Forecasting monthly rainfall using time series modeling and spectral analysis with Fourier model

Document Type : Research/Original/Regular Article

Authors

1 PhD student, Department of Water Engineering, Faculty of Agriculture, Tabriz University, Tabriz, Iran

2 Professor, Department of Water Engineering, Faculty of Agriculture, Tabriz University, Tabriz, Iran

Abstract

Extended Abstract

Introduction

Rainfall plays an important role in maintaining life on Earth and maintaining the balance of ecosystems. It is essential to understand its importance for various environmental, agricultural, and hydrological aspects. Agriculture relies heavily on rainfall for crop growth. Adequate rainfall ensures that the soil remains fertile and productive. Forecasted rainfall is critical for various sectors including disaster management and urban planning. Accurate forecasts enable individuals and organizations to make informed decisions that can reduce risks and increase productivity. This information allows them to effectively plan planting and harvesting schedules and ensure optimal crop performance. Time series models are statistical tools used to analyze and forecast data points collected over time. Common types of models include autoregressive (AR), moving average (MA), and autoregressive integrated moving average (ARIMA) models. These models use a chronological sequence of observations and allow analysts to identify patterns, trends, and seasonal changes that can provide future predictions. Therefore, the purpose of this research is to develop an integrated system to investigate and evaluate the precipitation trend and its changes in the statistical period of 24 years and use it to predict precipitation in the next 5 years. For this purpose, the rainfall data of three stations, Tabriz, Amol and Yazd, which have different climates, were used. On the other hand, for the evaluation of time series and forecasting operations, two models of Fourier series and auto-regression were used, and then the obtained results were analyzed with evaluation criteria and graphic diagrams.

Materials and Methods

In this study, the monthly rainfall time series of three stations in Tabriz, Amol, and Yazd during 24 years (2023-2000) was taken from the database of Mathematica software. First, the data were examined and by applying the outlier data processing, the data that had a significant difference were removed and the data that were not measured were completed with the interpolation method. After the data was completed and evaluated, it was divided into two parts, training and testing. Normally, the ratio of this division in the fields of hydrology and hydraulics is 70 to 30. For this purpose, 70% of the data (2016-2000) was assigned to training and 30% (2023-2017) to the test. Then, to predict monthly rainfall, two models, Fourier and autoregressive, which are based on time series, were used. These methods are described below. Fourier series is the mathematical representation of a periodic function as an infinite sum of sine and cosine functions. This concept is fundamental in various fields such as signal processing, physics, and engineering, and allows complex periodic signals to be analyzed. Auto-Regressive (AR) models work on the principle that the current value of a time series can be expressed as a linear combination of its past values with a random error. The performance comparison of the two models was evaluated using four criteria: root mean square error (RMSE), correlation coefficient (r), Nash Sutcliffe coefficient (NSE), and Wilmot coefficient (WI). The results showed.

Results and Discussion

The results showed that the Fourier model has an average error of 1.21, correlation coefficient of 0.87, Nash Sutcliffe coefficient of 0.74, and Wilmot coefficient of 0.91. The predictions made with the Fourier model were more reliable than the autocorrelation model. The graph related to the Fourier model in all three stations has almost the same trend as the real values and has less difference. But in some places, there are few fits. On the other hand, the graph of the AR model has a big difference from the real values, and in most of the points, it has a poor fit. Considering the scatter diagrams of Fourier model and AR, the scatter diagrams of Fourier model have less dispersion. Therefore, the coefficient of determination for the Fourier model in predicting the monthly rainfall of Tabriz is equal to 0.86, Amol is equal to 0.73, and Yazd is equal to 0.68. For the AR model, it is equal to 0.18 for Tabriz, 0.34 for Amol, and 0.48 for Yazd. These results show that the Fourier model has identified changes in precipitation trends better than the AR model and has provided more reliable predictions. Also, the predicted five-year trends by the Fourier model have more natural changes than the AR model, but on the contrary, the AR model has fewer fluctuations and has changes close to a straight line.

