A conformable fractional sparse robust grey RVFL model for energy time series forecasting

Yang Yue; Shaoyong Liu; Yang Yue; Shaoyong Liu

doi:10.3934/math.20261282

AIMS Mathematics

2026, Volume 11, Issue 10: 32628-32670. doi: 10.3934/math.20261282

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Research article

A conformable fractional sparse robust grey RVFL model for energy time series forecasting

Yang Yue ^,,
Shaoyong Liu

School of Big Data and Statistics, Tongling University, Tongling 244000, China

Received: 01 August 2026 Revised: 14 September 2026 Accepted: 22 September 2026 Published: 10 October 2026
MSC : 26A33, 62M10, 68T05

Forecasting short energy time series is challenging because limited modeling observations, nonlinear dynamics, and atypical residuals can destabilize conventional grey models. This study proposed a conformable fractional sparse robust grey random vector functional link model (CFSRGRVFL). The model embeded a complete random vector functional link (RVFL) forcing term, comprising an unpenalized direct time link and randomized hidden features, within a conformable fractional grey system to control temporal weighting and nonlinear representation. Coefficients were estimated using a bounded exponential-squared loss, while an $ \ell_1/\ell_2 $ regularizer was applied only to the hidden-output coefficients to promote sparse feature selection without suppressing the direct link. An alternating direction method of multipliers scheme, incorporating an $ \ell_1/\ell_2 $ proximal update and difference-of-convex majorization, was developed for numerical estimation. Bayesian optimization with a fixed seed set tuned the coupled hyperparameters using the complete modeling sequence without accessing forecasting observations. Experiments on five energy series against three neural grey benchmarks showed that CFSRGRVFL achieved the lowest forecasting mean absolute percentage error in all cases and ranks first on most out-of-sample metrics. These results supported the model's effectiveness within the tested short-series settings.
Citation: Yang Yue, Shaoyong Liu. A conformable fractional sparse robust grey RVFL model for energy time series forecasting[J]. AIMS Mathematics, 2026, 11(10): 32628-32670. doi: 10.3934/math.20261282

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Abstract

Forecasting short energy time series is challenging because limited modeling observations, nonlinear dynamics, and atypical residuals can destabilize conventional grey models. This study proposed a conformable fractional sparse robust grey random vector functional link model (CFSRGRVFL). The model embeded a complete random vector functional link (RVFL) forcing term, comprising an unpenalized direct time link and randomized hidden features, within a conformable fractional grey system to control temporal weighting and nonlinear representation. Coefficients were estimated using a bounded exponential-squared loss, while an $ \ell_1/\ell_2 $ regularizer was applied only to the hidden-output coefficients to promote sparse feature selection without suppressing the direct link. An alternating direction method of multipliers scheme, incorporating an $ \ell_1/\ell_2 $ proximal update and difference-of-convex majorization, was developed for numerical estimation. Bayesian optimization with a fixed seed set tuned the coupled hyperparameters using the complete modeling sequence without accessing forecasting observations. Experiments on five energy series against three neural grey benchmarks showed that CFSRGRVFL achieved the lowest forecasting mean absolute percentage error in all cases and ranks first on most out-of-sample metrics. These results supported the model's effectiveness within the tested short-series settings.

