Regional monthly rainfall prediction based on CEEMDAN-SSA-BiLSTM coupled modeling

Regional monthly rainfall prediction based on CEEMDAN-SSA-BiLSTM coupled modeling | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Regional monthly rainfall prediction based on CEEMDAN-SSA-BiLSTM coupled modeling Xianqi Zhang, He Ren, Jiawen Liu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3262470/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Accurate rainfall prediction plays a vital role in optimizing water resource management, reducing impacts on water resources and related water conservation and utilization.. This study combines the advantages of CEEMDAN model's ability to handle nonlinear and nonstationary data, SSA model to decompose and reconstruct the data to get the subsequence with spatio-temporal information, BiLSTM model to effectively learn the dependency relationship between the current data and the data of the previous moment, and to use the relationship to predict the rainfall in the future moments to construct the regional monthly rainfall prediction model of CEEMDAN-SSA-BiLSTM and applied it to predict monthly rainfall in Kaifeng City. The findings indicate that the proposed model is effective for accurately predicting monthly rainfall in the city of Kaifeng. Compared with the EMD-SSA-BiLSTM, CEEMDAN-BiLSTM, and BiLSTM models, the CEEMDAN-SSA-BiLSTM model achieves higher accuracy with an average absolute error (MAE) of 3.75, an average absolute percentage error (MAPE) of 5.44%, and a coefficient of determination (R 2 ) of 0.99. Furthermore, the decomposition of monthly rainfall time series signals helps in identifying and revealing cycles and trends in the series, thereby effectively improving the prediction accuracy of monthly rainfall. Biological sciences/Computational biology and bioinformatics/Computational models Earth and environmental sciences/Ecology Earth and environmental sciences/Hydrology Monthly rainfall forecasting CEEMDAN-SSA-BiLSTM data reconstruction Kaifeng City Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 1. Introduction Accurate rainfall prediction is crucial for various aspects, including the efficient utilization of regional water resources, disaster prevention and mitigation, and the protection of related water resources[ 1 ]. Rainfall is affected by a variety of factors, with significant uncertainty and randomness[ 2 ], herefore, improving the accuracy of rainfall prediction has become a focal point and hot topic among scholars both domestically and internationally. Scholars from both domestic and international backgrounds have conducted extensive research on rainfall prediction. Kala et al. addressed the nonlinear and nonsmooth challenges in rainfall prediction modeling by constructing a hybrid model based on adaptive noise (CEEMDAN) and long short-term memory (LSTM). The results show that the hybrid model exhibits better performance in predicting monthly rainfall time series compared to the LSTM model alone[ 5 ]. Lima et al. addressed the issue of missing values in real-time series data by using the Singular Spectrum Analysis (SSA) prediction algorithm, but the Singular Spectrum Analysis suffers from the problem of easy overfitting or loss of information, such as too much information; Mumtaz and others used the Singular SpectrumAnalysis (SSA) prediction algorithm to solve the problems of the actual time series, but there is a tendency to overfitting or loss of information[ 4 ]. Mumtaz and others combined the Complementary Ensemble Empirical Mode Decomposition (CEEMD) with Random Forest (RF) and Kernel Ridge Regression (KRR) algorithms to address the issue of non-stationarity in rainfall forecasting models. They achieved more accurate rainfall predictions compared to non-coupled models[ 7 ]. Johny, K proposed an integrated approach combining Long Short-Term Memory (LSTM) and Multivariate Empirical Mode Decomposition (MEMD) for monthly rainfall prediction. They further augmented the model with Time-Domain Intrinsic Cross-Correlation (TDICC) analysis algorithm[ 9 ]. Hu et al.used Markov chain model to realize the short-term prediction of precipitation from 1958 to 2016 at Menghai Hydrological Station, and the prediction accuracy is high, which shows that Markov chain has high accuracy in the prediction of precipitation in watersheds[ 3 ]. Wu et al.combined the artificial neural network (Aitificial Neural Netw orks (ANN)) with the Moving Average (MA), Singular Spectrum Analysis (SSA), and predicted the daily and monthly precipitation, and the results showed that the daily and monthly precipitation was predicted by the Aitificial Neural Netw ork (ANN), Average (MA) and Singular Spectrum Analysis (SSA), respectively, and predicted daily and monthly precipitation, and the results indicated that the combination of ANN and MA model yielded better prediction performance[ 8 ]. Fu and others proposed a short-term rainfall prediction based on the GUSS and BP-NN algorithms, which was able to complement the traditional linear and nonlinear rainfall prediction methods. and nonlinear rainfall prediction methods[ 6 ]. From the above, it can be seen that the research on rainfall prediction at home and abroad is mainly based on rainfall data statistics for regression analysis, or using a single model into the prediction, there are also two or more models coupled for prediction, in general the coupled model compared with a single model to be more accurate, better prediction effect. Under the dual influence of climate change and human activities, monthly rainfall has obvious uncertainty, nonlinearity, stochasticity, etc. The current research lacks a clear understanding of the intrinsic physical mechanisms underlying rainfall, which greatly reduces the accuracy of rainfall prediction.CEEMDAN has the advantage of dealing with nonlinear and non-stationary problems, SSA model has good global search capability and fast convergence, and BiLSTM model, known for its high prediction accuracy, effectively captures the relationship between current data, previous time steps, and future time steps.Therefore, the paper constructs a coupled model based on CEEMDAN-SSA-BiLSTM and applies it to rainfall prediction in Kaifeng City. 2. Theory and Methods 2.1 Complete Ensemble Empirical Modal Decomposition (CEEMDAN) The Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) method is based on the EMD algorithm improved, while borrowing the EEMD method by adding Gaussian noise and superimposed on each other several times and then averaged to achieve the purpose of canceling the noise[ 10 ]. This method effectively decomposes monthly rainfall time series data into multiple Intrinsic Mode Functions (IMFs) and a residual component.The calculation steps of the CEEMDAN method are as follows: (1) The rainfall time series and the Gaussian white noise series are superimposed on each other to form a new rainfall time series, which is subsequently subjected to EMD decomposition with a first order intrinsic modal component (IMF): $$im{\stackrel{-}{f}}_{1}\left(t\right)=\frac{\sum _{i=1}^{\text{n}}im{f}_{1}^{i}\left(t\right)}{\text{n}} \text{⑴}$$ Here, \(im{\stackrel{-}{f}}_{1}\left(t\right)\) denotes the average component of the first-order IMT component; \(m{f}_{1}^{i}\) denotes the ith IMF component obtained after the first decomposition; n denotes the maximum number of times white noise was added. (2) Calculate the residue of the first order decomposition: \({x}_{1}\left(t\right)=P\left(t\right)-im{\stackrel{-}{f}}_{1}\left(t\right)\) ⑵ (3) A Gaussian white noise sequence is added to the first-order residue \({x}_{1}\left(t\right)\) to obtain the new sequence to be decomposed \({x}_{1}^{{\prime }}\left(t\right)\) , which is then subjected to EDM decomposition to obtain the second-order IMT component: \(im{\stackrel{-}{f}}_{2}\left(t\right)=\frac{\sum _{i=1}^{\text{n}}im{f}_{2}^{i}\left(t\right)}{\text{n}}\) ⑶ (4) Repeat the above steps until the residuals are monotonic, at which point the precipitation time series is expressed as: \(\text{P}\left(\text{t}\right)=\sum _{\text{j}=1}^{\text{m}}im{\stackrel{-}{f}}_{j}\left(t\right)+{x}_{\text{k}}\left(\text{t}\right)\) ⑷ Here, \({x}_{\text{k}}\left(\text{t}\right)\) denotes the final residuals 2.2 Sparrow Search Algorithm (SSA) The Sparrow Search Algorithm (SSA) is an optimization algorithm that mimics the foraging behavior and predator evasion of sparrows for local and global search. In the context of the algorithm, the foraging process of sparrows serves as an analogy for the optimization process[ 19 ]. The algorithm includes three types of sparrows: explorers, joiners and scouts[ 22 ]. Explorers, characterized by higher fitness values, undertake the task of searching for food and sharing foraging areas and methods with other sparrows in the population; joiners generally follow the discoverers, monitor them and compete for food as a way to ensure the predation rate; and scouts send out alarm signals as soon as they find a predator, and all the sparrows engage in antipredator behavior[ 12 ]. Assuming the population of sparrows is n and the search space is d-dimensional, the position information of the sparrows can be abstracted as an n × d matrix. The formula for the finder to update the location is: $${X}_{i,j}^{t+1}=\left\{\begin{array}{c}{X}_{i,j}^{t}exp\left(\frac{-i}{\delta \bullet ite{r}_{max}}\right){ R}_{2}<S\\ {X}_{i,j}^{t}+QL{ R}_{2}\ge S\end{array}\right.