# The Method of Restoring Lost Information from Sensors Based on Auto-Associative Neural Networks
## 4 Results
At the same time, in the ANN (Figure 1),  
\[F_{2}(V):F_{1}^{-1}(X)+\varepsilon^{n}, \tag{12}\]  
where \(\varepsilon^{n}\) = (\(\varepsilon_{1}\), \(\varepsilon_{2}\),..., \(\varepsilon_{n}\)) is the training error. It is considered that the ANN has completed training if \(\varepsilon^{n}\) = \(\sigma^{n}\) where \(\sigma^{n}\) = (\(\sigma_{1}\), \(\sigma_{2}\),..., \(\sigma_{n}\)) insurmountable training error [45]. The hidden layer \(V\) of a smaller dimension amplifies the values that allow for the inverse mapping of \(F_{2}(V)\) with the smallest error \(\varepsilon^{n}\)\(\rightarrow\)\(\sigma^{n}\) and attenuates the values that introduce noise. Then \(F_{2}(V)\) at \(\varepsilon^{n}\) = \(\sigma^{n}\) is the main component of the set \(\mathbf{X}^{n}\) (Figure 6). Figure 6: The main component diagram for the set \(\mathbf{X}^{n}\) on a two-dimensional plane where black points are \(F_{2}(V)\) and white points are \(\mathbf{X}\): (**a**) Experiment I; (**b**) Experiment II; (**c**) Experiment III; (**d**) Experiment IV; (**e**) Experiment V; (**f**) Experiment VI (authors’ research, based on [45]). The test event \(y_{k}\) classification is possible on the error plane due to the deviation evaluation from the main component \(F_{2}(V)\). Since the training was performed on the \(\mathbf{Y}^{0}\) set, the deviation for \(\mathbf{Y}^{1}\) events is expected to be higher. On the error plane, \(\mathbf{Y}^{0}\) events will be concentrated in the zero zone, and \(\mathbf{Y}^{1}\) will be distributed throughout space. Therefore, classification on the error plane will be simpler than on the feature plane (Figure 7). Classification in the error space (Figure 7) can be implemented by separate methods, in particular, a threshold condition (Figure 8) of the form [45]:  
\[\left\{ \begin{aligned} & 1,\exists\varepsilon_{j}>\sigma_{j},e_{j} \in\varepsilon_{k},\\ & 0,\forall e_{j}>\sigma_{j},e_{j}\in\varepsilon_{k}.\end{aligned} \right. \tag{13}\]  
The errors in the second kind are class \(\mathbf{Y}^{0}\) outliers. A reduction in the emissions number is possible due to an increase in class \(\mathbf{Y}^{0}\) data--the main component of the set will be formed more accurately. At the same time, it is easier to increase the data amount on the correct values of TV3-117 TE parameters than on abnormal ones. It should be noted that Figures 7 and 8 show two-dimensional planes. When classifying in a multidimensional form, the error is much lower. The overtraining risk is extremely high when training ANNs of type \(F:\;X_{i}\to y_{i}\) on a similar set. Since the \(\mathbf{Y}^{1}\) data are much smaller than \(\mathbf{Y}^{0}\), the separation surfaces can be distributed as in Figure 9 [45]. Figure 8: Classification diagram on the error plane where the black points are \(\mathbf{Y}^{0}\) and the white points are \(\mathbf{Y}^{1}\) (authors’ research, based on [45]). Figure 7: The feature set diagram for one of the two-dimensional planes: (**b**) errors set for the plane (**a**) where the black points are \(\mathbf{Y}^{0}\) and the white points are \(\mathbf{Y}^{1}\) (authors’ research, based on [45]). The ANN loss function dynamics [48] are analyzed at the stages for the model training and validation (Figure 10). This analysis provides a graphical representation of the change in loss function over time (training epochs number) for the validation and training data sets. Figure 10 allows us to evaluate the training model process and its generalization ability. The obtained results of the loss function dynamics determined at the model validation and training stages allowed us to draw the following conclusions:  
1. The loss function dynamics made it possible to evaluate the model training effectiveness. A decrease in the loss function on the training data set indicates that the model is successfully training and identifying patterns in the data. The loss function on the validation data set also decreases at the same time, which indicates the elimination of the overfitting effect. 2.