Precipitation prediction based on CEEMDAN–VMD–BILSTM combined quadratic decomposition model

The occurrence of water and drought extremes has become increasingly frequent due to global climate change, leading to a severe water resources situation (Mohammadi et al. 2021). Accurate long-term precipitation prediction serves as a critical indicator for efficient water resource utilization. Precipitation, a key source of recharge for regional water resources, significantly impacts various aspects of regional life and production. Precipitation anomalies often result in destructive floods, highlighting the importance of precise precipitation prediction. Monthly precipitation series are influenced by diverse factors, such as the atmosphere, region, and environment, exhibiting substantial ambiguity, contingency, and uncertainty. The study of monthly precipitation is a complex problem involving multiple levels and orders (Radhakrishnan & Dinesh 2006). Scholars worldwide have made substantial efforts to enhance prediction accuracy and optimize prediction models, yielding fruitful results. In recent years, the rapid development of artificial intelligence and modern statistical methods has propelled their application in precipitation prediction. Integrating these advanced techniques with traditional precipitation occurrence models is crucial to improve the reliability of water resource management decisions in scientific research and business (Jones & Pittock 2002).

Consequently, extensive research has been conducted by scholars worldwide, resulting in significant advancements in accurate precipitation prediction. Hua (2022) conducted a study analyzing the spatial and temporal characteristics of precipitation levels in Jilin Province. The study utilized daily data on precipitation and conventional meteorological elements from 2016 to 2020 for 50 cities. The parameters of the support vector machine were optimized using the gray wolf optimization algorithm and differential evolutionary algorithm. The study aimed to predict the occurrence and level of precipitation for selected stations. In a separate study, Wu et al. (2022) proposed a combined CNN-Attention-BP model that integrated the attention mechanism, convolutional neural network (CNN), and BP neural network. This model was developed through a comprehensive analysis of statistical prediction models for precipitation. The researchers empirically analyzed summer precipitation for different climate types at Changchun, Baicheng, and Yanji stations from 1961 to 2020. Shen et al. (2020) utilized the long-short-term memory (LSTM) network to predict summer precipitation in China for the years 2014 and 2015. They employed the historical return data of the BCC-CSM seasonal climate prediction model system, along with monthly surface precipitation values provided by the National Meteorological Information Center. The researchers compared multiple methods and examined factors influencing the prediction results. In another study, Dong et al. (2020) compared the prediction accuracy of the LSTM network for monthly runoff using different feature inputs. They analyzed historical runoff data from Yingluo Gorge and Zamashk hydrological stations in the upper reaches of the Heihe River basin, as well as rainfall data from nearby meteorological stations. Han et al. (2022) identified that the LSTM-based precipitation prediction model is prone to overfitting and time lag. To address these issues, Hao-Ran (2022) proposed a depth-width prediction architecture that combines the advantages of width learning and depth learning. They established a CEEMD–LSTM–BLS single-factor monthly precipitation prediction model, considering the long-term memory function of LSTM and the noise elimination capability of complete ensemble empirical modal decomposition (CEEMD). The model was applied to analyze five representative stations in Hubei Province with diverse geographical and precipitation characteristics. In the field of data denoising, empirical modal decomposition (EMD) is a commonly used method (Norden et al. 1998). The ensemble empirical modal decomposition (EEMD) improves upon EMD by introducing Gaussian white noise and averaging the decomposition results to better reflect the original series' variation characteristics (Wu & Huang 2009). EEMD-based hybrid models are widely utilized in hydrological and meteorological forecasting. Complementary CEEMD (Xue et al. 2013) reduces signal reconstruction errors by adding white noise with zero mean and opposite amplitude to the original signal. Completely ensemble empirical modal decomposition with adaptive noise (CEEMDAN) addresses the slow decomposition of EEMD and the challenge of complete noise cancelation by decomposing the added white noise together (Torres et al. 2011). Chen et al. (2022) employed EMD, attention mechanism, and bidirectional long- and short-term memory (BILSTM) neural network in combination with an interpolation method for input data to enhance runoff prediction accuracy. Wei et al. (2017) applied the complete ensemble empirical modal decomposition (CEEMDAN) method to perform multi-scale analysis of flood precipitation in the Haihe River basin, identifying evolutionary patterns and selecting the best prediction model using methods such as nearest neighbor sampling regression model (NNBR), autoregressive model (AR), and neural network model. Zhang et al. (2021) achieved higher prediction accuracy for annual runoff using the CEEMDAN–ARMA model compared to a single ARIMA model. Kan et al. (2022) examined regional differences in climate change and circulation effects in Gansu Province using various methods, including least squares, complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN), cross wavelet transform, wavelet coherence, and Hurst index analysis. Luo et al. (2022) developed a coupled model based on CEEMDAN and long- and short-term memory neural network (LSTM) to predict monthly precipitation in Zhengzhou City.

