AccScience Publishing / IJOCTA / Online First / DOI: 10.36922/IJOCTA026290164
Cite this article
60
Views
Related Info Links
More by Authors Links
Journal Browser
Volume | Year
Issue
Search
News and Announcements
View All
RESEARCH ARTICLE

A weierstrass-based fractal feature framework with monte carlo parameter optimization for financial time-series forecasting

Rabha W. Ibrahim1,2* ,  Dumitru Baleanu3 ,  Soheil Salahshour4,5
Show Less
1 Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences SIMATS, Chennai, Tamil Nadu , India
2 Information and Communication Technology Research Group, Scientific Research Center, Al-Ayen University, Thi-Qar, Nasiriyah , Iraq
3 Department of Computer Science and Mathematics, Lebanese American University, Beirut , Lebanon
4 Advanced Computing Lab, Faculty of Engineering and Natural Sciences, Istanbul Okan University, Istanbul , Turkey
5 Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul , Turkey
Received: 16 July 2026 | Revised: 14 August 2026 | Accepted: 25 August 2026 | Published online: 6 November 2026
© 2026 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
Abstract

Stock-market forecasting is challenging because financial time series exhibit nonlinearity, volatility clustering, scale-dependent fluctuations, and irregular self-similar behavior. The goal of this research is to create a hybrid forecasting framework for improved stock-price prediction by combining fractal feature extraction, Monte Carlo parameter optimization, and Long Short-Term Memory (LSTM) neural networks. A Weierstrass-based fractal representation is used to describe multiscale oscillatory behavior in financial returns. Several fractal descriptors, including Weierstrass oscillatory features, generalized fractalintegral features, local roughness, volatility scaling, Hurst-exponent features, and fractal-dimension features, are extracted from moving windows of historical stock returns. These features are then incorporated as auxiliary input channels into an LSTM forecasting architecture. The fractal-parameter vector that minimizes validation-set forecasting loss is selected via Monte Carlo optimization. Using actual stock market data, the proposed Monte Carlo fractal-LSTM model is evaluated and compared with benchmark models such as naïve persistence, moving-average forecasting, classical LSTM, and non-optimized fractal- LSTM models. RMSE, MAE, MAPE, and directional accuracy are used to measure forecasting performance. The empirical findings show that the LSTM model’s capacity to represent nonlinear, memory-dependent, and multiscale market dynamics is enhanced by the addition of fractal features. The Monte Carlo fractal–LSTM model achieves lower forecasting errors and higher directional accuracy than the benchmark models in the out-of-sample validation experiment. These results imply that, in addition to raw return sequences, optimized fractal descriptors offer valuable additional information. The present investigation is unique in that it combines LSTM-based time-series forecasting and Monte Carlo parameter selection with Weierstrass-based fractal features. The proposed methodology offers an interpretable approach for incorporating roughness, self-similarity, volatility scaling, and memory effects into financial forecasting by connecting a mathematical fractal model with data-driven deep learning.

