Precipitation forecast based on CEEMD–LSTM coupled model

Precipitation is an important recharge method for regional water resources, and precipitation anomalies are the direct cause of regional droughts and floods. Therefore, accurate precipitation forecasting can provide technical support for the sustainable use of regional water resources, flood prevention and mitigation, and ecological environmental protection (Chen et al. 2017). Precipitation is affected by a variety of uncertainties and its process is very complex. Precipitation sequences are characterized by significant randomness, uncertainty and non-linearity (Peng et al. 2015), which makes precipitation prediction relatively inaccurate. In recent years, scholars in domestic and foreign countries have carried out many studies on precipitation forecasting and have achieved fruitful results. Kang et al. (2020) used the LSTM networks model to predict precipitation in Jingdezhen and compare the performance with other classical statistical methods and machine learning algorithms. The empirical evidence suggests that the LSTM method is applicable to precipitation forecast. Kumar & Samui (2019) used long-range raw data for time series analysis. Rainfall was predicted using the recurrent neural network (RNN) and LSTM training model. Asanjan et al. (2018) proposed LSTM and artificial neural network-based remote sensing precipitation estimation methods. Compared with the RNN method, the proposed LSTM performed better for correlation coefficient and root mean square error (RMSE) in prediction. Reddy et al. (2021) used a novel hybrid neural network–Emotional Artificial Neural Networks (EANN) to predict monthly surface runoff in tropical climates, with better prediction accuracy than the traditional Feed Forward Neural Network (FFNN) and Multivariate Adaptive Regression Splines (MARS) models. Nanda et al. (2016) used a wavelet-based nonlinear autoregressive neural network (NANN) for flood prediction in the upper Mohanadi River area in real time. Yaseen et al. (2018) combined the Adaptive Neuro Fuzzy Inference System (ANFIS) and Firefly Optimization Algorithm (FOA) for monthly precipitation prediction in Pahang River, Malaysia. The prediction results were excellent. Song et al. (2018) established a clustering-fuzzy Markov rainfall prediction model and improved the results. The annual precipitation prediction for Jiangsu province was achieved. Wang et al. (2017) proposed an improved Adaboost algorithm integrating Back Propagation (BP) neural network combined classification model for predicting daily rainfall in Jiangsu Province. Particle Swarm Optimization-Least Squares Support Vector Machine (PSO-LSSVM) based on particle swarm algorithm for medium-term and long-term precipitation prediction in the Altay region, Xinjiang, China by Meng (2016). Li & Guo (2017) applied a BP neural network approach for fitting regional precipitation. To enhance the prediction precision of the precipitation–runoff prediction model. Ren et al. (2016) took advantage of empirical mode decomposition (EMD) to handle nonlinear complex signals and used a coupled BP neural network to predict precipitation and runoff processes at the Chaoyang hydrological station on the Ling River. Deep learning is an improved neural network with the advantages of high learning ability, wide coverage, portability and adaptability. At the same time, deep learning also suffers from poor portability, large computational effort, complex model design and high hardware requirements. Most scholars in China and abroad are interested in deep learning prediction models by simply adjusting the model parameters or modifying part of the code to improve the model accuracy. There are fewer studies on processing time series to reduce their non-smoothness before prediction and then optimizing the model. CEEMD can effectively improve the smoothness of time sequences, which has high adaptability and solves the modal mixing problem of the EMD approach. LSTM neural networks effectively overcome the gradient disappearance or explosion problem of recurrent neural network RNNs, allowing RNNs to be implemented in processing long sequences. Therefore, this paper combines the advantages of CEEMD and LSTM to develop a coupled CEEMD–LSTM prediction model, and apply it to the monthly precipitation prediction in Zhengzhou City.

