117 
The i-th input vector with d-dimension is given to all 
input neurons. The j-th input neuron has the parameter. 
This parameter is given by the mean vector and the 
cocovariance matrix, where the cocovariance matrix is the 
matrix whose kl-th element is 
2 (xj-md Z,'0c f - m,)> 
to the Lyapunov function. 
3. REPRODUCTIVE CRITERIA OF NETWORK 
dE(w,</>) = ^ dE(w,</>) d Wj < Q 
dt d\V, dt 
PARAMETER 
given by the inverse matrix. The j-th input neuron outputs 
for the input vector, where the subscript denotes the 
transpose of the matrix. The input neuron does such 
output is called the radial basis function. The output is 
propagated to the output neurons through the synaptic 
weight. These are added up in the output neurons and 
We further extend CRBFN for adapting to the change of 
network environment. For this purpose, reproduction of 
parameter is considered. The update rule of parameter is 
given by 
5 F ,im) 
d m m 
is outputted. The function approximation by neural 
network is regarded as that the nonlinear function is 
approximated by the output of the network. For this 
purpose, the 
approximation by the RBFN is realized by decreasing the 
total of squared error function 
E(w, (p)m~ z E( Xi , w, </>) 
where 
E (x, w ’</>) = 
is the squared error function related with the each output 
neuron. That is, RBFN must get the synaptic weights, 
parameters of the j-th radial basis function by learning. 
A *m l = A m J 
instead of equation for CRBFN, where the mean vector 
depending on each input vector. If the initial value satisfy 
then the update rule for the parameter of RCRBFN is just 
same to the update rule for the parameter of CRBFN. It is 
found that the parameter which is given by 
Z?0t ’ ntjvfoc, ~ m At ) Wx,)' s( *Xi ’ ntm)} = 0 
The teaching signal can be detected as the convergence of 
the parameter. 
The algorithm of RCRBFN 
We have already proposed CRBFN, which is superior to 
RBFN. The learning algorithm of CRBFN is as follows; 
A 
A 
dE(w,(f)) 
w, = - £ . . 
k dE(w,tf>) 
m=- £ —^— 
d m. 
where the coefficient takes 1 or -1 depending on the sign 
STEP 1 
The synaptic weight is updated by equation, the 
parameters by equation and equation. 
STEP 2 
If the total of squared error function then stop 
the learning, otherwise go to next step. 
STEP 3 
For all radial basis function, the parameter is 
updated by equation when the coefficient 
increase from 0 gradually. 
A wj = £ (a j M - £ y v M /u t Wk) Wj 
of the synaptic weight is given by 
We showed that the squared error function is equivalent 
dE(w,</>) 
C7, 
= -8 
cr, 
STEP 4 
If the value satisfying the fixed point increases 
by bifurcation, the j-th radial basis function is 
reproduced as the p-th radial basis function. 
Then the synaptic weight, the parameters take 
over ones of the j-th radial basis function, the 
parameter is given by the added point newly by 
bifurcation. Return to step 1.