## 3 Results
### Results of the Preliminary Data Analysis
#### 3.2.3 Results of Neural Network Training  
According to the generally accepted neural network design technology, the entire general population is divided into training, testing and confirming sets in the ratio of 70:20:10% [52]. In our problem, we neglect a strict formal approach to determining the percentage ratio between training, testing and confirming sets due to a fairly small number of examples, which is acceptable [63], thereby dividing the entire initial set only into training and testing in the ratio of 90:10%. We performed multiple training and testing of neural networks for diagnosing Alzheimer's disease on limited raw data using one-layer and two-layer neural networks. Several results of one-layer neural networks training and testing are shown below (Figure 12, Table 6). Figure 11: The dependence of the number of neurons between the two layers (\(x\)—first hidden layer, \(y\)—second hidden layer). Figure 10: Schematic representation of the structure of a fully connected two-layer neural network and the number of unknown parameters. Figure 12a-e shows five neural networks, distinguished by five neurons. Figure 12f shows the training and testing errors for the neural network HL1-27 with the 27 neurons satisfying inequality (14). Figure 12 shows that the quadratic error becomes less than 10% in all cases. Therefore, we should compare the results of the training with the other criteria: Q\({}_{1}\)-average quadratic relative (%); Q\({}_{2}\)-average relative error; Q\({}_{3}\)-average quadratic relative (Test, %); Q\({}_{4}\)-average relative error (Test); Q\({}_{5}\)-quadratic train error; Q\({}_{6}\)-quadratic test error; Q\({}_{7}\)-R\({}^{2}\) train set; Q\({}_{8}\)-R\({}^{2}\) test set (Table 6). Comparing the performance by all the criteria in Table 6, it can be seen that the proposed neural network HL1-27 is better on almost all the criteria. At the same time, the best neural network training results from the multiple experiments we showed on both Figure 12a-e and the first seven rows of Table 6. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **Neural Network** & **Q\({}_{1}\)** & **Q\({}_{2}\)** & **Q\({}_{3}\)** & **Q\({}_{4}\)** & **Q\({}_{5}\)** & **Q\({}_{6}\)** & **Q\({}_{7}\)** & **Q\({}_{8}\)** \\ \hline HL1-5 & 1 layer with 5 neurons & 17.068 & 12.582 & 24.517 & 21.047 & 2.913 & 6.011 & 0.849 & 0.744 \\ HL1-10 & 1 layer with 10 neurons & 11.557 & 8.718 & 17.329 & 13.899 & 1.334 & 3.003 & 0.930 & 0.872 \\ HL1-15 & 1 layer with 15 neurons & 9.268 & 7.000 & 21.897 & 18.348 & 0.859 & 4.795 & 0.955 & 0.795 \\ HL1-20 & 1 layer with 20 neurons & 5.829 & 4.422 & 21.262 & 18.988 & 0.034 & 4.521 & 0.982 & 0.807 \\ HL1-25 & 1 layer with 25 neurons & 2.047 & 1.440 & 21.827 & 16.692 & 0.042 & 4.764 & 0.997 & 0.797 \\ HL1-27 & 1 layer with 27 neurons & 1.987 & 1.392 & 21.138 & 14.462 & 0.039 & 4.468 & 0.997 & 0.809 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of a trained different fully connected one-layer neural networks. Figure 12: Changing of the square errors at training and testing depending on iterations of calculation synaptic coefficients at fully connected one-layer neural networks with different numbers of neurons on the hidden layer (hereinafter the following designation will be used HL1 stands for neural network with 1 hidden layer): HL1-5 has 5 neurons (**a**); HL1-10 has 10 neurons (**b**); HL1-15 has 15 neurons (**c**); HL1-20 has 20 neurons (**d**), HL1-25 has 25 neurons (**e**), and fully connected one-layer neural network with predefined number of neurons HL1-27 has 27 neurons (**f**). In our experiments, we noticed that the quality and speed of neural network learning could be improved by increasing the number of neurons. The neural network with pre-defined number of neurons HL1-27 (see Figure 12f) was trained rather quickly, and the results were more robust compared to variants HL1-5, HL1-10, HL1-15, HL1-20, which often over-trained the neural network.