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

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.