$$ 5 Where, \({X}_{i,j}^{t}\) denotes the position of the ith sparrow in dimension j of the search space at the tth iteration; \(i\) denotes the sparrow number; \(ite{r}_{max}\) denotes the maximum number of iterations of the algorithm, \(\delta\) denotes the number of random numbers between (0, 1]; \(Q\) denotes the number of random numbers obeying a positively-targeted distribution; \(L\) denotes the all-one matrix of 1×d; \({R}_{2}\) denotes the warning value, \({R}_{2}\) ∈[0,1]; and \(S\) denotes the safety value. \(S\) ∈[0.5,1]. When \({R}_{2}<S\) , there is no predator in the foraging environment, and the discoverer conducts an extensive search in the region; when \({R}_{2}\ge S\) ,the scout detects the presence of a predator, and the group moves rapidly toward the safe region. The formula for updating the position of the joiner (sparrow) in the sparrow search algorithm can be given as: \({X}_{i,j}^{t+1}=\left\{\begin{array}{c}Qexp\left(\frac{{X}_{worst}^{t}-{X}_{x,j}^{t}}{{i}^{2}}\right) i>\frac{n}{2}\\ {X}_{P}^{t+1}+|{X}_{i,j}^{t}-{X}_{P}^{t+1}|{A}^{+}L i\le \frac{n}{2}\end{array}\right.\) ⑹ Where, \({X}_{P}^{t+1}\) denotes the location with the optimal adaptation controlled by the discoverer at the t+1st iteration; \({X}_{worst}^{t}\) denotes the location with the worst global adaptation; A denotes a 1×d matrix with elements of 1 and − 1 randomly assigned values. When \(i>n/2\) , the \(i\) th accession that failed to obtain food and has too low energy level needs to go to other regions to forage; when \(i\le n/2\) , the \(i\) th accession will follow the action of the discoverer's foraging center and randomly forage near the center position. Scouts make up about 10–20% of the population size, and their updated position is given by the formula: \({X}_{i,j}^{t+1}=\left\{\begin{array}{c}{X}_{best}^{t}+\beta |{X}_{i,j}^{t}-{X}_{best}^{t}| {f}_{i}>{f}_{g}\\ {X}_{i,j}^{t}+K\left[\frac{|{X}_{i,j}^{t}-{X}_{worst}^{t}|}{{(f}_{i}-{f}_{w})+\epsilon }\right] {f}_{i}={f}_{w}\end{array}\right.\) ⑺ Where, \({X}_{best}^{t}\) denotes the current global optimal position; \(\beta\) denotes a positively too distributed random number with mean 0 and variance 1 for controlling the step size; \(K\) denotes the step size control parameter of the sparrow's moving direction; \(\epsilon\) denotes the smallest constant avoiding the denominator to be 0; \({f}_{i}\) denotes the fitness value of the \(i\) th sparrow; \({f}_{g}\) , \({f}_{w}\) denote the current optimal and worst fitness values, respectively. When \({f}_{i}>{f}_{g}\) , the sparrow is at the edge of the population and prone to encounter predators; when \({f}_{i}={f}_{w}\) , the sparrow is at the center of the population and randomly approaches other sparrows; when \({f}_{i}<{f}_{g}\) , the scout does not act. 2.3 Bidirectional Long Short-Term Memory Network (BiLSTM) The Bi-directional Long Short-Term Memory (BiLSTM) network is an extension of the traditional Bi-directional Recurrent Neural Network (BiRNN) where the regular RNN units are replaced with LSTM units, consisting of a forward LSTM and a backward LSTM combined[ 21 ]. The BiLSTM network is capable of recursively responding to the hidden states of the start and end of a sequence during training. This allows for exploring the relationships between the current data, previous moments, and future moments[ 20 ], enabling a deeper understanding of reverse information. It effectively addresses the issue of long-term dependencies and improves the prediction accuracy of the model.[ 23 ]. The construction diagram of the Bidirectional Long Short-Term Memory (BiLSTM) neural network is shown in Fig. 1 . 2.4 CEEMDAN-SSA-BiLSTM model prediction process The flowchart of the CEEMDAN-SSA-BiLSTM rainfall prediction model, which combines CEEMDAN, SSA, and BiLSTM, is shown in Fig. 2 . The specific operational steps are as follows: Step 1 The rainfall information of n consecutive months is selected as input to the model. Step 2 Decompose the original rainfall sequence with the help of CEEMDAN model to obtain k components. Step 3 First, set the number of sparrow populations N, the maximum iteration count M, and the search range for parameter scopes. Then, choose mean square error (MSE) as the objective function for the optimization algorithm. Next, establish the SSA-BiLSTM model that combines the sparrow search algorithm with the bidirectional long short-term memory network (BiLSTM). Step 4 Input the SSA-BiLSTM prediction model for each component separately to get k prediction models . Step 5 Finally, add the predicted values of k prediction models corresponding to each month to obtain the predicted values of rainfall for each month. 3. Example Applications 3.1 Overview of the study area and data sources Kaifeng City is located in the east-central part of Henan Province, with a geographic location of 113°52′15"~115°15′42 "E, 34°11′45"~35°00′20 "N. It is 87.5 km wide from north to south and 125 km long from east to west, with a total area of 6444km², of which 546km² is urban area. Kaifeng City is situated in a warm temperate zone with a semi-humid continental monsoon climate. It experiences windy springs, hot summers, cool autumns, and cold winters with little snowfall.The intra-annual distribution of rainfall in the city is extremely uneven, with around 70% of the annual rainfall occurring between June and September, and the spatial distribution of rainfall is also uneven, generally increasing gradually from northwest to southeast[ 14 ].The location map of Kaifeng City is shown in Fig. 3 . In this study, the average values of measured monthly rainfall in five counties and four districts of Kaifeng City from 2000 to 2019 are selected as data information, in which the first 90% of the data set is used as the training month for rainfall prediction, and rolling prediction of the data is carried out sequentially to compare with the last 10% of the measured values. The monthly rainfall series plot for the study area is shown in Fig. 4 . 3.2 Data decomposition From the monthly rainfall sequence graph in Fig. 4 , it can be observed that the monthly rainfall sequence of Kaifeng City has randomness and instability, which can be considered as a nonlinear time series signal, and with the help of MATLAB software, the CEEMDAN decomposition model [ 15 ] was constructed to decompose the monthly rainfall data of Kaifeng City in the period of 2000–2019, and five modal components and one residual component were obtained. It can be seen from the image that the IMF1 component has the largest fluctuation, the highest frequency, and the shortest wavelength; As the decomposition level increases, the magnitude and trend terms of IMF2-IMF5 gradually decrease, the frequencies reduce, and the wavelengths become longer. The decomposed rainfall series shows better periodicity and smaller fluctuation, which helps the subsequent modeling prediction.The CEEMDAN decomposition results are shown in Fig. 5 . 3.3 Rainfall prediction The monthly rainfall in Kaifeng City from 2000 to 2019 is taken as the prediction sample (totaling 240 months), in which the first 90% of the data set (216 in total, numbered 1 ~ 216) is taken as the training period sample, and rolling prediction is carried out sequentially, and the next 10% (24 in total, numbered 217 ~ 240) is taken as the test period sample for comparing with rolling prediction value. Based on the previously described steps, the prediction of rainfall in Kaifeng City from 2000 to 2019 is carried out, and the prediction results obtained through a large number of tests are shown in Fig. 6 . As can be seen from the comparison chart in Fig. 6 , the CEEMDAN-SSA-BiLSTM coupled model in the prediction of rainfall in Kaifeng City, most of the predicted data and the actual measurements basically coincide with each other, and there are only a few data with large deviations, which indicates that the prediction accuracy of the model is better, and from this, it can be introduced that the model has a rationality. 