LINEAR STOCHASTIC DISTRIBUTED MODEL OF MONEY ACCUMULATION IN THE FORM OF A STATE SPACE
DOI:
https://doi.org/10.26577/JMMCS.2021.v110.i2.11Abstract
The article deals with the problem of the passive parametric identification of systems for modeling the evolution of money savings income and expenses of one household using a linear stochastic distributed model in the form of a state space taking into account white noises model of the investigated object dynamics’ and white noises of the linear model measuring system of a distributed type. The use of the finite difference method allowed reducing the solution of partial
differential equations to the solution of linear finite difference system with private derivatives to be reduced to the solution of a system of linear finite-difference and algebraic equations represented by models in the form of state space. It was proposed the use of a Kalman filtering algorithm for reliable evaluation of object behavior. The statement of the problem of estimating the coefficients of the equation of evolution of money savings income and expenses of one household is given. The structure of household income and expenses is described, taking into account additional additive white noise meters. An algorithm for numerical approbation of method for solving the problem of estimating the coefficients of an equation in the form of the state space for the evolution of money savings income and expenses of one household is considered. Calculations were carried out using the Matlab mathematical system based on statistical data for five years, taken from the site “Agency for Strategic planning and reforms of the Republic of Kazakhstan Bureau of National statistics”. The proposed method for solving the problem of coefficients assessment’s passive identification using the equations of money savings for one household in the form of a state space is sufficiently universal.
Key words: linear finite-difference equation, model in the form of a state space, evolution of one household money savings, passive identification, Kalman filter, prediction estimates, filtering estimates.
References
[2] Van Den Hof P., Wahlberg B., Weiland S., System Identification, 1st Edition, (Elsevier Science, 2004), 2088.
[3] Bensoussan A. et al., "Identification of linear dynamical systems and machine learning" , Optimization and Control (2020): 1-26, https://www.semanticscholar.org/paper/Identification-of-linear-dynamical-systems-and-
Bensoussan-Gelir/141e7250c785c093fec97b4cd51245c307c9e122.
[4] Ramirez R.I., Montejo L.A., "On the identification of damping from non-stationary free decay signals using modern signal processing techniques" , Int J Adv Struct Eng 7 (2015): 321–328, https://doi.org/10.1007/s40091-015-0096-3.
[5] Wagenmaker A., Jamieson K., "Active Learning for Identification of Linear Dynamical Systems" , Proceedings of Machine Learning Research vol. 125 (2020): 1-96, https://arxiv.org/abs/2002.00495.
[6] Diblik J., Dzhalladova I., Michalkova M. et al., "Modeling of applied problems by stochastic systems and their analysis using the moment equations" , Adv Differ Equ 152 (2013), https://doi.org/10.1186/1687-1847-2013-152.
[7] Xu L., "Machine learning and causal analyses for modeling financial and economic data" , Appl Inform 5, article number: 11 (2018). https://doi.org/10.1186/s40535-018-0058-5.
[8] Wang D., Gao Y., "Recursive maximum likelihood identification method for a multivariable controlled autoregressive moving average system" , IMA Journal of Mathematical Control and Information vol. 33 (2016): 1015-1031, https://doi.org/10.1093/imamci/dnv021.
[9] Sircar P., Dutta M.K., Mukhopadhyay S., "Signal parameter estimation of complex exponentials using fourth order statistics: additive Gaussian noise environment" , SpringerPlus 4, article number: 360 (2015), https://doi.org/10.1186/s40064-015-1131-3.
[10] Cui J., Li Y., Yan L., "Temporal variation for fractional heat equations with additive white noise" , Bound Value Probl 123 (2016), https://doi.org/10.1186/s13661-016-0630-7.
[11] Zhang E., Pintelon R., "Identification of dynamic errors-in-variables systems with guasi-stationary input and colored noise" , Automatica 123 (2020):1-10.
[12] Sharabati W., Xi B., "A neighborhood regression approach for removing multiple types of noises" , J Image Video Proc. 19 (2018), https://doi.org/10.1186/s13640-018-0259-9.
[13] Achdou Y., Buera F.J., Lasry J-M., Lions P-L., Moll B., "Partial differential equation models in macroeconomics" , Phil. Trans. R. Soc. A372:20130397 (2014), http://dx.doi.org/10.1098/rsta.2013.0397.
[14] Pareschi L., Toscani G., "Wealth distribution and collective knowledge: a Boltzmann approach" , Phil. Trans. R. Soc. A 372:20130396 (2014), http://dx.doi.org/10.1098/rsta.2013.0396.
[15] Albi G., Pareschi L., Zanella M., "Boltzmann-type control of opinion consensus through leaders" , Phil. Trans. R. Soc. A372:20140138 (2014), http://dx.doi.org/10.1098/rsta.2014.0138.
[16] Ozbilge E., Demir A., "Identification of the unknown diffusion coefficient in a linear parabolic equation via semigroup approach" , Adv Differ Equ 47 (2014), https://doi.org/10.1186/1687-1847-2014-47.