Keywords
Weierstrass function
Monte Carlo optimization
Long short-term memory networks
Financial time series
Hurst exponent
Fractal dimension
Stock market forecasting
Funding
None.
Conflict of interest
The authors have no relevant financial or nonfinancial interests to disclose.
References
  1. Khan MA, Ali H, Shabbir H, Noor F, Majid MD. Impact of macroeconomic indicators on stock market predictions: A cross country analysis. J Comput Biomed Inform. 2024;8(1). https://doi.org/10.56979/801/2024
  2. Geng X, Shi B. Application of deep learning-based social media data analysis in macroeconomic forecasting. Discov Artif Intell. 2026. https://doi.org/10.1007/s44163-026-01154-x
  3. Gupta A, Mishra R. A comprehensive study of stock market volatility: Types, determinants, and measurement methods. Int J Multidiscip Res. 2024;6(6):30054. https://doi.org/10.36948/ijfmr.2024.v06i06.30054
  4. Liu Z, Wang Y, Wu J, Chen K. A review of research on the relationship between macroeconomic variables and stock market returns. Highlights Bus Econ Manag. 2025;65:856-865. https://doi.org/10.54097/0nc9fa59
  5. Sakariyahu R, Paterson A, Chatzivgeri E, Lawal R. Chasing noise in the stock market: An inquiry into the dynamics of investor sentiment and asset pricing. Rev Quant Finance Account. 2024;62(1):135-169. https://doi.org/10.1007/s11156-023-01214-8
  6. Saravanos C, Kanavos A. Forecasting stock market volatility using social media sentiment analysis. Neural Comput Appl. 2025;37(17):10771-10794. https://doi.org/10.1007/s00521-024-10807-w
  7. Naidoo T, Moores-Pitt P, Muzindutsi PF. The effect of investor sentiment on stock market liquidity under changing market conditions: Evidence from South Africa. Cogent Econ Finance. 2025;13(1):2444395. https://doi.org/10.1080/23322039.2024.2444395
  8. Maurya PK, Bansal R, Mishra AK. Investor sentiment and its implication on global financial markets: A systematic review of literature. Qual Res Financ Mark. 2026;18(1):209-248. https://doi.org/10.1108/QRFM-04-2024-0087
  9. Saberironaghi M, Ren J, Saberironaghi A. Stock market prediction using machine learning and deep learning techniques: A review. AppliedMath. 2025;5(3):76. https://doi.org/10.3390/appliedmath5030076
  10. Peter M, Mirau S, Sinkwembe E, Kasumo C, Guambe C. Symmetric and asymmetric GARCH estimations of the impact of macroeconomic uncertainties on stock market dynamics in Tanzania. Future Bus J. 2025;11(1):203. https://doi.org/10.1186/s43093-025-00632-5
  11. Ayad A. Applications of fractal geometry in geosciences: A literature. Int Res J Multidiscip Scope. 2025;6:1472-1485. https://doi.org/10.47857/irjms.2025.v06i01.02581
  12. Nahapetyan AA. Fractal geometry and quantitative evaluation of the aesthetic appeal of ancient Armenian architecture monuments. Math Probl Comput Sci. 2025;63:42-53. https://doi.org/10.51408/1963-0130
  13. Uthayakumar P, Tharani S, Agathiyan A, Akgül A. Analysis on historical data of stock indices using fractal dimension methods by measuring the complexity. J Comput Appl Math. 2026;474:117008. https://doi.org/10.1016/j.cam.2025.117008
  14. Adjemi S, Biomy M, Ouchenane D, Zitouni S, Pham VT. Scaling laws and multifractal spectrum transformation under fractional calculus: Theory and AI-based identification. Fractals. 2026:2740002. https://doi.org/10.1142/S0218348X27400020
  15. Qiu Y, Liang Y. Progress on fractal dimensions of the Weierstrass function and Weierstrass-type functions. Fractal Fract. 2025;9(3):143. https://doi.org/10.3390/fractalfract9030143
  16. Diaz Palencia JL, Rahman SU, Reyes EG. Asymptotic analysis and explicit Weierstrass traveling waves for a Darcy-Forchheimer flow model. ZAMM J Appl Math Mech. 2026;106(8):e70512. https://doi.org/10.1002/zamm.70512
  17. Gepreel KA, El-Horbaty M, Aljohani HM, Akbar MA. Dynamics of optical solitons of the pure cubic derivative NLSE with multiplicative white noise using analytical schemes. AIP Adv. 2026;16(1).
  18. Demulder S, Knysh M, Rolph A. Krylov exponents and power spectra for maximal quantum chaos: An EFT approach. J High Energy Phys. 2026;2026(4):59.
  19. Guler E, Toda M. Weierstrass-type constructions, variational analysis and integral-free minimal immersions in Rⁿ. Electron Res Arch. 2026;34(3):1885-1899.
  20. Li Z, Yang Y. The localized nonlinear excitation on the doubly periodic backgrounds governed by Weierstrass-type elliptic waves of the fifth-order KdV equation. Chin Phys B. 2026.
  21. Shams M, Carpentieri B. Hybrid Caputo-type fractional parallel schemes for nonlinear elliptic PDEs with chaos- and bifurcation-based acceleration. Fractal Fract. 2026;10(3):142.
  22. Shams M, Velichko A, Carpentieri B. Finite-time contractivity profiling of a two-parameter parallel root-finding scheme via a kNN-LLE proxy. Mathematics. 2026;14(5):879.
  23. Ibrahim RW, AlSheikh MH. Model validation pipeline against longitudinal Alzheimer's biomarker data. Neuroinformatics. 2026;24(2):33.
  24. Ibrahim RW. Image edge detections based on fractional deformed harmonic univalent functions. Comput Math Model. 2026;2026:1-50. https://doi.org/10.1007/s10598-026-09714-7
  25. Ibrahim RW. Quantum-deformed growth geometry in the open unit disk, with applications in subordination, extremal functions, and phase transitions. Arab J Math Sci. 2026:1-17.
  26. Deng Q, Xu C, Lin J, Zhao Y. Bifurcation mechanism, speed feedback controller, and hybrid controller design in a delayed tumor-immune competitive model. AIP Adv. 2025;15(9).
  27. Xu C, Balci E. Hunting cooperation and gestation delay in a prey-predator model with fractional derivative. J Appl Anal Comput. 2026;16(2):1035-1053.
  28. Lin J, Xu C, Zhao Y, Deng Q. Hopf bifurcation and controller design for a predator-prey model with double delays. AIP Adv. 2025;15(7).
  29. Xu C, Pang Y, Liao M, Shen J, Liu Z. A stochastic approach to leptospirosis disease in human and vector populations: Analysis, dynamics and validation with worldwide real data. J Biol Syst. 2026;34(2):375-398.
  30. Hurst HE. Long-term storage capacity of reservoirs. Trans Am Soc Civ Eng. 1951;116(1):770-799.
Share
Back to top
An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing