Multifractal Analysis and ARFIMA Modeling in Predicting Organic Pollution of the Danube River

Abstract:

This paper presents an investigation into the complex dynamics and enhanced predictability of organic water pollution using a multifractal methodology. The study utilizes a time series dataset of water pollution measures, where time-series analysis and the Multi-Fractal Detrended Fluctuation Analysis (MFDFA) were individually applied to both Biochemical Oxygen Demand (BOD) and Chemical Oxygen Demand (COD) metrics. By analyzing and comparing the characteristics of each metric, the research establishes a deeper insight into the overall organic pollution dynamics, which informs the subsequent modeling. The MFDFA results were used to accurately estimate the fractal differentiation parameter (d) for the ARFIMA (AutoRegressive Fractionally Integrated Moving Average) model. The resulting forecasts for pollution were evaluated and compared against those obtained from the traditional ARIMA (AutoRegressive Integrated Moving Average) model and a basic ARFIMA fractal model. Comparative analysis, utilizing performance metrics such as Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE), conclusively demonstrates that the methodology integrating multifractal analysis and the ARFIMA fractal model provides an enhancement in predictive accuracy for water pollution time series for metrics where high multifractal evidence was detected through the MFDFA, thereby validating the utility of fractal methods in capturing the underlying complexity and long-memory dependencies of highly heterogeneous environmental processes.

• Periodical: Advances in Science and Technology (Volume 174)
• Pages: 109-119
• DOI: https://doi.org/10.4028/p-B8TQja
• Online since: April 2026
• Authors: Mykhailo Bordun*, Yaroslav Sokolovskyy
• Keywords: ARFIMA, ARIMA, Autoregressive Model, Biochemical Oxygen Demand, Chemical Oxygen Demand, Fractal Model, Long Memory, MFDFA, Multifractal Analysis, Time Series

References

[1] L. Lin, H. Yang, X. Xu, Effects of Water Pollution on Human Health and Disease Heterogeneity: A Review, Front. Environ. Sci. 10 (2022) 880246.

[2] R. Schwarzenbach, T. Egli, T. Hofstetter, U. Gunten, B. Wehrli, Global Water Pollution and Human Health, Ann Rev Environ Resour. 35 (2010) 109-136.

DOI: 10.1146/annurev-environ-100809-125342

[3] P. Chowdhary, R.N. Bharagava, S. Mishra, N. Khan, Role of Industries in Water Scarcity and Its Adverse Effects on Environment and Human Health, in: V. Shukla, N. Kumar (Eds.), Environmental Concerns and Sustainable Development, Springer, Singapore, 2020, pp.235-256.

DOI: 10.1007/978-981-13-5889-0_12

[4] F. Lyu, Research on the Current Status of Global Water Pollution and Corresponding Countermeasures, Highl. Sci. Eng. Technol. 141 (2025) 14-19.

[5] M.T.H. van Vliet, J. Thorslund, M. Strokal, N. Hofstra, M. Flörke, H. Ehalt Macedo, A. Nkwasa, T. Tang, S.S. Kaushal, R. Kumar, A. van Griensven, L. Bouwman, L.M. Mosley, Global river water quality under climate change and hydroclimatic extremes, Nat. Rev. Earth Environ. 4 (2023) 687-702.

DOI: 10.1038/s43017-023-00472-3

[6] Saad, F. Asari, S. Affandi, A. Zid, River pollution: a mini review of causes and effects, Journal of Tourism, Hospitality and Environment Management. 7 (2022) 139-151.

DOI: 10.35631/jthem.729011

[7] Y. Wen, G. Schoups, N. van de Giesen, Organic pollution of rivers: Combined threats of urbanization, livestock farming and global climate change, Sci. Rep. 7 (2017) 43289.

DOI: 10.1038/srep43289

[8] Information on https://fieldreport.caes.uga.edu/publications/C992/understanding-laboratory-wastewater-tests-i-organics-bod-cod-toc-og/

[9] A.H. Anusha, A. Jayatunga, Correlation between biochemical oxygen demand (BOD) and chemical oxygen demand (COD) for different industrial waste waters, J. Natl. Sci. Found. 21 (2017) 259-266.

DOI: 10.4038/jnsfsr.v21i2.8110

[10] Makki, B. Khudhair, Correlation Between BOD5 and COD for Al- Diwaniyah Wastewater Treatment Plants to Obtain the Biodigrability Indices, Pak. J. Biotechnol. 15 (2018) 423-427.

[11] P. Biedka, Biodegradation Kinetics of Organic Matter in Water from Sludge Dewatering after Autothermal Thermophilic Aerobic Digestion, Energies. 16 (2023) 203.

DOI: 10.3390/en16010203

[12] H. Mekaoussi, S. Heddam, N. Bouslimanni, S. Kim, M. Zounemat-Kermani, Predicting biochemical oxygen demand in wastewater treatment plant using advance extreme learning machine optimized by Bat algorithm, Heliyon. 9 (2023) e21351.

DOI: 10.1016/j.heliyon.2023.e21351

[13] Mr. Patil, Dr. Gidde, Prediction Of BOD And COD For Sewage Treatment Plant Using Artificial Neural Network Approach. Int. J. Environ. Sci. 11 (2025) 2090-2102.

DOI: 10.64252/byt1t116

[14] Y. Bai, M. Peng, M. Wang, A River Water Quality Prediction Method Based on Dual Signal Decomposition and Deep Learning, Water. 16 (2024) 3099.

