The Development of Neural Network Based System Identification ...

5.3 OFF-LINE BASED SYSTEM IDENTIFICATION FOR HMLP NETWORK 137
Table 5.5
Summaries of Error Statistics of HMLP Network Model
Error Statistics
System Responses RMSE RMSE (%) R 2
One-Step Ahead Prediction
p 0.0360 12.0944 0.9852
q 0.0082 3.6340 0.9984
5-Step Ahead Prediction
p 0.1199 28.7239 0.9175
q 0.0718 23.9955 0.9424
The robustness of the HMLP network’s model structure against perturbation of
weights is given in Table 5.6. Table 5.6 shows the prediction results of optimal network
structure for HMLP network with addition of Gaussian distributed random noise to the
optimal weights. The optimal weights is corrupted by random noise with zero mean
and standard deviation s of 0.01, 0.1, 0.3, 0.5, 0.7 and 0.9. For each noise levels, 300
sets of weights around the optimum weights are generated, which results in average
RMSE and R 2 values shown in Table 5.6. The average RMSE on the test data set for
various noise levels indicate that an exceptional prediction performance is achieved up
until random noise with standard deviation s = 0.7 added to the optimal weights.
Using the 300 set of weights generated, the 95% confidence intervals can be constructed
for the optimal weights using the standard statistical inference method. The
average range of the upper and lower output performance (NMCIW) for each noise
level case is also given in Table 5.6. As can be seen from the result, the weights set
added with s = 0.9 random noise provide a wide average confidence interval (23.9983%)
compared with the range of measurement between −1 rad/s to 1 rad/s. This indicates
the imprecise and high level of uncertainty to produce predictions that represent the
real output values.