The adaptive extended kalman filter approach for the Lotka-Volterra model and application to economic variables

Abstract

The main aim of this article is to extend on the application of the grey Lotka-Volterra model by Wu et al. [1] with a linear programming method. We used this method for estimating the parameters of behavioral variables under the criterion of the minimization of mean absolute percentage error (MAPE). Our empirical analysis indicates that the adaptive extended Kalman filter (EKF) approach performs far better compared to traditional Lotka-Volterra model in the prediction of the relevant parameters. Comparisons of empirical results with the linear pro-gramming method for parameter estimation of the grey Lotka-Volterra model demonstrate that the EKF approach has more powerful and efficient prediction performance.

Keywords

• Lotka-Volterra Model
• Extended Kalman Filter
• Parameters Estimation

References

1. REFERENCES
2. [1] Wu L, Liu S, Wang Y. Grey Lotka–Volterra model and its application. Technol Forecast Soc Change 2012;79:17201730. [CrossRef]
3. [2] Julong D. Introduction to grey system theory. J Grey Syst 1989;1:124.
4. [3] Cheng H, Wu XH. R&D investment, R&D stock and elasticity production of R&D stock—empirical study based on age/effectiveness function. Stud Sci Sci 2006;24:108113.
5. [4] Gatabazi P, Mba JC, Pindza E, Labuschagne C. Grey Lotka–Volterra models with application to cryptocurrencies adoption. Chaos Solit Fract 2019;122:4757. [CrossRef]
6. [5] Mao S, Gao M, Zhu M. The impact of R&D on GDP study based on grey delay Lotka-Volterra model. Grey Syst 2015;5:7488. [CrossRef]
7. [6] Tsai BH. Modelling energy consumption and carbon dioxide emissions of fossil fuels and nuclear energy using Lotka-Volterra equations Appl Ecol Environ Res 2022;20:14351455. [CrossRef]
8. [7] Zhang M, Guo H, Sun M, Liu S, Forrest J. A novel flexible grey multivariable model and its jmkoapplication in forecasting energy consumption in China. Energy 2022;239:122441. [CrossRef]

9. [8] Debnath P, Srivastava HM. Optimizing stock market returns during global pandemic using regression in the context of Indian stock market. J Risk Financial Manag 2021;14:386. [CrossRef]
10. [9] Debnath P, Srivastava HM. Optimal Returns in Indian Stock Market during Global Pandemic: A Comparative Study. J Risk Financial Manag 2021;14:592. [CrossRef]
11. [10] Goel NS, Maitra SC, Montroll EW. On the Volterra and other nonlinear models of interacting populations. Rev Modern Phys 1971;43:231. [CrossRef]
12. [11] Volterra V. Fluctuations in the abundance of a species considered mathematically. Nature 1926;118:558560. [CrossRef]
13. [12] Shim H, Fishwick PA. Visualization and Interaction Design for Ecosystem Modeling. In J. Sven Erik & F. Brian (Eds.), Encyclopedia of Ecology. Oxford: Academic Press; 2008. p. 36853693. [CrossRef]
14. [13] Křivan V. The Lotka-Volterra predator-prey model with foraging–predation risk trade-offs. Am Natur 2007;170:771782. [CrossRef]
15. [14] Křivan V, Cressman R. On evolutionary stability in predator--Prey models with fast behavioral dynamics. Evol Ecol Res 2009;11:227251.
16. [15] Sun X, Jin L, Xiong M. Extended Kalman filter for estimation of parameters in nonlinear state-space models of biochemical networks. PloS One 2008;3:e3758. [CrossRef]
17. [16] Özbek L, Efe M. An adaptive extended Kalman filter with application to compartment models. Commun Stat Simul Comput 2004;33:145158. [CrossRef]
18. [17] Özbek L, Aliev FA. Comments on adaptive fading Kalman filter with an application. Automatica 1998;34:16631664. [CrossRef]
19. [18] Liu S, Forrest J, Yang Y. A brief introduction to grey systems theory. In Proceedings of 2011 IEEE International Conference on Grey Systems and Intelligent Services 2011; p. 19. [CrossRef]
20. [19] Wang Z X, Dang YG, Liu SF. Unbiased grey Verhulst model and its application. Syst Eng Theory Pract 2009;29:138144. [CrossRef]
21. [20] Morris SA, Pratt D. Analysis of the Lotka–Volterra competition equations as a technological substitution model. Technol Forecast Soc Change 2003;70:103133. [CrossRef]
22. [21] Wu L, Wang Y. Estimation the parameters of Lotka–Volterra model based on grey direct modeling method and its application. Expert Syst Appl 2011;38:64126416. [CrossRef]
23. [22] Mao S, Zhang Y, Kang Y, Yuannong Mao, Y. Coopetition analysis in industry upgrade and urban expansion based on fractional derivative gray Lotka-Volterra model. Soft Comput 2021;25:11485–11507. [CrossRef]
24. [23] Liang Y, Li G. An empirical study on fixed asset investment, consumer price index and economic development: from the perspective of Xinjiang. Finance 2020;10:3037.
25. [24] Özbek L. Kalman Filtresi. Ankara: Akademisyen Yayınları; 2017. [Turkish]