Abstract

Precipitation forecasting is an important guide to the prevention and control of regional droughts and floods, the rational use of water resources and the ecological protection. The precipitation process is extremely complex and is influenced by the intersection of many variables, with significant randomness, uncertainty and non-linearity. Based on the advantages that complementary ensemble empirical modal decomposition (CEEMD) can effectively overcome modal aliasing, white noise interference, and the ability of long short-term memory (LSTM) networks to handle problems such as gradient disappearance. A CEEMD–LSTM coupled long-term and short-term memory network model was developed and adopted for monthly precipitation prediction of Zhengzhou City. The performance shows that the CEEMD–LSTM model has a mean absolute error of 0.056, a root mean square error of 0.153, a mean relative error of 2.73% and a Nash efficiency coefficient of 0.95, which is better than the CEEMD–Back Propagation (BP) neural network model, the LSTM model and the BP model in terms of prediction accuracies. This demonstrates its powerful nonlinear and complex process learning capability in hydrological factor simulation for regional precipitation prediction.

HIGHLIGHTS

  • Complementary ensemble empirical modal decomposition (CEEMD) is a relatively novel data preprocessing method that can effectively reduce the non-smoothness of time series.

  • Long short-term memory network (LSTM) as a prediction model is more adept at handling long time series.

  • The CEEMD–LSTM coupled has better nonlinear and complex process learning ability in hydrological factor simulation.

INTRODUCTION

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.

RESEARCH METHODOLOGY

CEEMD

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:
    formula
    (1)
  • (2)
    For , calculate the first order residual . The expressions appear below:
    formula
    (2)
  • (3)
    EMD realizes , until the first IMF (t) condition is satisfied and defines the overall average as IMF2 (t) with the following equation:
    formula
    (3)
  • (4)
    For , compute the kth order residual , that is:
    formula
    (4)
  • (5)
    Extract the of and calculate their overall mean and produce the of the objective function using the formula:
    formula
    (5)
  • (6)
    Steps (4), (5) are repeated until the residuals cannot be decomposed any further to obtain the ultimate residual R(t) as:
    formula
    (6)
    where K is the aggregate of . Thus, the goal signal is denoted as:
    formula
    (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

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).

Figure 1

Storage block structure of LSTM.

Figure 1

Storage block structure of LSTM.

In the diagram, i stands for input gate that determines how the input layer information is passed to the memory cell. f is for forgotten gate that determines how the historical information is retained. o for export gate determines how the memory module is passed to the next moment of storage block. c represents the memory cell. The symbol represents the addition operation between vectors. The symbol represents the dot product operation between vectors. stands for the sigmoid activation function. tanh stands for the hyperbolic tangent activation function. The formula is as shown below:
formula
(8)
formula
(9)
The signal needs go through the following processes from input to output:
formula
(10)
formula
(11)
formula
(12)
formula
(13)
formula
(14)
formula
(15)
where: ‘’ denotes the dot product operation between two vectors. Wf, Wi, Wo, Wc indicates the respective weight vectors of the forgotten gate, the input gate, the export gate and the memory cell. bf, bi, bo, bc represent the bias vectors of the forgotten gate, the input gate, the export gate and the memory cell, respectively. xt denotes the input to the network at time t.

CEEMD–LSTM COUPLED MODEL

Model steps

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.

Figure 2

Technical route of the coupled CEEMD–LSTM model.

Figure 2

Technical route of the coupled CEEMD–LSTM model.

Model verification

To measure the prediction precision of the coupled CEEMD–LSTM model, the mean relative error (MAPE), the Nash efficiency factor (NSE), the mean absolute error (MAE), and the RMSE between the predicted and actual rainfall data were used as assessment criteria. The specific equations are shown below:
formula
(16)
formula
(17)
formula
(18)
formula
(19)
where: is the actual value of precipitation at time i. is the predicted value of precipitation at time I, and N is the overall length of the precipitation series.

EXAMPLE APPLICATION

Regional profile

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.

Figure 3

Location of Zhengzhou City.

Figure 3

Location of Zhengzhou City.

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.

Figure 4

Monthly precipitation data for Zhengzhou City, 2010–2019.

