=Paper=
{{Paper
|id=Vol-3039/paper23
|storemode=property
|title=Simulation of gas consumption process based on the mathematical model in the form of cyclic random process considering the scale factors
|pdfUrl=https://ceur-ws.org/Vol-3039/paper23.pdf
|volume=Vol-3039
|authors=Serhii Lupenko,Iaroslav Lytvynenko,Nataliia Kunanets,Oleg Nazarevych,Grigorii Shymchuk,Volodymyr Hotovych
|dblpUrl=https://dblp.org/rec/conf/ittap/LupenkoLKNSH21
}}
==Simulation of gas consumption process based on the mathematical model in the form of cyclic random process considering the scale factors==
https://ceur-ws.org/Vol-3039/paper23.pdf
Simulation of gas consumption process based on the
mathematical model in the form of cyclic random process
considering the scale factors
Iaroslav Lytvynenkoa, Serhii Lupenkoa, Nataliia Kunanetsb, Oleg Nazarevycha, Grigorii
Shymchuka, Volodymyr Hotovycha
a
Ternopil Ivan Puluj National Technical University, 56, Ruska Street, Ternopil, 46001, Ukraine
b
Lviv Polytechnic National University, 12, Bandera Street, Lviv, 79013, Ukraine
Abstract
The article deals with the problem of constructing a new mathematical model of gas
consumption process in which, in contrast to the known mathematical model, in the form of an
additive combination of three components: cyclic random process, trend component, and
stochastic residue, the component in the form of a cyclic random process with taking into
account the scale factors, is introduced. Based on segmentation using the Caterpillar method,
ten components of singular decomposition are obtained. The sum of nine components of
singular decomposition forms the cyclic component, the cyclic random process. This
component takes into account the scale factors of gas consumption range on every segment
cycle. The trend component of mathematical model is the second component of singular
decomposition and stochastic residue, which is formed on the basis difference of values of the
studied gas consumption process and the sum of cyclic and trend components. In the research,
computer simulation of gas consumption process on the basis of the known model and the new
one is carried out and simulation errors with the real gas consumption process are estimated.
The computer simulation results are compared in accordance with the proposed mathematical
model and the known one. The use of the proposed mathematical model considering the cyclic
component as a random cyclic process with cyclic structure and scale factors allowed to
increase the accuracy of computer simulation which is evidenced by the obtained error
estimation results.
Keywords
Cyclic process, gas consumption process, statistical processing, segmentation, cyclic random
process.
1. Introduction
Gas consumption management is of particular urgency nowadays. Analysis, simulation and
prediction of gas consumption process are the vital problems set by gas companies which, in turn,
requires efficient hardware and software for gas consumption processing. Development of new software
complexes demands the improving of mathematical software, which involves the development of new
mathematical models and methods of gas consumption processing. This will ultimately allow to form
the forecast results based on the information obtained.
__________________________
ITTAP’2021: 1nd International Workshop on Information Technologies: Theoretical and Applied Problems, November 16–18, 2021,
Ternopil, Ukraine
EMAIL: iaroslav.lytvynenko@gmail.com (I. Lytvynenko); lupenko.san@gmail.com (S. Lupenko); nek.lviv@gmail.com
(N. Kunanets); taltek.te@gmail.com (O. Nazarevych), gorych@gmail.com (G. Shymchuk), gotovych@gmail.com
(V. Hotovych)
ORCID: 0000-0001-7311-4103 (I. Lytvynenko); 0000-0002-6559-0721 (S. Lupenko); 0000-0003-3007-2462 (N. Kunanets);
0000-0002-8883-6157 (O. Nazarevych), 0000-0003-2362-7386 (G. Shymchuk), 0000-0003-2143-6818 (V. Hotovych)
©️ 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
This research deals with creating a new mathematical model of gas consumption process using
stochastic approach in the form of additive combination of three components, one of which is a cyclic
random process that takes into account scale factors, other components are trend and stochastic residue.
