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This work is devoted to the procedure of the construction of an abstract cyclic functional relation, which summarizes and extends the known results for a cyclically correlated random process and a cyclic (cyclically distributed) random process to the case of arbitrary cyclic functional relations. Two alternative definitions of the abstract cyclic functional relation are given, and the fundamental properties of its cyclic and phase structures are presented. The theorem on the invariance of cyclicity attributes of an abstract cyclic functional relation to shifts of its argument, and which are determined by the rhythm function of this functional relation, is formulated and proved. This theorem gives the sufficient and necessary conditions that the rhythm function of an abstract cyclic functional relation must satisfy. By specifying the range of values and attributes of the cyclicity of an abstract cyclic functional relation, the definitions of important classes of cyclic functional relations are formulated. A deductive approach to building a wide system of taxonomies of classes of deterministic, stochastic, fuzzy and interval cyclic functional relations as potential mathematical models of cyclic signals is demonstrated. A comparative analysis of an abstract cyclic functional relation with the known mathematical models of cyclic signals was carried out. The results obtained in the article significantly expand and systematize the mathematical tools of the description of cyclic signals and are the basis for the development of effective model-based technologies for processing and computer simulation of signals with a cyclic space-time structure.

This work is devoted to the procedure for constructing of a cyclically correlated random process of a continuous argument as a mathematical model of cyclic signals in dynamic systems, which makes it possible to consistently describe cyclic stochastic signals, both with regular and irregular rhythms, not separating them, but complementing them within the framework of a single integrated model. The class of cyclically correlated random processes includes the subclass of cyclostationary (periodically) correlated random processes, which enable the use of a set of powerful methods of analysis and the forecasting of cyclic signals with a stable rhythm. Mathematical structures that model the cyclic, phase and rhythmic structures of a cyclically correlated random process are presented. The sufficient and necessary conditions that the structural function and the rhythm function of the cyclically correlated random process must satisfy have been established. The advantages of the cyclically correlated random process in comparison with other mathematical models of cyclic signals with a variable rhythm are given. The obtained results contribute to the emergence of a more complete and rigorous theory of this class of random processes and increase the validity of the methods of their analysis and computer simulation.

In this study is substantiated the new mathematical model of vector of electroencephalographic signals, registered under the conditions of multiple repetitions of the mental control influences of brain–computer interface operator, in the form of a vector of cyclic rhythmically connected random processes, which, due to taking into account the stochasticity and cyclicity, the variability and commonality of the rhythm of the investigated signals have a number of advantages over the known models. This new model opens the way for the study of multidimensional distribution functions; initial, central, and mixed moment functions of higher order such as for each electroencephalographic signal separately; as well as for their respective compatible probabilistic characteristics, among which the most informative characteristics can be selected. This provides an increase in accuracy in the detection (classification) of mental control influences of the brain–computer interface operators. Based on the developed mathematical model, the statistical processing methods of vector of electroencephalographic signals are substantiated, which consist of statistical evaluation of its probabilistic characteristics and make it possible to conduct an effective joint statistical estimation of the probability characteristics of electroencephalographic signals. This provides the basis for coordinated integration of information from different sensors. The use of moment functions of higher order and their spectral images in the frequency domain, as informative characteristics in brain–computer interface systems, are substantiated. Their significant sensitivity to the mental controlling influence of the brain–computer interface operator is experimentally established. The application of Bessel’s inequality to the problems of reducing the dimensions (from 500 to 20 numbers) of the vectors of informative features makes it possible to significantly reduce the computational complexity of the algorithms for the functioning of brain–computer interface systems. Namely, we experimentally established that only the first 20 values of the Fourier transform of the estimation of moment functions of higher-order electroencephalographic signals are sufficient to form the vector of informative features in brain–computer interface systems, because these spectral components make up at least 95% of the total energy of the corresponding statistical estimate of the moment functions of higher-order electroencephalographic signals.

