resampler based PF ap-proximation exhibits agreeable representation of the posterior described by the theoretically optimal Kalman filter. The soft resampling (with renormal-isation) can be considered as soft systematic resampling with (α, β) = (1, 0).

Therefore as observed in Fig. 5.14(a), decreasing the number of replications by reducing α will cause inaccuracies in posterior representation but the fea-ture can be used to address other PF problems, e.g., to increase the diversity of the particles when the weights of the particles near a target are very large [256].

Information retention

The proposed soft and soft systematic resamplers are analysed in terms of their ability to retain more information over successive time steps and thereby aid in a faster lock. For this test, the 2D linear Gaussian model described in section 5.6.2 is again used. Here, the particles are initialised with unit covari-ance and a mean that is 8m away from the actual target. This arrangement results in a particle set that is just near the actual target by the time filtering starts. This arrangement requires retention of the weight information over time for a faster lock. Fig. 5.15 illustrates how fast the target is detected by showing the drop in the rmse of the PF estimates for 10 time samples. It can be inferred that by appropriate reweighting and retaining more informa-tion contained in the lower weights, the soft and soft systematic resamplers lock onto the targets faster than the systematic resampler that regards all the particles as equally weighted. The target in this test is allowed to make sharp manoeuvres, hence it is highly probable that the target could move towards a low weight particle. By ensuring that not all low weight particles are lost all the time, the soft systematic resampler aids in faster lock than its soft counterpart. The soft redistributed variant exhibits almost similar performance to that of soft renormalised variant and hence is not shown.

2 4 6 8 10 0

2 4 6 8 10 12

Time [s]

rmse [m]

Soft systematic: (α, β)= (0.95, 4) Soft−renormalised

Systematic Kalman

Figure 5.15: rmse versus time samples. The number of particles I = 1024. The results are averaged over 100 instances, each having 10 time steps.

Accuracy

The accuracy of the various resamplers is tested for the same 2D linear Gaus-sian model. Fig. 5.16 shows the rmse in position for varying sensor noise variance σ² for 1024 particles. Again, the soft redistributed variant exhibits almost similar performance to that of the soft renormalised variant and hence not shown. It can be observed that the proposed techniques are comparable to state-of-the-art resamplers.

1 2 3 4 5 6 7 8 0

0.5 1 1.5 2

sensor noise variance σ² [m]

rmse [m]

Soft systematic: (α, β)= (0.95, 1) Soft systematic: (α, β)= (0.95, 4) Soft−renormalised

Systematic Multinomial Kalman

Figure 5.16: rmse versus the sensor noise variance σ². The number of particles is I = 1024. The results are averaged over 20 instances, each having 50 time steps.