Modeling Methodology: Dimension Reduction and Resampling Methods - Neural Networks: Methodology and Applications - page 229

Fig. 3.12. Generator φ 12 EMBED

The analysis of all results given in Figs. 3.11 and 3.12 illustrate the salient
features of the NeMo method,

• the generalization error is estimated accurately, even in complex cases

(large number of inputs + few examples);

• the bootstrap is used to automate the adjustment of the network to the

data by monitoring the termination of training.

Figures 3.11 and 3.12 show estimates of the generalization error very close
to the real values. The low error values correspond to training cycles performed
from databases with enough examples. For these cases, the estimated error on
the Y-axis is virtually equal to the actual error on the X-axis.
A slight overestimate should be noted for 4 cases out of 75 between val-
ues 0.01 and 0.02 for φ 8 EMBED (Fig. 3.11) and less precision for the more
complex φ 12 EMBED (Fig. 3.12). In the latter case, regression concerns a rela-
tion from

12 to

with a maximum of 1500 points to represent the relation.
There is an overestimate of the error for the low values and a underestimate
for values greater than 0.2. Nevertheless, and in spite of the large dimensions
of the input spaces, the relation of

R R

12 in R is correctly modeled using a few

R

hundred examples.

3.6.6 Conclusions

The above illustrative example shows that