The work is devoted to a problem of the rhythm function estimation of a cyclic random process, which is based on the least squares approximation methods instead of well-known interpolation approach. Analytical dependencies between errors of estimation of a discrete rhythm function and errors of segmentation of a cyclic random process into cycles and zones were constructed. This made it possible to develop a procedure for calculating and controlling errors of estimating rhythm function of a cyclic random process as certain functions of errors of the segmentation method. The general problem of least squares approximation of the rhythm function of a cyclic random process is formulated as a problem of optimal selection of a parametric function derived from a predetermined class of functions that satisfy the necessary and sufficient conditions of the rhythm function of a cyclic random process. New parametric classes of rhythm characteristics of cyclic random processes such as parametric monomials of degree k , parametric logarithmic functions and parametric exponential functions have been built. The advantage of considered method over well-known interpolation approach refers to the improvement of accuracy of rhythm function estimation and reduction of the rhythm function estimation parameters’ number. For example, in presented computer simulation experiment for the parametric class of monomials of degree 2, average value of the mean square errors for 500 simulations in the case of the interpolation is over 40 times higher than the corresponding value for approximation. Moreover, for that parametric class, the number of estimated parameters is almost equal to doubled number of considered cycles in the case of piecewise linear interpolation and is reduced to 1 for least square approximation. The results obtained in the work constitute the basis for improvement of rhythm-adaptive methods and spectral analysis of cyclic random processes, including the area of statistical methods for detecting hidden cyclic structures of the investigated cyclic stochastic signals with an irregular rhythm.

This article presents a mathematical model of cyclical economic processes, formulated as the sum of a deterministic polynomial function and a cyclic random process that simultaneously captures trend, stochasticity, cyclicity, and rhythm variability. Building on this stochastic framework, we propose rhythm-adaptive statistical techniques for estimating the probabilistic characteristics of the cyclic component; by adjusting to rhythm changes, these techniques improve estimation accuracy. We also introduce a forecasting procedure that constructs a system of rhythm-adaptive confidence intervals for future cycles. The effectiveness of the model and associated methods is demonstrated through a series of computational experiments using Federal Reserve Economic Data. Results show that the rhythm-adaptive forecasting approach achieves mean absolute errors less than half of those produced by a comparable non-adaptive method, underscoring its practical advantage for the analysis and prediction of cyclic economic phenomena.

This paper is devoted to the research of the isomorphic multidimensional cyclic structure and multidimensional phase structure of the cyclic random process (CRP) and to its formation method, which enables a rigorous formalization of intuitive ideas concerning cyclic stochastic motion. The fundamental properties of the cyclic random process and analytical dependencies between the multidimensional cyclic structure, multidimensional phase structure and rhythm structure of the CRP have been established. This work shows that the CRP is able to take into account the cyclicity of multidimensional distribution functions of cyclic signals as well as the variability in the rhythm of the investigated signals. A subclass of the CRP is the periodic random process, which allows for the use of classical processing methods of cyclic signals with a regular rhythm. Based on a series of experiments, significant advantages of the CRP as a mathematical model of electrocardiographic signals (ECG) compared to the periodic random process are shown.

The paper aims to develop an information system for human emotion recognition in streaming data obtained from a PC or smartphone camera, using different methods of modality merging (image, sound and text). The objects of research are the facial expressions, the emotional color of the tone of a conversation and the text transmitted by a person. The paper proposes different neural network structures for emotion recognition based on unimodal flows and models for the margin of the multimodal data. The analysis determined that the best classification accuracy is obtained for systems with data fusion after processing each channel separately and obtaining individual characteristics. The final analysis of the model based on data from a camera and microphone or recording or broadcast of the screen, which were received in the “live” mode, gave a clear understanding that the quality of the obtained results is highly dependent on the quality of the data preparation and labeling. This is directly related to the fact that the data on which the neural network is trained is highly qualified. The neural network with combined data on the penultimate layer allows a psycho-emotional state recognition accuracy of 0.90 to be obtained. The spatial distribution of emotion analysis was also analyzed for each data modality. The model with late fusion of multimodal data demonstrated the best recognition accuracy.

