algorithm framework was applied to the calibration data of the refrigerant data
set. Thereby 100 parallel runs of the GA were used with the same settings of
the parameters as described in section 7.1. The results
of the first step of the global algorithm are shown in figure
46. Thereby the ranking of the variables is shown as frequency of the variables
being present in the last population of the genetic algorithms. In the second
step, these variables enter the model according to their rank until the prediction
of the test data of a 20-fold random subsampling does not improve any more.
The iterative procedure stopped after the addition of 10 time points, which
are labeled in figure 46. As the labels are rounded
seconds of the time points, the most important time point "0" does
not represent the absolute beginning of the measurement, but 0.3 seconds after
the beginning of exposure to analyte.
figure 46: Frequency of the selection
of the time points for 100 parallel runs of the genetic algorithms. The 10 time
points selected by the algorithm are labeled additionally.
networks (10 input neurons, 8 hidden neurons, 1 output neuron, fully connected)
predicted the test data of the 20-fold random subsampling procedure (figure
45) with rather low rel. RMSE of 1.94% (R22) and 3.05% (R134a). The predictions
of the external validation data by these networks, which had been trained by
the complete calibration data set, were best of all methods used so far with
2.04% for R22 and 2.89% for R134a (see table
3 in section 7.4). Practically no gap between the
prediction of the calibration and validation data is noticeable indicating much
more stable models compared with the non-optimized networks. The predictions
of the validation data, which are shown in the true-predicted plots in figure
47, are not biased and hardly scattered.
figure 47: Predictions of the
validation data by neural networks optimized by the genetic algorithm framework.
of the predictions of the neural networks were further examined in respect to
the compositions of the predicted analyte concentrations. In figure
48, the absolute residuals of the predictions of the analyte concentrations
are plotted versus the corresponding analyte concentrations of the predicted
sample. The plot for R22 demonstrates that the absolute residual for the prediction
of R22 increases with an increasing concentration of R22, but is practically
randomly distributed along the axis representing the concentration of R134a.
For R134a the plot shows that the residuals of the predictions of R134 show
a higher dependency on the concentration of R134a than on the concentration
of R22. This means that the concentration of the interfering analyte does practically
not influence the prediction quality of the analyte of interest. Thus, it should
be possible that the system of the time-resolved measurements, the variable
selection and the calibration by neural networks can be extended to parallel
quantifications of even more analytes.
figure 48: Absolute residuals
of the predictions of the concentrations of R22 (top) and R134a (bottom) versus
the compositions of the corresponding samples.
use of many genetic algorithms is often limited by chance correlations of variables.
This can happen if variables are noisy, if the number of samples is limited
and if there are many variables to select. In that case, it can happen that
the GA models noise instead of information and consequently selects randomly
correlated variables. Therefore, a test similar to ,
was performed to investigate the robustness of the variable selection algorithm
proposed in this study. In this test, the number of variables is increased by
adding meaningless artificial variables, which contain only random numbers,
to the meaningful original variables. Then, the algorithm for the variable selection
is run using the increased amount of variables. A well performing algorithm
should not select any of the artificial random variables, which contain no meaningful
information. For this study, 40 random variables were added to the set of 40
original time points. The random variables were created by uniformly distributed
random numbers with the same variation as the original time points. The genetic
algorithm framework was used for this extended data set same way as described
before except of two parameters adapted for the increased data set: The population
size was increased to 100 resulting in about 120 generations until the convergence
criterion was reached and the parameter a was set to 1, which resulted in
approximately 6 variables being selected in single runs of the GA.
ranking after the first step of the algorithm is shown in figure
49. It is obvious that all random time points are ranked very low and no
random variable can be found among the most important 18 time points. The parallel
runs of multiple GA with different combinations of test and calibration data
seem to prevent the selection of randomly correlated variables whereas single
runs of the GA selected random variables evident by non-zero frequencies of
random variables in figure 49. Additionally, the
left side of figure 49 looks very similar to figure
46 demonstrating the reproducibility of the ranking of meaningful variables
when running the global algorithm repeatedly. The top 11 time-points are ranked
the same way as for the algorithm applied to the original data (figure
46). Consequently, the same 10 variables are selected in the second step
of the algorithm demonstrating the reproducibility of the selection of the variables
by the genetic algorithm framework.
figure 49: Frequency of selection
for 40 time points and 40 additional random variables (R1 R45) after the first
step of the genetic algorithm framework.