References

[1]	J. L. Deng, Control problems of grey systems, Syst. Control Lett., 1 (1982), 288–294. https://doi.org/10.1016/S0167-6911(82)80025-X doi: 10.1016/S0167-6911(82)80025-X
[2]	S. Liu, Y. Lin, Grey Systems: theory and applications, Berlin/Heidelberg: Springer, 2010. https://doi.org/10.1007/978-3-642-16158-2
[3]	S. Feng, Y. Ma, Z. Song, J. Ying, Forecasting the energy consumption of China by the grey prediction model, Energy Sources Part B Econ. Plan. Policy, 7 (2012), 376–389. https://doi.org/10.1080/15567240903330426 doi: 10.1080/15567240903330426
[4]	S. B. Tsai, Using grey models for forecasting China's growth trends in renewable energy consumption, Clean Technol. Environ. Policy, 18 (2016), 563–571. https://doi.org/10.1007/s10098-015-1017-7 doi: 10.1007/s10098-015-1017-7
[5]	X. He, Y. Wang, Y. Zhang, X. Ma, W. Wu, L. Zhang, A novel structure adaptive new information priority discrete grey prediction model and its application in renewable energy generation forecasting, Appl. Energy, 325 (2022), 119854. https://doi.org/10.1016/j.apenergy.2022.119854 doi: 10.1016/j.apenergy.2022.119854
[6]	J. Guo, L. Wu, Y. Mu, An optimized grey model for predicting non-renewable energy consumption in China, Heliyon, 9 (2023), e17037. https://doi.org/10.1016/j.heliyon.2023.e17037 doi: 10.1016/j.heliyon.2023.e17037
[7]	Y. Wang, X. He, Y. Zhou, Y. Luo, Y. Tang, G. Narayanan, A novel structure adaptive grey seasonal model with data reorganization and its application in solar photovoltaic power generation prediction, Energy, 302 (2024), 131833. https://doi.org/10.1016/j.energy.2024.131833 doi: 10.1016/j.energy.2024.131833
[8]	N. M. Xie, S. F. Liu, Discrete grey forecasting model and its optimization, Appl. Math. Model., 33 (2009), 1173–1186. https://doi.org/10.1016/j.apm.2008.01.011 doi: 10.1016/j.apm.2008.01.011
[9]	J. Cui, S. F. Liu, B. Zeng, N. M. Xie, A novel grey forecasting model and its optimization, Appl. Math. Model., 37 (2013), 4399–4406. https://doi.org/10.1016/j.apm.2012.09.052 doi: 10.1016/j.apm.2012.09.052
[10]	M. Cai, Non-homogeneous grey model NGM (1, 1) with initial value modification and its application, Proceedings of the 2010 2nd International Conference on Industrial and Information Systems, 2010,102–104. https://doi.org/10.1109/INDUSIS.2010.5565904
[11]	X. Ma, Z. Liu, Application of a novel time-delayed polynomial grey model to predict the natural gas consumption in China, J. Comput. Appl. Math., 324 (2017), 17–24. https://doi.org/10.1016/j.cam.2017.04.020 doi: 10.1016/j.cam.2017.04.020
[12]	L. Wu, S. Liu, L. Yao, S. Yan, D. Liu, Grey system model with the fractional order accumulation, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 1775–1785. https://doi.org/10.1016/j.cnsns.2012.11.017 doi: 10.1016/j.cnsns.2012.11.017
[13]	H. Duan, G. R. Lei, K. Shao, Forecasting crude oil consumption in China using a grey prediction model with an optimal fractional-order accumulating operator, Complexity, 2018 (2018), 3869619. https://doi.org/10.1155/2018/3869619 doi: 10.1155/2018/3869619
[14]	Y. Wang, P. Chi, R. Nie, X. Ma, W. Wu, B. Guo, Self-adaptive discrete grey model based on a novel fractional order reverse accumulation sequence and its application in forecasting clean energy power generation in China, Energy, 253 (2022), 124093. https://doi.org/10.1016/j.energy.2022.124093 doi: 10.1016/j.energy.2022.124093
[15]	X. Ma, W. Wu, B. Zeng, Y. Wang, X. Wu, The conformable fractional grey system model, ISA Trans., 96 (2020), 255–271. https://doi.org/10.1016/j.isatra.2019.07.009 doi: 10.1016/j.isatra.2019.07.009
[16]	W. Wu, X. Ma, Y. Zhang, W. Li, Y. Wang, A novel conformable fractional non-homogeneous grey model for forecasting carbon dioxide emissions of BRICS countries, Sci. Total Environ., 707 (2020), 135447. https://doi.org/10.1016/j.scitotenv.2019.135447 doi: 10.1016/j.scitotenv.2019.135447
[17]	W. Wu, X. Ma, B. Zeng, P. Zhang, A conformable fractional non-homogeneous grey forecasting model with adjustable parameters CFNGMA (1, 1, k, c) and its application, J. Grey Syst., 36 (2024), 1–12. Available from: https://jgrey.nuaa.edu.cn/EN/Y2024/V36/I2/1.
[18]	W. Gong, Q. An, Novel conformable fractional order unbiased kernel regularized nonhomogeneous grey model and its applications in energy prediction, Systems, 13 (2025), 527. https://doi.org/10.3390/systems13070527 doi: 10.3390/systems13070527
[19]	C. I. Chen, H. L. Chen, S. P. Chen, Forecasting of foreign exchange rates of Taiwan's major trading partners by novel nonlinear grey Bernoulli model NGBM (1, 1), Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 1194–1204. https://doi.org/10.1016/j.cnsns.2006.08.008 doi: 10.1016/j.cnsns.2006.08.008
