## 2 Materials and Methods
### Application and Analysis
**(A)** Machine failure occurrence (-); **(B)** concrete cover thickness (cm); **(C)** machine gravity center distortion (mm); **(D)** angular frequency of vertical self-excited vibrations (s\({}^{-1}\)); **(E)** angular frequency of horizontal self-excited vibrations (s\({}^{-1}\)); **(F)** amplitudes of oscillation (\(\upmu\)m); **(G)** foundation area (m\({}^{2}\)); **(H)** foundation volume (m\({}^{3}\)). The vertical axis represents the cumulative explained variance, while the horizontal axis corresponds to the individual principal components. The first direction responded to 60% of the variation at normalized evolutionary distances, and including the three primary components, we covered 90% of the total explained variance. The remaining principal components contributed with progressively declining variance. We think that this minor variation was negligible. To find an optimal neural network architecture, we first adopted an initial architecture and then tried to optimize it. The ANN that we propose has eight input and scaling neurons, eight principal components, three hidden layers, four perceptron neurons, and one probabilistic and output neuron. Figure 5 presents the adopted, optimal ANN architecture. Figure 4: Cumulative explained variance for the principal components (CEV—cumulative explained variance). Figure 5: The architecture of the ANN. The figure shows the network architecture, which includes the following parts: input neurons ([... ]), scaling neurons (green), principal components (blue), perceptron neurons (red), probabilistic neuron (yellow), and output neuron ([FFO]). Abbreviations: [MF]—machine failure occurrence; [CT]—cover thickness; [GC]—machine gravity center distortion; [VA]—angular frequency of vertical self-excited vibrations; [VB]—angular frequency of horizontal self-excited vibrations; [VC]—amplitudes of oscillation; [FA]—foundation area; [FV]—foundation volume; [FFO]—foundation failure occurrence. We used the Broyden-Fletcher-Goldfarb-Shanno algorithm to find a suitable training rate [26; 27; 28; 29; 30; 31] and the Brent method [32; 33; 34; 35] to obtain a quasi-Newton training direction step. We evaluated the influence of each input variable by eliminating training input selectively and examining the output results. The higher the input contribution nominal value was, the more substantial this variable was. The lower the input contribution nominal value was, the lower the contribution it had. We found out that the most influential were the foundation volume, machine gravity center distortion, the angular frequency of vertical self-excited vibrations of the machine, and concrete cover thickness. Figure 6 shows an evaluation of the input contributions. We introduced input selection by growing algorithm [36; 37; 38; 39]. To find out the optimal number of artificial neurons, we used an order selection algorithm [40; 41]. To evaluate the adopted approach and study the loss, we prepared a receiver operating characteristic (ROC) curve. This method is a graphical illustration of how well the classifier discriminates between the two different classes. This capacity of discrimination is measured by calculating the area under the curve (AUC) [42; 43; 44]. Figure 7 presents the receiver operating characteristic curve. Figure 6: Input contribution analysis. Figure 7: Receiver operating characteristic (ROC) curve graph. An ROC curve represents the calculated sensitivity and specificity for a group of different binary classifiers gathered by wavering the threshold value. The threshold value increases along with the output probability of the testing instances. The adopted classifier is random. The vertical axis corresponds to sensitivity (true positive rate), while the horizontal axis refers to specificity (false positive rate). The more the chart deviates to the upper left corner, the larger the AUC, and the better the model performs. In Figure 6, we can see that the graph is not entirely in the upper left corner, but is tilted toward it.