4 Discussion In order to verify the prediction effect of the CEEMDAN-SSA-BiLSTM model, four models, namely, CEEMDAN-SSA-BiLSTM, EMD-SSA-BiLSTM, CEEMDAN-BiLSTM, and BiLSTM, are selected in this study to predict the monthly rainfall in Kaifeng City in 2018–2019 in turn. In order to better quantify the prediction performance of the above four models, the root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and correlation coefficient (R^2) are introduced as the evaluation indexes in this study, The results of each performance metric are shown in Table 1 . The calculation formulas are as follows: \(\text{R}\text{M}\text{S}\text{E}=\sqrt{\frac{1}{n}\sum _{i=1}^{n}{({\text{P}}_{i}-{\text{P}}_{i}^{\text{*}})}^{2}}\) ⑻ \(\text{M}\text{A}\text{E}=\frac{1}{n}\sum _{i=1}^{n}|{\text{P}}_{i}-{\text{P}}_{i}^{\text{*}}|\) ⑼ \(\text{M}\text{A}\text{P}\text{E}=\frac{1}{n}\sum _{i=1}^{n}\left|\frac{{\text{P}}_{i}-{\text{P}}_{i}^{\text{*}}}{{\text{P}}_{i}}\right|\) ⑽ \({R}^{2}=1-\frac{\sum _{i=1}^{n}{({\text{P}}_{i}-{\text{P}}_{i}^{\text{*}})}^{2}}{\sum _{i=1}^{n}{({P}_{i}-\stackrel{-}{{P}_{i}})}^{2}}\) ⑾ Table 1 Predictive performance statistics of the four models 模型 MAE/mm RMSE/mm MAPE/% R² CEEMDAN-SSA-BiLSTM 3.75 4.23 5.44 0.99 EMD-SSA-BiLSTM 7.95 9.42 10.98 0.96 CEEMDAN-BiLSTM 9.15 10.66 21.36 0.95 BiLSTM 26.87 42.42 60.89 0.21 Where: \({\text{P}}_{i}\) denotes the measured value of monthly rainfall., \({\text{P}}_{i}^{\text{*}}\) denotes the predicted monthly rainfall; \(\stackrel{-}{{P}_{i}}\) denotes the mean of rainfall measurements. From the specific data aspects of the above table, it can be observed that the CEEMDAN-SSA-BiLSTM model constructed in this paper, compared with the other three models, reduces the root-mean-square error by 38.19, 6.43, and 5.19, respectively, and improves the correlation coefficient value by 0.78, 0.04, and 0.03, respectively, compared with the other three models.In conclusion, the CEEMDAN-SSA-BiLSTM model exhibits lower values for RMSE, MAE, and MAPE, and higher values for the correlation coefficient, indicating that its predictive performance is significantly better than the other models. Additionally, from the table, it can be observed that the overall predictive performance of the EMD-SSA-BiLSTM model is slightly higher than that of the CEEMDAN-BiLSTM model, with an increase in average absolute error, root mean square error, and average absolute percentage error by 1.2, 1.24, and 10.38, respectively. The main reason for this difference is that SSA optimization algorithm is a more flexible tool for the analysis, prediction, and feature extraction of monthly precipitation time series data. Furthermore, the performance of a single BiLSTM model is the lowest, which indicates that the overall predictive performance of a single model is lower than that of a coupled model. Overall, in terms of the predictive performance of the four models, the CEEMDAN-SSA-BiLSTM model performs better. By plotting the predicted values of the four models against the measured values, the comparative effectiveness of these models can be visually demonstrated. As can be seen from Fig. 7 , using a single BiLSTM model in monthly rainfall prediction can only roughly reflect the linear pattern of the rainfall sequence, but there is still a large error between the actual predicted value and the true value. Compared with the use of the CEEMDAN-BiLSTM model, the CEEMDAN-SSA-BiLSTM model with SSA optimization has better prediction results and greater accuracy.The results of the prediction made by the CEEMDAN-SSA-BiLSTM and EMD-SSA-BiLSTM models are more overlapped with the actual observations, which may not be discernible from Fig. 7 alone, and further The CEEMDAN-SSA-BiLSTM has a slightly higher prediction accuracy than the EMD-SSA-BiLSTM model, as can be seen from the scatter plot in Fig. 8 . By analyzing the Taylor diagrams of the prediction results of the four models shown in Fig. 9 , it can be concluded that the CEEMDAN-SSA-BiLSTM model constructed in this study exhibits higher correlation coefficients and smaller standard deviations compared to the other three models. This indicates that the prediction results of the CEEMDAN-SSA-BiLSTM model are more closely aligned with the observed values, demonstrating higher prediction accuracies. The use of a single BiLSTM model is less accurate in the prediction of regional monthly rainfall, which may be due to the fact that the direct prediction of a single time series cannot fully take into account the physical situation that affects rainfall. Due to the complexity of the mechanism of rainfall formation, it makes the single model prediction accuracy poor and cannot meet the actual prediction needs.CEEMDAN is an improved adaptive decomposition method based on the EMD algorithm. Compared to EMD, CEEMDAN has better performance in handling randomness and non-stationarity. CEEMDAN effectively improves the stationarity of time series and exhibits higher stability, addressing the mode mixing problem in EMD. By introducing adaptive adjustments to the added noise, CEEMDAN can accurately extract local features and periodic components from the data, resulting in more accurate and reliable decomposition results. Therefore, CEEMDAN has been widely used in time series analysis and prediction.The data are decomposed by CEEMDAN, and the BiLSTM optimal model parameters are further optimized by virtue of the good global search ability of the SSA model as well as the fast convergence, and the coupling forms the CEEMDAN-SSA-BiLSTM model with high prediction accuracy. 5 Conclusion In this paper, a monthly rainfall prediction model based on CEEMDAN-SSA-BiLSTM is constructed, and the following conclusions are obtained by analyzing the rainfall monitoring data of Kaifeng City, Henan Province, from January 2000 to December 2019, as follows: (1)The CEEMDAN model is a signal decomposition method with multiple advantages such as adaptivity, completeness, suppression of noise control and high computational efficiency.The SSA model is a model that does not require any assumptions or predefinition of the data, and the algorithm is adapted to the time series data of monthly rainfall in this paper.The BiLSTM model has the advantages of strong modeling ability, bi-directional flow of information, high accuracy, and robustness, etc. In this paper, based on the advantages of the three models, we construct the CEEMDAN-SSA-BiLSTM model and apply it to the monthly rainfall forecast of Kaifeng city in Henan province. The BiLSTM model has the advantages of strong modeling capability, bi-directional information flow, high accuracy and robustness. In this paper, the CEEMDAN-SSA-BiLSTM model is constructed based on the advantages of the three models and applied to the monthly rainfall prediction of Kaifeng City, Henan Province. The results show that the model is feasible for regional monthly rainfall prediction. (2)The CEEMDAN method is used to decompose the original rainfall series, and with the help of SSA optimization algorithm, the volatility of the rainfall series is reduced and the stability of the rainfall series is enhanced, which provides a good condition for model coupling. Compared to the other three models, the proposed CEEMDAN-SSA-BiLSTM model yielded smaller values for root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE), while achieving higher correlation coefficients. This indicates that the CEEMDAN-SSA-BiLSTM model exhibits superior performance in rainfall prediction. (3)The predictive accuracy of the model after CEEMDAN decomposition is significantly improved, and the combination of "decomposition-prediction-reconstruction" pattern has a great advantage in predicting rainfall. It can be seen that the signal decomposition method can help the prediction model better learn the inherent features of the rainfall sequence, such as cycles and trends, thereby gaining a better understanding of the underlying structure of the data, predicting future trends and patterns, and conducting missing value imputation and analysis interpretation, thereby enhancing the predictive accuracy of monthly rainfall. (4)The study only carried out a single-step prediction of monthly rainfall in the study area, while the sparrow algorithm in function optimization, it is more accurate than the traditional particle swarm algorithm and gray wolf algorithm, better ability to find the optimal, but it is also easy to be trapped in the local optimum and multi-dimensional function solving accuracy is poor and other problems. The next step is the focus of the research in terms of algorithm improvement and model transformation for each research area. Declarations Availability of data and materials Data and materials are available from the corresponding author upon request. Author contribution All authors contributed to the study conception and design. writing and editing: Xianqi Zhang and He Ren; preliminary data collection: Jiawen Liu. All authors read and approved the final manuscript. Funding This work was supported by the Key Scientific Research Project of Colleges and Universities in Henan Province (CN) [grant numbers 17A570004]. Ethical Approval Not applicable. Consent to Participate Not applicable. Consent to Publish Not applicable. Competing interests None. References Sit, M., Demiray, B. Z., Xiang, Z., Ewing, G. J., Sermet, Y., & Demir, I. (2020). A comprehensive review of deep learning applications in hydrology and water resources. Water Science and Technology, 82(12), 2635–2670. McMillan, H., Krueger, T., & Freer, J. (2012). Benchmarking observational uncertainties for hydrology: rainfall, river discharge and water quality.Hydrological Processes,26(26), 4078–4111. Hu, X., Huang, Q., & Shu, K. (2018). Precipitation Prediction In The Liusha River Basin Based On Markov Chain.Journal Of Water Resources Research,7(4), 398–403. Lima, G. A. R. (2016, October). Gap filling of precipitation data by SSA-singular spectrum analysis. 