[20]	J. Lu, W. Xie, H. Zhou, A. Zhang, An optimized nonlinear grey Bernoulli model and its applications, Neurocomputing, 177 (2016), 206–214. https://doi.org/10.1016/j.neucom.2015.11.032 doi: 10.1016/j.neucom.2015.11.032
[21]	Q. Xiao, M. Gao, X. Xiao, M. Goh, A novel grey Riccati-Bernoulli model and its application for the clean energy consumption prediction, Eng. Appl. Artif. Intell., 95 (2020), 103863. https://doi.org/10.1016/j.engappai.2020.103863 doi: 10.1016/j.engappai.2020.103863
[22]	W. Nie, H. Duan, A novel multivariable grey differential dynamic prediction model with new structures and its application to carbon emissions, Eng. Appl. Artif. Intell., 122 (2023), 106174. https://doi.org/10.1016/j.engappai.2023.106174 doi: 10.1016/j.engappai.2023.106174
[23]	X. Ma, Research on a novel kernel based grey prediction model and its applications, Math. Probl. Eng., 2016 (2016), 5471748. https://doi.org/10.1155/2016/5471748 doi: 10.1155/2016/5471748
[24]	X. Ma, Y. Hu, Z. Liu, A novel kernel regularized nonhomogeneous grey model and its applications, Commun. Nonlinear Sci. Numer. Simul., 48 (2017), 51–62. https://doi.org/10.1016/j.cnsns.2016.12.017 doi: 10.1016/j.cnsns.2016.12.017
[25]	X. Ma, Z. B. Liu, The kernel-based nonlinear multivariate grey model, Appl. Math. Model., 56 (2018), 217–238. https://doi.org/10.1016/j.apm.2017.12.010 doi: 10.1016/j.apm.2017.12.010
[26]	H. Duan, D. Wang, X. Pang, Y. Liu, S. Zeng, A novel forecasting approach based on multi-kernel nonlinear multivariable grey model: a case report, J. Clean. Prod., 260 (2020), 120929. https://doi.org/10.1016/j.jclepro.2020.120929 doi: 10.1016/j.jclepro.2020.120929
[27]	X. Ma, M. Xie, J. A. Suykens, A novel neural grey system model with Bayesian regularization and its applications, Neurocomputing, 456 (2021), 61–75. https://doi.org/10.1016/j.neucom.2021.05.048 doi: 10.1016/j.neucom.2021.05.048
[28]	Y. Hao, X. Ma, L. Song, Y. Xiang, A residual learning-based grey system model and its applications in Electricity Transformer's Seasonal oil temperature forecasting, Eng. Appl. Artif. Intell., 147 (2025), 110260. https://doi.org/10.1016/j.engappai.2025.110260 doi: 10.1016/j.engappai.2025.110260
[29]	X. Ma, Y. Hao, W. Li, An evolutionary features-based neural grey system model and its application, Appl. Math. Model., 145 (2025), 116126. https://doi.org/10.1016/j.apm.2025.116126 doi: 10.1016/j.apm.2025.116126
[30]	Y. H. Pao, G. H. Park, D. J. Sobajic, Learning and generalization characteristics of the random vector functional-link net, Neurocomputing, 6 (1994), 163–180. https://doi.org/10.1016/0925-2312(94)90053-1 doi: 10.1016/0925-2312(94)90053-1
[31]	B. Igelnik, Y. H. Pao, Stochastic choice of basis functions in adaptive function approximation and the functional-link net, IEEE Trans. Neural Netw., 6 (1995), 1320–1329. https://doi.org/10.1109/72.471375 doi: 10.1109/72.471375
[32]	K. K. Xu, H. X. Li, H. D. Yang, Kernel-based random vector functional-link network for fast learning of spatiotemporal dynamic processes, IEEE Trans. Syst. Man Cybern. Syst., 49 (2017), 1016–1026. https://doi.org/10.1109/TSMC.2017.2694018 doi: 10.1109/TSMC.2017.2694018
[33]	V. Kushwaha, N. M. Pindoriya, A SARIMA-RVFL hybrid model assisted by wavelet decomposition for very short-term solar PV power generation forecast, Renew. Energy, 140 (2019), 124–139. https://doi.org/10.1016/j.renene.2019.03.020 doi: 10.1016/j.renene.2019.03.020
[34]	M. A. Al-qaness, A. A. Ewees, H. Fan, L. Abualigah, A. H. Elsheikh, M. Abd Elaziz, Wind power prediction using random vector functional link network with capuchin search algorithm, Ain Shams Eng. J., 14 (2023), 102095. https://doi.org/10.1016/j.asej.2022.102095 doi: 10.1016/j.asej.2022.102095
[35]	W. Gong, G. Zong, An efficient and sparse kernelized grey RVFL network for energy forecasting, Systems, 14 (2026), 257. https://doi.org/10.3390/systems14030257 doi: 10.3390/systems14030257
[36]	M. Wang, K. Wang, Non-convex extreme learning machine with $L_1$ over $L_2$ norm regularization and exponential squared loss for sparse and robust regression, Signal Process., 244 (2026), 110524. https://doi.org/10.1016/j.sigpro.2026.110524 doi: 10.1016/j.sigpro.2026.110524
[37]	B. K. Natarajan, Sparse approximate solutions to linear systems, SIAM J. Comput., 24 (1995), 227–234. https://doi.org/10.1137/S0097539792240406 doi: 10.1137/S0097539792240406
[38]	M. Tao, Minimization of $L_1$ over $L_2$ for sparse signal recovery with convergence guarantee, SIAM J. Sci. Comput., 44 (2022), A770–A797. https://doi.org/10.1137/20M136801X doi: 10.1137/20M136801X
[39]	P. I. Frazier, A tutorial on Bayesian Optimization, arXiv, 2018. https://arXiv.org/abs/1807.02811
[40]	Energy Institute, Statistical Review of World Energy 2025, 2025. Available from: https://www.energyinst.org/statistical-review.

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