# Examining the Applicability of Wavelet Packet Decomposition on Different Forecasting Models in Annual Rainfall Prediction

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## Abstract

**:**

## 1. Introduction

## 2. Study Area and Methods

#### 2.1. Study Area

#### 2.2. Wavelet Packet Decomposition WPD

#### 2.3. Extreme Learning Machine (ELM)

#### 2.4. Back-Propagation Neural Network (BPNN)

#### 2.5. ARIMA

_{s}, where (p, d, q) represents the non-seasonal order and (P, D, Q)

_{s}denotes the seasonal order. The ARIMA [50] model can be expressed as:

#### 2.6. Framework of the Proposed Hybrid Model

#### 2.7. Evaluation Indicators

## 3. Results

#### 3.1. Decomposition Results

#### 3.2. Selection of Input Variable

#### 3.3. Model Development

- (1)
- ELM and BPNN models

- (2)
- ARIMA

- (3)
- WPD-ANN and WPD-ARIMA

#### 3.4. Comparative Analysis

#### 3.5. Discussion of Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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No. | Series | Input Variables |
---|---|---|

1 | original | q_{(t−1)}~q_{(t−9)} |

2 | WPD_{1} | q_{(t−1)}~q_{(t−12)} |

3 | WPD_{2} | q_{(t−1)}~q_{(t−11)} |

4 | WPD_{3} | q_{(t−1)}~q_{(t−11)} |

5 | WPD_{4} | q_{(t−1)}~q_{(t−12)} |

6 | WPD_{5} | q_{(t−1)}~q_{(t−12)} |

7 | WPD_{6} | q_{(t−1)}~q_{(t−10)} |

8 | WPD_{7} | q_{(t−1)}~q_{(t−10)} |

9 | WPD_{8} | q_{(t−1)}~q_{(t−9)} |

No. | Series | h | p-Value | t-Statistic | Critical Value |
---|---|---|---|---|---|

1 | Original | 1 | 0 | −7.928 | −3.489 |

2 | WPD_{1} | 0 | 0.288 | −2.586 | −3.506 |

3 | Diff (WPD_{1}) | 1 | 0.020 | −2.348 | −1.948 |

4 | WPD_{2} | 1 | 0 | −7.001 | −3.504 |

5 | WPD_{3} | 1 | 0.007 | −4.299 | −3.504 |

6 | WPD_{4} | 1 | 0 | −6.753 | −3.504 |

7 | WPD_{5} | 1 | 0.004 | −4.470 | −3.506 |

8 | WPD_{6} | 1 | 0 | −7.419 | −3.504 |

9 | WPD_{7} | 1 | 0 | −11.164 | −3.504 |

10 | WPD_{8} | 1 | 0 | −9.553 | −3.504 |

**Table 3.**Auto-regressive integrated moving average (ARIMA) models based on BIC(Bayes information criteria).

No. | Series | ARIMA | BIC |
---|---|---|---|

1 | Original | ARIMA (12,1,2) | 11.292 |

2 | WPD_{1} | ARIMA (9,1,3) | 4.026 |

3 | WPD_{2} | ARIMA (6,0,6) | 5.63 |

4 | WPD_{3} | ARIMA (7,0,5) | 5.494 |

5 | WPD_{4} | ARIMA (5,0,8) | 6.531 |

6 | WPD_{5} | ARIMA (2,0,7) | 3.076 |

7 | WPD_{6} | ARIMA (11,0,9) | 6.535 |

8 | WPD_{7} | ARIMA (12,0,5) | 6.435 |

9 | WPD_{8} | ARIMA (6,0,6) | 6.806 |

Model | Training | Testing | ||||||
---|---|---|---|---|---|---|---|---|

R | NSEC | RMSE | MAE | R | NSEC | RMSE | MAE | |

ARIMA | 0.415 | 0.139 | 129.978 | 101.046 | −0.304 | −0.535 | 175.136 | 141.295 |

WPD-ARIMA | 0.991 | 0.981 | 19.399 | 14.951 | 0.951 | 0.903 | 44.127 | 37.199 |

BPNN | 0.618 | 0.357 | 112.634 | 53.775 | 0.820 | 0.434 | 106.368 | 74.221 |

WPD-BPNN | 0.978 | 0.957 | 29.445 | 22.924 | 0.988 | 0.973 | 23.176 | 19.947 |

ELM | 0.628 | 0.394 | 109.308 | 85.583 | 0.819 | 0.649 | 83.698 | 78.656 |

WPD-ELM | 0.986 | 0.9712 | 23.687 | 19.091 | 0.997 | 0.973 | 23.069 | 19.051 |

Model | Index | Training(%) | Testing(%) |
---|---|---|---|

WPD-ARIMA & ARIMA | R(↑) | 138.81 | 413.25 |

NSEC(↑) | 607.65 | 268.63 | |

RMSE(↓) | 85.08 | 74.80 | |

MAE(↓) | 85.20 | 73.67 | |

WPD-BPNN & BPNN | R(↑) | 58.14 | 20.43 |

NSEC(↑) | 167.82 | 124.37 | |

RMSE(↓) | 73.86 | 78.21 | |

MAE(↓) | 57.37 | 73.12 | |

WPD-ELM & ELM | R(↑) | 56.91 | 21.66 |

NSEC(↑) | 146.35 | 49.89 | |

RMSE(↓) | 78.33 | 72.44 | |

MAE(↓) | 77.69 | 75.78 |

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**MDPI and ACS Style**

Wang, H.; Wang, W.; Du, Y.; Xu, D.
Examining the Applicability of Wavelet Packet Decomposition on Different Forecasting Models in Annual Rainfall Prediction. *Water* **2021**, *13*, 1997.
https://doi.org/10.3390/w13151997

**AMA Style**

Wang H, Wang W, Du Y, Xu D.
Examining the Applicability of Wavelet Packet Decomposition on Different Forecasting Models in Annual Rainfall Prediction. *Water*. 2021; 13(15):1997.
https://doi.org/10.3390/w13151997

**Chicago/Turabian Style**

Wang, Hua, Wenchuan Wang, Yujin Du, and Dongmei Xu.
2021. "Examining the Applicability of Wavelet Packet Decomposition on Different Forecasting Models in Annual Rainfall Prediction" *Water* 13, no. 15: 1997.
https://doi.org/10.3390/w13151997