Figure 4

Monthly precipitation data for Zhengzhou City, 2010–2019.

CEEMD decomposition

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.

Table 1

Monthly rainfall for the subseries obtained from CEEMD

MonthIMF1IMF2IMF3IMF4Trend
9.2421 −4.4768 −30.9308 −17.0119 43.2775 
12.8497 −10.0428 −13.7876 −15.9949 44.0756 
11.9753 −31.4036 4.6143 −14.8332 44.8472 
45.9327 −43.6064 22.4275 −13.5463 45.5925 
−9.5580 −27.7044 37.8044 −12.1542 46.3121 
−65.2499 11.3222 48.8978 −10.6764 47.0064 
64.0207 54.4764 53.8600 −9.1328 47.6757 
0.0396 58.7172 50.9657 −7.5431 48.3206 
15.7609 19.9526 39.7721 −5.9270 48.9414 
10 −44.0235 −21.8499 23.1391 −4.3043 49.5386 
11 −13.1458 −37.4853 4.7132 −2.6947 50.1126 
12 −7.9948 −29.5209 −12.0302 −1.1180 50.6638 
13 −15.5785 −11.8349 −24.1855 0.4062 51.1927 
14 7.2127 −5.3507 −29.0196 1.8580 51.6996 
15 −5.2959 −21.0212 −24.7858 3.2178 52.1851 
16 16.3519 −44.4836 −13.6835 4.4658 52.6495 
17 22.6093 −49.9602 0.9752 5.5824 53.0932 
18 −48.1522 −20.9039 16.0908 6.5486 53.5167 
19 −35.6850 24.6478 29.4713 7.3455 53.9204 
20 −20.7113 60.3860 39.1660 7.9545 54.3048 
21 88.3010 59.0977 43.5739 8.3573 54.6702 
22 −76.3995 9.5533 42.6896 8.5396 55.0171 
23 52.4842 −39.2336 37.6961 8.5075 55.3459 
24 −26.7537 −60.2625 29.9750 8.2843 55.6570 
25 −12.6486 −69.6964 20.8979 7.8962 55.9509 
26 −6.6990 −68.5877 11.6899 7.3688 56.2280 
27 14.6841 −57.8081 3.0084 6.7269 56.4887 
28 18.5968 −39.4871 −4.9381 5.9950 56.7335 
29 −24.5765 −14.6925 −11.9911 5.1974 56.9627 
30 −43.0512 13.8993 −17.9835 4.3586 57.1768 
31 25.8232 41.7457 −22.7486 3.5034 57.3762 
32 47.4265 56.4768 −26.1228 2.6581 57.5614 
33 −23.6402 47.9865 −28.0299 1.8508 57.7327 
34 −35.2271 23.2442 −28.7197 1.1119 57.8907 
35 −9.6202 −5.9428 −28.5440 0.4713 58.0357 
36 4.5516 −27.1995 −27.8784 −0.0418 58.1681 
37 7.8787 −32.8904 −27.0747 −0.4020 58.2884 
38 −3.1895 −20.5185 −26.2929 −0.5960 58.3970 
39 −30.4195 5.0542 −25.4003 −0.6285 58.4941 
40 −38.7208 29.9486 −24.1922 −0.5157 58.5801 
41 47.4763 39.6956 −22.4481 −0.2790 58.6552 
42 −49.3003 30.8154 −19.9928 0.0580 58.7197 
43 45.9170 18.3518 −16.9146 0.4722 58.7737 
44 −33.7082 7.0882 −13.3388 0.9415 58.8174 
45 −31.9545 −7.7264 −9.3147 1.4447 58.8510 
46 −1.8473 −25.9584 −4.9298 1.9611 58.8745 
47 17.4723 −44.0468 −0.2835 2.4698 58.8882 
48 −8.2834 −58.0831 4.5241 2.9504 58.8921 
49 −7.0132 −64.0481 9.3919 3.3831 58.8863 
50 7.9361 −62.7614 14.2070 3.7473 58.8710 
51 −14.7620 −59.0960 18.6931 4.0186 58.8463 
52 28.8523 −52.6118 22.3751 4.1721 58.8124 
53 −3.8079 −32.0414 24.6980 4.1820 58.7693 
54 −61.0447 −0.1191 25.1215 4.0249 58.7174 
55 −78.1479 35.0008 23.0975 3.6926 58.6569 
56 −66.1812 62.3381 18.0608 3.1944 58.5880 
57 73.3117 58.8422 9.7875 2.5476 58.5109 