2. Analysis of recent research
In the field of energy consumption, in particular, electricity and gas, various mathematical models
are used to describe them, and there are two approaches to their representation – deterministic and
stochastic one [1-13]. When studying gas consumption process, not only presenting it objectively in the
form of mathematical model is essential, but also the ability to draw a gas consumption forecast based
on it. In [14], a short-term forecast using neural networks and the ARMA model is considered. In [15],
a neural network algorithm is used to draw an electricity consumption model in a gas transmission
system. Monthly gas consumption for household consumers was studied in [16], where a multivariate
regression analysis was considered, which made it possible to show the dependence of gas consumption
on the average monthly temperature. The model based on machine forecast is presented in [17], which
enables the prediction of future gas consumption using a statistical sample. In [18], the description of
mathematical model of energy loads based on a linear periodic random process (stochastic approach) is
given, which allows to take into account the main reasons that cause the rhythmicity of the process. In
[20], a mathematical model of loads on gas transmission system in the form of a conditional linear
periodic random process is presented. An interesting approach to drawing mathematical model is
described in [21] where the additive model is considered as a sum of the deterministic annual trend and
stochastic balance as a stochastic-periodic process, which allows to take into account the periodic and
random nature of gas consumption and variable topology of consumers in the annual observation
interval. In this approach, a singular decomposition based on the use of the Caterpillar method was used
to obtain the components. The emphasis in this paper is on the use of stochastic residue for segmentation
of gas consumption process in the season features. However, this mathematical model did not suggest
simulation of gas consumption process, and therefore to draw a forecast.
3. Main part
In [21], a mathematical model of gas consumption process was considered, one of the components
of which was a stochastic-periodic random process. In [22] it was shown that the model in the form of
a cyclic random process, as a partial case, includes a model in the form of stochastic-periodic random
process.
We present the known mathematical model of the random cyclic gas consumption process ( , t )
as an additive model (1) which consists of three components. As it was mentioned above, the similar
approach was used in [21], however, we specify the components of the proposed mathematical model:
( , t ) = ( , t ) + f tr (t ) + f rem ( , t ), t W, Ω, Ω , (1)
where ( , t ) is a cyclic component, ftr (t ) is a trend function, f rem ( , t ) is a stochatic residue
function, W is a domain of determining, is a elementary event, is a space of elementary events,
Ω and are another space of elementary events and elementary event from that space respectively.
Since in practice we are dealing with discrete data, we present mathematical model (1) as follows:
(l ) = (l ) + f tr (l ) + f rem (l ), l W = D , (2)
where (l ) is an implementation of cyclic component of gas consumption process, f tr (l ) is a trend
function, f rem (l ) is a function of stochatic residue, l stands for discrete samples of gas consumption
process, D is a discrete domain of determining.
For obtaining the components of mathematical model (2) during processing of the real cyclic gas
consumption process (l ), l = 1, L we apply the SSA-Caterpillar method. This method is given in [23]
and describes the transformation of a one-dimensional time series into a multidimensional one, which
makes it possible to obtain components of a singular segmentation. Comparison of time series
processing methods using the Caterpillar-SSA method is presented in [24].
When applying the Caterpillar method, we obtain k implementations of components
f (l), k = 0, K −1, l = 1, L, where K = 10 , l stands for parts of gas consumption process during
k
2006-2019 years, L is the number of discrete implementation samples.
