Multiple Dynamic Trajectories Prediction - Visualization of Distribution of Optimal Solutions

4.3 Visualization of Distribution of Optimal Solutions

4.4.2 Multiple Dynamic Trajectories Prediction

For a dynamic trajectory prediction problem, the goal may be to estimate the initial position and velocity of a moving object using the measured data. For the trajectory prediction of multiple targets, here we assume that the target detection has been carried out in advance, and we focus on the estimation of the initial states of multiple moving objects. Through a learning process, the NSOMS may determine a most probable initial state of each target through repeatedly comparing the measured data with the predicted trajectories derived from the possible initial states stored in the neurons of the SOM.

In this application, the nonlinear dynamic equation describing the trajectory of the moving object and the measurement equation are as those described in Chapter 3. The learning algorithm for multiple trajectories prediction is organized as follows.

Algorithm for multiple trajectory prediction based on the SOMS: Predict an optimal initial state for the trajectory of every moving target using the measured position data.

Step 1: Set the stage of learning k = 0. Choose H number of niches, N number of neurons within each niche, and reference value Pr. Estimate the ranges of the possible position and velocity of the moving object, and randomly store the possible initial states w^h_j(0) into the neurons, where j = 1, . . . , N , h = 1, . . . , H.

Step 2: Send w^h_j(k) into the dynamic model, described in Eqs.(3.25) and (3.28), to compute p^h_j(k).

Step 3: For each neuron j of every niche, compute its output O_j^h(k):

O^h_j(k) =^Y^M where M is the number of the objects detected.

Find the winning neuron j^∗ with the minimum O^h_j∗(k):

O^h_j∗(k) = min

j O^h_j(k) (4.18)

Step 4: Update the weight vectors of the winning neuron j^∗ and its neighbors to every niche, and update the positions of neurons of the entire network.

Step 5: If ^P^q

i=1σ˜^h_i(k) < Pr for every niche, w^h_j^∗(k) is determined to be as an effective optimal solution with the duplicate optimal solutions excluded. The prediction process outputs the predicted optimal initial states to the dynamic model to derive the object trajectories. From Eqs.(4.6) and (4.7), the new w^h_j(k) will be randomly regenerated and then added into the set <^c.

Step 6: Check whether the number of iterations is smaller than a pre-specified maximum number of iterations. If it is not, let k = k + 1, and go to Step 2; otherwise, the prediction process is completed and output optimal states of all objects. The final network mapping provides the visualization of the distribution of the optimal states. In addition, during each stage of learning, we perform a number of learning to increase the SOM learning speed. This number of learning is set to be a large number in the initial stage of the learning process, such that the NSOMS may converge faster at the price of more oscillations, and decreased gradually to achieve smooth learning in later stages of learning.

To demonstrate the effectiveness of the proposed NSOMS and weight updating rule, we performed a series of simulations for dynamic trajectory prediction based on using the proposed NSOMS and two NSOMS without the proposed dynamic weight updating rule (named as SOMSO-1 and SOMSO-2 by using the SOMO’s and SOSENs’s weight updating rules, respectively, in place of that of the NSOMS). The trajectory to predict in the simulations was designed to emulate that of a missile. Its governing equations of motion in the 3D Cartesian coordinate system are as those described in Chapter 3. The ranges of the possible initial states w_j(0) were estimated to be

1.14 × 10⁶m ≤ x₁(0) ≤ 2.14 × 10⁶m

Within the ranges described in Eq.(4.19), the possible initial positions and velocities of the missile were selected and stored into the 1125 (5×225) neurons of the 2D SOM. We consider three targets to be detected in the following simulations. The parameters of NSOMS were set to be n₁ = 2, n₀ = 1, respectively. The additional adaptation term ε, described in Eq.(4.4), was set to be 0.1. For comparison, we set the same learning rate as those in the NSOMS described in Sect. 4.4,1, and several parameters of the SOMSO-1 and SOMSO-2 were adjusted via a trial-and-error process to yield salient performance.

The number of learning is set to be 20 during each stage of learning.

We first applied the SOMS, SOMSO-1, and SOMSO-2 for trajectory prediction with a good estimate of the initial state. Three ideal initial states of the missiles were assumed to be within the estimated range. The variance of the measurement noise was set to be

(15m)². Figure 4.7 shows the simulation results. The ideal and measured trajectories are shown in Figure 4.7(a). These three methods predicted the initial state quite well and thus resulted in very small estimated errors, except in the initial stage of the prediction, as shown in Figure 4.7(b)-(d) (the estimated initial state error is shown for illustration).

We observed that the NSOMS converged faster than the other methods did. Figure 4.8 shows the neighboring relationship of neurons of the best three niches using the NSOMS.

In Figure 4.8(a), from a random distribution of neurons in the beginning of the learning, the mapping structure gradually form three clusters along with the stage of learning.

