## 4 Conclusions  
This study proposes a novel method for detecting sudden shifts in emotional states. This method utilizes a unique model to interpret facial expressions related to emotional states, a geometric interpretation technique for facial areas, and a hyperplane classification method. Distinguished by its simplicity and transparency in feature extraction and classification, the proposed technique enables the accurate localization of groups exhibitingsudden emotional changes. These groups are identified from the external photo and video surveillance system, enhancing the technique's reliability. Our contribution is designed to accurately detect changes in emotional states through facial expressions, facilitating identifying abnormally behaving groups within a crowd--crucial for outdoor surveillance systems adhering to security protocols.  
Experimental testing confirmed the efficacy of this new technique. The developed method surpassed its counterparts by improving classification accuracy by 2.20% and 0.91% on the FER+small and ADFES-small datasets, respectively, precision by 2.30% on the FER+small dataset, completeness by 2.21% and 0.91% on the FER+small and ADFES-small datasets, and F1-score by 2.26% and 0.47% for the FER+small and ADFES-small datasets. It also decreased false positive and false negative rates by 0.55% and 0.23%, and 2.21% and 0.91% for the FER+small and ADFES-small datasets, respectively.  
In conclusion, the experimental testing using the developed software prototype validates the scientific claims of the proposed technique. Its implementation improves the reliability of abnormal behavior detection via facial expressions by 0.91-2.20%, depending on different emotions and environmental conditions. Moreover, it decreases the error probability in identifying sudden emotional shifts by 0.23-2.21% compared to existing counterparts.  
Future research will aim to improve the approach quantitatively and address the limitations discussed in the paper. Moreover, our further investigations will include a leave-one-speaker-group-out cross-validation strategy to evaluate the technique's ability to generalize across diverse conditions during video surveillance.  
Conceptualization, O.B.; methodology, O.B. and P.R.; software, O.K.; validation, O.K. and P.R.; formal analysis, O.K.; investigation, O.K.; resources, P.R.; data curation, O.K.; writing--original draft preparation, O.K.; writing--review and editing, P.R.; visualization, P.R.; supervision, I.K. and O.B.; project administration, I.K. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Ministry of Education and Science of Ukraine, state grant registration number 0121U112025, project title \"Development of information technology for making human-controlled critical and safety decisions based on mental-formal models of machine learning.\" This publication reflects the views of the authors only, and the Ministry of Education and Science of Ukraine cannot be held responsible for any use of the information contained therein.  
Not applicable.  
The experimental data used to support the findings of this study are available from the corresponding author upon request.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} ADFES & Amsterdam Dynamic Facial Expression Set \\ AI & Artificial Intelligence \\ CNN & Convolutional Neural Network \\ DL & Deep Learning \\ FACTS & Fairness, Accountability, Confidentiality, Transparency, and Safety \\ FACS & Facial Action Coding System \\ FER & Facial Expression Recognition \\ MDS & Multidimensional Scaling \\ ML & Machine Learning \\ RNN & Recurrent Neural Network \\ \end{tabular}