Conclusion

Precipitation forecasting is critical for various sectors including agriculture, natural disaster management, and climate adaptation. Accurate forecasts can significantly impact food security, infrastructure planning, and environmental conservation. Complete and abundant data has a significant impact on the accuracy of predictions and improves it. The Fourier model with the lowest error and the highest degree of correlation estimated acceptable forecasts for the precipitation of three stations, and on the other hand, the forecasts of the trend of the Fourier model for five years have acceptable changes and are somewhat similar to the trend of real precipitation. Fourier series assume that the data are periodic, which may not be true for all precipitation patterns. Many regions experience irregular rainfall distributions that do not fit well into periodic models, leading to inaccurate forecasts. Therefore, in this research, by integrating the probability distribution with the Fourier model, this problem was solved as much as possible, and accurate predictions of precipitation were made. While Fourier models can be effective for short-term forecasting, their performance degrades over longer periods. This limitation is due to the fact that the basic assumptions and periodic nature of the model are not maintained with the extension of the forecasting horizon, making it challenging to accurately predict precipitation beyond a certain time frame. Also, the effectiveness of the Fourier model depends on the quality and temporal resolution of the input data.

Keywords

Main Subjects

References

منابع
آذرم، برهان (1395). سری فوریه در علوم مهندسی مهندسی. ایده‌های نو در علم و فناوری، 1(3)، 39-52.
جوان صمدی، سحر، ترابی آزاد، مسعود، و کرمی خانیکی، علی (1394). پیش‌بینی دمای سطح آب با استفاده از تحلیل فوریه در دریای خزر. مجله علوم و فنون دریایی، 14(4)، 84-97. doi: 10.22113/jmst.2016.12040
رهسپارمنفرد، رضا، کاردار، سعید، و شهروز تهرانی، ایرج (1399). بررسی اصول طراحی ساختمان‌های مسکونی در اقلیم معتدل و مرطوب با رویکرد تهویه طبیعی (نمونه موردی: تحلیل بازشو شبیه‌سازی شده در بنای مسکونی در شهر آمل). علوم و تکنولوژی محیط زیست، 22(12)، 133-146.
صلاحی، برومند، و ملکی مرشت، رقیه (1394). پیش‌بینی و تحلیل تغییرات بارش‌های ماهانۀ شهرستان اردبیل با استفاده از مدل‌های آریما، اتورگرسیو و وینترز. آب و خاک، 29(5)، 1391-1450.  doi: 10.22067/jsw.v29i5.33792
غیور، حسنعلی، و عساکره، حسین (1384). کاربرد مدل‌های فوریه در تخمین دمای ماهانه و پیش‌بینی آینده آن، مطالعه موردی: دمای مشهد. پژوهش‌های جغرافیایی، 77، 83-99.
فتحیان، فرشاد، فاخری‌فرد، احمد، دین‌پژوه، یعقوب، و موسوی ندوشنی، سید‌سعید (1395الف). ارزیابی عملکرد مدل‌های سری زمانی خطی (ARMA) و غیرخطی آستانه‌ای (TAR) در مدل‌سازی جریان روزانه رودخانه (مطالعه موردی: رودخانه‌های حوضه بالادست سد زرینه رود). آب و خاک، 30(5)، 1440-1460.
فتحیان، فرشاد، فاخری‌فرد، احمد، دین‌پژوه، یعقوب، و موسوی ندوشنی، سید‌سعید (1395ب). آزمایش ایستایی و غیرخطی بودن سری‌های زمانی جریان روزانه بر اساس آزمون‌های آماری مختلف (مطالعه موردی: رودخانه‌های حوضه بالادست سد زرینه‌رود). آب و خاک، 30 (4)، 1009-1024.  doi: 10.22067/jsw.v30i4.44103
قدوسی، میثم، مرید، سعید، و دلاور، مجید (1393). مقایسه روش‌های کاهش روند در سری‌های زمانی دما و بارش. هواشناسی کشاورزی، 1(2)، 32-45.
مزارعی بهبهانی، امین، و جعفرزاده، محمدرضا (1392). تحلیل جریان ناپایدار در کانال‌های باز با استفاده از سری فوریه. مجله هیدرولیک، 18 (1)، 63-79. doi: 10.30482/jhyd.2022.344351.1605
 میرزایی، سیدیحیی، چیت‌سازان، منوچهر، و چینی‌پرداز، رحیم (1384). استفاده از تحلیل‌ سری زمانی در تعیین زمان تأخیر رواناب در حوضه جهان‌بین. آب و فاضلاب، 15(3)، 53-59.
 