Figure 4.8(b) shows how the neighborhood function D_i and F (w_i(k)) varied during the SOM learning process, and eventually they are very close to each other.

In the second set of simulations, we investigated their performances for the condition of a bad estimate of the initial state. The ranges of the possible initial states w_j(0) were estimated to be

In this simulation, three ideal initial states were assumed to be outside of the estimated range. The variance of the measurement noise was enlarged to be (50m)². The setting of all parameters was set to be the same as them in first simulation. Figure 4.9(a) shows the ideal and measured trajectories and Figure 4.9(b)-(d) the estimated initial error. From the results, the influence of the bad estimate on these methods was mostly at the initial stage of the prediction. After the transient, the NSOMS still managed to find the optimal

(a) Ideal and measured TBM trajectories

Figure 4.7 Simulation results for the multiple trajectories prediction using the NSOM, SOMSO-1, and SOMSO-2 with a good estimate of the initial state: (a) the ideal and measured TBM trajectories, (b)-(d) the estimated initial state error by using the NSOM, SOMSO-1, and SOMSO-2.

(Cont. )

(b) NSOMS

(d) SOMSO-2

Estimated position error (m) Estimated position error (m) Estimated position error (m)

(a) Projection result in 2D neuron space

(b) Final neighborhood function values

Figure 4.8 Final results obtained by the NSOM S for the multiple trajectories prediction: (a) projection result in 2D neuron space and (b) finial neighborhood function values.

:D_i : (F w_i( ))k :D_i*

initial states of all targets. Meanwhile, we also observed that the NSOMS converged very faster than the other methods did. As for the SOMSO-1 and SOMSO-2, they converged very slowly as the optimal initial state did not fall within the estimated range. In Figure 4.10(a), from a random distribution of neurons in the beginning of the learning, the mapping structure gradually form three clusters along with the stage of learning. Figure 4.10(b) shows how the neighborhood function D_i and F (w_i(k)) varied during the SOM learning process, and eventually they are very close to each other.

To further demonstrate the NSOMS search ability, we used four different network sizes and learning parameters to run these optimization algorithms for the dynamic trajectory prediction. We only consider one target for comparison. The ideal initial state of the missile was assumed to be within the estimated range, described in Eq.(4.19). Figure 4.11 shows how the population size affects for these algorithms. We observed that the NSOMS performs better than the SOMSO-1 and SOMSO-2. As Figure 4.11 shows, the SOMSO-1 and SOMSO-2 did not converge to the optimal state when the network size was very small. Figure 4.12 show the performance with the different learning parameters for the influence of the these algorithms under a fixed network size (1 × 225). As Figure 4.12 illustrates, we observed that the large learning parameter may speed up the learning of the NSOMS. Although a small learning parameter made the NSOMS converge slight slowly, it still converged faster than the SOMSO-1 and SOMSO-2 did. Table 4.2 shows that the different network (population) size and learning parameter affect the learning result for the NSOMS, SOMSO-1 and SOMSO-2 in detail. We calculated the RMS (Root-Mean-Square) value of the error between the ideal and predicted trajectories at k = 100 to evaluate their performance. The comparison results of average of repeated 30 times are listed in Table 4.2. We observed that the NSOMS performed better than the other methods did. Figure 4.13 shows the performance of 5 runs with the same initial weights

(a) Ideal and measured TBM trajectories

Figure 4.9 Simulation results for multiple trajectories prediction using the NSOM, SOMSO-1, and SOMSO-2 with a bad estimate of the initial state:

(a) the ideal and measured TBM trajectories, (b)-(d) the estimated initial state error by using the NSOM, SOMSO-1, and SOMSO-2. (Cont. )

(b) NSOMS

(d) SOMSO-2

Figure 4.9 Simulation results for multiple trajectories prediction using the NSOM, SOMSO-1, and SOMSO-2 with a bad estimate of the initial state:

(a) the ideal and measured TBM trajectories, (b)-(d) the estimated initial state error by using the NSOM, SOMSO-1, and SOMSO-2.

Estimated position error (m) Estimated position error (m) Estimated position error (m)

(a) Projection result in 2D neuron space

(b) Final neighborhood function values

Figure 4.10 Final results obtained by the NSOM S for the multiple trajectories prediction : (a) projection result in 2D neuron space and (b) finial neighborhood function values.

:D_i : (F w_i( ))k :D_i*

(a) H 1, N !3 3

(b) H 1, N !5 5

Figure 4.11 Performance for different network sizes using the NSOMS, SOMSO-1, and SOMSO-2. (Cont. )

Estimated position error (m) Estimated position error (m)

(d) H 1, N 20 20!

Figure 4.11 Performance for different network sizes using the NSOMS, SOMSO-1, and SOMSO-2.