[17] Li J., Wu JL., "On drift parameter estimation for mean-reversion type stochastic differential equations with discrete observations" , Adv Differ Equ 90 (2016), https://doi.org/10.1186/s13662-016-0819-1.
[18] Al-Mazrooei A., Al-Mutawa J., El-Gebeily M. et al., "Filtering and identification of a state space model with linear and bilinear interactions between the states" , Adv Differ Equ 176 (2012), https://doi.org/10.1186/1687-1847-2012-176.
[19] Boikov I.V., Krivulin N.P., "Identifikatsiia sistem, opisyvaemykh differentsial’nymi uravneniiami v chastnykh proizvodnykh [Identification of systems described by partial differential equations]" , Mater. VII mezhdunar. nauch. konf.
¾Matematicheskoe i komp’iuternoe modelirovanie estestvennonauchnykh i sotsial’nykh problem¿, Penza (2012): 23–27.
[20] Krivulin N.P., "Opredelenie parametrov fizicheskikh protsessov, opisyvaemykh differentsial’nymi uravneniiami v chastnykh proizvodnykh s peremennymi koeffitsientami [Determination of the parameters of physical processes described by partial differential equations with variable coefficients]" , Mater. VIII mezhdunar. nauch. konf. ¾Matematicheskoe i komp’iuternoe modelirovanie estestvennonauchnykh i sotsial’nykh problem¿, Penza (2014): 172–178.
[21] Boikov I.V., Krivulin N.P., "Parametricheskaia identifikatsiia sistem, matematicheskie modeli kotorykh opisyvaiutsia differentsial’nymi uravneniiami s proizvodnymi drobnykh poriadkov[ Parametric identification of systems whose
mathematical models are described by differential equations with derivatives of fractional orders]" , Metrologiia, No 9 (2013): 3–17.
[22] Boikov I.V., Krivulin N.P., "Parametricheskaia identifikatsiia ereditarnykh sistem s raspredelennymi parametrami [Parametric identification of distributed parameter hereditary systems]" , Izvestiia vysshikh uchebnykh zavedenii.
Tekhnicheskie nauki, No 2(26) (2013): 120–129.
[23] Ivanov D.V., "Identifikatsiia diskretnykh lineinykh dinamicheskikh sistem drobnogo poriadka s oshibkoi na vykhode [Identification discrete fractional order linear dynamic systems with output error]" , Mater. mezhdunar. sibirskoi konf. po upravleniiu i sviazi (SIBCON), Krasnoiarsk (2013): 1-3, doi: 10.1109 / SIBCON.2013.6693623.
[24] Vasil’ev V.A., Konev V.V., "Posledovatel’naia identifikatsiia lineinykh dinamicheskikh sistem v nepreryvnom vremeni po shumnym nabliudeniiam [Sequential identification of linear dynamical systems in continuous time from noisy observations]" , Problemy upravleniia i teorii informatsii, No 16(2) (1987): 101-112.
[25] Yerofeenko V.T., Kozlovskaia I.S., Uravneniia s chastnymi proizvodnymi i matematicheskie modeli v ekonomike [Equations with private derivatives and mathematical models in economics] (Minsk: Izd-vo Bel. gos. un-ta, 2004), 245.
[26] Chernavskii D.S., Popkov Iu.S., Rakhimov A.Kh., "Matematicheskie modeli tipologii semeinykh nakoplenii [Mathematical models of family savings typology]" , Ekonomika i matematicheskie metody, vol.30, is. 2 (1994): 98-106.
[27] Statistika urovnia zhizni [Statistics of the standard of living], accessed December 8, 2020, https://stat.gov.kz/official/industry/64/statistic/7.
[28] Kopchenova N.V., Maron I.A., Vychislitelnaia matematika v primerakh i zadachakh [Computational mathematics in examples and tasks] (M.: Nauka, 2009), 370.
[29] Eliseeva I.I. (Eds), Ekonometrika: Uchebnik [Econometrics: Textbook] (M.: Finansy i statistika. 2001), 344.
[30] Abdenova G.A., Voevoda A.A., "Otsenivanie parametrov i kharakteristik shumov nestatsionarnykh protsessov v stokhasticheskikh sistemakh, opisyvaemykh v prostranstve sostoianii [Estimation of parameters and characteristics of noises of non-serial processes in stochastic systems described in the space of states]" , Sb. nauch. trudov NGTU, No 3 (61) (2010): 11-18.
[31] Abdenova G.A., "Prognozirovanie znachenii urovnia vremennogo riada na osnove uravnenii filtra Kalmana [Prediction of time series level values based on Kalman filter equations]" , Polzunovskii vestnik, No 2 (2010): 4-6.
[32] Abdenov A., Abdenova G., Kulbaev D., "Estimation of equation coefficients of free and forced string vibrations in continuous medium with consideration of dynamic noise and measurement" , J. Materials Today: Procedings vol. 16 (2019): 336-342.
[33] Gorskii V.G., Adler Iu.P., Talalai A.M., Planirovanie promyshlennykh eksperimentov. Modeli dinamiki [Planning of industrial experiments. Models of dynamics] (M.: Metallurgiia, 1978), 112.
