# A Combined Model for Water Quality Prediction Based on VMD-TCN-ARIMA Optimized by WSWOA

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## Abstract

**:**

## 1. Introduction

- A new optimization algorithm WSWOA is proposed based on the WOA algorithm. It introduces the perturbation strategy of dual weight factors and the particle swarm search method, which enhances the update ability of the later population and improves the convergence speed of the algorithm.
- The adaptive algorithm WSWOA-VMD is used to process water quality data. The number of modes and penalty parameters of VMD decomposition are optimized through WSWOA to improve the input quality of the prediction model.
- A TCN-ARIMA combination model is proposed to achieve water quality prediction. The different types of modal components obtained via decomposition are input into the corresponding prediction algorithms, which improves the accuracy and efficiency of the model.

## 2. Methodology

#### 2.1. Parameter Optimization Model

#### 2.1.1. WOA

- Step 1: Encircling prey

- Step 2: Bubble net fishing

- Step 3: Random search

#### 2.1.2. WSWOA

- Encircling prey:

- Bubble net fishing:

- Random search:

#### 2.2. Feature Extraction Model

#### 2.2.1. VMD

#### 2.2.2. VMD Parameters Optimized Based on WSWOA

#### 2.3. Combined Forecasting Model

#### 2.3.1. Feature Classification

#### 2.3.2. Model Selection

#### 2.4. VMD-TCN-ARIMA Model Optimized by WSWOA

## 3. The Result of the Proposed Model

#### 3.1. Dataset Settings

_{2}), biochemical oxygen consumption (BOD), total nitrogen (TN), a total of 10 water quality characteristics. The resolution of the dataset varied from semi-monthly to monthly, with each measurement including water quality levels at three depths: surface, middle, and bottom.

#### 3.2. Evaluation Metrics

#### 3.3. Model Decomposition Results Based on WSWOA-VMD

#### 3.4. Water Quality Prediction Results Based on WSWOA-VMD-TCN-ARIMA

^{2}of this prediction is 0.9951 and 0.9957, which is close to 1. The RMSE is 0.0958 mg/L, the MAE is 0.0747 mg/L, and the SAMPE is 1.48%.

## 4. Analysis of Experimental Results

#### 4.1. Multi-Model Comparison

#### 4.1.1. The Influence of Using Optimization Algorithm WSWOA

#### 4.1.2. The Influence of Using Combined Model

- The influence of adding WSWOA-VMD

^{2}by 0.231. According to the results, VMD decomposition of raw water quality sequences can provide enough effective information for the prediction model and substantially improve the prediction accuracy.

- 2.
- The influence of using TCN-ARIMA

#### 4.2. Further Research

## 5. Discussion

## 6. Conclusions

- The results of decomposing the time series demonstrate that the VMD optimized by WSWOA can effectively suppress the mixing of modal components, extract various features from the original sequence, and enhance the accuracy of the subsequent prediction steps. Compared with other existing parameter optimization algorithms, this method has stronger global search capabilities and faster convergence speed, providing an effective new method for existing model parameter optimization technology.
- Comparisons with different forecasting models show that the combined model is more accurate and efficient in forecasting than a single model. It is demonstrated that the TCN-ARIMA combined model can effectively suit the modal components following VMD decomposition of the original data and achieve more precise and faster prediction results.
- The results of predicting multiple water quality characteristics demonstrate that the model in this study is distinct from the majority of models at this stage, which must be restricted to a fixed range of water quality areas and characteristics. The model proposed in this study can adaptively find the optimal model parameters for different types of time series data by utilizing the WSWOA, which removes the need for manual adjustments and greatly improves the generalizability and intelligence of the model.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 9.**Optimal fitness and average fitness curves of different algorithms: (

**a**) curves of average fitness; (

**b**) curves of optimal fitness.

Metrics | Formula | Explanation |
---|---|---|

RMSE (Root Mean Square Error) | $\sqrt{\frac{1}{N}{\displaystyle {\displaystyle \sum}_{i=1}^{N}}{\left({y}_{p}{}^{1}-{y}_{t}{}^{2}\right)}^{2}}$ | RMSE denotes the mean error, which is more sensitive to extreme values, and it can be used as the benchmark for the robustness test of the model. |

MAE (Mean Absolute Error) | $\frac{1}{N}{\displaystyle {\displaystyle \sum}_{i=1}^{N}}\left|{y}_{p}-{y}_{t}\right|$ | MAE is a linear score in which all individual differences are equally weighted on the mean. |

SMAPE (Symmetric Mean Absolute Percentage Error) | $\frac{100\%}{N}{\displaystyle {\displaystyle \sum}_{i=1}^{N}}\frac{\left|{y}_{p}-{y}_{t}\right|}{(\left|{y}_{p}\right|+\left|{y}_{t}\right|)/2}$ | SMAPE uses a percentage 0–100% to represent the error size of the model. |

NSE (Nash–Sutcliffeefficiency Coefficient) | $1-\frac{{{\displaystyle \sum}}_{i=1}^{N}{\left({y}_{p}-{y}_{t}\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{N}{\left({\overline{y}}_{t}-{y}_{t}\right)}^{2}}$ | NSE is widely utilized model performance indices in the water-related research domain, and the closer the value is to 1, the better the predictive ability of the model. |

