A novel modeling and prediction approach using Caputo derivative: An economical review via multi-deep assessment methodology

Nisa Özge Önal Tuğrul; Kamil Karaçuha; Esra Ergün; Vasil Tabatadze; Ertuğrul Karaçuha; Nisa Özge Önal Tuğrul; Kamil Karaçuha; Esra Ergün; Vasil Tabatadze; Ertuğrul Karaçuha

doi:10.3934/math.20241143

AIMS Mathematics

2024, Volume 9, Issue 9: 23512-23543. doi: 10.3934/math.20241143

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Research article

A novel modeling and prediction approach using Caputo derivative: An economical review via multi-deep assessment methodology

1.
Informatics Institute, Istanbul Technical University, Istanbul 34467, Turkiye
2.
Faculty of Electrical and Electronics Engineering, Istanbul Technical University, Istanbul 34467, Turkiye

Received: 05 April 2024 Revised: 28 June 2024 Accepted: 08 July 2024 Published: 06 August 2024
MSC : 15A09, 26A33, 44A10, 91-10

In this study, we proposed a novel modeling and prediction method employing both fractional calculus and the multi-deep assessment methodology (M-DAM), utilizing multifactor analysis across the entire dataset from 2000 to 2019 for comprehensive data modeling and prediction. We evaluated and reported the performance of M-DAM by modeling various economic factors such as current account balance (% of gross domestic product (GDP)), exports of goods and services (% of GDP), GDP growth (annual %), gross domestic savings (% of GDP), gross fixed capital formation (% of GDP), imports of goods and services (% of GDP), inflation (consumer prices, annual %), overnight interbank rate, and unemployment (total). The dataset used in this study covered the years between 2000 and 2019. The Group of Eight (G-8) countries and Turkey were chosen as the experimental domain. Furthermore, to understand the validity of M-DAM, we compared the modeling performance with multiple linear regression (MLR) and the one-step prediction performance with a recurrent neural network, long short-term memory (LSTM), and MLR. The results showed that in 75.04% of the predictions, M-DAM predicted the factors with less than 10% error. For the order of predictability considering the years 2018 and 2019, Germany was the most predictable country; the second group consisted of Canada, France, the UK, and the USA; the third group included Italy and Japan; and the fourth group comprised Russia. The least predictable country was found to be Turkey. Comparison with LSTM and MLR showed that the three methods behave complementarily.
- Caputo fractional derivative,
- deep assessment methodology,
- mathematical modeling,
- time series prediction
Citation: Nisa Özge Önal Tuğrul, Kamil Karaçuha, Esra Ergün, Vasil Tabatadze, Ertuğrul Karaçuha. A novel modeling and prediction approach using Caputo derivative: An economical review via multi-deep assessment methodology[J]. AIMS Mathematics, 2024, 9(9): 23512-23543. doi: 10.3934/math.20241143

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Abstract

In this study, we proposed a novel modeling and prediction method employing both fractional calculus and the multi-deep assessment methodology (M-DAM), utilizing multifactor analysis across the entire dataset from 2000 to 2019 for comprehensive data modeling and prediction. We evaluated and reported the performance of M-DAM by modeling various economic factors such as current account balance (% of gross domestic product (GDP)), exports of goods and services (% of GDP), GDP growth (annual %), gross domestic savings (% of GDP), gross fixed capital formation (% of GDP), imports of goods and services (% of GDP), inflation (consumer prices, annual %), overnight interbank rate, and unemployment (total). The dataset used in this study covered the years between 2000 and 2019. The Group of Eight (G-8) countries and Turkey were chosen as the experimental domain. Furthermore, to understand the validity of M-DAM, we compared the modeling performance with multiple linear regression (MLR) and the one-step prediction performance with a recurrent neural network, long short-term memory (LSTM), and MLR. The results showed that in 75.04% of the predictions, M-DAM predicted the factors with less than 10% error. For the order of predictability considering the years 2018 and 2019, Germany was the most predictable country; the second group consisted of Canada, France, the UK, and the USA; the third group included Italy and Japan; and the fourth group comprised Russia. The least predictable country was found to be Turkey. Comparison with LSTM and MLR showed that the three methods behave complementarily.