References
Arfken, G. (1985). Fourier Series. Ch. 14 in Mathematical Methods for Physicists, 3th. Orlando, FL: Academic Press, pp. 760-793.
Azaram, B. (2016). Fourier Series in engineering sciences. New Ideas in Science and Technology, 1(3), 39-52. [In Persian]
Asmat, A., Wahid, S. N. S., & Deni, S. M. (2021). Identifying rainfall patterns using Fourier series: A case of daily rainfall data in Sarawak, Malaysia. In Journal of Physics: Conference Series. IOP Publishing. doi:  10.1088/1742-6596/1988/1/012086
Banihabib, M.E., Bandari, R., & Valipour, M. (2020). Improving daily peak flow forecasts using hybrid Fourier-series autoregressive integrated moving average and recurrent artificial neural network models. Artificial Intelligence, 1(2), 17. doi:10.3390/ai1020017
Bouttier, F., & Marchal, H. (2024). Probabilistic short-range forecasts of high-precipitation events: optimal decision thresholds and predictability limits. Natural Hazards and Earth System Sciences, 24(8), 2793-2816. doi: 10.5194/nhess-24-2793-2024
Cinkus, G., Mazzilli, N., Jourde, H., Wunsch, A., Liesch, T., Ravbar, N., Chen, Z. & Goldscheider, N. (2023). When best is the enemy of good–critical evaluation of performance criteria in hydrological models. Hydrology and Earth System Sciences, 27(13), 2397-2411. doi: 10.5194/hess-27-2397-2023
Cardenas, C. E., Yang, J., Anderson, B. M., Court, L. E., & Brock, K.B. (2019). Advances in auto-segmentation. In Seminars in Radiation Oncology, 185-197. doi: 10.1016/j.semradonc.2019.02.001
Dinpashoh, Y., Fakheri-Fard, A., Khalili, K., Darbandi, S. (2015). Comparing nonlinear time series models and genetic programming for daily river flow forecasting (Case study: Barandouz-Chai River). Journal of Water and Soil Conservation, 22(1), 151-169. dor: 20.1001.1.23222069.1394.22.1.8.0
Dinpashoh, Y., & Shafaei, S. (2018). Analysis of drought characteristics of Tabriz (1951-2015). Water and Soil Science, 28(3), 117-130.  [In Persian]
Espeholt, L., Agrawal, S., Sønderby, C., Kumar, M., Heek, J., Bromberg, C., Gazen, C., Carver, R., Andrychowicz, M., Hickey, J., & Bell, A. (2022). Deep learning for twelve-hour precipitation forecasts. Nature Communications, 13(1), 1-10. doi: 10.1038/s41467-022-32483-x
Fathian, F., Fakheri-Fard, A., Dinpashoh, Y., & Mousavi Nadoushani, S.S. (2016a). Performance evaluation of linear (ARMA) and threshold nonlinear (TAR) time series models in daily river flow modeling (Case study: upstream basin rivers of Zarrineh Roud dam). Water and Soil, 30(5), 1440-1460. doi: 10.22067/jsw.v0i0.48161 [In Persian]
Fathian, F., Fakheri Fard, A., Dinpashoh, Y. & Mousavi Nadoshani, S.S. (2016b). Testing for Stationarity and Nonlinearity of Daily Streamflow Time Series Based on Different Statistical Tests (Case Study: Upstream Basin Rivers of Zarrineh Roud Dam). Water and Soil, 30(4), 1009-1024. doi: 10.22067/jsw.v30i4.44103 [In Persian]
Fathian, F., Fakheri Fard, A., Ouarda, T.B.M.J, Dinpashoh, Y., & Mousavi Nadoushani, S. (2019a) Modeling streamflow time series using nonlinear SETAR-GARCH models. Journal of Hydrology, 573, 82-97. doi.org/10.1016/j.jhydrol.2019.03.072
Fathian, F., Fakheri-Fard, A., Ouarda, T.B.M.J, Dinpashoh, Y., & Mousavi Nadoushani, S.  (2019b). Multiple streamflow time series modeling using VAR–MGARCH approach. Stochastic Environmental Research and Risk Assessment, 33, 407–425.  doi: 10.1007/s00477-019-01651-9
Foufoula-Georgiou, E., Guilloteau, C., Nguyen, P., Aghakouchak, A., Hsu, K. L., Busalacchi, A., Turk, F.J., Peters-Lidard, K., Oki, T., Duan, Q., Krajewski, w., Uijlenhoet, R., Barros, A., Kristetter, P., Logan, W., Hogue, T., Gupta, H., & Levizzani, V. (2020). Advancing precipitation estimation, prediction, and impact studies. Bulletin of the American Meteorological Society, 101(9), E1584. doi: 10.1175/bams-d-20-0014.1
