Entropy-Regularized Nonlinear Auto-Regressive Network with eXogenous Inputs (ER-NARX): A Mathematical Framework for Scalable and Robust Big Data Forecasting Using ITL and Fractional Dynamics

Authors

  • Zulfatri Aini Universitas Islam Negeri Sultan Syarif Kasim Riau
  • Tengku Reza Suka Alaqsa Universitas Islam Negeri Sultan Syarif Kasim Riau

DOI:

https://doi.org/10.55123/jomlai.v4i4.6689

Keywords:

Entropy Regularized , NARX , Mathematical Framework , Big Data , Forecasting , ITL , Fractional Dynamic

Abstract

This study proposes the Entropy-Regularized NARX (ER-NARX) model, which integrates nonlinear autoregressive modeling, entropy-based regularization, and information-theoretic learning for big data forecasting. The NARX model captures temporal dependencies between past outputs and exogenous inputs, while entropy regularization is incorporated to control the uncertainty of model predictions and prevent overfitting. The innovation of this model is its ability to control information flow through entropy regularization, which helps balance predictive accuracy with uncertainty, preventing the model from becoming overly deterministic. By combining these components, the ER-NARX model enhances the stability and robustness of the forecasts and improves its generalization to complex, high-dimensional data. Additionally, fractional dynamics are employed to model long-range memory effects in temporal data to enhancing the model's ability to handle datasets with extended dependencies. The resulting ER-NARX framework provides a mathematically grounded approach to big data forecasting improved performance in a computationally efficient manner. Future research may explore advanced entropy regularization techniques and apply the model to more diverse real-world data with intricate dependencies.

References

R. Sharipov, "Innovative Approaches To Data Analysis In Commercial IT Projects," Universum: технические науки, vol. 9, no. 4 (121), pp. 28-32, 2024.

N. Kashpruk, C. Piskor-Ignatowicz, and J. Baranowski, "Time series prediction in industry 4.0: A comprehensive review and prospects for future advancements," Applied Sciences, vol. 13, no. 22, p. 12374, 2023.

G. P. Papaioannou, C. Dikaiakos, A. Dramountanis, and P. G. Papaioannou, "Analysis and modeling for short-to medium-term load forecasting using a hybrid manifold learning principal component model and comparison with classical statistical models (SARIMAX, Exponential Smoothing) and artificial intelligence models (ANN, SVM): The case of Greek electricity market," Energies, vol. 9, no. 8, p. 635, 2016.

K. U. Apu, M. M. Rahman, A. B. Hoque, and M. Bhuiyan, "Forecasting Future Investment Value With Machine Learning, Neural Networks, And Ensemble Learning: A Meta-Analytic Study," Review of Applied Science and Technology, vol. 1, no. 02, pp. 01-25, 2022.

C. Sweeney, R. J. Bessa, J. Browell, and P. Pinson, "The future of forecasting for renewable energy," Wiley Interdisciplinary Reviews: Energy and Environment, vol. 9, no. 2, e365, 2020.

A. Saghir, K. D. Tran, and K. P. Tran, "Explainable Machine Learning based Control Charts for High-Dimensional Non-Stationary Time Series Data in IoT Systems: Challenges, Methods, and Future Directions," in Computational Techniques for Smart Manufacturing in Industry 5.0, CRC Press, 2025, pp. 166-184.

Z. Wang, T. Hong, H. Li, and M. A. Piette, "Predicting city-scale daily electricity consumption using data-driven models," Advances in Applied Energy, vol. 2, p. 100025, 2021.

A. Abdussamad, H. Daud, R. Sokkalingam, I. K. Khan, A. S. Azad, M. Zubair, and F. Hassan, "Regularized Stacked Autoencoder with Dropout-Layer to Overcome Overfitting in Numerical High-Dimensional Sparse Data," Journal of Advanced Research Design, vol. 129, no. 1, pp. 60-74, 2025.

I. M. Onyejekwe, "Modelling Annual Natural Gas Demand Forecasting Using Non-Linear Autoregressive with Exogenous Input (NARX) Neural Networks," Petroleum & Coal, vol. 67, no. 1, 2025.

W. Masmoudi, A. Djebli, and F. Moussaoui, "Evaluating LSTM and NARX neural networks for wind speed forecasting and energy optimization in Tetouan, Northern Morocco," Energy Exploration & Exploitation, pp. 01445987241309035, 2025.

J. Gawlikowski, C. R. N. Tassi, M. Ali, J. Lee, M. Humt, J. Feng, and X. X. Zhu, "A survey of uncertainty in deep neural networks," Artificial Intelligence Review, vol. 56, Suppl 1, pp. 1513-1589, 2023.

M. Maksimovic and I. S. Maksymov, "Quantum-cognitive neural networks: Assessing confidence and uncertainty with human decision-making simulations," Big Data Cogn. Comput., vol. 9, p. 12, 2025.

A. Liu and G. Van den Broeck, "Tractable regularization of probabilistic circuits," Advances in Neural Information Processing Systems, vol. 34, pp. 3558-3570, 2021.

B. Wu, S. Luo, and C. S. Suh, "A Comprehensive Review of Propagation Models in Complex Networks: From Deterministic to Deep Learning Approaches," arXiv preprint arXiv:2410.02118, 2024.

M. A. Hossain, "Artificial Intelligence-Driven Financial Analytics Models For Predicting Market Risk And Investment Decisions In US Enterprises," ASRC Procedia: Global Perspectives in Science and Scholarship, vol. 1, no. 01, pp. 1066-1095, 2025.

T. Wand, M. Heßler, and O. Kamps, "Memory effects, multiple time scales and local stability in Langevin models of the S&P500 market correlation," Entropy, vol. 25, no. 9, p. 1257, 2023.

T. R. S. Alaqsa, "Forecasting Electricity Consumption in Riau Province Using the Artificial Neural Network (ANN) Feed Forward Backpropagation Algorithm for the 2024-2027," Indonesian Journal of Electronics, Electromedical Engineering, and Medical Informatics, vol. 7, no. 1, 2025.

D. Stamenov, G. Abbiati, and T. Sauder, "Numerical estimation of generalized frequency response functions from time series data using NARX," SSRN 5231027, 2025.

H. Xu, J. Xuan, G. Zhang, and J. Lu, "Trust region policy optimization via entropy regularization for Kullback–Leibler divergence constraint," Neurocomputing, vol. 589, p. 127716, 2024.

Z. Duan, H. Klaudel, and M. Koutny, "ITL semantics of composite Petri nets," The Journal of Logic and Algebraic Programming, vol. 82, no. 2, pp. 95-110, 2013.

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Published

2025-12-20

How to Cite

Zulfatri Aini, & Tengku Reza Suka Alaqsa. (2025). Entropy-Regularized Nonlinear Auto-Regressive Network with eXogenous Inputs (ER-NARX): A Mathematical Framework for Scalable and Robust Big Data Forecasting Using ITL and Fractional Dynamics. JOMLAI: Journal of Machine Learning and Artificial Intelligence, 4(4), 180–189. https://doi.org/10.55123/jomlai.v4i4.6689

Issue

Vol. 4 No. 4 (2025): Desember 2025

Section

Articles

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