58 −55.3843 12.0059 −0.4218 1.7741 58.4261 
59 7.0180 −32.9493 −10.4004 0.8982 58.3335 
60 −1.9082 −38.2987 −17.8712 −0.0554 58.2334 
61 −2.2322 −25.7451 −20.7917 −1.0569 58.1259 
62 14.6495 −19.4594 −17.8376 −2.0634 58.0110 
63 −15.1243 −20.3922 −8.9299 −3.0423 57.8887 
64 14.1135 −3.9837 2.5727 −3.9615 57.7590 
65 −14.2453 20.9803 12.7308 −4.7877 57.6218 
66 17.3743 22.2275 18.1090 −5.4882 57.4773 
67 −19.6087 2.7810 17.3368 −6.0345 57.3255 
68 18.9221 −3.8436 11.4719 −6.4169 57.1665 
69 −17.2702 8.3319 2.3717 −6.6341 57.0008 
70 17.0745 14.3336 −6.9485 −6.6881 56.8285 
71 −6.4885 4.3223 −14.3025 −6.5812 56.6499 
72 −18.3388 −11.8749 −19.0353 −6.3163 56.4654 
73 −3.2605 −23.1933 −20.8258 −5.8955 56.2752 
74 13.8298 −27.7501 −19.4394 −5.3201 56.0798 
75 −10.5976 −25.3712 −15.1135 −4.5973 55.8796 
76 1.5097 −12.0077 −8.7254 −3.7515 55.6749 
77 −13.2738 13.6444 −1.3257 −2.8110 55.4661 
78 5.6025 36.2223 6.0260 −1.8042 55.2533 
79 34.9530 43.2920 12.2777 −0.7595 55.0369 
80 0.4096 28.9701 16.6083 0.2949 54.8170 
81 −38.8790 6.0456 19.0081 1.3315 54.5938 
82 31.5678 −11.7261 19.6680 2.3225 54.3676 
83 −14.4105 −25.9477 18.7783 3.2413 54.1386 
84 −10.7006 −41.5959 16.5200 4.0697 53.9068 
85 0.0893 −53.9238 13.0643 4.7977 53.6726 
86 −2.1829 −56.0531 8.5831 5.4169 53.4360 
87 −1.0165 −46.8070 3.3070 5.9192 53.1972 
88 2.3112 −30.2265 −2.3360 6.2948 52.9565 
89 10.2565 −10.1581 −7.8462 6.5337 52.7140 
90 −12.4290 13.5444 −12.7113 6.6259 52.4699 
91 11.8660 40.2606 −16.4123 6.5613 52.2244 
92 3.8802 58.3411 −18.4289 6.3299 51.9777 
93 −14.0417 53.0605 −18.3728 5.9241 51.7299 
94 17.3983 19.2595 −16.3889 5.3498 51.4813 
95 −19.1759 −21.9080 −12.7788 4.6306 51.2322 
96 0.0402 −46.2672 −7.8506 3.7950 50.9827 
97 14.6493 −52.1308 −1.9237 2.8721 50.7330 
98 −5.8894 −45.6590 4.5717 1.8932 50.4835 
99 −7.2949 −34.0511 10.9219 0.8900 50.2342 
100 4.2304 −20.7494 16.3397 −0.1061 49.9854 
101 13.8215 −4.1327 20.0370 −1.0631 49.7374 
102 −21.6876 15.9216 21.2265 −1.9507 49.4903 
103 −11.6017 30.3747 19.1294 −2.7468 49.2444 
104 31.2642 30.4360 13.2449 −3.4449 48.9998 
105 28.4638 11.6385 4.2828 −4.0419 48.7569 
106 −26.1515 −11.3793 −6.1499 −4.5351 48.5159 
107 14.8871 −19.3596 −16.2829 −4.9215 48.2768 
108 4.0758 −11.5005 −24.3171 −5.1983 48.0401 
109 2.7414 −3.7858 −28.4989 −5.3625 47.8058 
110 −2.2276 −3.7633 −27.6718 −5.4115 47.5742 
111 −7.7055 −10.0429 −21.9547 −5.3424 47.3456 
112 16.7960 −17.8190 −12.5445 −5.1525 47.1201 
113 −23.5497 −15.2590 −1.1504 −4.8389 46.8979 
114 30.0500 7.2520 10.6175 −4.3989 46.6794 
115 −35.4372 34.3735 21.4287 −3.8296 46.4646 
116 37.3833 34.3767 30.0145 −3.1284 46.2539 
117 −33.6829 2.4214 35.1064 −2.2923 46.0475 
118 25.2659 −33.4285 35.4358 −1.3187 45.8454 