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Vibration Signal Features Prediction of GIS Equipment Based on Improved Slime Mold Optimization Algorithm Optimizing CNN-BiLSTM.Traitement du Signal,39(4). Raval, M., Sivashanmugam, P., Pham, V., Gohel, H., Kaushik, A., & Wan, Y. (2021). Automated predictive analytics tool for rainfall forecasting.Scientific Reports,11(1), 17704. Guo, S., Wen, Y., Zhang, X., & Chen, H. (2023). Runoff prediction of lower Yellow River based on CEEMDAN–LSSVM–GM (1, 1) model.Scientific Reports,13(1), 1511. Liu, Y., Zhao, Q., Yao, W., Ma, X., Yao, Y., & Liu, L. (2019). Short-term rainfall forecast model based on the improved BP–NN algorithm.Scientific reports,9(1), 19751. Zhou, J., Xu, Z., & Wang, S. (2022). A novel hybrid learning paradigm with feature extraction for carbon price prediction based on Bi-directional long short-term memory network optimized by an improved sparrow search algorithm. Environmental Science and Pollution Research, 29(43), 65585–65598. Lu, J., Fu, Y., Yue, J., Zhu, L., Wang, D., & Hu, Z. (2022). Natural gas pipeline leak diagnosis based on improved variational modal decomposition and locally linear embedding feature extraction method. Process Safety and Environmental Protection, 164, 857–867. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3262470","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":226895524,"identity":"a842a351-a945-4ada-a510-dd70899a9dfb","order_by":0,"name":"Xianqi Zhang","email":"","orcid":"","institution":"Water Conservancy College, North China University of Water Resources and Electric Power","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Xianqi","middleName":"","lastName":"Zhang","suffix":""},{"id":226895525,"identity":"d0cd8321-4dce-4397-95cd-078e0c672826","order_by":1,"name":"He Ren","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA00lEQVRIiWNgGAWjYBACfv72gw8kDCSY+RkOHyBOi+SMM8kGFhU27JKNxxKI02JwIMFMouJMGr/B4TMGRLrswIE0iZtth6UNjp35eOMNg52cbgMBHYzNjYctZ7YdNpY8c3az5RyGZGOzAwS0MDMcSLwt2XY4me/G2W3SPEDeNkJa2BgSDKT/th2ub7j/5hlxWngYEowkJM6kMQscOMNGnBag8mQDiQobZsmGY8aWcwyI8Iv9eURUPrzxpsJOjqAWVCt5iI0aJC2k6hgFo2AUjIIRAQChpkliThn+rwAAAABJRU5ErkJggg==","orcid":"","institution":"Water Conservancy College, North China University of Water Resources and Electric Power","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"He","middleName":"","lastName":"Ren","suffix":""},{"id":226895526,"identity":"8e0713a5-58ee-47cd-8347-4d4b0029403f","order_by":2,"name":"Jiawen Liu","email":"","orcid":"","institution":"Water Conservancy College, North China University of Water Resources and Electric Power","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Jiawen","middleName":"","lastName":"Liu","suffix":""}],"badges":[],"createdAt":"2023-08-14 11:14:26","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3262470/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3262470/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":41957720,"identity":"60174ad8-62f1-42a8-aead-571fd7b1aa3e","added_by":"auto","created_at":"2023-08-22 20:45:09","extension":"jpeg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":129984,"visible":true,"origin":"","legend":"\u003cp\u003eConstruction diagram for Bidirectional Long Short-Term Memory Neural Network\u003c/p\u003e","description":"","filename":"floatimage1.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3262470/v1/fa1a85c99562905119f159a1.jpeg"},{"id":41957719,"identity":"1463340b-0f40-453f-8acd-b613f9d3158b","added_by":"auto","created_at":"2023-08-22 20:45:09","extension":"jpeg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":215910,"visible":true,"origin":"","legend":"\u003cp\u003eFlowchart of the CEEMDAN-SSA-BiLSTM Model\u003c/p\u003e","description":"","filename":"floatimage2.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3262470/v1/d7a26384709b5bf490cfe6bb.jpeg"},{"id":41957726,"identity":"d6196df7-a7c9-4e18-8130-bb92819294f0","added_by":"auto","created_at":"2023-08-22 20:45:09","extension":"jpeg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":3345319,"visible":true,"origin":"","legend":"\u003cp\u003eLocation map of the study area\u003c/p\u003e","description":"","filename":"floatimage3.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3262470/v1/18875429fa60ed636a3660c2.jpeg"},{"id":41957721,"identity":"451d7431-1314-471c-b96e-e717f9f6097b","added_by":"auto","created_at":"2023-08-22 20:45:09","extension":"jpeg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":1439676,"visible":true,"origin":"","legend":"\u003cp\u003ePlot of monthly rainfall series\u003c/p\u003e","description":"","filename":"floatimage4.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3262470/v1/29a05505fd0f167db9591e6e.jpeg"},{"id":41958200,"identity":"318e2974-924c-4ac0-a3be-c3401bc4cbd9","added_by":"auto","created_at":"2023-08-22 20:53:09","extension":"jpeg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":46618,"visible":true,"origin":"","legend":"\u003cp\u003eCEEMDAN decomposition diagram\u003c/p\u003e","description":"","filename":"floatimage5.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3262470/v1/1400006a395c20a6a37a4134.jpeg"},{"id":41958201,"identity":"be3a94ab-2863-4ddb-80e7-e4ac0d3480fa","added_by":"auto","created_at":"2023-08-22 20:53:09","extension":"jpeg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":873627,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of monthly rainfall forecasts\u003c/p\u003e","description":"","filename":"floatimage6.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3262470/v1/539b8ed3ecfdeec72dbed6bb.jpeg"},{"id":41957722,"identity":"d4698b2c-9e06-4697-bc43-3438585d358c","added_by":"auto","created_at":"2023-08-22 20:45:09","extension":"jpeg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":1099421,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of forecast results among the four models for the given period.\u003c/p\u003e","description":"","filename":"floatimage7.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3262470/v1/b7297078a0c5c61b953b9268.jpeg"},{"id":41957724,"identity":"0d00715e-10be-4347-8bd2-c128ebb36137","added_by":"auto","created_at":"2023-08-22 20:45:09","extension":"jpeg","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":998667,"visible":true,"origin":"","legend":"\u003cp\u003eScatterplot of Prediction Results for the Four Models in the Validation Period\u003c/p\u003e","description":"","filename":"floatimage8.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3262470/v1/bebe22a3097b9d2fd916098e.jpeg"},{"id":41958202,"identity":"7ef861e7-1be6-482b-a3f8-ec912f2c1c44","added_by":"auto","created_at":"2023-08-22 20:53:09","extension":"jpeg","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":1256390,"visible":true,"origin":"","legend":"\u003cp\u003eTaylor diagram of the prediction results of the four models\u003c/p\u003e","description":"","filename":"floatimage9.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3262470/v1/452d71da0c361546abb6ab12.jpeg"},{"id":44012576,"identity":"4b9c6141-f78c-4205-8595-186e126d8a4d","added_by":"auto","created_at":"2023-10-03 10:52:33","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1010080,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3262470/v1/5ff34f2e-0850-4231-9540-0cbfb75986ed.