Ghazi, B., & Jeihouni, E. (2022). Projection of temperature and precipitation under climate change in Tabriz, Iran. Arabian Journal of Geosciences, 15(7), 621. doi: 10.1007/s12517-022-09848-z
García-Feal, O., González-Cao, J., Fernández-Nóvoa, D., Astray Dopazo, G., & Gómez-Gesteira, M. (2022). Comparison of machine learning techniques for reservoir outflow forecasting. Natural Hazards and Earth System Sciences Discussions, 22, 1-27. doi: 10.5194/nhess-22-3859-2022
Ghayur, H., & Asakere, H. (2005). Application of Fourier models in estimating monthly temperature and its future forecasting, Case study: Mashhad temperature. Geographical Research, 77, 83-99. [In Persian]
Ghodousi, M., Murid, S., & Delavar, M. (2014). Comparison of detrending methods in temperature and precipitation time series. Agricultural Meteorology, 1(2), 32-45. [In Persian]
Hasanpour Kashani, M., & Dinpashoh, Y. (2012). Evaluation of the efficiency of different estimation methods for missing climatological data. Stochastic Environmental Research and Risk Assessment, 26, 59–71. doi: 10.1007/s00477-011-0536-y
Hodson, T.O. (2022). Root mean square error (RMSE) or mean absolute error (MAE): When to use them or not. Geoscientific Model Development Discussions, 2022, 1-10. doi: 10.5194/gmd-15-5481-2022
Hu, G., Zhao, L., Wu, X., Li, R., Wu, T., Xie, C., Qiao, Y., Shi, J., Li, W., & Cheng, G. (2016). New Fourier-series-based analytical solution to the conduction–convection equation to calculate soil temperature, determine soil thermal properties, or estimate water flux. International Journal of Heat and Mass Transfer, 95, 815-823. doi: /10.1016/j.ijheatmasstransfer.2015.11.078
Javan Samadi, S., Torabi Azad, M., & Karami Khaniki, A. (2015). Forecasting water surface temperature using Fourier analysis in the Caspian Sea. Journal of Marine Science and Technology, 14(4), 84-97. doi: 10.22113/jmst.2016.12040 [In Persian]
Jou, P.H., & Mirhashemi, S.H. (2023). Frequency analysis of extreme daily rainfall over an arid zone of Iran using the Fourier series method. Applied Water Science, 13(1), 16. doi: 10.1007/s13201-022-01823-z
Kaur, J., Parmar, K.S., & Singh, S. (2023). Autoregressive models in environmental forecasting time series: a theoretical and application review. Environmental Science and Pollution Research, 30(8), 19617-19641. doi: 10.1007/s11356-023-25148-9
Li, L. (1996). Use of Fourier series in the analysis of discontinuous periodic structures. Journal of the Optical Society of America A, 13(9), 1870-1876. doi: 10.1364/JOSAA.13.001870
Liang, L., Zhao, L., Gong, Y., Tian, F., & Wang, Z. (2012). Probability distribution of summer daily precipitation in the Huaihe basin of China based on Gamma distribution. Acta Meteorologica Sinica, 26(1), 72-84. doi: 10.1016/j.atmosres.2020.105221
Lima, A.O., Lyra, G.B., Abreu, M.C., Oliveira-Júnior, J.F., Zeri, M., & Cunha-Zeri, G. (2021). Extreme rainfall events over Rio de Janeiro State, Brazil: Characterization using probability distribution functions and clustering analysis. Atmospheric Research, 247, 105221. doi: 10.1016/j.atmosres.2020.105221
Mah Hashim, N., Mohd Deni, S., Shariff, S.S.R., Tahir, W., & Jani, J. (2016). Identification of seasonal rainfall peaks at Kelantan using Fourier Series. In ISFRAM 2015: Proceedings of the International Symposium on Flood Research and Management 2015, p 169-179. Springer Singapore. doi: 10.1007/978-981-10-0500-8_14
Mazarei Behbahani, A., & Jafarzadeh, M. (2023). Analysis of unsteady flow in open channels using Fourier series. Journal of Hydraulics, 18(1), 63-79. doi: 10.30482/jhyd.2022.344351.1605 [In Persian]