119 −15.1830 −55.3946 29.7342 −0.2047 45.6481 
120 −5.8931 −50.2479 16.7328 1.0525 45.4557 
MonthIMF1IMF2IMF3IMF4Trend
9.2421 −4.4768 −30.9308 −17.0119 43.2775 
12.8497 −10.0428 −13.7876 −15.9949 44.0756 
11.9753 −31.4036 4.6143 −14.8332 44.8472 
45.9327 −43.6064 22.4275 −13.5463 45.5925 
−9.5580 −27.7044 37.8044 −12.1542 46.3121 
−65.2499 11.3222 48.8978 −10.6764 47.0064 
64.0207 54.4764 53.8600 −9.1328 47.6757 
0.0396 58.7172 50.9657 −7.5431 48.3206 
15.7609 19.9526 39.7721 −5.9270 48.9414 
10 −44.0235 −21.8499 23.1391 −4.3043 49.5386 
11 −13.1458 −37.4853 4.7132 −2.6947 50.1126 
12 −7.9948 −29.5209 −12.0302 −1.1180 50.6638 
13 −15.5785 −11.8349 −24.1855 0.4062 51.1927 
14 7.2127 −5.3507 −29.0196 1.8580 51.6996 
15 −5.2959 −21.0212 −24.7858 3.2178 52.1851 
16 16.3519 −44.4836 −13.6835 4.4658 52.6495 
17 22.6093 −49.9602 0.9752 5.5824 53.0932 
18 −48.1522 −20.9039 16.0908 6.5486 53.5167 
19 −35.6850 24.6478 29.4713 7.3455 53.9204 
20 −20.7113 60.3860 39.1660 7.9545 54.3048 
21 88.3010 59.0977 43.5739 8.3573 54.6702 
22 −76.3995 9.5533 42.6896 8.5396 55.0171 
23 52.4842 −39.2336 37.6961 8.5075 55.3459 
24 −26.7537 −60.2625 29.9750 8.2843 55.6570 
25 −12.6486 −69.6964 20.8979 7.8962 55.9509 
26 −6.6990 −68.5877 11.6899 7.3688 56.2280 
27 14.6841 −57.8081 3.0084 6.7269 56.4887 
28 18.5968 −39.4871 −4.9381 5.9950 56.7335 
29 −24.5765 −14.6925 −11.9911 5.1974 56.9627 
30 −43.0512 13.8993 −17.9835 4.3586 57.1768 
31 25.8232 41.7457 −22.7486 3.5034 57.3762 
32 47.4265 56.4768 −26.1228 2.6581 57.5614 
33 −23.6402 47.9865 −28.0299 1.8508 57.7327 
34 −35.2271 23.2442 −28.7197 1.1119 57.8907 
35 −9.6202 −5.9428 −28.5440 0.4713 58.0357 
36 4.5516 −27.1995 −27.8784 −0.0418 58.1681 
37 7.8787 −32.8904 −27.0747 −0.4020 58.2884 
38 −3.1895 −20.5185 −26.2929 −0.5960 58.3970 
39 −30.4195 5.0542 −25.4003 −0.6285 58.4941 
40 −38.7208 29.9486 −24.1922 −0.5157 58.5801 
41 47.4763 39.6956 −22.4481 −0.2790 58.6552 
42 −49.3003 30.8154 −19.9928 0.0580 58.7197 
43 45.9170 18.3518 −16.9146 0.4722 58.7737 
44 −33.7082 7.0882 −13.3388 0.9415 58.8174 
45 −31.9545 −7.7264 −9.3147 1.4447 58.8510 
46 −1.8473 −25.9584 −4.9298 1.9611 58.8745 
47 17.4723 −44.0468 −0.2835 2.4698 58.8882 
48 −8.2834 −58.0831 4.5241 2.9504 58.8921 
49 −7.0132 −64.0481 9.3919 3.3831 58.8863 
50 7.9361 −62.7614 14.2070 3.7473 58.8710 
51 −14.7620 −59.0960 18.6931 4.0186 58.8463 
52 28.8523 −52.6118 22.3751 4.1721 58.8124 
53 −3.8079 −32.0414 24.6980 4.1820 58.7693 
54 −61.0447 −0.1191 25.1215 4.0249 58.7174 
55 −78.1479 35.0008 23.0975 3.6926 58.6569 
56 −66.1812 62.3381 18.0608 3.1944 58.5880 
57 73.3117 58.8422 9.7875 2.5476 58.5109 
58 −55.3843 12.0059 −0.4218 1.7741 58.4261 
59 7.0180 −32.9493 −10.4004 0.8982 58.3335 
