In this section, we describe the working our new flow. We divide the description into two parts.
First, using a simple example we gain more insight into the measurement update using our new flows. This will help understand the benefits and potential issues. Next, using a 1D example of a nonlinear process and measurement model, we present results for the GMPF filters compared to those of a DHF based on theexact flowand a SIR-PF with 500 particles. The particular
6.3 Numerical Results 175 number for the SIR-PF particles is chosen, because the filter seem to saturate around this value.
Increasing it further does not lead to any significant improvement in the performance.
First, we study the working of our two new flow equations. For that we consider the tri-modal form of the prior density,
p(xk+1|Zk) =
3
X
i=1
wiN(xk+1|µi, Pi) (6.40) wherew={0.6,0.25,0.15},µµµ ={−10,4,10}andP ={1,0.5,3}. Also the likelihood is given by,
p(zk+1|xk+1) = 1
√2πexp
"
−0.5
zk+1−x2k+1 20
2#
(6.41)
−200 −10 0 10 20
0.05 0.1 0.15 0.2 0.25
xk+1
p(xk+1|Zk)
(a)
−200 −10 0 10 20
0.1 0.2 0.3 0.4
xk+1 p(zk+1|xk+1)
(b)
−200 −10 0 10 20
1 2 3 4x 10−3
xk+1
p(xk+1|Zk+1)
(c)
Figure 6.1: (a) Prior density, (b) Likelihood and (c) the Posterior density for the toy model.
wherezk+1= 3. We see that the prior density is given by a tri-modal GMM, while the likeli-hood is bi-model for the given set of parameters. We generate 100 samples from the prior den-sity and apply equations (6.4), (6.28) and (6.31) separately to propagate the particles through the pseudo-time loop. We discretize theλinto 30 geometrically spaced points between 0 and 1.
Below, we plot the three flow values for the particles vs. the pseudo-time values.
176 6 Flow solution for the sum of Gaussians based prior densities
10−3 10−2 10−1 100
−100 250
λ f(xλ,λ)
(a)
10−3 10−2 10−1 100
−15
−10
−5 0 5
λ f(xλ,λ)
(b)
10−3 10−2 10−1 100
−20
−15
−10
−5 0 5
λ f(xλ,λ)
(c)
Figure 6.2: (a) Exact flow, (b)GMPF-1 and (c)GMPF-2 vs.λ
We see some interesting trends here. Theexact flowhas very large dynamic range when com-pared to the other two flows, thus suggesting a requirement for very fineλdiscretization. The flow values forGMPF-2 are more concentrated for all value ofλ, suggesting lesser variance of the particles. We see relatively large value of the flow for some particle, but they are not that many. Finally, we plot the posterior particles super-imposed on the posterior density.
6.3 Numerical Results 177
−200 −10 0 10 20
0.01
xk+1 p(xk+1|Zk+1)
True posterior GMPF−1 EF
(a)
−200 −10 0 10 20
0.01
xk+1
p(xk+1|Zk+1)
True posterior GMPF−2 EF
(b)
Figure 6.3: True posterior density with the particle spread for (a)GMPF-1 and (b)GMPF-2.
Here, we see the full power of our new flows. Theexact flowclearly fails to capture the mode 2, located at the x=4. Also, the spread about the two captured mode less than what is actual, suggesting the inadequacy of the flow equation to fully capture the multi-modal dynamics. On the other hand, the other flows nicely capture the full structure of posterior density, by not only spreading the particles around the three modes but also encapsulating the information regarding the variance of the individual modes.GMPF-2 is better than 1 as its mode variance is close to the actual.
In the next part of this section, we consider a univariate non-stationary growth model, similar to the one studied in [KD03].
xk+1= 0.5xk+ 25 xk
1 +x2k + 8 cos(1.2k) +wk (6.42) yk+1= x2k+1
20 +vk (6.43)
wherewkis the bi-modal process noise distribution, represented by a GMM such thatp(wk) = 0.5N(wk| −2,3) + 0.5N(wk|2,3), whilevkis the measurement noise given byN(vk|0,1).
We compare the performance of DHF based on the variants of our new flows, employing both resampling and merging for reducing the number of components. They are named here as DHF-GMPF-1-resamp, DHF-GMPF-1-merge, resamp and DHF-GMPF-2-merge. We use 3 components to represent the initial densityp(x0)i.e. M = 3. Mean and covari-ance for individual components are chosen such that the overall mean is zero and covaricovari-ance is 25. The number of particles per component,L, is chosen to be 50. We use 30 geometrically spacedλpoints between 0 and 1. Linearization of the process modelh(x)and the variable α(x)is carried about individual particles. In addition to the variants of our new flows, we also study the performance of the Gaussian prior basedexact flowwith 100 particles (DHF-EF), SIR particle filter with 500 particles (PF-500) and the Extended Kalman filter (EKF). We simulate the scenario for a total of 50 times, each running for 100 time instances. Increasing the number of particles beyond the values stated for the DHF-EF and SIR-PF, does not enhance the perfor-mance significantly. Also, the choice of parameters for our new flows is made in part to achieve a trade-off between the performance and the computational complexity. In Figure 6.4, we plot
178 6 Flow solution for the sum of Gaussians based prior densities
the time averaged root mean square error (RMSE) against the realizations (simulation index).
0 10 20 30 40 50
Realization 2
4 6 8
RMSE
DHF-GMPF-1-resamp DHF-GMPF-1-merge DHF-GMPF-2-resamp DHF-GMPF-2-merge DHF-EF SIR-PF-500
Figure 6.4: Time averaged RMSE vs. realizations
We note that the lowest error is given by the DHF-GMPF-2-merge, closely followed by PF-500.
This is followed by DHF-GMPF-2-resamp and other the variants of DHF-GMPF-1. There are three observations to be made here. First, the filter based on the inclusion of the cubic term in the flow leads to a better performance. Secondly, merging of components seems to be a better solution as compared to the resampling, and thirdly, differences between all GMPF versions and PF-500 are not significant. Next we find that DHF-EF, in comparison to theGMPFand PF, exhibits a large error. This further supports our claim, that a flow based on the single Gaussian prior assumption might be inadequate for scenarios involving multi modal densities. EKF fails to track the state and has the highest error amongst all. RMSE for the EKF is not shown in Figure 6.4.
Method Avg. RMSE Var. RMSE
DHF-GMPF-1-resamp 3.04 7.02
DHF-GMPF-1-merge 2.99 6.93
DHF-GMPF-2-resamp 2.97 5.67
DHF-GMPF-2-merge 2.77 5.42
DHF-EF 5.08 6.20
EKF 19.2 14.24
PF-500 2.86 3.74
Table 6.1: Comparison for different filters
Furthermore, we note that the lowest value for the error standard deviation is given by the par-ticle filter. DHF-GMPF-2-merge fares second, followed by the other variants of DHF-GMPF.
This shows that error variability for our newly proposed filters is slightly higher when compared to the standard particle filter. EKF is the quickest of all filters, taking 0.004 second per iteration.
Next comes the PF-500 which takes about 0.009 second, followed by the DHF-EF with 0.013
6.4 Conclusions 179