9.1.4.   Brute Force Variable Selection

The low number of 3 variables selected by the growing neural network framework for an optimal model also allows a comparison with the variable selection by a brute force method. According to expression (14) , there are 140556 different realizations for selecting 3 variables out of 53. For all these realizations neural networks (fully connected with 4 hidden and 1 output neuron) were trained using the calibration data set and then the mean error of the prediction of the validation data set for both analytes was calculated (similar to the prediction error shown in figure 4 for 2 variables of the refrigerant data set). This procedure was repeated 25 times using different initial weights for the neural networks (a higher number of runs is desirable but limited by the computing time). Among the 25 best networks in respect to the lowest mean prediction error of the validation set, only 1 combination of 3 variables was selected more than once. This selection was the same than the 3 time points selected by the growing networks, whereas the 23 other best selections all were different. Thus, the best combination of the variables highly depends on the initial weights of the training, which is an indication of a high correlation of the variables rendering many realizations of 3 variables very similar. Nevertheless, the 3 variables selected by the parallel growing network framework were the most frequently individually selected variables among the 25 selections by the brute force method confirming the variable selection quality of the parallel growing network framework.