# Research on the Application of CEEMD-LSTM-LSSVM Coupled Model in Regional Precipitation Prediction

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Research Methodology

#### 2.1. Complementary Ensemble Empirical Modal Decomposition (CEEMD)

- A pair of white noises with opposite signs and zero mean is randomly added to the original time series X
_{t}to obtain two new series M_{1}and M_{2}, one of which is denoted by ω_{t}.

- 2.
- The EMD algorithm is used to decompose M
_{1}and M_{2}to obtain two sets of IMF components and residual terms. - 3.
- Repeat the above steps N times, N = 0, 1, 2,…, and the eigenmodal components of the CEEMD decomposition can be obtained by taking the mean value of the overall 2N modal components generated.

_{j}denotes the IMF component of the decomposed sequence, 1 ≤ j ≤ m. The jth IMF component of the ith sequence is denoted by IMF

_{ij}, and R

_{i}denotes the residual term of the ith sequence.

#### 2.2. Long Short-Term Memory Neural Network (LSTM)

_{f}(f for forget) is the forget gating, and Z

_{i}(i for information) is the selection gating. The results obtained from the above two steps are summed to obtain the Ct transmitted to the next state. The output stage is mainly controlled by Z

_{0}and the Ct obtained in the previous stage is deflated (changed by a tanh activation function) to finally obtain the output ht of the current state. The specific calculation process is as follows.

_{t}and h

_{t−1}passed from the previous state are used as the current input, and the two are trained by LSTM splicing to obtain the following four states:

_{f}, Z

_{i}, and Z

_{0}represent a gating state by multiplying the splicing vector by the weight matrix and then converting it to a value between 0 and 1 by the activation function sigmoid. Z is taken as input data and converted to a value between 1 and −1 using the tanh activation function. The two activation functions are calculated as follows:

#### 2.3. Least Squares Support Vector Machine (LSSVM)

_{i}is the Lagrangian multiplier. When $\partial \mathrm{L}/\partial \mathsf{\omega}=0$, $\partial \mathrm{L}/\partial \mathrm{b}=0$, $\partial \mathrm{L}/\partial \mathsf{\xi}=0$ and $\partial \mathrm{L}/\partial {\mathsf{\alpha}}_{\mathrm{i}}=0$, it can represent the best conditions for KKT, and w and ξ in the model can be eliminated, resulting in the equation shown below:

_{i}, x

_{j}) and the RBF is chosen as the kernel function of the LSSVM, as shown below:

#### 2.4. Mann–Kendall Test

_{k}is the cumulative count of the number of times when the value at time i is greater than the value at time j. Under the assumption of random independence of the time series, a statistic can be defined as follows:

_{1}= 0, E(S

_{k}), and Var(S

_{k}) are the mean and variance of the cumulative count S

_{k}, respectively. When x

_{1}, x

_{2},…, x

_{n}are mutually independent and have the same continuous distribution, they can be calculated using the following formulas:

_{i}is a standardized normal distribution calculated in the order of the time series x

_{1}, x

_{2},…, x

_{n}. When a given significance level is α, according to the normal distribution table, if |UF

_{i}| > U

_{α}, a clear trend change in the sequence is indicated.

_{n}, x

_{n−1},…, x

_{1}, and repeating the above process while setting UB

_{k}= −UF

_{k}for k = n, n−1,…, 1 and UB

_{1}= 0, we can obtain a change point detection plot [24]. Among them, the value of UF is greater than zero, indicating that the sequence is in an uptrend, and vice versa. If the two curves of UF and UB intersect between the critical line, the moment corresponding to the intersection point is the time of the start of the mutation. The advantage of this method is that it is not only computationally simple but also able to determine the time when the change point begins and identify the change point region, making it a commonly used method for change point detection.