To explore new coupled models, we consider the strengths of different techniques. CEEMDAN effectively reduces the reconstruction error of time series, variational modal decomposition (VMD) addresses the non-smoothness of precipitation time series with high complexity and strong non-linearity, while the BILSTM model effectively captures long-term dependencies in time series. With these considerations, we construct a combined CEEMDAN–VMD–BILSTM prediction model for precipitation forecasting. This model follows a ‘primary decomposition-quadratic decomposition-prediction-reconstruction’ approach. The average relative error of the CEEMDAN–VMD–BILSTM combined prediction model is a low 1.69%, and the average absolute error of the CEEMDAN–VMD–BILSTM combined quadratic decomposition model is only 1.32 m. The Nash–Sutcliffe efficiency (NSE) coefficient reaches a high value of 0.92, indicating a superior prediction capability.

Empirical modal decomposition (EMD) is a signal decomposition technique developed by NASA. It is specifically designed to address the non-linearity and non-smoothness of signals, providing an adaptive spatio-temporal analysis method that ultimately achieves signal smoothing. EMD decomposes the signal into a sum of eigenmode functions (IMFs) and a trend term (Res), as shown in Equation (1). Each eigenmode function captures different time-scale features of the signal, while the trend term represents the overall signal trend. Building upon EMD and EEMD, Torres et al. (2011) proposed CEEMDAN, an enhanced signal decomposition method. CEEMDAN overcomes the modal mixing issue encountered in EMD by incorporating adaptive white noise multiple times during the decomposition process. This complete decomposition process of CEEMDAN ensures an accurate reconstruction of the original signal. Specifically, En(·) denotes the modal component of the nth stage generated by the EMD algorithm, while the nth modal component generated by the CEEMDAN algorithm is denoted as IMFn. The algorithm is implemented as follows.

• (1)
  The signal x(t) to be decomposed is added to a Gaussian white noise sequence with N times the mean value of 0 to construct the sequence x_i(t) to be decomposed for a total of N experiments (i = 1, 2, ⋯, N)

  

  (1)

  where ε is the Gaussian white noise weighting coefficient; δ_i (t) is the ith added white noise sequence.
• (2)
  The EMD algorithm is applied to decompose x_i (t) to obtain the first modal component (IMF) and the first unique residual component r₁ (t).

  

  (2)

  

  (3)
• (3)
  Adding noise to the residual components of the jth (j = 2, 3, ⋯, N) stage obtained after decomposition continues to apply EMD for decomposition.

  

  (4)

  

  (5)
• (4)
  Repeat (3) until the termination condition is satisfied. The termination criterion is that the number of residual signal extremum points does not exceed at most two. Finally, the original signal sequence is decomposed into N modal components and the residual term R(t).

  

  (6)

Compared with other time-frequency decomposition methods, VMD can effectively reduce the non-smoothness of reservoir water level time series with high complexity and strong non-linearity. The specific steps of VMD are as follows.

• (1)

  Constructing variational problems

• (2)

  Solving variational problems

• (3)

  Find each mode component and center frequency

Conventional recurrent neural networks commonly employ a logistic nonlinear activation function for recurrent learning. However, due to the derivative values of the logistic function being bounded between 0 and 1, these networks encounter challenges when processing sequences with long time intervals. Specifically, when the time interval is substantial, the gradient tends to approach 0 or become excessively large, resulting in the issues of gradient disappearance or gradient explosion. Consequently, traditional recurrent neural networks face difficulties in effectively handling long-interval information sequences.