The cyclic component is obtained by summing the components obtained on the basis of the
Caterpillar method, in particular, components: 0-1,3-9, component 2 is a component of the
trend f 2 (l ) :
1 9
(l ) = f k (l ) + f k (l ) , l = 1, L , (3)
k =0 k =3
f tr (l ) = f 2 (l ) , l = 1, L . (4)
The stochastic residue is obtained on the basis of the relation:
f rem (l ) = (l ) − ( (l ) + f tr (l )) , l = 1, L . (5)
Consider (l ) , the cyclic component of the mathematical model (1), which carries information
about the process of gas consumption in more detail, we present it as:
C
(l ) = f i (l ), l W , (6)
i =1
where C is the number of segments-cycles of the cyclic process of gas consumption, W is the domain
of determining the cyclic process of gas consumption, and the domain of its values, for the case of the
stochastic approach is the Hilbert space of random variables given on one probabilistic space
( (l ) Ψ = L 2 (Ω, P)). In the design (6), the segments-cycles f i (l ) of the cyclic gas consumption
process are determined by indicator functions, i.e.:
f i (l ) = (l ) I Wi (l ), i = 1, C, l W . (7)
The indicator functions, which allocate segments-cycles, are defined as:
1, l Wi ,
I Wi ( l ) = i = 1, C , (8)
0, l W i ,
where Wi is the domain of determining the indicator function, which in the case of a discrete signal,
i.e. W = D , is equal to a discrete set of samples:
Wi = li , j , j = 1, J , i = 1, C . (9)
The segmental cyclic structure D̂c is taken into account by the set of time samples {li } or li , j ,
i = 1, C , j = 1, J , where J is the number of discrete samples in the cycle. This notation of the
mathematical model (9) takes into account the rhythm of the cyclic gas consumption process due to the
continuous function of the rhythm T (l , n) namely:
I Wi (l ) = I Wi + n (l + T (l , n) ) , i = 1, C , n = 1, l W . (10)
In order to estimate the rhythm function T (l , n) , the segment structure of gas consumption process
(in this case the segment cyclic structure) was first determined as D ˆ = {l , i = 1, C} which is a set of
c i
time moments that correspond to the boundaries of the segments-cycles of gas consumption process. In
this case, the estimation of the segmental cyclic structure of gas consumption process can be performed
using the segmentation method presented in [25]. It has been shown before that segmentation of gas
consumption process is better not to carry out on the vertices, but on the depressions, which does not
allow "blurring" of statistical estimates after processing the studied implementation.
Consider the block diagram of the method of computer simulation of gas consumption process based
on the known mathematical model (1) described in [26] for the process of relief formations (Fig. 1).
After applying the Caterpillar method, the trend component and the stochastic residue enter the
computer simulation unit, except for the cyclic component. The cyclic component enters the block of
statistical processing where the estimation of mathematical expectation and variance is carried out on
the basis of the received information on the estimated rhythm function which is obtained on the basis
of information on segmental and rhythmic structure. The estimated rhythm function, statistical
estimates, trend component and stochastic balance are used for computer simulation of gas consumption
process implementation.
mˆ (l ), l W1
(lk ), lk W (lk ), lk W D c i
ˆ = l , i = 1, С Tˆ (l ,1), l W dˆ (l ), l W1
Singular
decomposition by
Estimation of
segment
Estimation of
rhythmic
Statistical
Simulation of ˆ1 (lk ), lk W
cyclic gas
processing of gas
the Caterpillar- structure of gas structure of gas consumption
consumption
SSA method consumption consumption process
rem (lk ), lk W
ftr (lk ), lk W
Figure 1: Block diagram of the method of computer simulation of gas consumption process (known
approach)
An example of result of gas consumption process computer simulation based on the block diagram
(see Fig. 1) is given in Fig. 2.
(lk ), lk W (lk ), lk W D c i
ˆ = l , i = 1, С Tˆ (l ,1), l W mˆ (l ), l W1 dˆ (l ), l W1
Singular Estimation of
segment
Estimation of
rhythmic
Statistical
Simulation of ˆ2 (lk ), lk W
decomposition by cyclic gas
processing of gas
the Caterpillar- structure of gas structure of gas consumption
consumption
SSA method consumption consumption process
W (l)
i
rem (lk ), lk W Determination of aver
i max Estimation of
ftr (lk ), lk W amplitude
maximums of gas scale factors of
consumption gas consumption
segments-cycles amplitude
Figure 2: Block diagram of the proposed method of computer simulation of gas consumption process
To adequately describe the real gas consumption process, it is also necessary to consider changes in
the load amplitude on the segments-cycles, which are caused by various factors, such as climate
(temperature, pressure, wind force and direction, humidity), changes in consumer topology [21]. For
this reason, it is considered in the proposed mathematical model.
