# Prediction of Rainfall Time Series Using the Hybrid DWT-SVR-Prophet Model

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## Abstract

**:**

^{2}values of 6.17, 3.3, and 0.75, respectively. The DSP model yields higher prediction accuracy than the three baseline models considered, with the prediction accuracy ranking as follows: DSP > SSP > Prophet > SVR. In addition, the DSP model is quite stable and can achieve good results when applied to rainfall data from various climate types, with RMSEs ranging from 1.24 to 7.31, MAEs ranging from 0.52 to 6.14, and R

^{2}values ranging from 0.62 to 0.75. The proposed model may provide a novel approach for rainfall forecasting and is readily adaptable to other time series predictions.

## 1. Introduction

## 2. Methodology

#### 2.1. Hybrid Model Based on DWT-SVR-Prophet

- (i)
- Data preparation and preprocessing: We construct a dataset with daily rainfall data from the National Meteorological Center. The validity and superiority of the model introduced in this paper are verified using this dataset. We preprocess the measured rainfall data to ensure the fitting effect of the applied machine learning model. This process is described in detail in Section 2.2.
- (ii)
- DWT processing: The rainfall time series are decomposed using DWT to obtain high-frequency subsequences with high randomness and volatility and low-frequency subsequences with high periodicity (see Section 2.3.1). This approach allows us to choose forecasting models based on the characteristics of each subseries.
- (iii)
- Hyperparameter optimization: The hyperparameters of the three methods in the coupled model are optimized to obtain the optimal prediction effect. Notably, the parameters of the DWT method are determined by referring to the previous literature, and a grid search method is used to set the hyperparameters of the SVR and Prophet models. The specific process of parameter selection is described in detail in Section 2.5.
- (iv)
- Rainfall prediction: The optimized SVR model and Prophet model are used to predict high-frequency subseries and low-frequency subseries, respectively. The prediction results for each subseries are summed to obtain the final prediction results.

#### 2.2. Data Preprocessing

_{norm}, P

_{i}, P

_{min}, and P

_{max}are the normalized, measured, minimum, and maximum values of rainfall, respectively.

#### 2.3. Methods Used in the DSP Model

#### 2.3.1. Discrete Wavelet Transform

#### 2.3.2. Support Vector Regression Model

#### 2.3.3. Prophet Model

#### 2.4. Hyperparameter Optimization

#### 2.4.1. DWT

#### 2.4.2. SVR

- (i)
- The radial basis function (rbf) is chosen as the kernel function of the SVR model. First, the ranges of values and search steps are set for the main parameters C and γ, and all parameter combinations within the given ranges are obtained $\left({\gamma}_{x},{C}_{y}\right),\left(x=1,2,\dots ,M;y=1,2,\dots ,N\right)$.
- (ii)
- All parameter combinations are applied to rainfall predictions, and the best parameter combination is selected based on effect evaluation $\left({\gamma}_{j},{C}_{k}\right)$.
- (iii)
- To ensure the stability of the search result, the adjacent interval of the optimal parameter combination is selected as the new search range $\gamma \in \left({\gamma}_{j-1},{\gamma}_{j+1}\right),C\in \left({C}_{k-1},{C}_{k+1}\right)$. Then, the search step size is reduced by a factor of 2 (or another multiple), and the optimal parameter combination is again obtained. If the result is unstable, the process is continued until a stable result, i.e., the optimal combination of parameters, is obtained.

#### 2.4.3. Prophet Model

- (i)
- Both linear and logistic “Growth” parameters, and additive and multiplicative “Seasonality mode” parameters are considered.
- (ii)
- The monthly period term is summed with the “Add_seasonality” function in the Prophet model, with “period” = 30.5. Then, the initial range of “Year_seasonality” and “Seasonality_prior_scale” is set to $\left[\mathrm{1,100}\right]$ with a step size of 5.
- (iii)
- The “Changepoint_prior_scale” parameter has a range of $\left[\mathrm{0.01,20}\right]$, and the corresponding step size is 0.5.

#### 2.5. Evaluation Metrics

^{2}) are chosen as evaluation indicators, and the formulae are shown below:

#### 2.6. Open-Source Libraries

## 3. Results

#### 3.1. The DSP Model Provides Accurate Predictions of Rainfall

^{2}= 0.6757.

^{2}were 3.8163, 2.0920, and 0.4334, respectively. The model can simulate the basic trend of A3, and the prediction results are within the acceptable range. The main parameters of the Prophet model are shown in Table 4.

