The small
number of 3 variables allows an efficient sensitivity analysis [95],[101]
to give a little insight into the "black box" of neural networks.
Thereby the values of two time points were systematically varied between the
measured minimum and maximum sensor responses of these time points with the
third time point kept constantly in the middle of the corresponding range of
the measured sensor responses. The trained neural nets described above were
fed with the values of the 3 time points. The concentrations calculated by the
neural nets are plotted versus the two varying time points in figure
62.
The top
row of this figure shows the prediction of the concentrations of methanol (left)
and ethanol (right) depending on the sensor responses of the time points 10
s (x-axis) and 30 s (y-axis). The prediction of methanol is determined
by the ensemble of both time points. This is in accordance with figure
15, which demonstrates that the sorption process of methanol has come to
a steady state after 10 s. Hence, a high sensor response after 10 s caused by
a high concentration of methanol automatically induces a high sensor response
after 30 s. The top right plot of figure 62 shows
that the prediction of ethanol is practically not influenced by the time point
10 s explainable by the variance of this time point being mainly caused by the
sorption of methanol (see figure 15).
Nevertheless, the prediction of ethanol is nearly linearly correlated with the
sensor response of the time point 30 s. The bottom row of figure
62 shows the prediction of the concentration depending on the signal of
the time points 10 s (x-axis)
and 130 s (y-axis). The plane
parallel to the y-axis demonstrates that the prediction of methanol is practically
independent from the time point 130 s. However, the prediction of ethanol highly
depends on the sensor response of the time point 130 s. The plots of figure
62 top right and bottom right are nearly identical except of the higher
dynamics of time point 130 s.
figure 62: Predicted concentrations
of the sensitivity analysis versus the sensor responses of the two time points
varied.
In summary,
it may be said that the prediction of methanol depends on the combination of
the time points 10 s and 30 s whereas the prediction of ethanol depends on the
time points 30 s and on the time points 130 s. The similar dependencies of the
predictions of ethanol on the time points 30 s and 130 s indicate that the time
point 130 s could be rendered unnecessary by calculating the concentrations
of methanol using the sensor response of the time point 10 s and by calculating
the concentration of ethanol by the ratio of the sensor responses after 10 s
and 30 s. Thus, neural networks of the topology 2-4-1 were
trained using only the time points 10 s and 30 s. These small networks perform
quite well predicting both analytes with relative errors of only 2.22% and 2.70%
(see table 1). The only small deterioration
of the predictive ability is overcompensated by the fact that the measurement
time can be reduced from 130 s to 30 s. In addition, the shortening of the time
with the polymer being exposed to the analyte results in the sorption of less
ethanol into the polymers. Therefore, the desorption of ethanol needs less time
to be completed additionally shortening the time needed between 2 measurements.
Ph. D. Thesis