In the new design of mathematical model (1), the cyclic component (6) takes into account the
segments-cycles of cyclic gas consumption process as multiplicative components considering the
indicator functions and scale factors of gas consumption amplitude, i.e.
f i (l ) = (l ) Wi (l ) I Wi (l ), i = 1, C, l W . (11)
In formula (11), an additional component Wi (l ) that reflects the scale factors of gas consumption
amplitude in each segment-cycle of the cyclic process, are introduced:
, l Wi ,
W (l ) = i i = 1, C , (12)
0, l Wi .
i
where i stands for the scale factors of gas consumption amplitude at every i -segment-cycle, are
determined as follows:
i max
i = , i = 1, C , (13)
aver
where i max is the maximum value of gas consumption range at i -segment-cycle (determined at the
stage of segmentation of cyclic gas consumption process), aver is the average value of gas
consumption range (the maximum value of estimation range of mathematical expectation, is determined
at the stage of statistical processing of cyclic gas consumption). The block diagram of the proposed
approach to gas consumption process simulation will look as in Fig. 2.
Consider and compare the results of computer simulation based on two approaches (two
mathematical models, known and the proposed one).
(l )
120000
100000
80000
60000
40000
20000
0
2007 2010 2013 2016 2019
Figure 3: Fragment of input implementation of cyclic gas consumption process
f tr (l ) f rem (l )
37000 30000
35000 20000
10000
33000
0
31000
-10000
29000 -20000
27000 -30000
25000 -40000
𝑙, 𝑦𝑒𝑎𝑟 -50000 l, year
23000
2006 2008 2010 2012 2014 2016 2006 2008 2010 2012 2014 2016
a) b)
Figure 4: Estimated components of the mathematical model: a) trend component; b) stochastic
residue
70000 (l )
60000
50000
40000
30000
20000
10000
0 𝑙, 𝑦𝑒𝑎𝑟
2007 2010 2013
a) b)
Figure 5: Fragments of the studied implementation of gas consumption process for the case of
segmentation by depressions: a) cyclic component; b) results of segmentation of the cyclic component
into segments-cycles (on the abscissa axis the data are given in conventional units, the specified
number of samples)
374
372
370
368
366
364
362
360
358
356
354
352
350
226 590 954 1318 1682
Figure 6: Fragments of the result of estimated rhythm function (piecewise linear interpolation) of
cyclic component of gas consumption process (based on segmentation into cycles by depressions)
Having obtained the segment structure D̂c and estimating the rhythmic structure (discrete rhythm
function T (l , n) ) by the methods proposed in [27], the methods of statistical processing were applied
taking into account the rhythm function [27], while the estimation of mathematical expectation was
determined:
1 M
M n=1
(l + T (l , n)), l W1 = l1 , l2 ) ,
mˆ T ( l ,n ) (l ) = (13)
where l1 0 in the general case, l1 ,l 2 are the discrete time samples which correspond to the beginning
and end of the first segment-cycle, M is the number of cycles.
And the estimation of variance was determined as follows:
M 2
dˆT ( l ,n ) (l ) = (l + T (l , n )) − mˆ T ( l ,n ) (l + T (l , n )) , l W1 = l1 , l2 ) .
1
(14)
M n=1
Applying the methods of statistical processing, we obtained statistical estimates of probabilistic
ˆ T ( l ,n ) (l ), l W1 and variance dˆT ( l ,n ) (l ), l W1 based on
characteristics (mathematical expectation m
the rhythm function T (l , n) of the cyclic component of gas consumption process. Examples of the
obtained estimates are given in Fig. 7.
