## 3 Results
### Results of the Preliminary Data Analysis
#### 3.2.3 Results of Neural Network Training
We propose to consider nine variations of two-layer neural networks with different numbers of neurons on first and second hidden layers. The network structures are chosen so that the number of neurons on the first and the second hidden layer form a cross of gridlines with steps of five neurons--selected combinations of neuron number on the first and second layers are highlighted by black points (Figure 13, points 1-9). We also propose to consider several variations of neural networks whose structures are comparable to dependence (see, Figure 11). Selected eight combinations of neuron number on first and second are highlighted by red points (Figure 13, points a-h). This is necessary to assess the quality and ability to recognize and predict our neural network, the structure of which is determined according to the decision's root and convolution matrices. The dotted line in Figure 13 divides all neural networks into two types: in the upper left corner, there are neural networks with fewer neurons on the first layer than on the second layer. In the lower right area, on the contrary, there are neural networks with more neurons on the first layer than on the second layer. The network HL2-10-10 refers to both a set of nine neural networks (see Figure 13, point 5) chosen by varying the number of variables on the first and second layer of five neurons (see Figure 13, black points 1-9), and a set of eight predefined neural networks (Figure 13, point d) that contain a number of synaptic coefficients comparable to a decision-root-based neural network (see Figure 8). Therefore, out of nine neural networks, Figure 14 and Table 7 show the training and testing results of eight networks: HL2-5-5-5 (Figure 14a); HL2-5-10 (Figure 14b); HL2-5-15 (Figure 14c); HL2-10-5 (Figure 14d); HL2-10-15 (Figure 14e); HL2-15-5 (Figure 14f); HL2-15-10 (Figure 14g); and HL2-15-15 (Figure 14h). Figure 13: Selected two-layers neural networks with different numbers of neurons on first and second hidden layers (hereinafter the following designation will be used–HL2 stands for two hidden layers, the next number after the hyphen stands for the number of neurons on the first hidden layer, and the last number stands for the number of neurons on the second hidden layer, respectively). Figure 14: Changing of the square errors at training and testing depending on iterations of calculation synaptic coefficients at two-layers neural networks with different numbers of neurons on the hidden layers: HL2-5-5 (**a**); HL2-5-10 (**b**); HL2-5-15 (**c**); HL2-10-5 (**d**); HL2-10-15 (**e**); HL2-15-5 (**f**); HL2-15-10 (**g**); HL2-15-15 (**h**). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **Neural Network** & \(\mathbf{Q_{1}}\) & \(\mathbf{Q_{2}}\) & \(\mathbf{Q_{3}}\) & \(\mathbf{Q_{4}}\) & \(\mathbf{Q_{5}}\) & \(\mathbf{Q_{6}}\) & \(\mathbf{Q_{7}}\) & \(\mathbf{Q_{8}}\) \\ \hline HL2-5-5 & 2 layers with 5 neurons on first layer and & 4.132 & 2.755 & 16.518 & 12.429 & 0.145 & 2.210 & 0.996 & 0.884 \\  & 2 layers with 5 neurons on first layer and & 4.003 & 2.922 & 17.234 & 11.922 & 0.160 & 2.970 & 0.992 & 0.873 \\ HL2-5-10 & 2 layers with 5 neurons on first layer and & 4.581 & 3.163 & 17.720 & 15.341 & 0.210 & 3.140 & 0.989 & 0.866 \\ HL2-10-5 & 2 layers with 10 neurons on first layer and & 0.715 & 0.480 & 13.094 & 8.362 & 0.005 & 1.714 & 0.999 & 0.934 \\ HL2-10-15 & 2 layers with 10 neurons on first layer and & 0.775 & 0.458 & 16.823 & 9.647 & 0.006 & 2.830 & 0.999 & 0.888 \\ HL2-15-5 & 2 layers with 15 neurons on second layer & 0.665 & 0.479 & 9.918 & 7.110 & 0.004 & 0.797 & 0.999 & 0.958 \\ HL2-15-10 & 2 layers with 15 neurons on first layer and & 0.904 & 0.548 & 14.145 & 10.509 & 0.007 & 1.621 & 0.999 & 0.915 \\ HL2-15-15 & 2 layers with 15 neurons on first layer and & 1.010 & 0.654 & 12.276 & 9.647 & 0.008 & 1.221 & 0.999 & 0.936 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results of a trained different fully connected two-layer neural networks.