The authors developed and substantiated the original methods of arterial oscillography, which were implemented in the developed Oranta-AO information system. The methods of application to the arterial oscillogram registered at measurement of arterial pressure gives the possibility to carry out the supplementary systematic assessment of health, functional state of cardiovascular system, its reserve possibilities etc. The authors also developed an Expert System (based on machine-learning methods) for the differential diagnosis of risks of heart, lung, mental illness and prognosis of some blood parameters. Oranta-AO software system was created based on research results due to methods and algorithms that were innovate. For the mathematical modeling of arterial oscillograms used cyclic random processes. Methods of arterial oscillograms processing based on its model in the form of a cyclic random process was developed. The method of evaluation of the rhythm function of arterial oscillograms and statistical methods for estimating the probabilistic characteristics of arterial oscillograms were developed. To solve the clustering problem, the Python k -means and k -means++ algorithm were used. Oranta-AO information system consists of three interrelated parts: mobile application, computing kernel and web system. Computing kernel and web system are deployed on AWS servers and have been tested already. The developed environment aims to be integrated into every new model of electronic meters in the world. Certification (EN 62304:2014, ISO 13485: 2018) in Ukraine is completed, PCT priority is completed. The next step will be to establish cooperation with manufacturers of electronic pressure monitors, patenting and certification in world.

The paper presents the axiomatic-deductive strategy of organizing the content of an academic discipline with the help of ontological approach in the e-learning systems in the field of information technologies. The authors have taken into account that the necessary property of the system of axiomatic statements is their consistency. On the basis of axiomatic-deductive strategy, new approaches to the formation of the discipline content are proposed. It is proved that the system of true statements of an academic discipline is based on its terminology-conceptual apparatus, in particular, axiomatic statements. The developed mathematical structures that describe the axiomatic-deductive substrategy of the organization of the academic discipline general statements and the taxonomically oriented substrategy of the deployment of the academic discipline content are presented in the article. This ensures the transition from the content form of representation of the set of statements of the academic discipline to its presentation by means of artificial languages of mathematical logic. The use of descriptive logic ensures the formalization of the procedure for displaying an axiomatic informal system in an axiomatic formal system. The mathematical structures describe and detail the abstract logical-semantic core of the academic discipline in the form of a group of axiomatic systems. It is noted that the basic core of the content of academic discipline contains its basic concepts and judgments. This ensures a strictly logical transition from abstract general concepts and statements to the concepts and assertions of the lower level of universality and abstraction. It is noted that in order to accommodate the content of an academic discipline is advisable to develop a taxonomically oriented sub-strategy based on the multiple application of operations of general concept division. The mathematical structures allow for analysis of a generalized structure of interactions between the verbal level of the description of the academic discipline subject area, the formal level of description of the subject area and the description of the subject area at the level of computer ontology, which is implemented through the formalization, interpretation, encoding and decoding in the computer-ontology development environment. As an example of the application of the proposed axiomatic-deductive strategy, the elements of the glossary and taxonomies of the concepts of the discipline \"Computer Logic\", which are embodied in the Protégé environment with the help of OWL ontology description language have been developed.

The new methods of statistical analysis of heart rhythm were developed based on its generalized mathematical model in a form of random rhythm function, that allows to increase the informativeness and detailed analysis of heart rhythm in cardiovascular information systems. Three information criteria (BIC, AIC and AICc) were used to determine the cumulative distribution functions that best describe the sample and to assess the unknown parameters of distributions. The usage of the rhythm function to analyse heart rhythm allows to consider much better its time structure that is the basis to improve the accuracy of diagnosis of cardiac rhythm.