References

[1]	N. I. Sapankevych, R. Sankar, Time series prediction using support vector machines: A survey, IEEE Comput. Intell. Mag., 4 (2009), 24–38. https://doi.org/10.1109/MCI.2009.932254 doi: 10.1109/MCI.2009.932254
[2]	J. Bai, S. Ng, Forecasting economic time series using targeted predictors, J. Econom., 146 (2008), 304–317. https://doi.org/10.1016/j.jeconom.2008.08.010 doi: 10.1016/j.jeconom.2008.08.010
[3]	J. D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57 (1989), 357–384. https://doi.org/10.2307/1912559 doi: 10.2307/1912559
[4]	Z. Wang, H. Zhao, M. Zheng, S. Niu, X. Gao, L. Li, A novel time series prediction method based on pooling compressed sensing echo state network and its application in stock market, Neural Networks, 164 (2023), 216–227. https://doi.org/10.1016/j.neunet.2023.04.031 doi: 10.1016/j.neunet.2023.04.031
[5]	T. Van Gestel, J. A. Suykens, D. E. Baestaens, A. Lambrechts, G. Lanckriet, B. Vandaele, et al., Financial time series prediction using least squares support vector machines within the evidence framework, IEEE Trans. Neural Networks, 12 (2001), 809–821. https://doi.org/10.1109/72.935093 doi: 10.1109/72.935093
[6]	I. Kaastra, M. Boyd, Designing a neural network for forecasting financial and economic time series, Neurocomputing, 10 (1996), 215–236. https://doi.org/10.1016/0925-2312(95)00039-9 doi: 10.1016/0925-2312(95)00039-9
[7]	V. Zarnowitz, L. A. Lambros, Consensus and uncertainty in economic prediction, J. Political Econ., 95 (1987), 591–621. https://doi.org/10.1086/261473 doi: 10.1086/261473
[8]	R. E. Lucas Jr, On the mechanics of economic development, J. Monetary Econ., 22 (1998), 3–42. https://doi.org/10.1016/0304-3932(88)90168-7 doi: 10.1016/0304-3932(88)90168-7
[9]	H. White, Economic prediction using neural networks: The case of IBM daily stock returns, IEEE 1988 Int. Conf. Neural Networks, 2 (1988), 451–458. https://doi.org/10.1109/ICNN.1988.23959 doi: 10.1109/ICNN.1988.23959
[10]	X. Pang, Y. Zhou, P. Wang, W. Lin, V. Chang, An innovative neural network approach for stock market prediction, J. Supercomput., 76 (2020), 2098–2118. https://doi.org/10.1007/s11227-017-2228-y doi: 10.1007/s11227-017-2228-y
[11]	J. Furman, S. Robert, AI and the economy, Innovation Policy Econ., 19 (2019), 161–191. https://doi.org/10.1086/699936 doi: 10.1086/699936
[12]	W. Kristjanpoller, C. M. Marcel, A hybrid volatility forecasting framework integrating GARCH, artificial neural network, technical analysis and principal components analysis, Expert Syst. Appl., 109 (2018), 1–11. https://doi.org/10.1016/j.eswa.2018.05.011 doi: 10.1016/j.eswa.2018.05.011
[13]	M. L. Shen, C. F. Lee, H. H. Liu, P. Y. Chang, C. H. Yang, Effective multinational trade forecasting using LSTM recurrent neural network, Expert Syst. Appl., 182 (2021), 115199. https://doi.org/10.1016/j.eswa.2021.115199 doi: 10.1016/j.eswa.2021.115199
[14]	S. Nosratabadi, A. Mosavi, P. Duan, P. Ghamisi, F. Filip, S. S. Band, et al., Data science in economics: Comprehensive review of advanced machine learning and deep learning methods, Mathematics, 8 (2020), 1799. https://doi.org/10.3390/math8101799 doi: 10.3390/math8101799