^{2}of the DSP model are 6.1704, 3.2901, and 0.7518, respectively, indicating that among the studied models, the DSP model provides the best prediction of the basic trend of daily rainfall at station 57348. The predicted rainfall is generally similar to the actual rainfall, with satisfactory prediction accuracy.

#### 3.2. The Prediction Accuracy of the DSP Model Is Higher Than That of the Baseline Models

^{2}was improved by 67.4%. The results above verify the superiority of the DSP model.

^{2}values of 4.3486, 2.5343, and 0.6754, respectively. These findings indicate why the DSP model is superior to the SSP model in terms of peak rainfall prediction.

#### 3.3. The DSP Model Displays Outstanding Stability

^{2}values range from 0.6217 to 0.7518. We select R

^{2}as the evaluation index to further evaluate the performance of each model in different cases of rainfall prediction (see Figure 11). We found that most of the R

^{2}values of the DSP model fluctuated slightly while remaining high at different stations. The R

^{2}values of the SSP model were not stable and fluctuated considerably. Additionally, the R

^{2}values of the Prophet and SVR models were consistently low. Therefore, compared with other baseline models, the DSP model displays the stronger generalization ability and yields stable prediction results.

## 4. Discussion

#### 4.1. The DSP Model Achieves Accurate Forecasts of Rainfall Time Series

#### 4.2. The DSP Model Effectively Captures the Detailed Features of Rainfall Time Series

#### 4.3. Generalization of DSP Models

^{2}of 0.3214. Although the prediction accuracy at this station is low, the trend of the prediction results is similar to the actual trend, and the predictions provide reference significance to some extent. At different stations, the DSP model produced reliable predictions, and the prediction accuracy was higher than that of the compared models. The above results fully verify that the DSP model provides outstanding prediction accuracy and stability in applications involving rainfall time series prediction. For other fields (e.g., the energy field, transportation field, and others), the complex time series of interest have the same characteristics as rainfall time series and contain both periodic and nonlinear variations. Theoretically, the DSP model can potentially achieve good performance in these fields and is worthy of further evaluation and application.

#### 4.4. Disadvantages and Direction

## 5. Conclusions

^{2}values of 6.17, 3.29, and 0.75, respectively. The DSP model significantly improves the prediction accuracy, and the results are compared with those of the three baseline methods. Notably, the performances of the methods rank as follows: DSP > SSP > Prophet > SVR. Compared with the SSP model, the RMSE and MAE of the DSP model are reduced by 46.3% and 55.9%, respectively, and R

^{2}is improved by 67.4%, verifying that the DSP model is an excellent prediction model. Moreover, the DSP model displays excellent prediction capability for peak rainfall events. The detailed feature prediction results of the DSP model display well, with RMSE = 4.35, MAE = 2.53, and R

^{2}= 0.68. The DSP model also exhibits good generalization capability. The model achieves reliable predictions of rainfall time series with different features. The calculated RMSEs ranged from 1.24 to 7.31, the MAEs ranged from 0.52 to 6.14, and most R

^{2}values ranged from 0.62 to 0.75. The DSP model can be applied in other fields in which time series are also periodic and nonstationary (e.g., the transportation and energy fields). Thus, the prediction performance of the DSP model deserves further study in additional applications.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 8.**Comparison of the predicted and true rainfall time series using the DSP, SSP, SVR, and Prophet models at station 57348.

**Figure 9.**Comparison of the predicted and true characteristics of the different components (D1 and residual terms).

**Figure 10.**Comparison of the predicted and true values using the DSP model at different stations: (

**a**) Station 59855; (

**b**) Station 56018; (

**c**) Station 58345; (

**d**) Station 54823.

Station (ID) | Rainfall (mm) | Rainfall Frequency (%) | Climatic Type | ||
---|---|---|---|---|---|

Per Year | Per Month | Daily Maximum | |||

Hainan (59855) | 1972.2 | 182.6 | 253.1 | 41.4 | Northern tropics |

Jiangsu (58345) | 1362.7 | 126.2 | 154.8 | 36.0 | Northern subtropics |

Chognqi (57348) | 881.9 | 81.7 | 113.6 | 35.8 | Mid-subtropics |

Shandon (54823) | 642.5 | 59.5 | 127.1 | 20.7 | Southern temperate |

Qinghai (56018) | 466.3 | 43.2 | 31.3 | 40.4 | Plateau climate |

Model | Parameters | Parameters Description | Default Value | Optimization Method |
---|---|---|---|---|

