The efficiency of cyclic signal modeling algorithms is significantly determined by their mathematical models. Using a mathematical model in the form of a cyclic random process and taking into account the obtained statistical information in the form of statistical estimates of mathematical expectation and variance, as well as the estimated discrete rhythm function of the studied cyclic signal, a computer modeling method was developed. This method allows modeling a wide range of signals (of different physical nature), which have a repetitive (cyclic) structure. Based on the information about the estimated rhythmic structure (discrete rhythm function) of the known signal, or by setting (simulating) the rhythm of unfolding in time of the cyclic signal, which can be constant (periodic) and variable (cycle lengths are not equal to each other), the method allows modeling signals with different rhythms. Computer modeling, methods of statistical processing, cyclic signal, cyclic random process
Computer simulation of various cyclic signals and their processing using digital systems is an important task. The simulation allows determining the possibilities of known and created methods of processing cyclic signals at different stages of signal analysis, testing, for example, them using simulated implementations. In addition, the simulation of cyclic signals allows the training of newly created decision-making systems (diagnostic or prognostic), as well as methods for recognizing biomedical images through their training and testing.
Stochastic periodic processes (periodic random processes), such as periodically correlated random
process and periodically distributed random process with different types of distributions, can be
distinguished as mathematical models used for modeling cyclic signals, taking into account the
stochastic approach. They are widely used in the stochastic modeling paradigm for mathematical
models of cyclic signals [
The model of a cyclic random process is substantiated and applied as a mathematical model of various signals with cyclic spatiotemporal structure in different diagnostics information systems (medical diagnostics, technical diagnostics, and forecasting), as well as in systems of analysis and forecasting of mechanical and economic cyclic processes [23, 24].
This work is devoted to the development of a method of computer modeling of cyclic signals of different physical nature on the basis of information obtained by statistical processing methods on the probabilistic characteristics of cyclic random processes, based on a mathematical model in the form of cyclic random process in medicine, technology, and economics.
2021 Copyright for this paper by its authors.
Modeling of cyclic signals allows to carry out training of the created systems of decision-making (diagnostic or prognostic), methods of recognition of biomedical images by their training and testing.
Often for the needs of testing and analysis of various algorithms in practice, it is necessary to have a large base of representative implementations of cyclic signals. Therefore, for these purposes, it is convenient to have tools (methods and algorithms) for computer modeling, which allow you to simulate the desired implementation of cyclic signals. Thus in cyclic signals which are modeled, if the mathematical model allows, those parameters and characteristics to which the investigated method (algorithm) reacts and which need to be investigated, to carry out an estimation by the created methods are put. In the formation of real cyclic signals, their implementation should be preprocessed by an expert who, using his own experience, will determine in advance representative cycles and certain parameters and characteristics, which will then be taken into account in cyclic signals modeled, analyzed, and determined by research methods.
In computer simulation, the input data are the characteristics and parameters of the morphological and rhythmic nature of cyclic signals. Figure 1 shows a block diagram of the general algorithm of the method of modeling cyclic signals using two approaches – deterministic and stochastic.
(t), t W
C Z mˆ (t), t W1 dˆ (t), t W1
f1(t), t W1 fd (t), (t), Stest (t), t W Tˆ(t, n), t W, n Z
rhythmic structure r o
Z аоб ,n Z tW ,n __ ,Z __ ,1 W = ,t ,j ˆ=,,,it)(1nTСi ˆ=,,,it)(n1TСij Ti
durations r Formation of o cycles durations
ˆ T
Dˆc = tˆi , i = 1,С
Tij
____ Dˆz = tˆi , i = 1, С, j = 1, Z j
Features of the form of cyclic signals are taken into account in the block of morphological characteristics and parameters. Here, a representative cyclic signal cycle can be introduced f1(t), t W , using a deterministic approach to simulation of a cyclic signal. Implementations of statistical estimates of mathematical expectation can be or are introduced mˆ (t), t W and dispersions dˆ (t), t W , using a stochastic approach to simulating the cyclic signal. In addition, to add additional stochasticity to the simulated data, if necessary, you can add white noise with a normal distribution law, as an additive component of the simulation model. In addition to morphological characteristics, the modeling takes into account the parameters of segmental structures, namely, the number of cycles C and zones Z , as well as the characteristics and parameters of the rhythm in the corresponding block on the block diagram.
The following data can be entered in the rhythm generation unit (simulation): discrete rhythmic structure, if known, for cases of identified segmental cyclic structure T (ti , n), ti W, n Z, i = 1, C and the identified segmental zone structure T (ti , n), ti W, n Z , i = 1, C, j = 1, Z ; sets of j j samples of segmental cyclic structure Dc = ti , i = 1, C or multiple samples of the segmental zone structure Dz = ti , i = 1, C, j = 1, Z .
j
In the case where certain durations of the respective segment segments are known Ti , i = 1, C or zone segments Ti , i = 1, C, j = 1, Z or an estimate of the value of the period Tˆ (stable rhythm) then j the rhythm can be set taking into account these parameters. The input data for modeling the implementation of the cyclic signal is a continuous rhythm function, which can be evaluated in the block "estimation of rhythmic structure", taking into account one of the developed methods, or if it is known in advance, can be entered in the block characteristics and rhythm parameters.