[15]	S. Athey, The impact of machine learning on economics, In: The economics of artificial intelligence: An agenda, Chicago: University of Chicago Press, 2018.
[16]	A. Charpentier, R. Elie, C. Remlinger, Reinforcement learning in economics and finance, Comput. Econ., 62 (2023), 425–462. https://doi.org/10.1007/s10614-021-10119-4 doi: 10.1007/s10614-021-10119-4
[17]	O. Claveria, E. Monte, S. Torra, Economic forecasting with evolved confidence indicators, Econ. Model., 93 (2020), 576–585. https://doi.org/10.1016/j.econmod.2020.09.015 doi: 10.1016/j.econmod.2020.09.015
[18]	A. Seck, International technology diffusion and economic growth: Explaining the spillover benefits to developing countries, Struct. Change Econ. Dyn., 23 (2012), 437–451. https://doi.org/10.1016/j.strueco.2011.01.003 doi: 10.1016/j.strueco.2011.01.003
[19]	P. P. Combes, G. Laurent, Z. Yanos, Urban economics in a historical perspective: Recovering data with machine learning, Reg. Sci. Urban Econ., 2021 (2021), 103711. https://doi.org/10.1016/j.regsciurbeco.2021.103711 doi: 10.1016/j.regsciurbeco.2021.103711
[20]	W. Chen, H. Xu, L. Jia, Y. Gao, Machine learning model for Bitcoin exchange rate prediction using economic and technology determinants, Int. J. Forecast., 37 (2021), 28–43. https://doi.org/10.1016/j.ijforecast.2020.02.008 doi: 10.1016/j.ijforecast.2020.02.008
[21]	R. Van Eyden, M. Difeto, R. Gupta, M. E. Wohar, Oil price volatility and economic growth: Evidence from advanced economies using more than a century's data, Appl. Energy, 233 (2019), 612–621. https://doi.org/10.1016/j.apenergy.2018.10.049 doi: 10.1016/j.apenergy.2018.10.049
[22]	H. Ghoddusi, G. G. Creamer, N. Rafizadeh, Machine learning in energy economics and finance: A review, Energy Econ., 81 (2019), 709–727. https://doi.org/10.1016/j.eneco.2019.05.006 doi: 10.1016/j.eneco.2019.05.006
[23]	Y. Yue, L. He, G. Liu, Modeling and application of a new nonlinear fractional financial model, J. Appl. Math., 2013 (2013), 1–9. https://doi.org/10.1155/2013/325050 doi: 10.1155/2013/325050
[24]	E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuous-time finance, Physica A, 284 (2000), 376–384. https://doi.org/10.1016/S0378-4371(00)00255-7 doi: 10.1016/S0378-4371(00)00255-7
[25]	M. M. Meerschaert, E. Scalas, Coupled continuous time random walks in finance, Physica A, 370 (2006), 114–118. https://doi.org/10.1016/j.physa.2006.04.034 doi: 10.1016/j.physa.2006.04.034
[26]	O. Marom, E. Momoniat, A comparison of numerical solutions of fractional diffusion models in finance, Nonlinear Anal. Real, 10 (2009), 3435–3442. https://doi.org/10.1016/j.nonrwa.2008.10.066 doi: 10.1016/j.nonrwa.2008.10.066
[27]	J. Korbel, Y. Luchko, Modeling of financial processes with a space-time fractional diffusion equation of varying order, Fract. Calc. Appl. Anal., 19 (2016), 1414–1433. https://doi.org/10.1515/fca-2016-0073 doi: 10.1515/fca-2016-0073
[28]	V. E. Tarasov, V. V. Tarasova, Long and short memory in economics: Fractional-order difference and differentiation, Int. J. Manag. Soc. Sci., 5 (2016), 327–334. https://doi.org/10.21013/jmss.v5.n2.p10 doi: 10.21013/jmss.v5.n2.p10
[29]	V. V. Tarasova, V. E. Tarasov, Economic interpretation of fractional derivatives, Prog. Fract. Differ. Appl., 1 (2017), 1–6. https://doi.org/10.18576/pfda/030101 doi: 10.18576/pfda/030101