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Regional monthly rainfall prediction based on CEEMDAN-SSA-BiLSTM coupled modeling","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eAccurate rainfall prediction is crucial for various aspects, including the efficient utilization of regional water resources, disaster prevention and mitigation, and the protection of related water resources[\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. Rainfall is affected by a variety of factors, with significant uncertainty and randomness[\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e], herefore, improving the accuracy of rainfall prediction has become a focal point and hot topic among scholars both domestically and internationally. Scholars from both domestic and international backgrounds have conducted extensive research on rainfall prediction. Kala et al. addressed the nonlinear and nonsmooth challenges in rainfall prediction modeling by constructing a hybrid model based on adaptive noise (CEEMDAN) and long short-term memory (LSTM). The results show that the hybrid model exhibits better performance in predicting monthly rainfall time series compared to the LSTM model alone[\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. Lima et al. addressed the issue of missing values in real-time series data by using the Singular Spectrum Analysis (SSA) prediction algorithm, but the Singular Spectrum Analysis suffers from the problem of easy overfitting or loss of information, such as too much information; Mumtaz and others used the Singular SpectrumAnalysis (SSA) prediction algorithm to solve the problems of the actual time series, but there is a tendency to overfitting or loss of information[\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. Mumtaz and others combined the Complementary Ensemble Empirical Mode Decomposition (CEEMD) with Random Forest (RF) and Kernel Ridge Regression (KRR) algorithms to address the issue of non-stationarity in rainfall forecasting models. They achieved more accurate rainfall predictions compared to non-coupled models[\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e]. Johny, K proposed an integrated approach combining Long Short-Term Memory (LSTM) and Multivariate Empirical Mode Decomposition (MEMD) for monthly rainfall prediction. They further augmented the model with Time-Domain Intrinsic Cross-Correlation (TDICC) analysis algorithm[\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. Hu et al.used Markov chain model to realize the short-term prediction of precipitation from 1958 to 2016 at Menghai Hydrological Station, and the prediction accuracy is high, which shows that Markov chain has high accuracy in the prediction of precipitation in watersheds[\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. Wu et al.combined the artificial neural network (Aitificial Neural Netw orks (ANN)) with the Moving Average (MA), Singular Spectrum Analysis (SSA), and predicted the daily and monthly precipitation, and the results showed that the daily and monthly precipitation was predicted by the Aitificial Neural Netw ork (ANN), Average (MA) and Singular Spectrum Analysis (SSA), respectively, and predicted daily and monthly precipitation, and the results indicated that the combination of ANN and MA model yielded better prediction performance[\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. Fu and others proposed a short-term rainfall prediction based on the GUSS and BP-NN algorithms, which was able to complement the traditional linear and nonlinear rainfall prediction methods. and nonlinear rainfall prediction methods[\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eFrom the above, it can be seen that the research on rainfall prediction at home and abroad is mainly based on rainfall data statistics for regression analysis, or using a single model into the prediction, there are also two or more models coupled for prediction, in general the coupled model compared with a single model to be more accurate, better prediction effect. Under the dual influence of climate change and human activities, monthly rainfall has obvious uncertainty, nonlinearity, stochasticity, etc. The current research lacks a clear understanding of the intrinsic physical mechanisms underlying rainfall, which greatly reduces the accuracy of rainfall prediction.CEEMDAN has the advantage of dealing with nonlinear and non-stationary problems, SSA model has good global search capability and fast convergence, and BiLSTM model, known for its high prediction accuracy, effectively captures the relationship between current data, previous time steps, and future time steps.Therefore, the paper constructs a coupled model based on CEEMDAN-SSA-BiLSTM and applies it to rainfall prediction in Kaifeng City.\u003c/p\u003e"},{"header":"2. Theory and Methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Complete Ensemble Empirical Modal Decomposition (CEEMDAN)\u003c/h2\u003e \u003cp\u003eThe Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) method is based on the EMD algorithm improved, while borrowing the EEMD method by adding Gaussian noise and superimposed on each other several times and then averaged to achieve the purpose of canceling the noise[\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. This method effectively decomposes monthly rainfall time series data into multiple Intrinsic Mode Functions (IMFs) and a residual component.The calculation steps of the CEEMDAN method are as follows:\u003c/p\u003e \u003cp\u003e(1) The rainfall time series and the Gaussian white noise series are superimposed on each other to form a new rainfall time series, which is subsequently subjected to EMD decomposition with a first order intrinsic modal component (IMF):\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$im{\\stackrel{-}{f}}_{1}\\left(t\\right)=\\frac{\\sum _{i=1}^{\\text{n}}im{f}_{1}^{i}\\left(t\\right)}{\\text{n}} \\text{⑴}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eHere,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(im{\\stackrel{-}{f}}_{1}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e denotes the average component of the first-order IMT component; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(m{f}_{1}^{i}\\)\u003c/span\u003e\u003c/span\u003e denotes the ith IMF component obtained after the first decomposition; n denotes the maximum number of times white noise was added.\u003c/p\u003e \u003cp\u003e(2) Calculate the residue of the first order decomposition:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({x}_{1}\\left(t\\right)=P\\left(t\\right)-im{\\stackrel{-}{f}}_{1}\\left(t\\right)\\)\u003c/span\u003e \u003c/span\u003e ⑵\u003c/p\u003e \u003cp\u003e(3) A Gaussian white noise sequence is added to the first-order residue \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{1}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e to obtain the new sequence to be decomposed \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{1}^{{\\prime }}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e, which is then subjected to EDM decomposition to obtain the second-order IMT component:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(im{\\stackrel{-}{f}}_{2}\\left(t\\right)=\\frac{\\sum _{i=1}^{\\text{n}}im{f}_{2}^{i}\\left(t\\right)}{\\text{n}}\\)\u003c/span\u003e \u003c/span\u003e ⑶\u003c/p\u003e \u003cp\u003e(4) Repeat the above steps until the residuals are monotonic, at which point the precipitation time series is expressed as:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\text{P}\\left(\\text{t}\\right)=\\sum _{\\text{j}=1}^{\\text{m}}im{\\stackrel{-}{f}}_{j}\\left(t\\right)+{x}_{\\text{k}}\\left(\\text{t}\\right)\\)\u003c/span\u003e \u003c/span\u003e ⑷\u003c/p\u003e \u003cp\u003eHere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({x}_{\\text{k}}\\left(\\text{t}\\right)\\)\u003c/span\u003e\u003c/span\u003e denotes the final residuals\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Sparrow Search Algorithm (SSA)\u003c/h2\u003e \u003cp\u003eThe Sparrow Search Algorithm (SSA) is an optimization algorithm that mimics the foraging behavior and predator evasion of sparrows for local and global search. In the context of the algorithm, the foraging process of sparrows serves as an analogy for the optimization process[\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]. The algorithm includes three types of sparrows: explorers, joiners and scouts[\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e]. Explorers, characterized by higher fitness values, undertake the task of searching for food and sharing foraging areas and methods with other sparrows in the population; joiners generally follow the discoverers, monitor them and compete for food as a way to ensure the predation rate; and scouts send out alarm signals as soon as they find a predator, and all the sparrows engage in antipredator behavior[\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eAssuming the population of sparrows is n and the search space is d-dimensional, the position information of the sparrows can be abstracted as an n \u0026times; d matrix.\u003c/p\u003e \u003cp\u003eThe formula for the finder to update the location is:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$${X}_{i,j}^{t+1}=\\left\\{\\begin{array}{c}{X}_{i,j}^{t}exp\\left(\\frac{-i}{\\delta \\bullet ite{r}_{max}}\\right){ R}_{2}\u0026lt;S\\\\ {X}_{i,j}^{t}+QL{ R}_{2}\\ge S\\end{array}\\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{i,j}^{t}\\)\u003c/span\u003e\u003c/span\u003e denotes the position of the ith sparrow in dimension j of the search space at the tth iteration; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(i\\)\u003c/span\u003e\u003c/span\u003e denotes the sparrow number; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(ite{r}_{max}\\)\u003c/span\u003e\u003c/span\u003e denotes the maximum number of iterations of the algorithm, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\delta\\)\u003c/span\u003e\u003c/span\u003e denotes the number of random numbers between (0, 1]; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(Q\\)\u003c/span\u003e\u003c/span\u003e denotes the number of random numbers obeying a positively-targeted distribution; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(L\\)\u003c/span\u003e\u003c/span\u003e denotes the all-one matrix of 1\u0026times;d; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({R}_{2}\\)\u003c/span\u003e\u003c/span\u003e denotes the warning value,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({R}_{2}\\)\u003c/span\u003e\u003c/span\u003e\u0026isin;[0,1]; and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S\\)\u003c/span\u003e\u003c/span\u003e denotes the safety value. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S\\)\u003c/span\u003e\u003c/span\u003e\u0026isin;[0.5,1]. When\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({R}_{2}\u0026lt;S\\)\u003c/span\u003e\u003c/span\u003e, there is no predator in the foraging environment, and the discoverer conducts an extensive search in the region; when\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({R}_{2}\\ge S\\)\u003c/span\u003e\u003c/span\u003e,the scout detects the presence of a predator, and the group moves rapidly toward the safe region.\u003c/p\u003e \u003cp\u003eThe formula for updating the position of the joiner (sparrow) in the sparrow search algorithm can be\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003egiven as:\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({X}_{i,j}^{t+1}=\\left\\{\\begin{array}{c}Qexp\\left(\\frac{{X}_{worst}^{t}-{X}_{x,j}^{t}}{{i}^{2}}\\right) i\u0026gt;\\frac{n}{2}\\\\ {X}_{P}^{t+1}+|{X}_{i,j}^{t}-{X}_{P}^{t+1}|{A}^{+}L i\\le \\frac{n}{2}\\end{array}\\right.\\)\u003c/span\u003e \u003c/span\u003e ⑹\u003c/p\u003e \u003cp\u003eWhere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{P}^{t+1}\\)\u003c/span\u003e\u003c/span\u003e denotes the location with the optimal adaptation controlled by the discoverer at the t+1st iteration; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{worst}^{t}\\)\u003c/span\u003e\u003c/span\u003e denotes the location with the worst global adaptation; A denotes a 1\u0026times;d matrix with elements of 1 and \u0026minus;\u0026thinsp;1 randomly assigned values. When \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(i\u0026gt;n/2\\)\u003c/span\u003e\u003c/span\u003e, the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(i\\)\u003c/span\u003e\u003c/span\u003eth accession that failed to obtain food and has too low energy level needs to go to other regions to forage; when \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(i\\le n/2\\)\u003c/span\u003e\u003c/span\u003e, the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(i\\)\u003c/span\u003e\u003c/span\u003eth accession will follow the action of the discoverer's foraging center and randomly forage near the center position.\u003c/p\u003e \u003cp\u003eScouts make up about 10\u0026ndash;20% of the population size, and their updated position is given by the formula:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({X}_{i,j}^{t+1}=\\left\\{\\begin{array}{c}{X}_{best}^{t}+\\beta |{X}_{i,j}^{t}-{X}_{best}^{t}| {f}_{i}\u0026gt;{f}_{g}\\\\ {X}_{i,j}^{t}+K\\left[\\frac{|{X}_{i,j}^{t}-{X}_{worst}^{t}|}{{(f}_{i}-{f}_{w})+\\epsilon }\\right] {f}_{i}={f}_{w}\\end{array}\\right.\\)\u003c/span\u003e \u003c/span\u003e ⑺\u003c/p\u003e \u003cp\u003eWhere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{best}^{t}\\)\u003c/span\u003e\u003c/span\u003e denotes the current global optimal position; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\beta\\)\u003c/span\u003e\u003c/span\u003e denotes a positively too distributed random number with mean 0 and variance 1 for controlling the step size; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(K\\)\u003c/span\u003e\u003c/span\u003e denotes the step size control parameter of the sparrow's moving direction; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\epsilon\\)\u003c/span\u003e\u003c/span\u003e denotes the smallest constant avoiding the denominator to be 0; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({f}_{i}\\)\u003c/span\u003e\u003c/span\u003e denotes the fitness value of the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(i\\)\u003c/span\u003e\u003c/span\u003eth sparrow; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({f}_{g}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({f}_{w}\\)\u003c/span\u003e\u003c/span\u003e denote the current optimal and worst fitness values, respectively. When \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({f}_{i}\u0026gt;{f}_{g}\\)\u003c/span\u003e\u003c/span\u003e, the sparrow is at the edge of the population and prone to encounter predators; when \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({f}_{i}={f}_{w}\\)\u003c/span\u003e\u003c/span\u003e, the sparrow is at the center of the population and randomly approaches other sparrows; when \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({f}_{i}\u0026lt;{f}_{g}\\)\u003c/span\u003e\u003c/span\u003e, the scout does not act.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3 Bidirectional Long Short-Term Memory Network (BiLSTM)\u003c/h2\u003e \u003cp\u003eThe Bi-directional Long Short-Term Memory (BiLSTM) network is an extension of the traditional Bi-directional Recurrent Neural Network (BiRNN) where the regular RNN units are replaced with LSTM units, consisting of a forward LSTM and a backward LSTM combined[\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e]. The BiLSTM network is capable of recursively responding to the hidden states of the start and end of a sequence during training. This allows for exploring the relationships between the current data, previous moments, and future moments[\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e], enabling a deeper understanding of reverse information. It effectively addresses the issue of long-term dependencies and improves the prediction accuracy of the model.[\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]. The construction diagram of the Bidirectional Long Short-Term Memory (BiLSTM) neural network is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e2.4 CEEMDAN-SSA-BiLSTM model prediction process\u003c/h2\u003e \u003cp\u003eThe flowchart of the CEEMDAN-SSA-BiLSTM rainfall prediction model, which combines CEEMDAN, SSA, and BiLSTM, is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. The specific operational steps are as follows:\u003c/p\u003e \u003cp\u003eStep 1 The rainfall information of n consecutive months is selected as input to the model.\u003c/p\u003e \u003cp\u003eStep 2 Decompose the original rainfall sequence with the help of CEEMDAN model to obtain k components.\u003c/p\u003e \u003cp\u003eStep 3 First, set the number of sparrow populations N, the maximum iteration count M, and the search range for parameter scopes. Then, choose mean square error (MSE) as the objective function for the optimization algorithm. Next, establish the SSA-BiLSTM model that combines the sparrow search algorithm with the bidirectional long short-term memory network (BiLSTM).\u003c/p\u003e \u003cp\u003eStep 4 Input the SSA-BiLSTM prediction model for each component separately to get k prediction models .\u003c/p\u003e \u003cp\u003eStep 5 Finally, add the predicted values of k prediction models corresponding to each month to obtain the predicted values of rainfall for each month.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"3. Example Applications","content":"\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Overview of the study area and data sources\u003c/h2\u003e \u003cp\u003eKaifeng City is located in the east-central part of Henan Province, with a geographic location of 113\u0026deg;52\u0026prime;15\"~115\u0026deg;15\u0026prime;42 \"E, 34\u0026deg;11\u0026prime;45\"~35\u0026deg;00\u0026prime;20 \"N. It is 87.5 km wide from north to south and 125 km long from east to west, with a total area of 6444km\u0026sup2;, of which 546km\u0026sup2; is urban area. Kaifeng City is situated in a warm temperate zone with a semi-humid continental monsoon climate. It experiences windy springs, hot summers, cool autumns, and cold winters with little snowfall.The intra-annual distribution of rainfall in the city is extremely uneven, with around 70% of the annual rainfall occurring between June and September, and the spatial distribution of rainfall is also uneven, generally increasing gradually from northwest to southeast[\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e].The location map of Kaifeng City is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn this study, the average values of measured monthly rainfall in five counties and four districts of Kaifeng City from 2000 to 2019 are selected as data information, in which the first 90% of the data set is used as the training month for rainfall prediction, and rolling prediction of the data is carried out sequentially to compare with the last 10% of the measured values.