60 −1.9082 −38.2987 −17.8712 −0.0554 58.2334 
61 −2.2322 −25.7451 −20.7917 −1.0569 58.1259 
62 14.6495 −19.4594 −17.8376 −2.0634 58.0110 
63 −15.1243 −20.3922 −8.9299 −3.0423 57.8887 
64 14.1135 −3.9837 2.5727 −3.9615 57.7590 
65 −14.2453 20.9803 12.7308 −4.7877 57.6218 
66 17.3743 22.2275 18.1090 −5.4882 57.4773 
67 −19.6087 2.7810 17.3368 −6.0345 57.3255 
68 18.9221 −3.8436 11.4719 −6.4169 57.1665 
69 −17.2702 8.3319 2.3717 −6.6341 57.0008 
70 17.0745 14.3336 −6.9485 −6.6881 56.8285 
71 −6.4885 4.3223 −14.3025 −6.5812 56.6499 
72 −18.3388 −11.8749 −19.0353 −6.3163 56.4654 
73 −3.2605 −23.1933 −20.8258 −5.8955 56.2752 
74 13.8298 −27.7501 −19.4394 −5.3201 56.0798 
75 −10.5976 −25.3712 −15.1135 −4.5973 55.8796 
76 1.5097 −12.0077 −8.7254 −3.7515 55.6749 
77 −13.2738 13.6444 −1.3257 −2.8110 55.4661 
78 5.6025 36.2223 6.0260 −1.8042 55.2533 
79 34.9530 43.2920 12.2777 −0.7595 55.0369 
80 0.4096 28.9701 16.6083 0.2949 54.8170 
81 −38.8790 6.0456 19.0081 1.3315 54.5938 
82 31.5678 −11.7261 19.6680 2.3225 54.3676 
83 −14.4105 −25.9477 18.7783 3.2413 54.1386 
84 −10.7006 −41.5959 16.5200 4.0697 53.9068 
85 0.0893 −53.9238 13.0643 4.7977 53.6726 
86 −2.1829 −56.0531 8.5831 5.4169 53.4360 
87 −1.0165 −46.8070 3.3070 5.9192 53.1972 
88 2.3112 −30.2265 −2.3360 6.2948 52.9565 
89 10.2565 −10.1581 −7.8462 6.5337 52.7140 
90 −12.4290 13.5444 −12.7113 6.6259 52.4699 
91 11.8660 40.2606 −16.4123 6.5613 52.2244 
92 3.8802 58.3411 −18.4289 6.3299 51.9777 
93 −14.0417 53.0605 −18.3728 5.9241 51.7299 
94 17.3983 19.2595 −16.3889 5.3498 51.4813 
95 −19.1759 −21.9080 −12.7788 4.6306 51.2322 
96 0.0402 −46.2672 −7.8506 3.7950 50.9827 
97 14.6493 −52.1308 −1.9237 2.8721 50.7330 
98 −5.8894 −45.6590 4.5717 1.8932 50.4835 
99 −7.2949 −34.0511 10.9219 0.8900 50.2342 
100 4.2304 −20.7494 16.3397 −0.1061 49.9854 
101 13.8215 −4.1327 20.0370 −1.0631 49.7374 
102 −21.6876 15.9216 21.2265 −1.9507 49.4903 
103 −11.6017 30.3747 19.1294 −2.7468 49.2444 
104 31.2642 30.4360 13.2449 −3.4449 48.9998 
105 28.4638 11.6385 4.2828 −4.0419 48.7569 
106 −26.1515 −11.3793 −6.1499 −4.5351 48.5159 
107 14.8871 −19.3596 −16.2829 −4.9215 48.2768 
108 4.0758 −11.5005 −24.3171 −5.1983 48.0401 
109 2.7414 −3.7858 −28.4989 −5.3625 47.8058 
110 −2.2276 −3.7633 −27.6718 −5.4115 47.5742 
111 −7.7055 −10.0429 −21.9547 −5.3424 47.3456 
112 16.7960 −17.8190 −12.5445 −5.1525 47.1201 
113 −23.5497 −15.2590 −1.1504 −4.8389 46.8979 
114 30.0500 7.2520 10.6175 −4.3989 46.6794 
115 −35.4372 34.3735 21.4287 −3.8296 46.4646 
116 37.3833 34.3767 30.0145 −3.1284 46.2539 
117 −33.6829 2.4214 35.1064 −2.2923 46.0475 
118 25.2659 −33.4285 35.4358 −1.3187 45.8454 
119 −15.1830 −55.3946 29.7342 −0.2047 45.6481 
120 −5.8931 −50.2479 16.7328 1.0525 45.4557 
Figure 5