EMD can break down a complex signal into a range of sub signals, called Intrinsic Mode Functions (IMFs), each with a unique frequency component, which can characterize the input signal on different scales (Xu et al. 2009). Nevertheless, the EMD can suffer from modal aliasing when the components of a frequency segment of the signal are not continuous or when there is intermittent signal or noise-based interference, which can destroy the physical meaning implied by each IMF and reduce the accuracy of the decomposition (Roushangar & Alizadeh 2019). In order to avoid modal aliasing during EMD decomposition, Wu & Huang (2009) developed an ensemble empirical modal decomposition (EEMD), which is based on the principle of adding a uniformly distributed white noise background to the signal, where signals of various scales will be mapped to the white noise background region accordingly. However, the white noise remains after the pooled average and cannot be ignored after reconstruction (Roushangar et al. 2021). To solve the problems of EEMD, Yeh et al. (2010) suggested the CEEMD, which can perfectly solve the modal mixing phenomenon and has high adaptivity. As with EEMD, the analysis is also aided by the addition of white noise. The operator is first defined, given its initial signal, and the jth mode is generated by EMD. denotes a zero-mean Gaussian white noise of standard variance N(0, 1), i= 1,…, I. the coefficients allow the signal-to-noise ratio to be chosen at each stage. Let be the target signal, the CEEMD decomposition steps are as follows (Zhao et al. 2015):

• (1)
  The calculation method of IMF₁ (t) is the same as that of EEMD, that is to say, different noises are used to realize the calculation. Through the EMD repeated decomposition process I times, the total average value is calculated and defined as the IMF₁ (t) of the target signal. The formula is:

  

  (1)
• (2)
  For ⁠, calculate the first order residual ⁠. The expressions appear below:

  

  (2)
• (3)
  EMD realizes ⁠, until the first IMF (t) condition is satisfied and defines the overall average as IMF₂ (t) with the following equation:

  

  (3)
• (4)
  For ⁠, compute the kth order residual ⁠, that is:

  

  (4)
• (5)
  Extract the of and calculate their overall mean and produce the of the objective function using the formula:

  

  (5)
• (6)
  Steps (4), (5) are repeated until the residuals cannot be decomposed any further to obtain the ultimate residual R(t) as:

  

  (6)

  where K is the aggregate of ⁠. Thus, the goal signal is denoted as:

  

  (7)

It can be seen that the residual noise of the CEEMD remains at a small level, regardless of the number of integration averages. In a sense, CEEMD can save computational time while guaranteeing small residual noise interference.

LSTM is a variation of recurrent neural networks (RNN), proposed by Hochreiter & Schmidhuber (1997), which overcomes the gradient disappearance or explosion problem of RNN. Like the basic architecture of most neural networks, the LSTM also has a three-layer structure, namely an input layer, an export layer and a hidden layer, and its storage block structure is shown in Figure 1 (Alizadeh et al. 2019).

CEEMD decomposition results in several IMF components and a trend term, which do not all contribute equally to the precipitation prediction results. Therefore, the IMF component and the trend term can be approximated as drivers of precipitation prediction.

The details of the coupled CEEMD–LSTM model are as shown below:

• (1)

  CEEMD decomposition. CEEMD decomposition of the original precipitation data yields multiple IMF elements and a trend term.

• (2)

  Data standardization. If precipitation data are directly used as input data for prediction, large errors will arise. We therefore normalized the CEEMD decomposition volume and transformed the data to the range [0, 1].

• (3)

  Determine the training dataset and the prediction dataset. The CEEND decomposition of monthly precipitation data in Zhengzhou City from 2010 to 2017 was used as the training dataset for the LSTM, and the CEEMD decomposition from 2018 to 2019 was used as the prediction dataset.

• (4)

  LSTM training. The input parameters of the LSTM are continuously adjusted so that the LSTM is adequately trained for the training dataset to ensure that the error is at a low level and to improve the prediction accuracy.

• (5)

  LSTM prediction. Prediction of individual CEEMD decomposition quantities using the trained training dataset.

• (6)

  Data reconstruction. The predicted values of each CEEMD component were inverse normalized and the model was reconstructed to obtain the precipitation prediction results for 2018–2019. The entire model building process is presented as Figure 2.

Zhengzhou City is the capital of Henan Province and is situated in the central north of Henan Province, belonging to the lower reaches of the Yellow River. The terrain of Zhengzhou City is high in the southwest and low in the northeast, showing a ladder-like downward trend. The average altitude is 108 meters. The highest terrain is located in the Yuzhai Mountain, and the altitude is 1,512.4 meters. The lowest terrain is located in the eastern alluvial plain area, and the altitude is 80 meters. The annual average temperature of Zhengzhou is 15.6 °C, the monthly average maximum temperature is 25.9 °C, and the monthly average minimum temperature is −2.15 °C. The average annual rainfall of Zhengzhou City is 542.15 mm.

