## 3 Materials and Methods
### Formal Formulation of the Evaluation Problem
The average value is determined from the data obtained at different periods \(t_{u}\):  
\[\widetilde{ve_{\delta}}(t_{u})=\frac{1}{2}(ve(R)_{\delta}(t_{u})+\;ve(L)_{ \delta}(t_{u})),\;\left(u=\overline{1,l};\;\delta=\overline{1,5}\right) \tag{2}\]  
Since the data for various noise characteristics are measured in different units, normalization is necessary for comparison purposes. Such normalization is effectively carried out using fuzzy set theory and membership functions. This allows the measurements to be adapted to changing conditions and provides a more flexible and accurate interpretation of different noise levels based on their impact on the individual. In doing so, it enables the consideration of not only specific numerical values but also factors that influence the perception of and efficiency of response to these phenomena. For instance, a harmonic S-spline is used with parameters \(a_{\delta}\) and \(b_{\delta}\), defined by the following formula:  
\[we_{\delta}(t_{u})=\left\{ \begin{array}{ll}0,&\widetilde{v\epsilon_{\delta}}(t _{u})<a_{\delta};\\ \frac{1}{2}+\frac{1}{2}cos\bigg{(}\frac{b_{\delta}-\widetilde{v\epsilon_{ \delta}}(t_{u})}{b_{\delta}-a_{\delta}}\cdot\pi\bigg{)};&a_{\delta}\leq \widetilde{v\epsilon_{\delta}}(t_{u})\leq b_{\delta};\\ 1,&\widetilde{v\epsilon_{\delta}}(t_{u})>b_{\delta}.\end{array} \right. \tag{3}\]  
where \(we_{\delta}(t_{u})\in[0;1]\) are normalized estimates of the factors of psychophysiological load on the flight instructor with regard to noise measurement \(\delta=\overline{1,5}\) in different time periods \(t_{u}\) (\(u=\overline{1,t}\)). Moreover, the parameters \(a_{\delta}\) and \(b_{\delta}\) change depending on the noise measurement. The system analyst can change them. Experimental studies were conducted on real data [26; 27], and the parameters were adjusted to the following: \(a_{1}=81.46\); \(b_{1}=97.32\); \(a_{2}=\)0.00583; \(b_{2}=0.0138\); \(a_{3}=\)0.0311; \(b_{3}=0.0786\); \(a_{4}=0.845\); \(b_{4}=4.08\); \(a_{5}=4.84\); \(b_{5}=31.7\). The harmonic S-spline was chosen for normalization due to its ability to ensure smooth transitions between noise levels, corresponding to real psychophysiological perception. It considers the nonlinear nature of human response to noise and allows modeling of the saturation effect, where further increases in noise have little impact on the perceived effect. Furthermore, the S-spline is highly adaptable, integrates with fuzzy sets, and is widely used in psychoacoustic research, making it effective for assessing the workload of flight instructors. Model for Predicting the Individual Psychophysiological Workload of Flight Instructors in the Working Environment--\(M_{\text{ave}}\)  
As described above, the input data are normalized and come in different time periods \(t_{u}\): \(we_{\delta}(t_{u})\), \(we_{\delta}(t_{u}),\dots,\;we_{\delta}(t_{u})\in[0;1]\), (\(u=\overline{1,t}\); \(\delta=\overline{1,5}\)). The input data are directed to 1, and the maximum adverse load of the working environment is considered. Thus, the model's input for prediction includes normalized input estimates for the respective measurement periods during the flight phases. In this regard, there is an opportunity not only to derive average values and convolutional knowledge but also to track the dynamics and predict the level of individual psychophysiological workload in the work environment for the future period \(t_{l+1}\). In the scope of the current study, there is a direct relationship between the noise level and the psychophysiological workload of the flight instructor. Therefore, it is appropriate to use regression approaches. For example, simple linear regression looks for a linear relationship between variables, which may be effective when the dependencies in the data are indeed linear or nearly linear. In this case, without losing generality, simple linear regression is used.