section, uniform fully connected feedforward neural networks are used for the
calibration of the refrigerant data set with the calibration data set used for
the training of the neural networks and the validation data set predicted by
the trained networks. Separate networks were used for both analytes. For the
training, the SCG algorithm with early stopping was used (see section
2.7.3). More details of the implementation of the networks and the learning
algorithms are described in the sections 2.7.2 and
2.7.3. Different network topologies were systematically
investigated by varying the number of hidden layers and the number of neurons
in the hidden layer. The best topology for both analytes evaluated by the lowest
crossvalidation error of the calibration data set are fully connected networks
with 6 hidden neurons organized in 1 hidden layer. Twenty networks of this topology
were trained using different random initial weights and the network with the
lowest crossvalidation error was used for the prediction of the validation data.
According to table 2, the validation data
were predicted with relative RMSE of 2.18% (R22) respectively 3.26% (R134a).
The predictions of the validation data (see figure 43)
do not show any type of nonlinearity in conformity with the Wald-Wolfowitz Runs
test and the Durbin-Watson statistics. Among the different calibration methods
used for the refrigerant data set so far, the neural networks can model the
nonlinear relationships between the concentrations of the analytes and the time-resolved
sensor signals best resulting in the lowest errors of prediction of the validation
data. Yet, the calibration data were predicted with significantly lower relative
RMSE of 1.47% (R22) respectively 2.62% (R134a) indicating an overfitting. According
to equation (13), the amount of 247
adjustable parameters of this network topology seems to be quite high for the
calibration problem with many redundant or unused links resulting in overoptimistic
low calibration errors but a decreased generalization ability. Thus, different
methods like a variable selection and an optimization of the network topology
are used in the next sections expecting that the reduction of the number of
parameters further improves the generalization ability.
figure 43: Predictions of the
validation data by fully connected neural networks.