The output of the bidirectional long and short-term memory network (BILSTM) is determined through the collaboration of two LSTM layers. The forward LSTM layer performs computations in a forward direction, starting from the initial moment and progressing to the final moment. On the other hand, the reverse LSTM layer conducts computations in a reverse direction, starting from the final moment and moving back to the initial moment. Both layers follow the same computational process. Ultimately, the outputs of the forward and reverse layers at each moment are combined to yield the corresponding output for that moment (Chen et al. 2022).

Using MATLAB software, we initially constructed the CEEMDAN decomposition model to decompose the monthly precipitation data of Fuzhou City spanning from 2000 to 2020. As the decomposition progressed, the amplitudes of IMF₂–IMF₇ and the trend term gradually diminished, the frequencies decreased, and the wavelengths became larger. Subsequently, we employed the MATLAB software to construct a VMD decomposition model, decomposing the IMF₁ component into five components and trend terms, as shown in Figure 6. Through the combined application of CEEMDAN and VMD, the monthly precipitation series underwent a reduction in volatility and improvement in non-stationarity.

Following the secondary decomposition, the precipitation time series in Fuzhou City exhibited enhanced smoothness and significantly reduced volatility. Moreover, the prediction errors for IMF₁ through IMF₇ progressively decreased, indicating an improved prediction effect with the secondary decomposition reconstruction.

The results in Table 1 show that IMF1 exhibits a small RMSE, MAE and a coefficient of determination close to 1, which can be attributed to its quadratic decomposition of the VMD. As we move from predicting IMF₁ to IMF₇, both the RMSE and MAE demonstrate a consistent downward trend. For IMF₂–IMF₇, there is a gradual reduction in the RMSE (from 0.7284 to 0.026316) and MAE (from 0.7488 to 0.25985). Additionally, the coefficient of determination progressively approaches 1, increasing from 0.89339 to 0.99222. Notably, IMF1 achieves relatively favorable levels of RMSE and MAE due to VMD's quadratic decomposition, resulting in improved prediction accuracy compared to the other components.

Table 1

Errors and coefficients of IMF1–IMF7 and trend term forecasts

Sequence .	RMSE .	MAE .	.
IMF1	0.2108	0.2072	0.96365
IMF2	0.7284	0.7488	0.89339
IMF3	0.6693	0.5406	0.85274
IMF4	0.4583	0.3943	0.93585
IMF5	0.2333	0.21022	0.95999
IMF6	0.16316	0.15985	0.97759
IMF7	0.026316	0.025985	0.99222
Residuals	0.33433	0.25898	0.90367

Sequence .	RMSE .	MAE .	.
IMF1	0.2108	0.2072	0.96365
IMF2	0.7284	0.7488	0.89339
IMF3	0.6693	0.5406	0.85274
IMF4	0.4583	0.3943	0.93585
IMF5	0.2333	0.21022	0.95999
IMF6	0.16316	0.15985	0.97759
IMF7	0.026316	0.025985	0.99222
Residuals	0.33433	0.25898	0.90367

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The results presented in Table 2 indicate that the combined CEEMDAN–VMD–BILSTM quadratic decomposition model achieves a low level of prediction error. The maximum relative error is 4.78%, the minimum is 0.00%, and the average relative error is 1.69%. These findings demonstrate that the model exhibits accurate predictions with a low relative error, a high success rate, and overall high prediction quality.

In order to assess the effectiveness of the CEEMDAN–VMD–BILSTM quadratic decomposition combination model, predictions were made using alternative models including the CEEMD–LSTM model, CEEMD–BILSTM model, and CEEMDAN–BILSTM model. The prediction results of the CEEMDAN–VMD–BILSTM quadratic decomposition combination model were then compared with those of the other models, as depicted in Table 3.

