In
recent literature, several approaches are reported to solve the different problems
of single runs of GA:

1.Massart and Leardi [98],[256]
use a very refined algorithm for the variable selection, which is based on parallel
runs of many GA with different combinations of test and calibration data. Then
a validation step is performed to find the best variable subset. The GA is a
hybrid algorithm using a stepwise backward elimination of variables to find
the smallest possible subset of variables. Although this approach is very promising,
Jouan-Rimbau et al. [255] showed that this algorithm
is still partly subject to chance correlation.

2.In [99] Leardi et al. use
100 runs of GA with the same calibration and test data sets. The final model
is obtained by adding systematically the variables, which are ranked according
to the frequency of selection of the GA runs and by using the combination with
the smallest error of prediction. In [97] this algorithm
is modified by the different GA runs learning from each other.

3.In [126] the predictions
are averaged by several models found by different GA runs. Yet, the average
prediction was not better than the prediction by a single model.

4.In [254] 10 runs of GA are
performed by using different calibration and test data subsets. The final model
uses all variables, which were selected at least 5 times, whereby this limit
is rather arbitrary.

The genetic
algorithm framework proposed in this work picks several elements of the studies
mentioned above and is presented in the flow diagram in figure
44. The framework can be divided into three steps. The first step consists
of multiple parallel runs of the GA presented in section
2.8.9 and in section 7.1 using different calibration
and test data subsets (yellow boxes in the flow diagram). Variables, which are
represented higher than average in the final population of each GA run, are
collected over all GA runs and are ranked according to the frequency of appearance
in the final populations. The second step of the framework finally selects the
variables in an iterative procedure by adding the variables to the neural network
model according to their rank in a stepwise procedure. The neural network is
evaluated by the use of different calibration and test data subsets (green boxes
in figure 44). The RMSE of prediction of all test
data sets is compared with the RMSE of the previous model. If the RMSE is lower
(see section 10.2), the last variable is accepted and
the procedure is repeated adding the next important variable until the predictions
are not improved any more.

In the third step, the final
neural network topology is determined. First, the number of hidden neurons of
a single hidden layer is optimized in an iterative procedure, which is shown
in figure 45. Starting with fully connected neural
networks with 1 hidden neuron additional fully connected neurons are added until
the error of prediction of the test data doesn't improve any more, whereby the
l different test data subsets are generated
by a data subsampling procedure. Finally, this neural network topology is trained
with the complete data set several times, and the neural net with the smallest
error of crossvalidation should be used as final optimized model and should
be validated by an external data set not used during the complete variable selection
algorithm.

In all
three major steps of the framework, the complete data set is split several times
into a calibration (75 %) and a test (25 %) subset, which was done by a random
subsampling procedure (see section 2.4) resulting in
rather pessimistic predictions of the test data. Consequently, according to
expression (16) models are
preferred, which are more predictive and which yield a better interpolation.

As already
stated in section 2.8.5, the choice of ain the fitness
function (16) influences the numbers of variables being selected during
each run of a GA. A too high value of a ignores partly the accuracy of the neural
nets and ends in only few variables being selected. Consequently, there might
be too few variables selected in the first step to be added to the neural net
in the second step. This problem can be recognized by all variables with a ranking
higher than "0" being used for the neural net in the second step.
On the other side, a too low value of a results in too
many variables being selected. This can be detected by the absence of a differentiation
of the variables in the ranking. An empirical way to select an optimal a
is based on running a single GA with different values of a and on choosing
that a, which results in the selection of the number of variables
expected to be needed for the calibration. A good choice to start with is setting
a to "1" for these single runs
of the GA. Yet, preliminary studies showed that the parallel runs of the GA
make the framework quite robust towards the choice of a and to the population
size, which is suggested to be set to the number of variables to select from.
Although the framework seems to be complex on the first sight, this robustness
renders the algorithm quite user-friendly.

figure 44:
Flow chart of the genetic algorithm framework.

figure 45: Optimization of the
number of hidden neurons. This figure is a detailed flow chart of the blue box
of the genetic algorithm framework shown in figure 44.