As shown by the data in the Zhengzhou Water Resources Bulletin, Zhengzhou is subject to a temperate continental climate, with large variations in monthly precipitation. Zhengzhou has very little water resources per capita and is a city with a serious water shortage. Precipitation, as an important source of water recharge for Zhengzhou, directly affects the natural ecological environment of the region. The location of Zhengzhou City is shown in Figure 3.

Influenced by atmospheric circulation, the nature of the subsurface and topography, precipitation in Zhengzhou is mainly concentrated in June to September, with monthly precipitation showing uneven distribution, randomness, volatility and non-linearity. CEEMD is ideally suited to handle non-stationary time series data, while LSTM has considerable potential for learning from long series data. The results prove that the coupled CEEMD–LSTM model is used to forecast monthly precipitation for Zhengzhou City. As a specific process, the precipitation data were decomposed using CEEMD, on the basis of which the precipitation was predicted using LSTM. Monthly precipitation data for Zhengzhou from 2010 to 2019 are shown in Figure 4.

The CEEMD model for monthly precipitation measurements in Zhengzhou City from 2010 to 2019 was applied to decompose. Moreover, after continuous experimentation, the most satisfactory decomposition results were obtained when the noise variance was 0.1, the number of noises was 80, the number of realizations was 500, and the maximum number of allowed screening iterations was 5,000. The decomposition of CEEMD is presented as Figure 5, and the monthly rainfall of each subseries is calculated as shown in Table 1.

It can be seen from Figure 5 that the monthly precipitation data for Zhengzhou (2010–2019) are decomposed as four IMF subcomponents and a trend term. The frequency and amplitude of the IMF1 and IMF2 components are still at a high level, but the fluctuations are significantly lower compared to the original data, and the smoothness is significantly improved. the frequency and amplitude of IMF3–IMF4, the trend term, gradually decrease, and the series gradually tends to be smooth. The trend term shows an upward trend in precipitation for the first 48 months, followed by a slow decline.

Monthly precipitation data from 2010 to 2017 in Zhengzhou City were used as the training set, and monthly precipitation data from 2018 to 2019 were used as the prediction samples. After continuous attempts, the parameters corresponding to the optimal LSTM model are: learning rate = 0.002, maximum iteration number 800, gradient threshold 1, hidden nodes 300, initial memory, initial output is 0. The training process for LSTM is presented in Figure 6.

From Figure 6, it can be seen that the loss value of training reduces rapidly within the first 150 steps, indicating a rapid learning rate. The input layer is passed linearly through the activation function before the LSTM unit, so the LSTM unit can effectively capture the long-term time dependence in the time series. The RSME and the training loss values both plateaud at 400 steps and fell rapidly to very low levels, indicating that the LSTM is well trained. The prediction results are shown in Figure 7.

Figure 7 shows that, after the CEEMD decomposition, the smoothness of the time series of precipitation in Zhengzhou City is improving and the volatility is significantly decreasing. The prediction errors from IMF1 to IMF4 are becoming lower and lower, which suggests that the training effect is gradually getting better. The trend term is the component with the longest wavelength and frequency after CEEMD decomposition, and thus has the best prediction effect.

As can be seen from Table 2, the maximum relative error, the minimum relative error, and the mean relative error of IMF1 are all larger, which is directly related to the larger frequency of IMF1. The relative errors of the predictions from IMF1 to the trend term show a stepwise decrease, where the minimum relative error of the trend term predictions reaches 0 and the mean relative error is 0.48, indicating a very good fit. Combined with Figure 7, it can be seen that the predicted value of the trend term basically matches that of the true value. Therefore, after preprocessing of CEEMD, the prediction effect of LSTM for IMF1–IMF4 and trend terms were significantly improved. The predictions for IMF1 to IMF4 and trend terms were rebuilt using this information and were compared with the true precipitation values for Zhengzhou, the results are given in Table 3.

As shown in Table 3, the forecast error of the coupled CEEMD–LSTM model is at a low level, where the maximum relative error is 7.38%, the minimum is 0.00% and the mean relative error is 2.73%. The Nash efficiency coefficient is 0.95, which is very close to 1. The performance indicates that the model has a low relative error in prediction, a high pass rate and good prediction quality.