\u003c/p\u003e \u003cp\u003eThe monthly rainfall series plot for the study area is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Data decomposition\u003c/h2\u003e \u003cp\u003eFrom the monthly rainfall sequence graph in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e, it can be observed that the monthly rainfall sequence of Kaifeng City has randomness and instability, which can be considered as a nonlinear time series signal, and with the help of MATLAB software, the CEEMDAN decomposition model [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] was constructed to decompose the monthly rainfall data of Kaifeng City in the period of 2000\u0026ndash;2019, and five modal components and one residual component were obtained. It can be seen from the image that the IMF1 component has the largest fluctuation, the highest frequency, and the shortest wavelength; As the decomposition level increases, the magnitude and trend terms of IMF2-IMF5 gradually decrease, the frequencies reduce, and the wavelengths become longer. The decomposed rainfall series shows better periodicity and smaller fluctuation, which helps the subsequent modeling prediction.The CEEMDAN decomposition results are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Rainfall prediction\u003c/h2\u003e \u003cp\u003eThe monthly rainfall in Kaifeng City from 2000 to 2019 is taken as the prediction sample (totaling 240 months), in which the first 90% of the data set (216 in total, numbered 1\u0026thinsp;~\u0026thinsp;216) is taken as the training period sample, and rolling prediction is carried out sequentially, and the next 10% (24 in total, numbered 217\u0026thinsp;~\u0026thinsp;240) is taken as the test period sample for comparing with rolling prediction value.\u003c/p\u003e \u003cp\u003eBased on the previously described steps, the prediction of rainfall in Kaifeng City from 2000 to 2019 is carried out, and the prediction results obtained through a large number of tests are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAs can be seen from the comparison chart in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e, the CEEMDAN-SSA-BiLSTM coupled model in the prediction of rainfall in Kaifeng City, most of the predicted data and the actual measurements basically coincide with each other, and there are only a few data with large deviations, which indicates that the prediction accuracy of the model is better, and from this, it can be introduced that the model has a rationality.\u003c/p\u003e \u003c/div\u003e"},{"header":"4 Discussion","content":"\u003cp\u003eIn order to verify the prediction effect of the CEEMDAN-SSA-BiLSTM model, four models, namely, CEEMDAN-SSA-BiLSTM, EMD-SSA-BiLSTM, CEEMDAN-BiLSTM, and BiLSTM, are selected in this study to predict the monthly rainfall in Kaifeng City in 2018\u0026ndash;2019 in turn.\u003c/p\u003e \u003cp\u003eIn order to better quantify the prediction performance of the above four models, the root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and correlation coefficient (R^2) are introduced as the evaluation indexes in this study, The results of each performance metric are shown in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. The calculation formulas are as follows:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\text{R}\\text{M}\\text{S}\\text{E}=\\sqrt{\\frac{1}{n}\\sum _{i=1}^{n}{({\\text{P}}_{i}-{\\text{P}}_{i}^{\\text{*}})}^{2}}\\)\u003c/span\u003e \u003c/span\u003e ⑻\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\text{M}\\text{A}\\text{E}=\\frac{1}{n}\\sum _{i=1}^{n}|{\\text{P}}_{i}-{\\text{P}}_{i}^{\\text{*}}|\\)\u003c/span\u003e \u003c/span\u003e ⑼\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\text{M}\\text{A}\\text{P}\\text{E}=\\frac{1}{n}\\sum _{i=1}^{n}\\left|\\frac{{\\text{P}}_{i}-{\\text{P}}_{i}^{\\text{*}}}{{\\text{P}}_{i}}\\right|\\)\u003c/span\u003e \u003c/span\u003e ⑽\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({R}^{2}=1-\\frac{\\sum _{i=1}^{n}{({\\text{P}}_{i}-{\\text{P}}_{i}^{\\text{*}})}^{2}}{\\sum _{i=1}^{n}{({P}_{i}-\\stackrel{-}{{P}_{i}})}^{2}}\\)\u003c/span\u003e \u003c/span\u003e ⑾\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003ePredictive performance statistics of the four models\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003e模型\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMAE/mm\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eRMSE/mm\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMAPE/%\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eR\u0026sup2;\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCEEMDAN-SSA-BiLSTM\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.75\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4.23\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e5.44\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.99\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eEMD-SSA-BiLSTM\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e7.95\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e9.42\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e10.98\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.96\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCEEMDAN-BiLSTM\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e9.15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e10.66\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e21.36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.95\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBiLSTM\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e26.87\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e42.42\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e60.89\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.21\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eWhere: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{P}}_{i}\\)\u003c/span\u003e\u003c/span\u003e denotes the measured value of monthly rainfall., \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\text{P}}_{i}^{\\text{*}}\\)\u003c/span\u003e\u003c/span\u003e denotes the predicted monthly rainfall; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\stackrel{-}{{P}_{i}}\\)\u003c/span\u003e\u003c/span\u003e denotes the mean of rainfall measurements. From the specific data aspects of the above table, it can be observed that the CEEMDAN-SSA-BiLSTM model constructed in this paper, compared with the other three models, reduces the root-mean-square error by 38.19, 6.43, and 5.19, respectively, and improves the correlation coefficient value by 0.78, 0.04, and 0.03, respectively, compared with the other three models.In conclusion, the CEEMDAN-SSA-BiLSTM model exhibits lower values for RMSE, MAE, and MAPE, and higher values for the correlation coefficient, indicating that its predictive performance is significantly better than the other models.\u003c/p\u003e \u003cp\u003eAdditionally, from the table, it can be observed that the overall predictive performance of the EMD-SSA-BiLSTM model is slightly higher than that of the CEEMDAN-BiLSTM model, with an increase in average absolute error, root mean square error, and average absolute percentage error by 1.2, 1.24, and 10.38, respectively. The main reason for this difference is that SSA optimization algorithm is a more flexible tool for the analysis, prediction, and feature extraction of monthly precipitation time series data. Furthermore, the performance of a single BiLSTM model is the lowest, which indicates that the overall predictive performance of a single model is lower than that of a coupled model.\u003c/p\u003e \u003cp\u003eOverall, in terms of the predictive performance of the four models, the CEEMDAN-SSA-BiLSTM model performs better. By plotting the predicted values of the four models against the measured values, the comparative effectiveness of these models can be visually demonstrated.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAs can be seen from Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e, using a single BiLSTM model in monthly rainfall prediction can only roughly reflect the linear pattern of the rainfall sequence, but there is still a large error between the actual predicted value and the true value. Compared with the use of the CEEMDAN-BiLSTM model, the CEEMDAN-SSA-BiLSTM model with SSA optimization has better prediction results and greater accuracy.The results of the prediction made by the CEEMDAN-SSA-BiLSTM and EMD-SSA-BiLSTM models are more overlapped with the actual observations, which may not be discernible from Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e alone, and further The CEEMDAN-SSA-BiLSTM has a slightly higher prediction accuracy than the EMD-SSA-BiLSTM model, as can be seen from the scatter plot in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eBy analyzing the Taylor diagrams of the prediction results of the four models shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e, it can be concluded that the CEEMDAN-SSA-BiLSTM model constructed in this study exhibits higher correlation coefficients and smaller standard deviations compared to the other three models. This indicates that the prediction results of the CEEMDAN-SSA-BiLSTM model are more closely aligned with the observed values, demonstrating higher prediction accuracies.