## 3 Results and Discussion
### Validation of the Model and Methods for Determining Emotional States  
First, the method of the geometric interpretation of facial expressions was applied to the training dataset. The input data of the method were 110 unique images of human faces obtained from the ADFES dataset. As a result of applying the method, the matrix of normalized values \(\textbf{X}^{\prime}=\left(x^{\prime}{}_{ijk}\right)\); \(i=\overline{1,7}\)--facial features of a person; \(j=\overline{1,110}\)--objects of a training dataset; and \(k=\overline{1,5}\)--researched emotions was received. Preparation of weights was conducted according to the hyperplane classification of emotional state by facial expressions. The input data of the method were a matrix of normalized values \(\textbf{X}^{\prime}=\left(x^{\prime}{}_{ijk}\right)\), \(i=\overline{1,7}\), \(j=\overline{1,110}\), and \(k=\overline{1,5}\). The first step in applying the method of hyperplane classification was the mapping of the matrix in the visualization of normalized values \(\textbf{X}^{\prime}=\left(x^{\prime}{}_{ijk}\right)\in\mathbb{R}^{7}\), \(i=\overline{1,7}\), \(j=\overline{1,110}\), \(k=\overline{1,5}\) in two-dimensional space, i.e., \(\mathbb{R}^{7}\rightarrow\mathbb{R}^{2}\). To do this, we solved the optimization  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \multicolumn{5}{c}{**Sadness**} & \multicolumn{5}{c}{**Anger**} \\ \hline
**Face** & **Eyes** & **Eyebrows** & **Lips** & **Face** & **Eyes** & **Eyebrows** & **Lips** \\ \hline
0.5 & 0.2 & 0.5  
problem according to the evolutionary algorithm. To effectively find the values of \(\mathbf{W}\) weights, the ADFES training set of 110 human faces was reduced to a subset of fifty-five faces, i.e., eleven models were left, each of which depicts five emotions: Anger, Fear, Joy, Neutral, and Sadness. As a result of solving the optimization problem, matrix \(\mathbf{X}^{\prime}=\left(x^{\prime}{}_{ijk}\right)\in\mathbb{R}^{7}\), \(i=\overline{1,7}\), \(j=\overline{1,110}\), was mapped to matrix \(\mathbf{X}^{*}=\left(x^{*}{}_{ijk}\right)\in\mathbb{R}^{2}\), \(i=\overline{1,2}\), \(j=\overline{1,55}\), \(k=\overline{1,5}\). Reflection \(\mathbb{R}^{7}\rightarrow\mathbb{R}^{2}\) for five emotional states, Anger, Fear, Joy, Neutral, and Sadness, is visualized in Figure 11.  
From Figure 11, we see that the values of matrix \(\mathbf{X}^{*}=\left(x^{*}{}_{ijk}\right)\in\mathbb{R}^{2}\), obtained by the method of the geometric interpretation of facial expressions, were stably grouped by different emotional states. Figure 11 also shows a dividing line that was manually drawn on the coordinate plane, which clearly distinguishes the target emotional state class (in this case, the emotional state \"Fear\") from all other emotion groups. This result confirms the ability of the proposed model (1) to be used to classify emotional states. Later, this dividing line was used in the hyperplane classification method to build a hyperplane in \(\mathbb{R}^{7}\). The next step of the method of hyperplane classification was inverse reflection \(\mathbb{R}^{2}\rightarrow\mathbb{R}^{7}\). To do this, we solved the optimization problem according to the evolutionary algorithm. Here we found matrix \(\mathbf{X}^{*L}=\left(x^{*L}{}_{11}\right)\in\mathbb{R}^{7}\), \(i=\overline{1,7}\), for one dividing line built on seven points. As a result, the dividing line \(L^{2}\in\mathbb{R}^{2}\) was mapped to hyperplane \(L^{7}\in\mathbb{R}^{7}\) with coordinates \(\mathbf{X}^{*L}\). The constructed system (8) was laid out in the first line, according to block 5 of the method of hyperplane classification. So, due to applying a method of hyperplane  
Figure 11: An example of visualization of input data for five emotional states (Anger, Fear, Joy, Neutral, and Sadness) in two-dimensional space.