As observed in Table 3, the combined CEEMDAN–VMD–BILSTM quadratic decomposition model consistently outperforms the other three models in terms of average relative errors for precipitation prediction in Fuzhou City. Figure 9 further illustrates that the combined model exhibits superior precipitation prediction results compared to the other models. The combined CEEMDAN–VMD–BILSTM quadratic decomposition model demonstrates enhanced accuracy in predicting precipitation, while the coupled model effectively captures the realistic changes in the precipitation series with reasonable detail. Additionally, the trend and periodicity of the combined CEEMDAN–VMD–BILSTM quadratic decomposition model's prediction results align closely with the actual data.

Here, is the predicted value at time i; is the measured value at time i; is the mean value of the measured value.

Table 4 presents the comparative results of the model metrics. The MAE of CEEMD–LSTM, CEEMD–BILSTM, CEEMDAN–BILSTM, and CEEMDAN–VMD–BILSTM are 7.93, 6.04, 3.52, and 1.32 m. The NSE is 0.75, 0.78, 0.76, and 0.92, respectively, yielding the following analytical insights: as the single model is optimized towards the combined model, incorporating BILSTM treatment and more comprehensive VMD decomposition, the MAE index decreases, and the NSE approaches 1. This overall improvement indicates a higher preference for predictions. However, the single CEEMD–LSTM, CEEMD–BILSTM, and CEEMDAN–BILSTM models exhibit poor prediction performance, which can be attributed to the complexity, stochasticity, and ambiguity of precipitation sequences. The coupled CEEMDAN–VMD–BILSTM model stands out as optimal in terms of error reduction due to the combined use of CEEMDAN decomposition, VMD decomposition, and the BILSTM model, resulting in enhanced prediction performance.

In this study, we propose the CEEMDAN–VMD–BILSTM model for precipitation prediction in Fuzhou City and compare it with similar models used in previous studies (Özger et al. 2020; Serencam et al. 2022). The findings indicate that decomposition algorithms, wavelet noise reduction, and neural networks can be applied to other forecasting domains. Whether it is precipitation prediction, wind speed prediction, or drought prediction, these models demonstrate promising results. Our research builds upon CEEMDAN, effectively reducing time series reconstruction errors, VMD, effectively addressing non-smoothness in complex and nonlinear precipitation time series, and bidirectional long-short-term memory (BILSTM) model, effectively capturing long-term dependencies in time series. The constructed combined CEEMDAN–VMD–BILSTM quadratic decomposition model offers detailed and accurate predictions, reflecting the true variations in precipitation series with improved performance.

• (1)

  The BILSTM model, with its incorporation of cell states and gate structure for information control, effectively captures long-term dependencies in time series. Complementing this, CEEMDAN reduces reconstruction errors in time series, while VMD mitigates non-smoothness in complex and nonlinear precipitation time series. By combining adaptive noise-CEEMDAN, variational modal decomposition (VMD), and bidirectional long-short-term memory (BILSTM), a comprehensive model is constructed, yielding more accurate predictions and reflecting the true variations of precipitation series in greater detail. The model demonstrates a low average relative error of 1.69% and exhibits a higher level of prediction.

• (2)

  A combined CEEMDAN–VMD–BILSTM quadratic decomposition model is developed and applied for urban monthly precipitation prediction. With an average relative error of 1.69% and a coefficient of determination close to 1, the model outperforms the CEEMD–LSTM, CEEMD–BILSTM, and CEEMDAN–BILSTM models in terms of prediction accuracy. Through the application of BILSTM processing and a more comprehensive VMD decomposition, the model achieves diminishing MAE index values and closer NSE values to 1. These findings confirm the feasibility of the combined CEEMDAN–VMD–BILSTM quadratic decomposition model for monthly precipitation prediction.

• (3)

  It is important to acknowledge that the current model is data-based, and further enhancement is required to incorporate a deeper understanding of the underlying physical mechanisms. Future research endeavors should focus on strengthening the investigation of physical mechanisms within the model. This will contribute to a more comprehensive understanding and improve the model's performance in future predictions.

Data and materials are available from the corresponding author upon request.

All authors contributed to the study conception and design. X. Z. and J. S. wrote and edited the article. G. Z. edited the chart; Y. X. and H. C. collected the preliminary data. All authors read and approved the final manuscript.

This work was supported by the Key Scientific Research Project of Colleges and Universities in Henan Province (CN) [grant numbers 17A570004].

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.