\u003c/p\u003e \u003cp\u003eThe use of a single BiLSTM model is less accurate in the prediction of regional monthly rainfall, which may be due to the fact that the direct prediction of a single time series cannot fully take into account the physical situation that affects rainfall. Due to the complexity of the mechanism of rainfall formation, it makes the single model prediction accuracy poor and cannot meet the actual prediction needs.CEEMDAN is an improved adaptive decomposition method based on the EMD algorithm. Compared to EMD, CEEMDAN has better performance in handling randomness and non-stationarity. CEEMDAN effectively improves the stationarity of time series and exhibits higher stability, addressing the mode mixing problem in EMD. By introducing adaptive adjustments to the added noise, CEEMDAN can accurately extract local features and periodic components from the data, resulting in more accurate and reliable decomposition results. Therefore, CEEMDAN has been widely used in time series analysis and prediction.The data are decomposed by CEEMDAN, and the BiLSTM optimal model parameters are further optimized by virtue of the good global search ability of the SSA model as well as the fast convergence, and the coupling forms the CEEMDAN-SSA-BiLSTM model with high prediction accuracy.\u003c/p\u003e"},{"header":"5 Conclusion","content":"\u003cp\u003eIn this paper, a monthly rainfall prediction model based on CEEMDAN-SSA-BiLSTM is constructed, and the following conclusions are obtained by analyzing the rainfall monitoring data of Kaifeng City, Henan Province, from January 2000 to December 2019, as follows:\u003c/p\u003e \u003cp\u003e(1)The CEEMDAN model is a signal decomposition method with multiple advantages such as adaptivity, completeness, suppression of noise control and high computational efficiency.The SSA model is a model that does not require any assumptions or predefinition of the data, and the algorithm is adapted to the time series data of monthly rainfall in this paper.The BiLSTM model has the advantages of strong modeling ability, bi-directional flow of information, high accuracy, and robustness, etc. In this paper, based on the advantages of the three models, we construct the CEEMDAN-SSA-BiLSTM model and apply it to the monthly rainfall forecast of Kaifeng city in Henan province. The BiLSTM model has the advantages of strong modeling capability, bi-directional information flow, high accuracy and robustness. In this paper, the CEEMDAN-SSA-BiLSTM model is constructed based on the advantages of the three models and applied to the monthly rainfall prediction of Kaifeng City, Henan Province. The results show that the model is feasible for regional monthly rainfall prediction.\u003c/p\u003e \u003cp\u003e(2)The CEEMDAN method is used to decompose the original rainfall series, and with the help\u003c/p\u003e \u003cp\u003eof SSA optimization algorithm, the volatility of the rainfall series is reduced and the stability of the rainfall series is enhanced, which provides a good condition for model coupling. Compared to the other three models, the proposed CEEMDAN-SSA-BiLSTM model yielded smaller values for root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE), while achieving higher correlation coefficients. This indicates that the CEEMDAN-SSA-BiLSTM model exhibits superior performance in rainfall prediction.\u003c/p\u003e \u003cp\u003e(3)The predictive accuracy of the model after CEEMDAN decomposition is significantly improved, and the combination of \"decomposition-prediction-reconstruction\" pattern has a great advantage in predicting rainfall. It can be seen that the signal decomposition method can help the prediction model better learn the inherent features of the rainfall sequence, such as cycles and trends, thereby gaining a better understanding of the underlying structure of the data, predicting future trends and patterns, and conducting missing value imputation and analysis interpretation, thereby enhancing the predictive accuracy of monthly rainfall.\u003c/p\u003e \u003cp\u003e(4)The study only carried out a single-step prediction of monthly rainfall in the study area, while the sparrow algorithm in function optimization, it is more accurate than the traditional particle swarm algorithm and gray wolf algorithm, better ability to find the optimal, but it is also easy to be trapped in the local optimum and multi-dimensional function solving accuracy is poor and other problems. The next step is the focus of the research in terms of algorithm improvement and model transformation for each research area.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAvailability of data and materials\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eData and materials are available from the corresponding author upon request.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contribution\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAll authors contributed to the study conception and design. writing and editing: Xianqi Zhang and He Ren; preliminary data collection: Jiawen Liu. All authors read and approved the final manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis work was supported by the Key Scientific Research Project of Colleges and Universities in Henan Province (CN) [grant numbers 17A570004].\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eEthical Approval\u0026nbsp;\u003c/strong\u003eNot applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConsent to Participate\u0026nbsp;\u003c/strong\u003eNot applicable.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConsent to Publish\u0026nbsp;\u003c/strong\u003eNot applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests\u0026nbsp;\u003c/strong\u003eNone.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eSit, M., Demiray, B. Z., Xiang, Z., Ewing, G. J., Sermet, Y., \u0026amp; Demir, I. (2020). 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Process Safety and Environmental Protection, 164, 857\u0026ndash;867.\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Monthly rainfall, forecasting, CEEMDAN-SSA-BiLSTM, data reconstruction, Kaifeng City","lastPublishedDoi":"10.21203/rs.3.rs-3262470/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3262470/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAccurate rainfall prediction plays a vital role in optimizing water resource management, reducing impacts on water resources and related water conservation and utilization.. This study combines the advantages of CEEMDAN model's ability to handle nonlinear and nonstationary data, SSA model to decompose and reconstruct the data to get the subsequence with spatio-temporal information, BiLSTM model to effectively learn the dependency relationship between the current data and the data of the previous moment, and to use the relationship to predict the rainfall in the future moments to construct the regional monthly rainfall prediction model of CEEMDAN-SSA-BiLSTM and applied it to predict monthly rainfall in Kaifeng City. The findings indicate that the proposed model is effective for accurately predicting monthly rainfall in the city of Kaifeng. Compared with the EMD-SSA-BiLSTM, CEEMDAN-BiLSTM, and BiLSTM models, the CEEMDAN-SSA-BiLSTM model achieves higher accuracy with an average absolute error (MAE) of 3.75, an average absolute percentage error (MAPE) of 5.44%, and a coefficient of determination (R\u003csup\u003e2\u003c/sup\u003e) of 0.99. Furthermore, the decomposition of monthly rainfall time series signals helps in identifying and revealing cycles and trends in the series, thereby effectively improving the prediction accuracy of monthly rainfall.\u003c/p\u003e","manuscriptTitle":"Regional monthly rainfall prediction based on CEEMDAN-SSA-BiLSTM coupled modeling","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-08-22 20:45:04","doi":"10.21203/rs.3.rs-3262470/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"fef73654-8f11-42e8-bc3a-9daf53fab47f","owner":[],"postedDate":"August 22nd, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":24092144,"name":"Biological sciences/Computational biology and bioinformatics/Computational models"},{"id":24092145,"name":"Earth and environmental sciences/Ecology"},{"id":24092146,"name":"Earth and environmental sciences/Hydrology"}],"tags":[],"updatedAt":"2023-10-03T10:44:24+00:00","versionOfRecord":[],"versionCreatedAt":"2023-08-22 20:45:04","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3262470","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3262470","identity":"rs-3262470","version":["v1"]},"buildId":"7rjqhiLT3MXkJMwkYKINL","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}