DOI: 10.3390/w16213099

[15] S.-S. Baek, J. Pyo, J.A. Chun, Prediction of Water Level and Water Quality Using a CNN-LSTM Combined Deep Learning Approach, Water. 12 (2020) 3399.

DOI: 10.3390/w12123399

[16] K. Sun, L. Hu, J. Sun, X. Cao, Enhancing groundwater level prediction accuracy at a daily scale through combined machine learning and physics-based modeling, J. Hydrol. Reg. Stud. 50 (2023) 101577.

DOI: 10.1016/j.ejrh.2023.101577

[17] S. Sheng, K. Lin, Y. Zhou, H. Chen, Y. Luo, S. Guo, C.Y. Xu, Exploring a multi-output temporal convolutional network driven encoder-decoder framework for ammonia nitrogen forecasting, J. Environ. Manag. 342 (2023) 118232.

DOI: 10.1016/j.jenvman.2023.118232

[18] Q. Tian, W. Luo, L. Guo, Water quality prediction in the Yellow River source area based on the DeepTCN-GRU model, J. Water Process Eng. 59 (2024) 105052.

DOI: 10.1016/j.jwpe.2024.105052

[19] J. Song, H. Meng, Y. Kang, M. Zhu, Y. Zhu, J.A. Zhang, A method for predicting water quality of river basin based on OVMD-GAT-GRU, Stoch. Environ. Res. Risk Assess. 38 (2024) 339-356.

DOI: 10.1007/s00477-023-02584-0

[20] X. Wang, Y. Wang, P. Yuan, L. Wang, D. Cheng, An adaptive daily runoff forecast model using VMD-LSTM-PSO hybrid approach, Hydrol. Sci. J. 66 (2021) 1488-1502.

DOI: 10.1080/02626667.2021.1937631

[21] M. He, S. Wu, B. Huang, C. Kang, F. Gui, Prediction of Total Nitrogen and Phosphorus in Surface Water by Deep Learning Methods Based on Multi-Scale Feature Extraction, Water. 14 (2022) 1643.

DOI: 10.3390/w14101643

[22] K. Liu, Y. Chen, X. Zhang, An Evaluation of ARFIMA (Autoregressive Fractional Integral Moving Average) Programs. Axioms. 6 (2017) 1-16.

DOI: 10.3390/axioms6020016

[23] M. Bordun, O. Mokrytska, Development of software and algorithmic equipment for prediction of river water pollution using fractal analysis methods, CDS. 6 (2024) 178-188.

DOI: 10.23939/cds2024.01.178

[24] M. Bordun, O. Salapak, Mathematical models for the analysis and forecasting of river water pollution using the multifractal method, CDS. 7 (2025) 310-318.

DOI: 10.23939/cds2025.01.310

[25] Y. Sokolovskyy, M. Levkovych, O. Mokrytska, Y. Kaspryshyn, Mathematical Modeling of Nonequilibrium Physical Processes, Taking into Account the Memory Effects and Spatial Correlation, 2019 9th International Conference on Advanced Computer Information Technologies (ACIT), 2019, 56-59.

DOI: 10.1109/acitt.2019.8780011

[26] Y. Sokolovskyy, K. Drozd, T. Samotii, I. Boretska, Fractional-Order Modeling of Heat and Moisture Transfer in Anisotropic Materials Using a Physics-Informed Neural Network, Materials 17 (2024) 4753.

DOI: 10.3390/ma17194753

[27] H. Kireitseva, I. Tsyhanenko-Dziubenko, K. Sheliah, T. Levytska, V. Kalenska, Mathematical forecasting of spatio-temporal dynamics of hydroecological parameters of river ecosystems using integrally-modified Streeter-Phelps model, Environmental Problems 10 (2025) 309–318.

DOI: 10.23939/ep2025.03.309

[28] G. Trokhymenko, N. Magas, O. Shumilova, V. Klochko, Analysis of surface water quality indicators in the Dnipro-Bug estuary region in the first months after the destruction of the Kakhovka hydroelectric power station dam, Environmental Problems 8 (2023) 231–240.

DOI: 10.23939/ep2023.04.231

[29] O. Stepanova, S. Shara, Development of water protection measures in European countries: Relevance for Ukraine, Environmental Problems 9 (2024) 249–253.

[30] Information on https://data.gov.ua/dataset/surface-water-monitoring

[31] D. Safitri, Mustafid, D. Ispriyanti, Sugito, Gold price modeling in Indonesia using ARFIMA method, J. Phys.: Conf. Ser. 1217 (2019) 012087.

DOI: 10.1088/1742-6596/1217/1/012087

[32] H. Al-Tameemi, M. Jabbar, A. Bader, BOD: COD Ratio as Indicator for Wastewater and Industrial Water Pollution, Int. J. Spec. Educ. 37 (2022) 1-18.

[33] X. Zhang, G. Zhang, L. Qiu, B. Zhang, Y. Sun, Z. Gui, Q. Zhang, A Modified Multifractal Detrended Fluctuation Analysis (MFDFA) Approach for Multifractal Analysis of Precipitation in Dongting Lake Basin, China, Water. 11 (2019) 891.

DOI: 10.3390/w11050891

[34] L. Ding, Y. Luo, Y. Lin, Y. Huang, Revisiting the Relations Between Hurst Exponent and Fractional Differencing Parameter for Long Memory, Phys. A: Stat. Mech. Appl. 566 (2021) 125603.

DOI: 10.1016/j.physa.2020.125603