[30]	Z. Hu, X. Tu, A new discrete economic model involving generalized fractal derivative, Adv. Differ. Equ., 2015 (2015), 65. https://doi.org/10.1186/s13662-015-0416-8 doi: 10.1186/s13662-015-0416-8
[31]	N. Laskin, Fractional market dynamics, Physica A, 287 (2000), 482–492. https://doi.org/10.1016/S0378-4371(00)00387-3 doi: 10.1016/S0378-4371(00)00387-3
[32]	T. Škovránek, I. Podlubny, I. Petráš, Modeling of the national economies in state-space: A fractional calculus approach, Econ. Model., 29 (2012), 1322–1327. https://doi.org/10.1016/j.econmod.2012.03.019 doi: 10.1016/j.econmod.2012.03.019
[33]	E. Karaçuha, V. Tabatadze, K. Karaçuha, N. Ö. Önal, E. Ergün, Deep Assessment Methodology using fractional calculus on mathematical modeling and prediction of gross domestic product per capita of countries, Mathematics, 8 (2020), 633. https://doi.org/10.3390/math8040633 doi: 10.3390/math8040633
[34]	V. V. Tarasova, V. E. Tarasov, Exact discretization of an economic accelerator and multiplier with memory, Fractal Fract., 1 (2017), 6. https://doi.org/10.3390/fractalfract1010006 doi: 10.3390/fractalfract1010006
[35]	I. Tejado, E. Perez, D. Valerio, Economic growth in the European Union modelled with fractional derivatives: First results, Bull. Pol. Acad. Sci., Tech. Sci., 66 (2018), 455–465. https://doi.org/10.24425/124262 doi: 10.24425/124262
[36]	I. Tejado, E. Perez, D. Valerio, Fractional calculus in economic growth modelling of the group of seven, Fract. Calc. Appl. Anal., 22 (2019), 139–157. https://doi.org/10.1515/fca-2019-0009 doi: 10.1515/fca-2019-0009
[37]	I. Tejado, D. Valerio, E. Perez, N. Valerio, Fractional calculus in economic growth modelling: The Spanish and Portuguese cases, Int. J. Dyn. Control, 5 (2017), 208–222. https://doi.org/10.1007/s40435-015-0219-5 doi: 10.1007/s40435-015-0219-5
[38]	J. T. Machado, M. E. Mata, Pseudo phase plane and fractional calculus modeling of western global economic downturn, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 396–406. https://doi.org/10.1016/j.cnsns.2014.08.032 doi: 10.1016/j.cnsns.2014.08.032
[39]	I. Tejado, E. Perez, D. Valerio, Fractional derivatives for economic growth modelling of the group of twenty: Application to prediction, Mathematics, 8 (2020), 50. https://doi.org/10.3390/math8010050 doi: 10.3390/math8010050
[40]	J. Blackledge, Application of the fractal market hypothesis for modelling macroeconomic time series, ISAST Trans. Electron. Signal Process., 2 (2008), 89–110. https://doi.org/10.21427/D7091P doi: 10.21427/D7091P
[41]	S. Dadras, H. R. Momeni, Control of a fractional-order economical system via sliding mode, Physica A, 389 (2010), 2434–2442. https://doi.org/10.1016/j.physa.2010.02.025 doi: 10.1016/j.physa.2010.02.025
[42]	H. Wang, Research on application of fractional calculus in signal real-time analysis and processing in stock financial market, Chaos Soliton Fract., 128 (2019), 92–97. https://doi.org/10.1016/j.chaos.2019.07.021 doi: 10.1016/j.chaos.2019.07.021
[43]	M. Pavlíčková, I. Petráš, A note on time series data analysis using a fractional calculus technique, In: Proceedings of the 2014 15th international carpathian control conference, 2014,424–427. https://doi.org/10.1109/CarpathianCC.2014.6843640
[44]	I. Petráš, J. Terpák, Fractional calculus as a simple tool for modeling and analysis of long memory process in industry, Mathematics, 7 (2019), 511. https://doi.org/10.3390/math7060511 doi: 10.3390/math7060511
[45]	H. Jahanshahi, S. S. Sajjadi, S. Bekiros, A. A. Aly, On the development of variable-order fractional hyperchaotic economic system with a nonlinear model predictive controller, Chaos Soliton Fract., 144 (2021), 110698. https://doi.org/10.1016/j.chaos.2021.110698 doi: 10.1016/j.chaos.2021.110698
[46]	N. Ö. Önal Tuğrul, C. Başer, E. Ergün, K. Karaçuha, V. Tabatadze, S. Eker, et al., Modeling of mobile and fixed broadband subscriptions of countries with fractional calculus, Transp. Telecommun. J., 23 (2022), 1–10. https://doi.org/10.2478/ttj-2022-0001 doi: 10.2478/ttj-2022-0001
[47]	N. Ö. Önal, K. Karacuha, E. Karacuha, A comparison of fractional and polynomial models: Modelling on number of subscribers in the Turkish mobile telecommunications market, Int. J. Appl. Phys. Math., 10 (2020), 41–48. https://doi.org/10.17706/ijapm.2020.10.1.41-48 doi: 10.17706/ijapm.2020.10.1.41-48
[48]	N. Ö. Önal Tuğrul, E. Ergün, D. C. Köseoğlu, K. Karaçuha, K. Şimşek, E. Karaçuha, Modeling of telecommunication revenue as a percentage of gross domestic product's for countries with fractional calculus, J. Cognit. Syst., 6 (2021), 28–34. https://doi.org/10.52876/jcs.911144 doi: 10.52876/jcs.911144
[49]	K. Karaçuha, S. A. Sağlamol, E. Ergün, N. Ö. Önal Tuğrul, K. Şimşek, E. Karaçuha, Mathematical modeling of European countries' telecommunication investments, El-Cezeri J. Sci. Eng., 9 (2022) 1028–1037. https://doi.org/10.31202/ecjse.1053776 doi: 10.31202/ecjse.1053776
[50]	N. Ö. Önal, K. Karacuha, E. Karacuha, Modelling on economic growth and telecommunication sector of Turkey using a fractional approach including error minimizing, AIP Conf. Proc., 2471 (2022), 020018. https://doi.org/10.1063/5.0082688 doi: 10.1063/5.0082688
[51]	N. Ö. Önal, K. Karaçuha, G. H. Erdinè, B. B. Karaçuha, E. Karaçuha, A mathematical approach with fractional calculus for the modelling of children's physical development, Comput. Math. Methods Med., 2019 (2019), 3081264. https://doi.org/10.1155/2019/3081264 doi: 10.1155/2019/3081264
[52]	E. Karaçuha, N. Ö. Önal, E. Ergün, V. Tabatadze, H. Alkaş, K. Karaçuha, Ö. Tontus, N.V.N. Nu, Modeling and prediction of the COVID-19 cases with Deep Assessment Methodology and fractional calculus, IEEE Access, 8 (2020), 164012–164034. https://doi.org/10.1109/ACCESS.2020.3021952 doi: 10.1109/ACCESS.2020.3021952
[53]	E. Karaçuha, E. Ergün, N. Ö. Önal Tuğrul, K. Karaçuha, V. Tabatadze, Analyzing Response Efficiency to COVID-19 and Underlying Factors of the Outbreak With Deep Assessment Methodology and Fractional Calculus, IEEE Access, 9 (2021), 157812–157824. https://doi.org/10.1109/ACCESS.2021.3129904 doi: 10.1109/ACCESS.2021.3129904
[54]	OECD data statistic, 2021. Available from: https://stats.oecd.org/.
[55]	The world bank, world bank open data, 2021. Available from: https://data.worldbank.org/.

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