# Stochastic Model for Drought Forecasting in the Southern Taiwan Basin

^{*}

## Abstract

**:**

^{2}) were over 0.80 at each station, and the root-mean-square error and mean absolute error were sufficiently low, indicating that the ARIMA model is effective and adequate for our stations. Finally, we employed the ARIMA model to forecast future drought conditions from 2019 to 2022. The results yielded relatively low SPI values in southern Taiwan in future summers. In summary, we successfully constructed an ARIMA model to forecast drought. The information in this study can act as a reference for water resource management.

## 1. Introduction

## 2. Study Area

## 3. Methodology

#### 3.1. SPI

#### 3.2. Time Series Model

#### 3.2.1. Nonseasonal ARIMA Model

#### 3.2.2. Seasonal ARIMA (SARIMA) Model

_{S}, where (p,d,q) is the nonseasonal part of the model and (P,D,Q)

_{S}is the seasonal part. This can be expressed as follows:

#### 3.3. ARIMA Model Development

#### 3.3.1. Model Identification

_{t}is the observed data and ${\widehat{y}}_{t}$ is the predicted value.

#### 3.3.2. Parameter Estimation

#### 3.3.3. Diagnostic Checking

_{k}is the sample autocorrelation at lag k. The statistical Q values are compared with the critical value with the degree of freedom at a 5% significance level. If the calculated values are less than the critical value, this means the residuals of the model are in accordance with white noise.

## 4. Results and Discussion

#### 4.1. Model Identification

#### 4.2. Parameter Estimation

#### 4.3. Diagnostic Checking

#### 4.4. Model Validation

^{2}value, the better the performance of the model. According to our results, the R

^{2}values in each station were greater than 0.8, and the root-mean-square error (RMSE) and mean absolute error (MAE) were also sufficiently low. Therefore, the SARIMA model used to predict drought index in this study is reasonably precise.

#### 4.5. Forecasting

## 5. Conclusions

^{2}) at each station (all over 0.80) and low values for the RMSE and MAE, which implies that the model is adequately precise at each basin. Finally, we used the ARIMA model to forecast the future drought conditions from 2019 to 2022. The forecasting results demonstrate that the SPI value is relatively low in the summer of 2020, which implies that there may be a water shortage in southern Taiwan. This phenomenon may be related to climate change, which leads to an enhancement of the Pacific high extending westward, thus affecting the paths of typhoons. In addition, the Pacific high dominates the summer climate in Taiwan, and if its intensity continues to increase, this will reduce precipitation in Taiwan in the future. However, the detailed evolution mechanism still remains to be discussed in the future.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Autocorrelation function (ACF) and partial autocorrelation function (PACF) plots for model selection at the PR1 station in the Pozi River basin.

**Figure 4.**The histogram (

**left column**) and normal probability plot (

**right column**) of residuals at the PR1 station in the Pozi River basin.

**Figure 6.**Comparison of observed data with predicted values using the best seasonal ARIMA (SARIMA) model at each basin.

**Figure 7.**Drought forecasting for the period 2018–2022 using the seasonal ARIMA model at each basin.

Basin | Station | Short Name |
---|---|---|

Pozi River basin | Zhang Nao Liao-2 | PR1 |

Bazhang River basin | Da Hu Shan | BR1 |

Jishui River basin | Guan Zi Ling-2 | GR1 |

Yanshui River basin | Qi Ding | YR2 |

Erren River basin | Gu Ting Keng | RR2 |

Gaoping River basin | Ping Dong-5 | KR3 |

Linbian River basin | Nan Han | LR2 |

**Table 2.**The Akaike information criterion (AIC) and Schwarz–Bayesian criterion (SBC) parameters of each station for selected candidate autoregressive integrated moving average (ARIMA) models.