## 3. Overview of the Study Area

## 4. Model Building

#### 4.1. Modeling Process

- CEEMD decomposition: The rainfall time series are decomposed by CEEMD to obtain K eigenmodal components and one residual term (trend term).
- Data pre-processing: If the decomposed data are used directly as the input term of the prediction, it will generate large errors, so the entire sample data needs to be normalized.
- To determine the training and prediction sets, the monthly precipitation data of Zhoukou City from 1978–2017 are used as the training data set of the prediction model after decomposition by the CEEMD method; similarly, the monthly precipitation data from 2017–2022 are used as the prediction data set of the forecast after decomposition by CEEMD. Then the Permutation Entropy of the IMF component is calculated, dividing the high and low frequencies.
- Model training: Each set of training input data and prediction data is put into the model for training, and the model input parameters are continuously adjusted so that the model is fully trained on the training data set to ensure that the error is at a low level and to improve the prediction accuracy.
- Model prediction: The prediction is performed using the coupled model, and then the results are inverse normalized and reconstructed, and the final prediction is obtained by superimposing the data according to the principle of superimposing the data at the same moment.
- Accuracy evaluation: The evaluation indicators of the model are calculated and compared with other selected models.

#### 4.2. Model Evaluation Indicators

^{2}) are used as evaluation indicators, where RMSE and MAE are prediction error measures. The smaller the value of these two performance indicators, the closer they are to the actual value. In addition, the closer the value of R

^{2}to 1, the better the fit of the prediction model. The specific equations are as follows:

_{i}is the actual value of precipitation at moment i, $\stackrel{\wedge}{{\mathrm{y}}_{\mathrm{i}}}$ is the predicted value of precipitation at moment i, and N is the total length of the precipitation series.

#### 4.3. Model Input

#### 4.4. Model Training

## 5. Analysis and Discussion

^{2}is 0.932. The smaller values of these two performance indicators, RSME and MAE, indicate that the predicted value is closer to the actual value and the error of prediction is smaller; the closer R

^{2}to 1, the better the fit of the prediction model. Therefore, the coupled model developed in the article has the highest accuracy compared to the selected comparison model and can be used for the prediction of actual precipitation in the study area.

^{2}of the CEEMD-LSTM-LSSVM model is closest to 1, which indicates that the model predictions are more consistent with the observed precipitation values.

## 6. Monthly Precipitation Forecasts

## 7. Conclusions

^{2}was increased by 0.08, 0.14, and 0.47, respectively. This indicates that the coupled model proposed in this paper is applicable to monthly precipitation prediction, improves the accuracy of prediction, and can be applied to data analysis and prediction in related fields as a time-frequency domain analysis method.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 6.**CEEMD decomposition results of month−by−month precipitation in Zhoukou City for 1978–2022.

**Figure 10.**Comparison of prediction results of the CEEMD-LSTM-LSSVM model with those of other models. (

**a**) Prediction results of CEEMD-LSTM-LSSVM compared with CEEMD-LSTM. (

**b**) Comparison of prediction results of CEEMD-LSTM-LSSVM with LSTM. (

**c**) Comparison of prediction results of CEEMD-LSTM-LSSVM and LSSVM.

**Figure 11.**Scatterplot of observed and predicted precipitation values. (

**a**) CEEMD-LSTM-LSSVM. (

**b**) CEEMD-LSTM. (

**c**) LSTM. (

**d**) LSSVM.

Rainfall Sequence | Min Value (mm) | Max Value (mm) | Average Value (mm) | Standard Deviation (mm) | Skewness | Kurtosis |
---|---|---|---|---|---|---|

Original sequence | 0.00 | 512.06 | 69.46 | 79.13 | 5.66 | 2.16 |

IMF1 | −194.27 | 201.66 | −5.62 | 52.25 | 1.51 | −0.12 |

IMF2 | −154.73 | 150.99 | −5.62 | 50.29 | 0.04 | 0.14 |

IMF3 | −96.53 | 92.15 | 0.21 | 28.46 | 0.36 | 0.25 |

IMF4 | −29.78 | 32.25 | −0.61 | 12.43 | −0.23 | 0.16 |

IMF5 | −27.40 | 35.63 | 0.47 | 9.21 | 1.10 | −0.51 |

IMF6 | −7.70 | 15.21 | −0.71 | 4.34 | −0.56 | 0.07 |

IMF7 | −11.31 | 18.74 | 4.85 | 10.38 | −1.41 | −0.18 |

IMF8 | −0.82 | 0.68 | −0.32 | 0.45 | −0.85 | 0.64 |

Trend | 61.87 | 97.31 | 76.81 | 10.33 | −1.11 | 0.32 |

Rainfall Sequence | Displacement Entropy | Rainfall Sequence | Displacement Entropy | Displacement Entropy | Displacement Entropy |
---|---|---|---|---|---|