McCuen, R.H., Knight, Z., & Cutter, A.G. (2006). Evaluation of the Nash–Sutcliffe efficiency index. Journal of Hydrologic Engineering, 11(6), 597-602. doi: 10.1061/(ASCE)1084-0699(2006)11:6(597)
Mirzaei, S., Chitsazan, M., & Chinipardaz, R. (2004). Using time series analysis to determine runoff delay time in Jahanbin Basin. Journal of Water and Wastewater, 15(3), 53-59. [In Persian]
Parviz, L., & Ghorbanpour, M. (2024). Assimilation of PSO and SVR into an improved ARIMA model for monthly precipitation forecasting. Scientific Reports, 14(1), 12107. doi: 10.1038/s41598-024-63046-3
Pollock, D. S. G., Green, R. C., & Nguyen, T. (Eds.). (1999). Handbook of Time Series Analysis, Signal Processing, and Dynamics. Elsevier.
Rahsparmanfard, R. Kardar, S., & Shahroz Tehrani, I. (2019). Studying the principles of designing residential buildings in temperate and humid climates with a natural ventilation approach (Case study: Simulated opening analysis in a residential building in Amol city). Environmental Science and Technology, 22(12), 133-146. [In Persian]
Ratner, B. (2009). The correlation coefficient: Its values range between + 1/− 1, or do they? Journal of Targeting, Measurement, and Analysis for Marketing, 17(2), 139-142. doi: 10.1057/jt.2009.5
Salahi, B., & Maleki Marsht, R. (2015). Forecasting and analyzing monthly precipitation changes in Ardabil city using ARIMA, autoregressive and Winters models. Water and Soil, 29(5), 2012-2013. doi: 10.22067/jsw.v29i5.33792 [In Persian]
Salas, J. D., Tabios III, G. Q., & Bartolini, P. (1985). Approaches to multivariate modeling of water resources time series 1. JAWRA Journal of the American Water Resources Association, 21(4), 683-708. doi: 10.1111/j.1752-1688.1985.tb05383.x
Singh, P., Joshi, S.D., Patney, R.K., & Saha, K. (2017). The Fourier decomposition method for nonlinear and non-stationary time series analysis. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473(2199), 20160871. doi: 10.1098/rspa.2016.0871
Soleymanpour, R., Parsaee, N., & Banaei, M. (2015). Climate comfort comparison of vernacular and contemporary houses of Iran. Procedia-Social and Behavioral Sciences, 201, 49-61. doi: 10.1016/j.sbspro.2015.08.18
Wang, K., Li, J., Wang, W., Zhang, Z., Wang, X., Wang, Q., Yeh, T.C.J. and Hao, Y. (2023). Calibrating a model of depth to water table using Fourier series and Simpson numerical integration. Journal of Hydrology, 620, 129516. doi: 10.1016/j.jhydrol.2023.129516
Wang, Q., & Yuan, H. (2017). Failure rate prediction based on AR model and residual correction. In 2017 Second International Conference on Reliability Systems Engineering (ICRSE), p1-5. IEEE. doi: 10.1109/ICRSE.2017.8030786
Wirsing, K. (2020). Time frequency analysis of wavelet and Fourier transform. Wavelet Theory, 3-20.
Wolfram Language (2008). WeatherData. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WeatherData.html
Wu, K., & Woo, M. K. (1989). Estimating annual flood probabilities using Fourier series method. JAWRA Journal of the American Water Resources Association, 25(4), 743-750. doi: 10.1111/j.1752-1688.1989.tb05388.x
Yan, C.W., Foo, S.Q., Trinh, V.H., Yeung, D.Y., Wong, K.H., & Wong, W.K. (2024). Fourier Amplitude and Correlation Loss: Beyond Using L2 Loss for Skillful Precipitation Nowcasting. Advances in Neural Information Processing Systems, 37, 100007-100041. doi:10.48550/arXiv.2410.23159
Zhang, Y., Shi, D., He, D., & Shao, D. (2021). Free Vibration Analysis of Laminated Composite Double‐Plate Structure System with Elastic Constraints Based on Improved Fourier Series Method. Shock and Vibration, 2021(1), 8811747. doi: 10.1155/2021/8811747
Water and Soil Management and Modelling
Volume 5, Issue 3
September 2025
Pages 140-155

Files

History

  • Receive Date: 22 February 2025
  • Revise Date: 26 May 2025
  • Accept Date: 30 May 2025

Share

How to cite

Statistics