CEEMD decomposition on monthly precipitation for Zhengzhou (2010–2019).

Figure 5

CEEMD decomposition on monthly precipitation for Zhengzhou (2010–2019).

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.

Precipitation forecast

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.

Figure 6

Training loss values and root mean square error for LSTM.

Figure 6

Training loss values and root mean square error for LSTM.

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

Forecast results for IMF1–IMF4 and trend terms.

Figure 7

Forecast results for IMF1–IMF4 and trend terms.

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.

Table 2

Relative errors of forecasts for IMF1–IMF4 and trend terms

QuantitiesMaximum relative error (%)Minimum relative error (%)Mean relative error (%)
IMF1 6.86 3.63 4.39 
IMF2 6.30 1.23 2.92 
IMF3 3.49 0.52 2.08 
IMF4 2.91 0.22 1.93 
Trend item 0.92 0.00 0.48 
QuantitiesMaximum relative error (%)Minimum relative error (%)Mean relative error (%)
IMF1 6.86 3.63 4.39 
IMF2 6.30 1.23 2.92 
IMF3 3.49 0.52 2.08 
IMF4 2.91 0.22 1.93 
Trend item 0.92 0.00 0.48 
Table 3

Errors in monthly precipitation forecasts for 2017–2019 in Zhengzhou

MonthTrue value/mmPredicted value/mmAbsolute error/mmRelative error/%
97 14.2 14.23 − 0.03 0.21 
98 5.4 5.56 − 0.16 2.96 
99 20.7 21.35 − 0.65 3.14 
100 49.7 50.76 − 1.06 2.13 
101 78.4 80.40 − 2.00 2.55 
102 63.0 61.33 1.67 2.65 
103 84.4 84.40 0.00 0.00 
104 120.5 122.50 − 2.00 1.66 
105 89.1 89.10 0.00 0.00 
106 0.3 0.29 0.01 3.33 
107 22.6 22.32 0.28 1.24 
108 11.1 11.00 0.10 0.90 
109 12.9 12.55 0.35 2.74 
110 8.5 9.10 − 0.60 7.08 
111 2.3 2.30 0.00 0.07 
112 28.4 26.79 1.61 5.68 
113 2.1 2.16 − 0.06 2.67 
114 90.2 96.48 − 6.28 6.96 
115 63.0 66.17 − 3.17 5.03 
116 144.9 144.08 0.82 0.57 
117 47.6 47.64 − 0.04 0.07 
118 71.8 68.16 3.64 5.08 
119 4.6 4.66 − 0.06 1.38 
120 7.1 7.62 − 0.52 7.38 
Mean relative error (%)  2.73  
Nash efficiency factor  0.95  
MonthTrue value/mmPredicted value/mmAbsolute error/mmRelative error/%
97 14.2 14.23 − 0.03 0.21 
98 5.4 5.56 − 0.16 2.96 
99 20.7 21.35 − 0.65 3.14 
100 49.7 50.76 − 1.06 2.13 
101 78.4 80.40 − 2.00 2.55 
102 63.0 61.33 1.67 2.65 
103 84.4 84.40 0.00 0.00 
104 120.5 122.50 − 2.00 1.66 
105 89.1 89.10 0.00 0.00 
106 0.3 0.29 0.01 3.33 
107 22.6 22.32 0.28 1.24 
108 11.1 11.00 0.10 0.90 
109 12.9 12.55 0.35 2.74 
110 8.5 9.10 − 0.60 7.08 
111 2.3 2.30 0.00 0.07 
112 28.4 26.79 1.61 5.68 
113 2.1 2.16 − 0.06 2.67 
114 90.2 96.48 − 6.28 6.96 
115 63.0 66.17 − 3.17 5.03 
116 144.9 144.08 0.82 0.57 
117 47.6 47.64 − 0.04 0.07 
118 71.8 68.16 3.64 5.08 
119 4.6 4.66 − 0.06 1.38 
120 7.1 7.62 − 0.52 7.38 
Mean relative error (%)  2.73  
Nash efficiency factor  0.95  

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.

Figure 8

Prediction results and errors of the coupled CEEMD–LSTM model.

Figure 8

Prediction results and errors of the coupled CEEMD–LSTM model.

DISCUSSION

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.

