Page 111 - Innovations in Intelligent Machines

UAV Path Planning Using Evolutionary Algorithms 101
selection of centers, which is the most generalized form of a RBFN since the
location of the centers undergo a supervised learning process along with the
rest of the network’s free parameters.
The standard process is to select the input vectors in the training set as
RBFN centers. In this case results M=NR, where NR is the number of train-
ing data. For large training sets (resulting in large M values) this choice is
expected to increase storage requirements and CPU cost. Additionally, the
M=NR choice could lead to over-fitting and/or bad generalization of the
network. The proposed solution [49, 45] is the selection of M<NR and con-
sequently the search for sub-optimal solutions, which will provide a better
generalizing capability to the network.
As far as training is concerned, there are two different approaches, the
direct and the iterative learning. In our case the first approach was adopted.
The direct learning process is based on a matrix formulation of the governing
equations of RBF network. The presentation of the network with the NR input
patterns allows the formulation of a (NR × M) matrix H, which becomes
square in the special case when NR=M. Each line in the interpolation matrix
H corresponds to a learning example and each column to a RBFN center. The
output unit values result in the form of the matrix product:

H (NR × M) w (M × 1) = yy (NR × 1) , (29)
where yy is the desired output vector as provided by the training dada set,
and w is the synaptic weights vector, which consists of M unknowns to be
computed.
A possible way for inverting H is through the Gram-Schmidt technique.
H is first decomposed as
H = QR, (30)
with Q and R being (NR × M) and (M × M) matrices respectively, where R
is upper triangular and
T
Q Q = diag (1, 1,... , 1) . (31)
After the computation of Q and R matrices, the weights vector can be com-
puted using back-substitution in

T
R(M × M)w(M × 1) = Q (M × NR)yy(NR × 1) (32)
There are several reasons why one should choose RBFN as the approxi-
mation model; Haykin [49] offers comparative remarks for RBFNs and Multi-
layer Perceptrons (MLPs). However, the main reason for choosing RBFNs is
their compatibility with the adopted local approximation strategy, as it is
described in the subsequent section. The use of relatively small numbers of
training patterns i.e. small networks, helps creating local range RBFNs. That
in turn allows the inversion of matrix H to use almost negligible CPU time
and the approximation error is kept very small. We should keep in mind that