# USE OF TRIANGULAR MODELS OF NON-STATIONARY PROCESSES IN MODELING VARIABILITY OF HEART RHYTHM

Keywords: heart rhythm, non-stationary random process, electrocardio signal, correlation function, triangular model, simulation modeling

### Abstract

The subject matter is a mathematical model describing the process of heart rhythm variability, which is based on the use of triangular models of non-stationary random processes in the Hilbert space. The goal of the research is to develop a mathematical model of non-stationary processes of cardiac activity based on a triangular model. This research was the basis for the development of a Matlab model that implements the proposed method for analyzing heart rate variability. Tasks are: to give a description of the variability heart rate as a non-stationary process in Hilbert space in terms of correlation functions; to research the possibility of constructing a correlation and spectral theory of a non-stationary process using triangular models; to synthesize the mathematical model of nonstationary process on the basis of correlation theory for solving mathematical processing and forecasting tasks on the basis of ECG data. Using the proposed mathematical method, we implemented the Matlab model of a heart signal generator, which allowed us to synthesize an ECG with different variability parameters in noisy conditions. Methods of mathematical statistics, simulation modeling, theory of random processes and control theory are used in this work. Results of this research are as follows: 1) It has been shown that the new approach to the description of the HRV as a random process in the application of the triangular model in the Hilbert space made it possible to obtain expressions for the correlation function. 2) The imitation simulation showed the sensitivity of the method within the 5% error rate under the conditions of different types of influence on HRV. The qualitative assessment of the possibilities of the proposed models to generate artificial ECG provided the possibility of visual analysis by the cardiologist of the identity of the interpretation of real ECG records. The identities of modeling results were checked on time samples of electrocardiographs of 7 patients from open PhysioNet cardiographic libraries on samples with the duration T = 10 s. 3) The standard low-frequency oscillations and "white noise" barrier are clearly differentiated on the applied correlation function by the distribution of spectral density power within the frequency range of 0,15–0,3 Hz. Conclusion. The simulation results confirmed the correctness of the theoretical conclusions about the possibility of using models based on the representation of non-stationary processes in a triangular Hilbert space.

### Author Biographies

Olena Akhiiezer, National Technical University "Kharkiv Polytechnic Institute"
PhD (Engineering Sciences), Associate Professor, Professor at the Department of Computer Mathematics and Data Analysis
Olha Dunaievska, National Technical University "Kharkiv Polytechnic Institute"
PhD (Engineering Sciences), Associate Professor at the Department of Computer Mathematics and Data Analysis
Mykhailo Shyshkin, National Technical University "Kharkiv Polytechnic Institute"
PhD (Engineering Sciences), Associate Professor, Associate Professor at the Department of Industrial and Biomedical Electronics
Olha Butova, National Technical University "Kharkiv Polytechnic Institute"
PhD (Engineering Sciences), Associate Professor, Associate Professor at the Department of Industrial and Biomedical Electronics
Anton Rohovyi, National Technical University "Kharkiv Polytechnic Institute"
PhD (Engineering Sciences), Associate Professor, Associate Professor at the Department of Strategic Management

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