For this TBM tracking application, the SOM needs to tackle the spatio-temporal data, instead of the spatial data it usually deals with. Therefore, a dynamic model that describes the behavior of the TBM is included in the intelligent radar predictor. By combining the SOM with the dynamic model, the SOM is able to tackle the spatio-temporal radar data. Figure 2.3 shows the structure and operation in the SOM. A 2D SOM is used for illustration in Figure 2.3, and the SOM can also be three-dimensional or other according to the applications. Each time a certain number of new measured position data v are sent in from the radar system, the SOM is triggered to operate. And, it will gradually converge to an optimal prediction along with the increase of the measurement data and learning time. In Figure 2.3, for each neuron j in the SOM, it contains a vector of a possible initial state wj and generates an output sj. By sending wj to the dynamic model, sj is computed as the difference between v and the predicted trajectory pj. Of all the neurons, the neuron j∗ with the smallest output sj∗ is chosen as the winner. When the weight of this winning neuron j∗ differs from that of the previous winner ˆj∗ (i.e., wj∗ 6= wˆj∗), the weights of ˆj∗ and its neighbors will be updated in a manner that moves these weight vectors toward neuron j∗, as shown in Figure 2.4(a). When j∗ is the same as ˆj∗ (i.e., wj∗ = wˆj∗), the weights will then be updated so as to let the weight vectors form more and more compact clusters centering at neuron j∗, as shown in Figure 2.4(b). Under successful learning, the SOM will finally converge to a predicted optimal initial state.
Several parameters need to be determined in implementing this SOM, including the learning rate, topological neighborhood function, and number of radar data used for sj computation. The selection of the learning rate η depends on the closeness of wj∗(k) and wˆj∗(k). When they are different from each other, we intend to speed up the learning
!
(a) wj* wˆj*
(b) wj* wˆj*
Figure 2.4 The movement of the weight vector in the two-dimensional space: (a) wj* w and (b)ˆj* !wj* w .ˆj*
j*
w
ˆj*
w
j*
w
ˆj*
w
process and choose η(k) in the kth stage of learning to be close to 1. And when they almost coincide, we slow down the learning gradually and determine η(k) according to Eq.(2.1):
η(k) =
η0(1 − k/τ ), f or k ≤ τ0 < τ
η1(1 − τ0/τ ), f or k > τ0 (2.1)
where η0 and η1 are constants smaller than 1, and τ and τ0 time constants. Other types of functions can also be used, for instance,
η(k) = η1· e−k/τ + η0 (2.2)
Because the weight updating also includes the neighbors of the winning neuron, the topo-logical neighborhood function hj∗ needs to be chosen. We adopt the Gaussian neighbor-hood function for hj∗(k) [17]:
hj∗(k) = exp(−d2j,j∗
2σ2) (2.3)
where dj,j∗ is a lateral connection distance between neural j and j∗, and σ the width. For the sake of efficiency in computing sj, not all the accumulated measured radar data will be used to compare with the predicted trajectory. Under such selection of the learning rate and neighborhood function, they will force the minimization of the difference between the weight of the winning neuron and those corresponding to every neuron within its neighborhood in each learning cycle. The learning in the algorithm will thus converge eventually.
Based on the discussions above, we developed the SOM learning algorithm. Before the description of the algorithm, we first introduce a simplified dynamic model of the TBM.
With the model, the SOM can obtain pj by sending wj into it. This dynamic model is formulated as
x(k + 1) = A(k)x(k) + Γ(k)ξ(k) (2.4)
v(k) = C(k)x(k) + µ(k) (2.5)
where
x(k) : n-dimensional state vector at the kth stage
A(k) : n × n transition matrix
Γ(k) : n × r input distribution matrix
ξ(k) : r-dimensional random input vector
v(k) : m-dimensional output vector
C(k) :m × n observation matrix
µ(k) : m-dimensional random disturbance vector
with ξ(k) and µ(k) assumed to be white Gaussian with the following properties:
E[ξ(k)] = 0 (2.6)
E[ξ(j)ξ(k)t] = Qδjk (2.7)
E[µ(k)] = 0 (2.8)
E[µ(j)µ(k)t] = Rδjk (2.9)
E[ξ(j)µ(k)t] = 0 (2.10)
where E[·] stands for the expectation function, Q and R the covariance matrix of the input noise and output noise, respectively, and δjk the Kronecker delta function. In using the dynamic model, the SOM is not necessarily aware of its statistical properties. By contrast, the Kalman filter needs to know the noise distribution in the dynamic model and also a guess on the system’s initial state for trajectory prediction. As the covariance matrices Q and R may be uncertain and varying in noisy, unknown environments, their estimated values are possibly imprecise, even incorrect. Thus, the Kalman filter may not be that effective under such circumstances. The reason that the SOM is more robust to the uncertainty of the dynamic model than the Kalman filter and why it does not require a guess on the initial state may be because it contains a large number of self-organizing neurons in the network. Via learning, these neurons provide many different directions to search for the optimal initial state.
In responding to the three variables, the launching position, velocity, and acceleration of the TBM, a 3D SOM is used for trajectory prediction. The SOM learning algorithm is organized as follows:
SOM Learning Algorithm: Predict an optimal initial state for an incoming TBM in a real-time manner using the measured position radar data.
Step 1: Set the stage of learning k = 0. Estimate the ranges of the possible initial position, velocity, and acceleration of the TBM, and randomly store the possible initial states wj(0) into the neurons, where j = 1, . . . , N3, N ×N ×N the total number of neurons
in the 3D space. Select neuron ˆj∗ in the center of the neuron space as the winning neuron.
Step 2: Send wj(k) into the dynamic model, described in Eqs.(2.4)-(2.5), to compute pj(k).
Step 3: For each neuron j, compute its output sj as the difference between the measured position data v(k) and pj(k):
Step 4: Update the weights of the previous winning neuron ˆj∗ and its neighbors within hˆj∗(k) using the following two rules:
If j∗ 6= ˆj∗, then wj(k + 1) = wj(k) + η(k)hˆj∗(k)(wj∗(k) − wˆj∗(k)) (2.13) If j∗ = ˆj∗, then wj(k + 1) = wj(k) + η(k)hˆj∗(k)(wj∗(k) − wj(k)) (2.14)
where η(k) is the learning rate described in Eq.(2.1) and hˆj∗(k) the neighborhood function in Eq.(2.3).
Step 5: Check whether the difference between wj∗(k) of the winning neuron j∗and wj(k) corresponding to every neuron j within hj∗(k) is smaller than a prespecified value ²:
maxj kwj(k) − wj∗(k)k < ², j ∈ hj∗(k). (2.15)
If Eq.(2.15) does not hold, let k = k + 1, and when k is smaller than a prespecified maximum value, go to Step 2; otherwise, the prediction process is completed and output the optimal initial state to the dynamic model to derive the TBM trajectory. Note that the value of ² is empirical according to the demanded resolution in learning, and we chose it very close to zero.