Figure 8 shows the training and prediction of the coupled CEEMD–LSTM model for 120 months of precipitation data in Zhengzhou City from 2010 to 2019. It can be seen that the prediction model works well, with relative errors fluctuating above and below 1%, and also shows excellent learning ability for sudden changes in the location of precipitation data, with no peak lags.

To validate the accuracy of the coupled CEEMD–LSTM model, the LSTM model, BP model and CEEMD–BP model were used to predict the precipitation data of Zhengzhou City respectively, the prediction results were compared. The comparison results are shown in Table 4.

As we can see from Table 4, except for the BP model, the accuracy of the predictions in all three models is within a reasonable range. Among them, the prediction relative error of the coupled CEEMD–LSTM model is controlled within 10%, and the mean relative error is also lower than that of the other models, with significantly better prediction results than the other models. It can also be seen that the precision of the coupled models is generally higher than the uncoupled models, and the coupled models CEEMD–LSTM and CEEMD–BP have higher accuracies, with an average relative error of less than 10%, which is at a low level. The prediction ability of the CEEMD–LSTM model is 9.34 times higher than that of the LSTM model, indicating that the CEEMD decomposition is more beneficial to the training of the LSTM neural network. To understand the prediction effect of the four models more intuitively, the prediction effect of the CEEMD–LSTM model is plotted against others, as shown in Figure 9. The results of comparing the RMSE, the MAE, and mean relative error of the training and prediction values of the four prediction models are shown in Table 5.

Table 5 shows that the MAE, RMSE, mean relative error and Nash efficiency coefficient of the coupled CEEMD–LSTM model for predicting precipitation in Zhengzhou are better than that of the other three models. Figure 9 shows that the CEEMD–LSTM coupled model predicts precipitation very well, outperforming the CEEMD–BP model, LSTM model and BP model.

The above comparison showed that the trend and period of the prediction results of the coupled CEEMD–LSTM model basically match that of the real data. Compared with the other three models, all indicators are at a lower level, indicating that CEEMD–LSTM has great potential for precipitation forecasting. In addition, the precision of the LSTM model is better than that of the BP model, which is due to the LSTM model's stronger learning ability for long-term precipitation data.

1. Decomposition results of CEEMD for monthly precipitation in Zhengzhou (2010–2019) solves the problem of large reconstruction errors arising from the addition of a single white noise to the EEMD decomposition by adding white noise of opposite amplitude and mean value of 0 to each other. It effectively overcomes modal confusion and white noise interference, greatly improves the smoothness of the original sequence and makes the sequence have a certain regularity, which provides a good basis for the LSTM long-term and short-term memory network to make predictions. LSTM can overcome the gradient disappearance or explosion problem of RNN, and its accuracy is better than that of the traditional RNN model, which is very reasonable for long-time precipitation sequence prediction.

2. A coupled CEEMD–LSTM model was constructed and applied to the monthly precipitation prediction of Zhengzhou City. The results suggest that the coupled CEEMD–LSTM model fits the locations of sudden changes in precipitation data better than the traditional single LSTM neural network model, and the coupled model can reflect the real changes of the series more reasonably in details. The MAE of 0.056, the RMSE of 0.153 and the mean relative error of 2.73 of the model prediction results are all at a low level. The Nash efficiency coefficient is 0.95, which is very close to 1. Its prediction accuracy is higher than that of the CEEMD–BP model, LSTM model and BP model. This suggests that the CEEMD–LSTM model is feasible for monthly precipitation prediction and can be used effectively for time series analysis in hydrology and related fields to mitigate the risk of climate extremes.

3. Although the overall prediction precision of the established coupled CEEMD–LSTM model is high, the relative error in the prediction of the high-frequency component IMF1 is large. In addition, the prediction model does not take into account the physical mechanism of monthly precipitation variability, and the neural network parameters need to be artificially adjusted to find the optimum, which are directions and priorities for further research.

This work is financially supported by the Technology Research Center of Water Environment Governance and Ecological Restoration Academician Workstation of Henan Province, Program for Science & Technology Innovation Talents in Universities of Henan Province (No. 15HASTIT049).

All relevant data are included in the paper or its Supplementary Information.