DWT | Wavelet name | Wavelet basis function | - | From previous research |

Level | Wavelet decomposition level | - | ||

SVR | Kernel | Kernel function | rbf | Grid search |

C | Penalty coefficient | 1 | ||

γ | Kernel function coefficient | auto | ||

Prophet | Growth | Function in the trend model | linear | Grid search |

Changepoint_prior_scale | Trend flexibility | 0.05 | ||

Year_seasonality | Year flexibility | 10 | ||

Seasonality_prior_scale | Seasonality flexibility | 10 | ||

Seasonality mode | Model learning style | additive |

Parameter | Parameter Values |
---|---|

kernel function | rbf |

C (penalty variable) | 1024 |

γ (kernel function parameter) | 0.03125 |

Parameter | Parameter Values |
---|---|

Growth | Linear |

Changepoint_prior_scale | 1 |

Year_seasonality | 9 |

Seasonality_prior_scale | 60 |

Seasonality mode | Additive |

Model | Advantages | Disadvantages |
---|---|---|

SVR | It displays good generalization ability and is suitable for nonlinear prediction | It has limitations for general data |

Prophet | It can effectively fit the trend and period variations of time series | Poor fitting ability for complex models |

DSP | It can extract linear and nonlinear features and fit each component using dominance models, respectively | The parameter selection of DWT affects the prediction accuracy and requires additional optimization |

SSP | It can extract linear and nonlinear features and fit each component using dominance models, respectively | The residual term is difficult to fit and requires high model performance |

**Table 6.**Comparison of the prediction accuracy of the DSP, SSP, SVR, and Prophet models for rainfall at station 57348.

Metric | DSP | SSP | SVR | Prophet |
---|---|---|---|---|

RMSE | 6.1704 | 9.1679 | 14.1779 | 12.1362 |

MAE | 3.2901 | 4.3931 | 9.5772 | 4.9510 |

R^{2} | 0.7518 | 0.4492 | −0.3061 | 0.0348 |

**Table 7.**Comparison of the prediction accuracy evaluation indexes for the different models at five stations in different climate regions.

Station | Metric | DSP | SSP | SVR | Prophet |
---|---|---|---|---|---|

59855 | RMSE | 7.3116 | 13.5961 | 19.9489 | 13.0716 |

MAE | 3.3035 | 7.4914 | 15.2908 | 4.4126 | |

R^{2} | 0.6632 | −0.1647 | −1.5074 | −0.0766 | |

56018 | RMSE | 1.2364 | 2.0519 | 2.9356 | 1.9336 |

MAE | 0.5197 | 0.9194 | 2.7219 | 0.9101 | |

R^{2} | 0.6330 | −0.0107 | −1.0688 | 0.1024 | |

58345 | RMSE | 8.8391 | 13.2151 | 17.0824 | 13.7878 |

MAE | 6.1431 | 8.0558 | 12.7209 | 7.1224 | |

R^{2} | 0.6217 | 0.1543 | −0.4131 | 0.0794 | |

54823 | RMSE | 4.8553 | 4.9367 | 9.7700 | 5.7756 |

MAE | 2.3643 | 2.8757 | 8.3983 | 2.4138 | |

R^{2} | 0.3214 | 0.2985 | −1.7476 | 0.0398 | |

57348 | RMSE | 6.1704 | 9.1679 | 14.1779 | 12.1362 |

MAE | 3.2901 | 4.3931 | 9.5772 | 4.9510 | |

R^{2} | 0.7518 | 0.4492 | −0.3061 | 0.0348 |

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## Share and Cite

**MDPI and ACS Style**

Li, D.; Ma, J.; Rao, K.; Wang, X.; Li, R.; Yang, Y.; Zheng, H.
Prediction of Rainfall Time Series Using the Hybrid DWT-SVR-Prophet Model. *Water* **2023**, *15*, 1935.
https://doi.org/10.3390/w15101935

**AMA Style**

Li D, Ma J, Rao K, Wang X, Li R, Yang Y, Zheng H.
Prediction of Rainfall Time Series Using the Hybrid DWT-SVR-Prophet Model. *Water*. 2023; 15(10):1935.
https://doi.org/10.3390/w15101935

**Chicago/Turabian Style**

Li, Dongsheng, Jinfeng Ma, Kaifeng Rao, Xiaoyan Wang, Ruonan Li, Yanzheng Yang, and Hua Zheng.
2023. "Prediction of Rainfall Time Series Using the Hybrid DWT-SVR-Prophet Model" *Water* 15, no. 10: 1935.
https://doi.org/10.3390/w15101935