Estimated position error (m) Estimated position error (m)

(a) _w( )k !0.2

(b) _w( )k !0.4

Figure 4.12 Performance for different parameters using the NSOMS, SOMSO-1, and SOMSO-2. (Cont. )

Estimated position error (m) Estimated position error (m)

(d) _w( )k !0.8

Figure 4.12 Performance for different parameters using the NSOMS, SOMSO-1, and SOMSO-2.

Estimated position error (m) Estimated position error (m)

Table 4.2: Comparison results for NSOMS, SOMSO-1, and SOMSO-2 on the dynamic trajectory prediction.

Mean and Standard Deviation of RMS Values (m) Network

25.981 1.233 1.056 10²!1.016 10² 44.012 35.543 25.711 0.786 25.838 0.752 25.834 0.761 25.775 1.075 25.834 1.258 25.845 1.281

for the influence of three algorithms under a quite small network size (H = 1, N = 2 × 2).

Figure 4.14 shows the performance of 5 runs with the different initial weights. From the results shown in Table 4.2 and Figures 4.11-4.14, the NSOMS was more robust and converged faster than the other two algorithms did.

We also performed simulations based on using the RCS-PSM. We modified the in-ternal parameters of the PSM method to enhance its search abilities. However, it was not that straightforward to determine its parameters properly, and the process was time-consuming. The SOMSO-1, SOMSO-2, and RCS-PSM might not be that effective under such circumstances that the ranges of the possible initial states may be uncertain and varying in noisy, unknown environments. We thus conclude that the NSOMS performed better than the SOMSO-1, SOMSO-2, and RCS-PSM for this dynamic trajectory predic-tion applicapredic-tion, and the proposed dynamic weight updating rule was effective.

As a summary, in this chapter, a niching SOM-based search algorithm has been pro-posed for identification and visualization of multiple optimal solutions. Through reducing the network size greatly for search in the high-dimensional space, we have also proposed a niche weight updating rule to raise the learning efficiency. The final network struc-ture allows us to easily classify the optimal solutions into clusters, thus yielding useful information for solution selection.

(a) SOMSO-1

(b) SOMSO-2

Figure 4.13 Performance of 5 runs with the same initial weights using the NSOMS, SOMSO-1, and SOMSO-2.

Estimated position error (m) Estimated position error (m) Estimated position error (m)

(a) SOMSO-1

(b) SOMSO-2

Figure 4.14 Performance of 5 runs with the different initial weights using the NSOMS, SOMSO-1, and SOMSO-2.

Estimated position error (m) Estimated position error (m) Estimated position error (m)

Chapter 5 Conclusion

In this dissertation, we have proposed an SOM-based search algorithm (SOMS), which can be used for both static and dynamic functions in real time. To achieve high learn-ing efficiency for system parameters in different worklearn-ing ranges, we have also proposed a new SOM weight updating rule. An intelligent radar predictor for trajectory estimation is first developed for application. With a simplified target dynamic model, the unsupervised SOM in the predictor can achieve salient prediction in noisy, unknown environments. The SOM is more robust to the uncertainty of the dynamic model than Kalman filter and GA.

Furthermore, the SOM’s search abilities have been adequately exploited in a multimodal domain. A new niche method (deterministic competition) to extend the ability of the SOM-based search algorithm has been proposed for identification of multiple optimal so-lutions. In order to reduce the network size, another new SOM weight updating rule is proposed to enhance the learning efficiency. With the dynamic weight updating, the NSOMS converges faster than other algorithms such as SOMO, KSOM-ES, SOSENs and RCS-PSM. Moreover, a new adaptive mapping model is proposed to visualize the distri-bution and structure of the optimal solutions into the 2D neuron space. In our proposed

NSOMS, only two learning parameters need to be determined in the weight and position updating rules. The applications of the proposed NSOMS on both function optimization in a multimodal domain and dynamic trajectory predictions involving multiple targets have clearly demonstrated its effectiveness.

5.1 Future Research

In this dissertation, by combining the SOM with the dynamic model, the SOM is able to tackle the spatiotemporal data. The SOM has been applied to search optimal parameters for dynamic systems. To further exploit its search ability, in one of the future works, we will apply the NSOMS for system identification and control problems. Because the current searching process includes the weight updating rules and the parameters of learn-ing, it is not easy to appropriate the learning rate, number of neurons, and termination criteria. Although these parameters can be selected through a trial-and-error process, the time response of the learning affects the performance of the dynamic systems for system identification and control problems. Thus, we will also discuss the convergence issue in details. As the SOM also possesses an appealing feature in responding to distinct proper-ties exhibited by the input data through forming several corresponding clusters, another worthwhile future work will be to extend the proposed NSOMS for a wide application such as image processing, speaker recognition, machine learning, and others.

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