# The Development of a Hybrid Wavelet-ARIMA-LSTM Model for Precipitation Amounts and Drought Analysis

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## Abstract

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## 1. Introduction

## 2. Data and Methodology

#### 2.1. Study Area

#### 2.2. Site Precipitation Observations

#### 2.3. Time Series Models on Monthly Precipitation

#### 2.3.1. Autoregressive Integrated Moving Average (ARIMA)

#### 2.3.2. Long Short-Term Memory Method (LSTM)

#### 2.3.3. Discrete Wavelet Transformation (DWT)

#### 2.3.4. Development of Wavelet-ARIMA-LSTM (W-AL) Model

#### 2.3.5. Evaluation Metrics

^{2}), which provides a way to assess the results of the same model on different data, are adopted to assess the performance of the models. Smaller RMSE and MAE values indicate better model performance, and larger R

^{2}values indicate better model performance. The criteria are defined as follows:

#### 2.4. Grey System Models on Drought Events

#### 2.4.1. China Z-Index (CZI)

#### 2.4.2. Grey Prediction Models

## 3. Results and Discussions

#### 3.1. Time Series Analysis of Monthly Precipitation Amounts

^{2}from 0.578 to 0.728. For Linjiang, the RMSE values ranged from 35.772 to 42.739, MAE from 21.712 to 29.439, and R2 from 0.626 to 0.738. For Qian Gorlos, the RMSE values ranged from 27.064 to 37.535, MAE from 19.111 to 21.712, and R

^{2}from 0.366 to 0.670. These results indicated that the best RMSE and MAE values were found for Qian Gorlos, while the best R

^{2}value was found for Linjiang. However, RMSE and MAE values have limitations; that is, the same algorithm model that is utilized to predict monthly precipitation at different stations, cannot reflect the fitting effect of the model at different stations. Because the dimensions of the data are different at different stations, it is impossible to directly compare the predicted values, and it is impossible to determine the stations for which the model performs better. In contrast, R2 converts the predicted results into accuracies, and the results all fall between 0 and 1. For the prediction accuracies of different stations, it is possible to employ R

^{2}to compare and determine which stations perform better. Based on this, it can be found that the fitting effect of the model was best in Linjiang with humid and semi-humid climate type and was worst in Qian Gorlos with arid and semi-arid climate type which revealed the model predicted better in humid region and worse in arid region.

#### 3.2. Grey System Analysis for Drought Events

#### 3.2.1. Identification of Drought Events

#### 3.2.2. Projections of Drought Events

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Time series of the observed monthly precipitation data during the training period of 1967–1997 and testing period of 1998–2017.

**Figure 6.**Observed versus forecasted monthly precipitation data for the ARIMA, LSTM and W-AL models for the test period from 1998 to 2017.

**Figure 7.**Scatter plots of observed and forecasted monthly precipitation for the W-AL model for the test period from 1998 to 2017.

Station | Longitude (°E) | Latitude (°N) | Altitude (m) | Climatic Type |
---|---|---|---|---|

Linjiang | 126.92 | 41.80 | 332.7 | Humid and semi-humid |

Changchun | 125.22 | 43.90 | 236.8 | Semi-arid and semi-humid |

Qian Gorlos | 124.87 | 45.08 | 136.2 | Arid and semi-arid |

**Table 2.**Classification of drought categories for the meteorological drought indices China Z-Index (CZI) [27].

Drought Category | CZI |
---|---|

No drought | −0.842 ≤ CZI |

Slight drought | −1.037 ≤ CZI < −0.842 |

Moderate drought | −1.645≤CZI < −1.037 |

Heavy drought | CZI < −1.645 |

**Table 3.**Evaluation standards of the posterior-variance test [38].

Grade | C | P |
---|---|---|

Good | <0.350 | >0.950 |

Pass | <0.500 | >0.800 |

Unconvincing pass | <0.650 | >0.700 |

Fail | ≥0.650 | ≤0.700 |

**Table 4.**Obtained error statistics for the ARIMA, LSTM and wavelet-ARIMA-LSTM (W-AL) models for the test period from 1998 to 2017.

