multivariate calibration, neural networks using all time points of the calibration
data were trained set and subsequently predicted the validation data set. The
root mean square errors of the crossvalidation and of the prediction of the
independent validation data set are listed in the first row of table
5. The concentrations of the test samples were predicted with a relative
RMSE of 3.32% for methanol and of 4.11% for ethanol. The fully connected networks
used for the calibration consisted of 53 input neurons, 4 hidden neurons and
1 output neuron. In order to improve the calibration, the parallel growing neural
network framework introduced in chapter 8 was applied to
the calibration data. Thereby the growing network algorithm was repeated 200
times for each analyte. In contrast to chapter 8, the frequencies
of the variables being selected after the first step of the algorithm are combined
for both analytes by summing up the individual frequencies of each analyte.
The corresponding plot (figure 60) shows that the
variables in three time intervals are prominent: The beginning of exposure to
analyte (10-45 s), when exposure to analyte has ended and exposure to synthetic
air starts again (130-140 s), and additionally hundred seconds after the start
of exposure to synthetic air (220-230 s). The first two intervals are easily
interpretable as in accordance with figure
15 the sensor responses for both analytes differ most during these time
intervals. The time interval around 220 s might be considered as reference signal
with practically no analyte remaining to compensate possible drifts of the baseline.
Number of Time Points
table 5: Relative RMSE in
% for the prediction by the neural networks using different time points.
figure 60: Ranking of the time
points after the first step of the parallel framework for both analytes together.
model obtained by a stepwise addition of time points (step 2 in figure
53) uses only the 3 time points 10 s, 30 s and 130 s (topology 3-4-1). According
to table 5 the prediction errors are significantly
lower compared with the fully connected networks for both, the crossvalidated
calibration data and for the validation data with excellent low errors of 1.97%
for methanol and 2.37 % for ethanol. The third time interval is not used
by this network and consequently seems not to contain significant additional
information not covered by the 3 other sensor signals. Using reproduced measurements
of the single analytes in the same concentration range, a standard deviation
of the signals was calculated with 0.62% for methanol and 0.98 % for ethanol.
These errors are caused by the noise of the spectrometer, inaccuracies of the
gas mixing station and fluctuations of the temperature and thus also exist for
the measurements of the mixtures. The only moderate increase of the errors for
the calibration and prediction shows the potential of the calibration and variable
selection by the growing neural network framework. In figure
61 the true-predicted plots of the validation data and the calibration data
are shown. The predictions of all concentrations are characterized by very small
standard deviations and by the absence of systematic errors.
figure 61: Predictions of the
calibration and validation data by the optimized neural networks using only
three time points.