Station | Model | AIC | SBC | Station | Model | AIC | SBC | Station | Model | AIC | SBC |
---|---|---|---|---|---|---|---|---|---|---|---|

PR1 | SARIMA(1,1,0)(1,0,4)_{12} | 348.0881 | 381.4784 | YR2 | SARIMA(1,1,0)(1,0,3)_{12} | 288.9087 | 318.1252 | KR3 | SARIMA(1,1,0)(2,0,2)_{12} | 356.0616 | 385.2781 |

SARIMA(1,1,0)(2,0,2)_{12} | 345.4032 | 374.6197 | SARIMA(1,1,0)(2,0,2)_{12} | 288.6429 | 317.8594 | SARIMA(1,1,0)(2,0,4)_{12} | 351.2744 | 388.8385 | |||

SARIMA(1,1,0)(2,0,3)_{12} | 347.0453 | 380.4356 | SARIMA(1,1,0)(2,0,3)_{12} | 277.1375 | 310.5278 | SARIMA(1,1,0)(3,0,2)_{12} | 355.4204 | 388.8107 | |||

SARIMA(1,1,0)(2,0,4)_{12} | 346.5433 | 384.1074 | SARIMA(1,1,0)(2,0,4)_{12} | 283.4290 | 320.9931 | SARIMA(1,1,0)(3,0,3)_{12} | 351.1653 | 388.7294 | |||

SARIMA(1,1,0)(3,0,2)_{12} | 346.7269 | 380.1172 | SARIMA(1,1,0)(3,0,3)_{12} | 279.1007 | 316.6648 | SARIMA(1,1,0)(4,0,2)_{12} | 352.6365 | 390.2005 | |||

SARIMA(1,1,0)(3,0,4)_{12} | 314.7873 | 356.5252 | SARIMA(1,1,0)(4,0,4)_{12} | 271.8073 | 317.7189 | SARIMA(1,1,0)(4,0,3)_{12} | 352.7802 | 394.5180 | |||

BR1 | SARIMA(1,1,0)(1,0,4)_{12} | 333.3618 | 366.7521 | RR2 | SARIMA(1,1,0)(2,0,2)_{12} | 194.4071 | 223.6236 | LR2 | SARIMA(1,1,0)(2,0,3)_{12} | 354.4606 | 387.8509 |

SARIMA(1,1,0)(2,0,4)_{12} | 332.7244 | 370.2284 | SARIMA(1,1,0)(2,0,4)_{12} | 184.9914 | 222.5554 | SARIMA(1,1,0)(2,0,4)_{12} | 351.3671 | 388.9312 | |||

SARIMA(1,1,0)(3,0,1)_{12} | 338.9115 | 368.1280 | SARIMA(1,1,0)(3,0,2)_{12} | 195.0079 | 228.3982 | SARIMA(1,1,0)(3,0,3)_{12} | 339.6911 | 377.2552 | |||

SARIMA(1,1,0)(3,0,2)_{12} | 331.8391 | 365.2294 | SARIMA(1,1,0)(4,0,0)_{12} | 190.8300 | 220.0465 | SARIMA(1,1,0)(3,0,4)_{12} | 338.4294 | 380.1673 | |||

SARIMA(1,1,0)(3,0,3)_{12} | 328.9278 | 366.4919 | SARIMA(1,1,0)(4,0,1)_{12} | 189.1966 | 222.5869 | SARIMA(1,1,0)(4,0,3)_{12} | 341.4640 | 383.2019 | |||

SARIMA(1,1,0)(3,0,4)_{12} | 319.4850 | 361.2228 | SARIMA(1,1,0)(4,0,2)_{12} | 191.1358 | 228.6999 | SARIMA(1,1,0)(4,0,4)_{12} | 339.7495 | 358.6611 | |||