R^{2} (Coefficient of Determination) | $\frac{{\left({{\displaystyle \sum}}_{i=1}^{N}\left({\overline{y}}_{t}-{y}_{t}\right)\ast \left({\overline{y}}_{p}-{y}_{p}\right)\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{N}{\left({\overline{y}}_{t}-{y}_{t}\right)}^{2}\ast {{\displaystyle \sum}}_{i=1}^{N}{\left({\overline{y}}_{p}-{y}_{p}\right)}^{2}}$ | R^{2} represents the goodness of model fitting. The closer the value is to 1, the better the fit between the fitted value and the actual value, and the better the model performance. |

^{1}${y}_{p}$ is the predicted value.

^{2}${y}_{t}$ is the true value.

Time Step T | Learning Rate LR | ${\mathit{c}}_{\mathit{v}}$ | Classification | |
---|---|---|---|---|

IMF_{1} | 7 | 0.105 | 1.000 | complex |

IMF_{2} | 7 | 0.099 | 0.857 | complex |

IMF_{3} | 7 | 0.052 | 0.594 | complex |

IMF_{4} | 8 | 0.073 | 0.401 | complex |

IMF_{5} | 5 | 0.114 | 0.225 | complex |

IMF_{6} | 4 | 0.325 | 0.158 | simple |

IMF_{7} | 4 | 0.097 | 0.076 | simple |

IMF_{8} | 4 | 0.032 | 0.000 | simple |

Model | Abbreviation | RMSE/mg | MAE/mg | SMAPE | NSE | R^{2} | TIME |
---|---|---|---|---|---|---|---|

BP | #1 | 0.936 | 0.684 | 12.86% | 0.518 | 0.723 | 1.89 |

LSTM | #2 | 0.890 | 0.675 | 12.61% | 0.564 | 0.756 | 7.65 |

ARIMA | #3 | 1.139 | 0.843 | 15.32% | 0.487 | 0.691 | 0.82 |

TCN | #4 | 0.872 | 0.629 | 11.98% | 0.582 | 0.763 | 6.82 |

VMD-TCN | #5 | 0.163 | 0.121 | 2.39% | 0.985 | 0.994 | 42.44 |

EMD-TCN | #6 | 0.495 | 0.372 | 7.31% | 0.866 | 0.935 | 32.39 |

CEEMDAN-TCN | #7 | 0.572 | 0.426 | 8.47% | 0.822 | 0.901 | 34.58 |

VMD-LSTM-ARIMA | #8 | 0.158 | 0.123 | 2.38% | 0.986 | 0.994 | 44.42 |

VMD-TCN-ARIMA | #9 | 0.096 | 0.075 | 1.48% | 0.995 | 0.996 | 31.38 |

CEEMDAN-IWOA-BP | #10 | 0.632 | 0.414 | 8.26% | 0.828 | 0.911 | 11.96 |

VMD Optimal Solutions | RMSE | MAE | SMAPE | NSE | R^{2} | ||
---|---|---|---|---|---|---|---|

K | σ | ||||||

DOsat | 9 | 1190 | 1.467 | 1.146 | 1.40% | 0.985 | 0.994 |

SS | 8 | 1277 | 0.436 | 0.349 | 8.17% | 0.987 | 0.994 |

turbidity | 8 | 1013 | 0.293 | 0.239 | 9.55% | 0.988 | 0.995 |

Salinity | 9 | 2082 | 0.185 | 0.146 | 0.48% | 0.989 | 0.996 |

Pheophytin | 7 | 1761 | 0.113 | 0.092 | 13.09% | 0.986 | 0.995 |

SD | 9 | 1311 | 0.098 | 0.072 | 3.02% | 0.980 | 0.994 |

SiO_{2} | 9 | 1769 | 0.041 | 0.032 | 6.09% | 0.985 | 0.995 |

BOD | 9 | 1569 | 0.033 | 0.027 | 7.66% | 0.985 | 0.993 |

TN | 9 | 1149 | 0.023 | 0.019 | 5.51% | 0.983 | 0.993 |

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## Share and Cite

**MDPI and ACS Style**

Zuo, H.; Gou, X.; Wang, X.; Zhang, M.
A Combined Model for Water Quality Prediction Based on VMD-TCN-ARIMA Optimized by WSWOA. *Water* **2023**, *15*, 4227.
https://doi.org/10.3390/w15244227

**AMA Style**

Zuo H, Gou X, Wang X, Zhang M.
A Combined Model for Water Quality Prediction Based on VMD-TCN-ARIMA Optimized by WSWOA. *Water*. 2023; 15(24):4227.
https://doi.org/10.3390/w15244227

**Chicago/Turabian Style**

Zuo, Hongyu, Xiantai Gou, Xin Wang, and Mengyin Zhang.
2023. "A Combined Model for Water Quality Prediction Based on VMD-TCN-ARIMA Optimized by WSWOA" *Water* 15, no. 24: 4227.
https://doi.org/10.3390/w15244227