Mathematical modeling and methods for processing heart signals based on cyclic random processes and vectors

Abstract

Анотація. У роботі розглянуто підхід до моделювання сигналів серця електричної, магнітної та акустичної (механічної) природи на основі моделей теорії циклічних випадкових функцій, а саме, з використанням циклічного випадкового процесу та вектора циклічних ритмічно пов'язаних випадкових процесів. Наведено структури статистичних оцінок ймовірнісних характеристик досліджуваних сигналів серця, а також результати їх спектрального аналізу. Обґрунтовано інформативні ознаки в комп'ютерних системах функціональної діагностики стану серця на основі запропонованих у роботі їх математичних моделей та методів.

Isomorphic Multidimensional Structures of the Cyclic Random Process in Problems of Modeling Cyclic Signals with Regular and Irregular Rhythms

“…Formulas ( 30)-( 37) are the foundation for CRP models, computer simulations and hardware generation (formation) [53,54].…”

“…In contrast, if for a random process ξ, there exists a numerical function T(t, n), t ∈ R, n ∈ Z with all the above-mentioned properties (( 51) and ( 52)) and if the equalities in (53) are true for any k ∈ N, then it is a CRP.…”

“…Let us demonstrate the process of the statistical estimation of some one-dimensional and two-dimensional moment functions of cyclic stochastic signals for the task of biometric authentication by ECG signals. We will conduct this study within the framework of two mathematical models of the ECG, namely, in the form of the periodic [32,40,[63][64][65] and CRP, which will enable us to demonstrate the advantages of the CRP as a more general random process over the periodic random process. We use ECG signals (see Figures 5 and 6) that were registered in the first and second lead from a conditionally healthy patient at rest over a long observation period.…”

“…The simplest non-stationary probabilistic models of cyclic signals are additive, multiplicative and additive-multiplicative models, which somehow combine a stationary random process and periodic deterministic functions. Much more complex probabilistic models of cyclic signals are non-stationary random processes, such as cyclostationary (periodic) and almost-cyclostationary (almost periodic) random processes [10][11][12][13][14][15][16][17][18][19][20][21], periodic Markov random processes and chains [22][23][24][25][26][27], stochastic difference and differential equations with periodic solutions [28][29][30], and linear periodic random processes [31,32].…”

“…This approach has been widely used for modeling cyclic signals of various natures [48][49][50][51][52]. As part of this approach, a conditional cyclic random process was developed, which enabled a consistent mathematical description of cyclic signals with double stochasticity, namely, with simultaneous stochasticity of their cyclic and rhythmic structures [53]. The developed models, methods and software tools for processing cyclic signals are organized in the form of a computer ontology, which is built on the basis of an axiomatic deductive strategy for the systematization of knowledge in modern intellectualized information systems [54,55].…”