ˆ 2T ( l ,n ) (l ), l W1
60000 m ˆ
18000000 d 2T ( l ,n ) (l ), l W1
50000 16000000
14000000
40000 12000000
10000000
30000
8000000
20000 6000000
4000000
10000 2000000
𝑙, 𝑑𝑎𝑦 0
0
0 100 200 300 400 0 100 200 300 400
а) b)
Figure 7: Estimation of mathematical expectation and variance based on the estimated rhythm
function of cyclic component of gas consumption process (segmentation into cycles by depressions):
a) estimation of mathematical expectation; b) estimation of variance
Taking into account the obtained statistical estimates, carry out the computer simulation of cyclic
components of gas consumption process implementations on the basis of two mathematical models (see
Fig. 8).
(l ), 1 (l ) (l ), 2 (l )
70000 70000 2 (l )
60000 1 (l ) 60000
(l ) (l )
50000 50000
40000 40000
30000 30000
20000 20000
10000 10000
0 𝑙, 𝑦𝑒𝑎𝑟 0 𝑙, 𝑦𝑒𝑎𝑟
2007 2010 2013 2016 2019 2022 2007 2010 2013 2016 2019 2022
a) b)
Figure 8: Results of computer simulation of the cyclic component of gas consumption process based
on two mathematical models: a) implementation of the cyclic component of gas consumption process
of real data and simulated on the basis of a known model; b) implementation of the cyclic component
of gas consumption process of real data and simulated on the basis of a proposed model
1 (l ) + f tr (l ) 2 (l ) + ftr (l )
90000 100000
80000 90000
80000
70000
70000
60000
60000
50000 50000
40000 40000
30000 𝑙, 𝑦𝑒𝑎𝑟 30000 𝑙, 𝑦𝑒𝑎𝑟
2007 2010 2013 2016 2019 2007 2010 2013 2016 2019
a) b)
Figure 9: Results of computer simulation of the cyclic component of gas consumption process and the
trend component based on two mathematical models: a) implementation of the simulated cyclic
component of gas consumption process based on the known model and trend component; b)
implementation of the simulated cyclic component of gas consumption process based on the
proposed model and trend component
110000 mod 1 (l ) = 1 (l ) + f tr (l ) + f rem (l ) 120000
mod 2 (l ) = 2 (l ) + ftr (l ) + f rem (l )
100000 110000
90000 100000
80000 90000
70000 80000
60000 70000
60000
50000 50000
40000 40000
30000 30000
20000 20000
10000 10000
0 𝑙, 𝑦𝑒𝑎𝑟 0 𝑙, 𝑦𝑒𝑎𝑟
2007 2010 2013 2016 2019 2007 2010 2013 2016 2019
a) b)
Figure 10: Results of computer simulation of gas consumption process based on two mathematical
models: a) implementation of simulated cyclic component of gas consumption process based on a
known model, trend component and stochastic residue; b) implementation of simulated cyclic
component of gas consumption process based on the proposed model, trend component and
stochastic balance
The root mean square absolute and relative errors of computer simulation of gas consumption
process were determined by the formulas:
1 L
2 q (k )
q (k ) = (l ) − mod q (l )) ; q (k ) =
L l =1 1 L
, k = 1, L , q = 1,2 ,
(15)
mod q (l ) 2
L l =1
where mod q (l ) is the value of simulated gas consumption process implementation based on two
mathematical models, q = 1,2 ; (l ) is the value of actual gas consumption implementation process;
L is the number of implementations samples, l = 1, L ; k is the sample for absolute and relative errors,
respectively, k = 1, L . The results of the obtained absolute and relative errors are shown in the Fig. 11.
