# The Method of Restoring Lost Information from Sensors Based on Auto-Associative Neural Networks
## 4 Results  
After developing the ANN training algorithm, the next research stage involves a numerical experiment simulating the lost information restoration in the failure event of one of the standard sensors of the TV3-117 TE. The input variables of this model for helicopters TE consist of the atmospheric conditions--\(\rho\) is the air density, \(P_{N}\) is the pressure, \(T_{N}\) is the temperature, and \(h\) is the flight altitude--alongside onboard parameters: \(T_{G}^{*}\) is the gas temperature in the compressor turbine front, \(n_{FT}\) is the free turbine rotor speed, \(n_{TC}\) is the gas generator rotor r.p.m., and \(G_{T}\) is the fuel consumption, calculated according to [37, 38, 39] and normalized to absolute values using gas-dynamic similarity principles. It is presumed that atmospheric conditions remain constant throughout the analysis. Detailed analysis and preprocessing of input data (Table 1) are described in [37, 38, 39]. At the first stage of input data preprocessing, the Fisher-Pearson test value [40] is calculated based on the observed frequencies \(m_{1}\), \(\ldots\), \(m_{r}\) (summing row by row the probability for each measured value result) and its comparison with the critical values of the Fisher-Pearson test \(\chi_{2}\) with the freedom degrees number \(r-k-1\). With the freedom degrees number \(r-k-1\) = 13 and \(\alpha\) = 0.05, the random variable \(\chi_{2}\) = 3.588 did not exceed the critical value of 22.362, which means that the normal distribution law hypothesis can be accepted and the samples are homogeneous. The conclusion about the study sample homogeneity is confirmed based on the Fisher-Snedecor test [41]. The obtained result analysis shows that the larger dispersion ratio \(\sigma_{\text{max}}^{2}\) to \(\sigma_{\text{min}}^{2}\), with the smaller one being 1.28 (less than the standardized critical value \(F_{critical}\) = 3.44). Therefore, the sample training is homogeneous. To guarantee the training and testing data sets' representativeness, a cluster analysis was performed on the initial data (Table 1), resulting in eight distinct class identifications (Figure 5a). After applying the randomization procedure, test and training (control) samples were formed in a 2:1 ratio (67 and 33%, respectively). The training and test sample (Figure 5b) clustering process [42, 43] showed that both of them, like the control sample, consisted of eight classes. The distances between the clusters practically coincide in each of the considered samples, which indicates the test and training samples' representativeness [37, 38, 39, 40]. Thus, the resulting sample sizes were as follows: training--256 elements, control--172 elements (67% of the training sample), and test--84 elements (the training sample is 33%). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Number**} & **Free Turbine** & **Gas Generator** & **Gas Temperature in the Compressor** & **Fuel** \\  & **Rotor Speed** & **Rotor r.p.m** & **Turbine Front** & **Consumption** \\ \hline \(1\) & 0.943 & 0.929 & 0.932 & 0.952 \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\ \(58\) & 0.985 & 0.983 & 0.899 & 0.931 \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\ \(144\) & 0.982 & 0.980 & 0.876 & 0.925 \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\ \(256\) & 0.981 & 0.973 & 0.953 & 0.960 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Training sample fragment (authors’ research, published in [37, 38, 39]). After the input data analysis and preprocessing, the ANN is directly trained. The ANN model is configured using the deep training libraries TensorFlow and Keras. In this model, the transmitter uses the ReLU activation function and the receiver uses the sigmoid function. The ANN is implemented using these typical deep learning Python libraries [44]. Table 2 shows the main ANN parameters for its training.