Table 4

Comparison of prediction results between the CEEMD–LSTM model and other models

MonthCEEMD–LSTM
CEEMD–BP
BP
LSTM
True value/mmPredicted value/mmRelative error/%Predicted value/mmRelative error/%Predicted value/mmRelative error/%Predicted value/mmRelative error/%
97 14.2 14.23 0.21 15.32 7.89 16.30 14.79 12.23 13.87 
98 5.4 5.56 2.96 4.53 16.11 8.56 58.52 6.56 21.48 
99 20.7 21.35 3.14 22.33 7.87 31.23 50.87 23.15 11.84 
100 49.7 50.76 2.13 52.66 5.96 55.36 11.39 50.76 2.13 
101 78.4 80.40 2.55 82.64 5.41 100.56 28.27 80.40 2.55 
102 63 61.33 2.65 62.63 0.59 81.22 28.92 61.26 2.76 
103 84.4 84.40 0.00 84.22 0.21 94.20 11.61 84.78 0.45 
104 120.5 122.50 1.66 120.60 0.08 142.00 17.84 110.06 8.66 
105 89.1 89.10 0.00 89.21 0.12 69.50 22.00 99.06 11.18 
106 0.3 0.29 3.33 0.33 10.00 0.60 100.00 0.11 63.33 
107 22.6 22.32 1.24 23.65 4.65 33.32 47.43 45.65 101.99 
108 11.1 11.00 0.90 12.64 13.87 20.00 80.18 26.36 137.48 
109 12.9 12.55 2.74 12.92 0.15 13.55 5.01 12.55 2.74 
110 8.5 9.10 7.08 8.48 0.18 10.10 18.84 10.14 19.29 
111 2.3 2.30 0.07 3.50 52.16 3.30 43.54 4.23 83.91 
112 28.4 26.79 5.68 26.88 5.36 26.79 5.68 23.79 16.24 
113 2.1 2.16 2.67 2.56 21.93 5.16 145.53 2.56 21.93 
114 90.2 96.48 6.96 100.48 11.40 113.48 25.81 100.48 11.40 
115 63 66.17 5.03 63.66 1.05 76.17 20.91 70.75 12.30 
116 144.9 144.08 0.57 143.92 0.68 150.08 3.57 150.92 4.15 
117 47.6 47.53 0.14 48.49 1.87 59.35 24.69 50.35 5.78 
118 71.8 68.16 5.08 66.56 7.30 68.16 5.08 72.16 0.50 
119 4.6 4.66 1.38 5.50 19.63 14.34 211.63 6.63 44.21 
120 7.1 7.62 7.38 8.39 18.13 10.39 46.30 6.24 12.13 
Mean relative error (%) 2.73 8.86 42.85 25.51 
MonthCEEMD–LSTM
CEEMD–BP
BP
LSTM
True value/mmPredicted value/mmRelative error/%Predicted value/mmRelative error/%Predicted value/mmRelative error/%Predicted value/mmRelative error/%
97 14.2 14.23 0.21 15.32 7.89 16.30 14.79 12.23 13.87 
98 5.4 5.56 2.96 4.53 16.11 8.56 58.52 6.56 21.48 
99 20.7 21.35 3.14 22.33 7.87 31.23 50.87 23.15 11.84 
100 49.7 50.76 2.13 52.66 5.96 55.36 11.39 50.76 2.13 
101 78.4 80.40 2.55 82.64 5.41 100.56 28.27 80.40 2.55 
102 63 61.33 2.65 62.63 0.59 81.22 28.92 61.26 2.76 