80000 1 (k ) 60000 2 (k )
70000
50000
60000
40000
50000
40000 30000
30000 20000
20000
10000
10000
0 𝑘, 𝑦𝑒𝑎𝑟 0 𝑘, 𝑦𝑒𝑎𝑟
2007 2010 2013 2016 2019 2007 2010 2013 2016 2019
a) b)
Figure 11: Results of determining the absolute error of computer simulation of gas consumption
process on the basis of two mathematical models: a) on the basis of a known model; b) based on the
proposed model
2.0 1 (k ) 1.6 2 (k )
1.8 1.4
1.6
1.4 1.2
1.2 1.0
1.0 0.8
0.8 0.6
0.6
0.4 0.4
0.2 0.2
0 𝑘, 𝑦𝑒𝑎𝑟 0 𝑘, 𝑦𝑒𝑎𝑟
2007 2010 2013 2016 2019 2007 2010 2013 2016 2019
a) b)
Figure 12: Results of determining the relative error of computer simulation of gas consumption
process on the basis of two mathematical models: a) on the basis of a known model; b) based on the
proposed model
4. Discussion of obtained results
Based on the obtained statistical estimates of the cyclic component (mathematical expectation and
variance), computer simulation of cyclic components of gas consumption process implementations on
the basis of two mathematical models was performed. After that, the trend component (see Fig. 9) and
the stochastic residue (see Fig. 10) were added to the simulated cyclic components. The obtained results
of computer simulation of gas consumption process implementations on the basis of the proposed
mathematical model (see Fig. 10, b) taking into account scale factors allow to obtain a smaller relative
modeling error (see Fig. 12, b) in comparison with the results on the basis of known mathematical
model (see Fig. 10, a), which allows more accurate computer simulation, and hence making a forecast
based on this approach.
5. Conclusions
In the research, a new mathematical model in the form of an additive combination of three
components, a cyclic random process taking into account the scale factors, a trend component and a
stochastic residue, are developed. A new method of computer simulation of gas consumption process
based on a proved mathematical model is proposed. The application of the proposed model taking into
account the scale factors allowed to increase the accuracy of computer modeling, as evidenced by the
results of the estimated errors in comparison with the known mathematical model.
In further research the values of scale factors obtained on the basis of aggregated data of climatic
indicators in the developed mathematical model will be taken into account and comparative analysis of
computer simulation of gas consumption based on a three-component mathematical model is going to
be conducted.
6. References
[1] W. A. Gardner, A. Napolitano, L. Paura, Cyclostationarity: Half a century of research, Signal
Processing. – 2005. – № 86 (2006). – P. 639-697. https://doi.org/10.1016/j.sigpro.2005.06.016.
[2] H. L. Hurd, Periodically Correlated Random Sequences : Spectral Theory and Practice, The
University of North Carolina at Chapel Hill Hampton University.
[3] P. Kochel, Periodically stationary Markovian decision models, Elektron. Informationsverarb.
Kybernet. – 1980. – No. 16. – P. 553-567 (in German).
[4] A. R. Nematollahi, A. R. Soltani, Discrete time periodically correlated Markov processes,
Probability and Mathematical Statistics. – 2000. – No. 20 (1). – P. 127-140.
[5] E. Ghysels, R. E. McCulloch, R. S. Tsay, Bayesian Inference for a General Class of Periodic
Markov Switching Models, 1993.
[6] I. Javorskyj, R. Yuzefovych, I. Kravets, I. Matsko, Methods of periodically correlated random
processes and their generalizations. In: Cyclostationarity: Theory and Methods, 73–93 (2014).
[7] E. Ghysels, On the Periodic Structure of the Business Cycle, Cowles Foundation, Yale Universiti.
– 1992. – No. 1028. https://doi.org/10.2307/1392085.
[8] Bittanti S. Markovian representations of cyclostationary processes, in: L. Gerencser, P.E. Caines
(Eds.) / S. Bittanti, F. Lorito, S. Strada // Topics in Stochastic Systems : Modelling, Estimation and
Adaptive Control. – Springer, Berlin, Germany. – 1991. – Vol. 161. – P. 31-46.
[9] H.L. Hurd, A.G. Miamee, Periodically Correlated Random Sequences: Spectral Theory and
Practice, Wiley, New York, 2006.
[10] A. Napolitano, Asymptotic Normality of statisticalfunction estimators for generalized almost-
cyclostationary processes, in: Proceedings of XIII European Signal Processing Conference
(EUSIPCO 2005), Antalya, Turkey, 2005.