IMF1 | 1.8 | IMF4 | 0.85 | IMF7 | 0.68 |

IMF2 | 1.35 | IMF5 | 0.78 | IMF8 | 0.62 |

IMF3 | 1.08 | IMF6 | 0.72 | Trend | 0 |

Rainfall Sequence | Number of Nodes in the Hidden Layer | Number of Training Sessions | Learning Rate |
---|---|---|---|

IMF1 | 200 | 230 | 0.01 |

IMF2 | 200 | 200 | 0.01 |

IMF3 | 150 | 150 | 0.01 |

IMF4 | 150 | 110 | 0.008 |

IMF5 | 130 | 100 | 0.008 |

IMF6 | 130 | 60 | 0.006 |

IMF7 | 100 | 50 | 0.005 |

IMF8 | 100 | 35 | 0.005 |

Trend | 60 | 25 | 0.005 |

Portion Size | Average Relative Error (%) | Portion Size | Average Relative Error (%) | Portion Size | Average Relative Error (%) |
---|---|---|---|---|---|

IMF1 | 94.59 | IMF4 | 5.4 | IMF7 | 2.85 |

IMF2 | 25.95 | IMF5 | 5.52 | IMF8 | 0.49 |

IMF3 | 6.74 | IMF6 | 4.27 | Trend | 0.1 |

Models | RSME (mm) | MAE (mm) | R^{2} |
---|---|---|---|

CEEMD-LSTM-LSSVM | 16.77 | 13.07 | 0.932 |

CEEMD-LSTM | 23.66 | 17.86 | 0.864 |

LSTM | 27.77 | 22.46 | 0.813 |

LSSVM | 38.92 | 31.06 | 0.633 |

**Table 6.**Prediction results of the coupled CEEMD-LSTM-LSSVM model of monthly precipitation in Zhoukou City from 2023 to 2025.

Year | January | February | March | April | May | June | July | August | September | October | November | December |
---|---|---|---|---|---|---|---|---|---|---|---|---|

2023 | 16.26 | 29.38 | 43.30 | 54.81 | 64.20 | 93.66 | 216.10 | 181.49 | 89.77 | 68.21 | 27.18 | 12.29 |

2024 | 15.80 | 27.69 | 41.25 | 53.09 | 65.96 | 115.81 | 270.31 | 134.74 | 89.82 | 57.40 | 24.52 | 14.48 |

2025 | 19.76 | 31.65 | 44.89 | 56.92 | 71.65 | 126.32 | 266.02 | 117.23 | 88.56 | 52.17 | 23.12 | 15.36 |

HistAvg | 16.05 | 20.37 | 34.34 | 50.21 | 88.21 | 87.01 | 197.68 | 146.43 | 83.92 | 56.85 | 33.33 | 19.07 |

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**MDPI and ACS Style**

Chen, J.; Guo, Z.; Zhang, C.; Tian, Y.; Li, Y.
Research on the Application of CEEMD-LSTM-LSSVM Coupled Model in Regional Precipitation Prediction. *Water* **2023**, *15*, 1465.
https://doi.org/10.3390/w15081465

**AMA Style**

Chen J, Guo Z, Zhang C, Tian Y, Li Y.
Research on the Application of CEEMD-LSTM-LSSVM Coupled Model in Regional Precipitation Prediction. *Water*. 2023; 15(8):1465.
https://doi.org/10.3390/w15081465

**Chicago/Turabian Style**

Chen, Jian, Zhikai Guo, Changhui Zhang, Yangyang Tian, and Yaowei Li.
2023. "Research on the Application of CEEMD-LSTM-LSSVM Coupled Model in Regional Precipitation Prediction" *Water* 15, no. 8: 1465.
https://doi.org/10.3390/w15081465