GR1 | SARIMA(1,1,0)(1,0,3)_{12} | 238.5829 | 267.7994 | ||||||||

SARIMA(1,1,0)(1,0,4)_{12} | 240.4575 | 273.8478 | |||||||||

SARIMA(1,1,0)(2,0,2)_{12} | 238.4253 | 267.6418 | |||||||||

SARIMA(1,1,0)(3,0,2)_{12} | 239.3507 | 272.7410 | |||||||||

SARIMA(1,1,0)(3,0,3)_{12} | 241.0885 | 278.6526 | |||||||||

SARIMA(1,1,0)(4,0,4)_{12} | 211.4762 | 257.3878 |

Pozi River Basin: SARIMA(1,1,0)(3,0,4)_{12} | ||||
---|---|---|---|---|

Model Parameters | Variables in the Model | |||

Value of Parameter | Standard Error | t-Statistic | p-Value | |

constant | 0.0020 | 0.0112 | 0.1765 | 0.8599 |

$\varnothing $_{1} | −0.0903 | 0.0355 | −2.5469 | 0.0109 |

Φ_{1} | −0.2841 | 0.0286 | −9.9461 | 0.0000 |

Φ_{2} | −0.2889 | 0.0203 | −14.2303 | 0.0000 |

Φ_{3} | −0.7950 | 0.0195 | −40.7233 | 0.0000 |

Θ_{1} | −0.4530 | 0.0366 | −12.3870 | 0.0000 |

Θ_{2} | 0.1171 | 0.0360 | 3.2491 | 0.0012 |

Θ_{3} | 0.6851 | 0.0304 | 22.5332 | 0.0000 |

Θ_{4} | −0.6389 | 0.0307 | −20.8422 | 0.0000 |

_{1}= nonseasonal AR parameter; Φ

_{1}, Φ

_{2}, Φ

_{3}, Φ

_{4}= seasonal AR parameters; Θ

_{1}, Θ

_{2}, Θ

_{3}, Θ

_{4}= seasonal MA parameters.

LBQ Test | |||||||
---|---|---|---|---|---|---|---|

Station | PR1 | BR1 | GR1 | YR2 | RR2 | KR3 | LR2 |

Test statistic (Q) | 55.33 | 58.41 | 73.58 | 53.91 | 41.43 | 37.89 | 48.18 |

Critical value | 89.39 | 89.39 | 89.39 | 89.39 | 89.39 | 89.39 | 89.39 |

Station | Model | Performance Measures | ||
---|---|---|---|---|

R_{2} | RMSE | MAE | ||

PR1 | ARIMA(1,1,0)(3,0,4)_{12} | 0.8775 | 0.3220 | 0.2159 |

BR1 | ARIMA(1,1,0)(3,0,4)_{12} | 0.8869 | 0.3628 | 0.2291 |

GR1 | ARIMA(1,1,0)(4,0,4)_{12} | 0.8938 | 0.3241 | 0.2120 |

YR2 | ARIMA(1,1,0)(2,0,3)_{12} | 0.8445 | 0.3123 | 0.2060 |

RR2 | ARIMA(1,1,0)(2,0,4)_{12} | 0.8602 | 0.3482 | 0.2239 |

KR3 | ARIMA(1,1,0)(3,0,3)_{12} | 0.8337 | 0.3699 | 0.2339 |

LR2 | ARIMA(1,1,0)(3,0,3)_{12} | 0.8213 | 0.3714 | 0.2281 |

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## Share and Cite

**MDPI and ACS Style**

Yeh, H.-F.; Hsu, H.-L.
Stochastic Model for Drought Forecasting in the Southern Taiwan Basin. *Water* **2019**, *11*, 2041.
https://doi.org/10.3390/w11102041

**AMA Style**

Yeh H-F, Hsu H-L.
Stochastic Model for Drought Forecasting in the Southern Taiwan Basin. *Water*. 2019; 11(10):2041.
https://doi.org/10.3390/w11102041

**Chicago/Turabian Style**

Yeh, Hsin-Fu, and Hsin-Li Hsu.
2019. "Stochastic Model for Drought Forecasting in the Southern Taiwan Basin" *Water* 11, no. 10: 2041.
https://doi.org/10.3390/w11102041