[11] E. Broszkiewicz-Suwaj, A. Makagon, R. Weron, A. Wy"oman´ska, On detecting and modeling
periodic correlation in financial data, Physica A Statist. Mech. Appl. 336 (1–2) (2004) 196–205.
[12] H.L. Hurd, A. Miamee, Periodically Correlated Random Sequences, Spectral Theory and Practice.
Wiley, New Jersey (2007).
[13] I. Javorskyi, Mathematical models and analsys of stochastic oscillations. Lviv, Karpenko Physico-
Mechanical Institute (2013). (in Ukrainian)
[14] R. Gutiérrez, A. Nafidi, R. Gutiérrez Sánchez, Forecasting total natural-gas consumption in Spain
by using the stochastic Gompertz innovation diffusion model, Applied Energy 80(2005) 115–124
[15] G. A. Darbellay, M. Slama, Forecasting the short-term demand for electricity. Do neural networks
stand a better chance?, International Journal of Forecasting 16(2000) 71–83.
[16] M. Thaler, I. Grabec, A. Poredoš, Prediction of energy consumption and risk of excess demand in
a distribution system, Physica A 355(2005) 46–53.
[17] T. Gelo, Econometric modelling of gas demand (In Croatian), Economic Review, 57(2006) 80–
96.
[18] P. Potočnik, M. Thaler, E. Govekar, I. Grabec, A. Poredos, Forecasting risks of natural gas
consumption in Slovenia, Energy Policy 35(2007) 4271–4282.
[19] M. Pryimak Analiz enerhonavantazhen iz vykorystanniam liniinykh vypadkovykh protsesiv.
Scientific Journal of the Ternopil state technical university 1999, Vol 4, No 1, pp. 84-87. [in
Ukrainian].
[20] B. Marchenko, N. Mulyk, M. Fryz, Kharakterystychna funktsiia umovnoho liniinoho
vypadkovoho protsesu yak matematychnoi modeli hazospozhyvannia Electronics and control
systems, 2006, No 3 (9), pp. 40–46. [in Ukrainian].
[21] O. Nazarevych Vydilennia richnoho trendu yak adytyvnoi skladovoi chasovoho riadu
hazospozhyvannia Scientific Journal of the Ternopil national technical university 2011, Vol 16,
No 4, pp. 201-209. [in Ukrainian].
[22] S. Lupenko, Cyclic functions and their classification in the tasks of modeling cyclic signals and
oscillation system, Measuring and computing method in technological processes. - Khmelnytskyi,
2005. - #1. - P. 277-185.
[23] Nina Golyandina, Anatoly Zhigljavsky, Singular Spectrum Analysis for Time Series,
SpringerBriefs in Statistics. Springer-Verlag Berlin Heidelberg. – 2020. P. 146.
[24] N.E. Golyandina, V.V Nekrutkin, A.A. Zhigljavsky, Analysis of Time Series Structure: SSA and
Related Techniques /. – Boca Raton: Chapman&Hall / CRC, 2000. – 305 p.
[25] I.V. Lytvynenko, The method of segmentation of stochastic cyclic signals for the problems of their
processing and modeling. Journal of Hydrocarbon Power Engineering, Oil and Gas Measurement
and Testing. 2017, Vol. 4, No. 2, pp. 93–103.
[26] Volodymyr Hutsaylyuk, Iaroslav Lytvynenko, Pavlo Maruschak, Volodymyr Dzyura, Georg
Schnell, and Hermann Seitz, A new method for modeling the cyclic structure of the surface
microrelief of titanium alloy ti6al4v after processing with femtosecond pulses Materials
(Basel). 2020 Nov; 13(21): 4983. doi:10.3390/ma13214983.
[27] I. Lytvynenko, A. Horkunenko, O. Kuchvara, Y. Palaniza, Methods of processing cyclic signals in
automated cardiodiagnostic complexes, in: Proceedings of the 1st International Workshop on
Information-Communication Technologies and Embedded Systems, Mykolaiv, Ukraine, 2019,
pp. 116-127.