103 84.4 84.40 0.00 84.22 0.21 94.20 11.61 84.78 0.45 
104 120.5 122.50 1.66 120.60 0.08 142.00 17.84 110.06 8.66 
105 89.1 89.10 0.00 89.21 0.12 69.50 22.00 99.06 11.18 
106 0.3 0.29 3.33 0.33 10.00 0.60 100.00 0.11 63.33 
107 22.6 22.32 1.24 23.65 4.65 33.32 47.43 45.65 101.99 
108 11.1 11.00 0.90 12.64 13.87 20.00 80.18 26.36 137.48 
109 12.9 12.55 2.74 12.92 0.15 13.55 5.01 12.55 2.74 
110 8.5 9.10 7.08 8.48 0.18 10.10 18.84 10.14 19.29 
111 2.3 2.30 0.07 3.50 52.16 3.30 43.54 4.23 83.91 
112 28.4 26.79 5.68 26.88 5.36 26.79 5.68 23.79 16.24 
113 2.1 2.16 2.67 2.56 21.93 5.16 145.53 2.56 21.93 
114 90.2 96.48 6.96 100.48 11.40 113.48 25.81 100.48 11.40 
115 63 66.17 5.03 63.66 1.05 76.17 20.91 70.75 12.30 
116 144.9 144.08 0.57 143.92 0.68 150.08 3.57 150.92 4.15 
117 47.6 47.53 0.14 48.49 1.87 59.35 24.69 50.35 5.78 
118 71.8 68.16 5.08 66.56 7.30 68.16 5.08 72.16 0.50 
119 4.6 4.66 1.38 5.50 19.63 14.34 211.63 6.63 44.21 
120 7.1 7.62 7.38 8.39 18.13 10.39 46.30 6.24 12.13 
Mean relative error (%) 2.73 8.86 42.85 25.51 

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

Prediction error assessment of the CEEMD–LSTM model and others

Predictive modelsMean absolute error/mmRoot mean square error/mmMean relative error/%Nash efficiency factor
CEEMD–LSTM 0.056 0.153 2.73 0.95 
CEEMD–BP 0.096 0.581 8.86 0.90 
LSTM 0.152 0.967 25.51 0.83 
BP 0.197 0.998 42.85 0.78 
Predictive modelsMean absolute error/mmRoot mean square error/mmMean relative error/%Nash efficiency factor
CEEMD–LSTM 0.056 0.153 2.73 0.95 
CEEMD–BP 0.096 0.581 8.86 0.90 
LSTM 0.152 0.967 25.51 0.83 
BP 0.197 0.998 42.85 0.78 
Figure 9

Comparison of prediction results of the CEEMD–LSTM model with other models.

Figure 9

Comparison of prediction results of the CEEMD–LSTM model with other models.

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.

CONCLUSION

  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.

ACKNOWLEDGEMENT

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).

DATA AVAILABILITY STATEMENT

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

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