Station | Model | RMSE (mm) | MAE (mm) | R^{2} |
---|---|---|---|---|

Changchun | ARIMA | 38.698 | 25.156 | 0.578 |

LSTM | 34.571 | 29.313 | 0.663 | |

W-AL | 31.086 | 22.215 | 0.728 | |

Linjiang | ARIMA | 42.739 | 29.439 | 0.626 |

LSTM | 38.994 | 27.864 | 0.728 | |

W-AL | 35.772 | 27.233 | 0.738 | |

Qian Gorlos | ARIMA | 37.535 | 21.712 | 0.366 |

LSTM | 34.509 | 20.420 | 0.464 | |

W-AL | 27.064 | 19.111 | 0.670 |

**Table 5.**Obtained RMSE statistics for the W-AL models at different ratio training:test for the test period from 1998 to 2017.

Station | Model | 60:40 Ratio | 70:30 Ratio | 80:20 Ratio | 90:10 Ratio |
---|---|---|---|---|---|

Changchun | ARIMA | 38.698 | 46.753 | 48.834 | 47.230 |

LSTM | 34.571 | 32.725 | 31.905 | 32.913 | |

W-AL | 31.086 | 29.975 | 29.913 | 31.339 | |

Linjiang | ARIMA | 42.739 | 48.537 | 63.155 | 46.458 |

LSTM | 38.994 | 37.795 | 36.538 | 37.801 | |

W-AL | 35.772 | 34.788 | 34.669 | 36.408 | |

Qian Gorlos | ARIMA | 37.535 | 46.265 | 36.170 | 37.161 |

LSTM | 34.509 | 28.762 | 27.185 | 25.188 | |

W-AL | 27.064 | 26.380 | 25.355 | 21.822 |

Station | GM (1, 1) | DGM (1, 1) | ||
---|---|---|---|---|

a | b | β1 | β2 | |

Changchun | −0.264 | 9.195 | 1.303 | 10.607 |

Linjiang | −0.213 | 14.746 | 1.225 | 16.955 |

Qian Gorlos | −0.185 | 14.686 | 1.201 | 16.323 |

Station | Model | Actual Drought | Predicted Drought | Average Relative Error |
---|---|---|---|---|

Changchun | GM (1, 1) | 2001, 2011 | 2001, 2012 | 0.022 |

DGM (1, 1) | 2001, 2014 | 2002, 2013 | 0.027 | |

Linjiang | GM (1, 1) | 2011, 2017 | 2007, 2017 | 0.191 |

DGM (1, 1) | 2002, 2014 | 2006, 2015 | 0.195 | |

Qian Gorlos | GM (1, 1) | 2004, 2007 | 2004, 2012 | 0.067 |

DGM (1, 1) | 2004, 2007 | 2004, 2012 | 0.084 |

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**MDPI and ACS Style**

Wu, X.; Zhou, J.; Yu, H.; Liu, D.; Xie, K.; Chen, Y.; Hu, J.; Sun, H.; Xing, F.
The Development of a Hybrid Wavelet-ARIMA-LSTM Model for Precipitation Amounts and Drought Analysis. *Atmosphere* **2021**, *12*, 74.
https://doi.org/10.3390/atmos12010074

**AMA Style**

Wu X, Zhou J, Yu H, Liu D, Xie K, Chen Y, Hu J, Sun H, Xing F.
The Development of a Hybrid Wavelet-ARIMA-LSTM Model for Precipitation Amounts and Drought Analysis. *Atmosphere*. 2021; 12(1):74.
https://doi.org/10.3390/atmos12010074

**Chicago/Turabian Style**

Wu, Xianghua, Jieqin Zhou, Huaying Yu, Duanyang Liu, Kang Xie, Yiqi Chen, Jingbiao Hu, Haiyan Sun, and Fengjuan Xing.
2021. "The Development of a Hybrid Wavelet-ARIMA-LSTM Model for Precipitation Amounts and Drought Analysis" *Atmosphere* 12, no. 1: 74.
https://doi.org/10.3390/atmos12010074