For the stochastic case of computer simulation of cyclic signal implementations, stochastic, independent cycles are modeled and sequentially combined:
In the case of simulation taking into account the segments-zones:
C ( , t) = U i ( , t), Ω, t W .
i=1
C Z ( , t) = UU i ( , t), Ω, t W .
i=1 j=1 j
Taking into account the relationship between segmental cyclic and segmental band structures of cyclic signal implementations:
Z i ( , t) = U i ( , t), Ω, t W .
j=1 j Computer simulation of a sequence of stochastic, equivalent, independent cycles:
i (, t) = ( yi (t1), gi (, t1)), t1 W1, i = 1, C, t W , where yi (t1) – is a scale transformation function that takes into account rhythmic structures; gi ( , t1) – is a set of independent basic (representative) cycles: (1) (2) (3) (4)
In such modeling, the attributes are the implementation of statistical estimates (mathematical expectation and variance) or the implementation of a representative cycle of the cyclic signal (in the case of a deterministic approach to modeling the cyclic signal): mg (t1) = mˆ (t1 + T (t1, n)), d g (t1) = dˆ (t1 + T (t1, n)), n Z;
mg (t1) = f1(t1 + T (t1, n)), n Z.
In block diagram (see Fig. 1) the characteristics of rhythm (rhythmic structures) can be obtained by taking into account the segmental structures of the set either or Dz = ti , i = 1, C, j = 1, Z .
j
In addition, we can take into account the variable rhythm by specifying either the appropriate sets of cycle durations Ti , i = 1,С or sets of duration zones Tij , j = 1, Z , i = 1, С on the cycle. Here Dc = ti , i = 1, C we take into account the stable rhythm by entering the value of the period Tˆ .
In this part of the study, we consider examples of the results of the application of the method of computer modeling of cyclic signals, taking into account different rhythmic structures in different areas (Figures 2 - 5).
Mathematical expectation and variance estimates were used to simulate the implementation of the electrocardiographic signal (Figure 2). Estimation of the mathematical expectation of the electrocardiographic cycle is the input data for modeling – as a representative cycle that characterizes a particular pathology. In this case it is hypertrophy of the right and left ventricles. The estimation of the mathematical expectation determines the morphology of the cycle corresponding to the pathology, and the variation of the values on the cycle when simulating the implementation of the electrocardiographic signal is determined by the estimation of the variance. The rhythm of unfolding the values of the implementation of the electrocardiographic signal in time is set by the rhythmic structure (discrete rhythm function) given in Figure 3, a. The result of simulation the implementation of the electrocardiographic signal is presented in Figure 3, b.
t, s 1 0,06 0,05 0,04 0,03 0,02 0,01 0 dˆ (t), mV 2 0,3 0,2 0,1 0 -0,1 -0,2 -0,3 mˆ (t ), mВ 0 0,2 0,4 0,6 0,8 0 0,2 0,4 0,6 0,8 a) b) Figure 2: Input data for computer simulation of cyclic signals (electrocardiographic signals), implementation of estimates of mathematical expectation and variance: a) Assessment of mathematical expectation electrocardiographic signal (II lead), diagnosis – hypertrophy of the right and left ventricles; b) Assessment of variance electrocardiographic signal (5) (6) t, s 1
T (t,1), s 1 dˆ (t), c. u.2
Proposed by the authors model resembles the classical autoregressive model for the analysis of a noisy signal. However, the new model differs from the autoregression one in that it allows taking into account both the morphology of the simulated signals (by taking into account statistical estimates: mathematical expectation, variance) of the cyclic process and information about the rhythm of its deployment which can be constant (periodic) and variable (differ from each other among particular cycles).
Comparison of the mathematical apparatus offered in this work with others needs separate research. For example, when comparing with mathematical apparatus of wavelets, the result of modeling using wavelets will depend on the correctness of the choice of "mother wavelet". That is, it should be close to the shape of the simulated cycle. There is no such shortcoming in the proposed method of modeling, because the shape (morphology) of the signal cycle is given arbitrarily and is determined by estimating the mathematical expectation and variance.
The developed method allows modeling a wide range of signals of different physical nature, which have a repetitive (cyclic) structure. Based on the information about the estimated rhythmic structure (that is discrete rhythm function) of the known signal, or by setting (simulating) the rhythm of time development of the cyclic signal, which can be constant (periodic signal) and variable (cycle lengths are not equal to each other), the proposed method allows simulation of signals with a different rhythm.
This method allows considering both a rhythm of signal development in time and features of morphology of a signal representative cycle.
The developed method based on a mathematical model in the form of a cyclic random process allows computer modeling of cyclic signals that can be used in intelligent information systems in economics, mechanics, and medicine. Also, this method is especially important at the testing stage of new systems for processing signals of such nature.
This paper concerns the method of modeling cyclic signals, the simulation results of which are sufficiently interpreted in this paper. To confirm the results obtained in this work, the authors have used mathematical statistics. However, it is not possible to place all the obtained results only in one work. In further research, it is planned to conduct computer modeling of cyclic signals in the field of energy consumption and to investigate the absolute and relative errors of modeling, in particular gas consumption and energy consumption signals.