# 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

- Dai, A. Increasing drought under global warming in observations and models. Nat. Clim. Chang.
**2013**, 3, 52–58. [Google Scholar] [CrossRef] - Habibi, B.; Meddi, M.; Torfs, P.J.; Remaoun, M.; Van Lanen, H.A. Characterisation and prediction of meteorological drought using stochastic models in the semi-arid Chéliff–Zahrez basin (Algeria). J. Hydrol. Reg. Stud.
**2018**, 16, 15–31. [Google Scholar] [CrossRef] - Tsakiris, G. Drought Risk Assessment and Management. Water Resour. Manag.
**2017**, 31, 3083–3095. [Google Scholar] [CrossRef] - Belayneh, A.; Adamowski, J.; Khalil, B.; Ozga-Zielinski, B. Long-term SPI drought forecasting in the Awash River Basin in Ethiopia using wavelet neural network and wavelet support vector regression models. J. Hydrol.
**2014**, 508, 418–429. [Google Scholar] [CrossRef] - Bordi, I.; Sutera, A. Drought monitoring and forecasting at large scale. In Methods and Tools for Drought Analysis and Management 2007; Springer: Dordrecht, The Netherlands, 2017; pp. 3–27. [Google Scholar]
- Mishra, A.K.; Singh, V.P. Drought modeling—A review. J. Hydrol.
**2011**, 403, 157–175. [Google Scholar] [CrossRef] - Mossad, A.; Alazba, A.A. Drought forecasting using stochastic models in a hyper-arid climate. Atmosphere
**2015**, 6, 410–430. [Google Scholar] [CrossRef] - Panu, U.S.; Sharma, T.C. Challenges in drought research: Some perspectives and future directions. Hydrol. Sci. J.
**2002**, 47, S19–S30. [Google Scholar] [CrossRef] - Li, J.; Zhou, S.; Hu, R. Hydrological drought class transition using SPI and SRI time series by loglinear regression. Water Resour. Manag.
**2016**, 30, 669–684. [Google Scholar] [CrossRef] - Park, S.; Im, J.; Jang, E.; Rhee, J. Drought assessment and monitoring through blending of multi-sensor indices using machine learning approaches for different climate regions. Agric. For. Meteorol.
**2016**, 216, 157–169. [Google Scholar] [CrossRef] - Durdu Ömer, F. Application of linear stochastic models for drought forecasting in the Büyük Menderes river basin, western Turkey. Stoch. Environ. Res. Risk Assess.
**2010**, 24, 1145–1162. [Google Scholar] [CrossRef] - Bazrafshan, O.; Salajegheh, A.; Bazrafshan, J.; Mahdavi, M.; Fatehi Maraj, A. Hydrological drought forecasting using ARIMA models (Case study: Karkheh Basin). Ecopersia
**2015**, 3, 1099–1117. [Google Scholar] - Mahmud, I.; Bari, S.H.; Rahman, M.T.U. Monthly rainfall forecast of Bangladesh using autoregressive integrated moving average method. Environ. Eng. Res.
**2016**, 22, 162–168. [Google Scholar] [CrossRef][Green Version] - Karthika, K.; Thirunavukkarasu, V.; Karthika, M. Forecasting of meteorological drought using ARIMA model. Indian J. Agric. Res.
**2017**, 51, 103–111. [Google Scholar] [CrossRef][Green Version] - Rahmat, S.N.; Jayasuriya, N.; Bhuiyan, M.A. Short-term droughts forecast using Markov chain model in Victoria, Australia. Theor. Appl. Climatol.
**2017**, 129, 445–457. [Google Scholar] [CrossRef] - Agboola, A.H.; Gabriel, A.J.; Aliyu, E.O.; Alese, B.K. Development of a fuzzy logic based rainfall prediction model. Int. J. Eng. Technol.
**2013**, 3, 427–435. [Google Scholar] - Jalalkamali, A.; Moradi, M.; Moradi, N. Application of several artificial intelligence models and ARIMAX model for forecasting drought using the Standardized Precipitation Index. Int. J. Environ. Sci. Technol.
**2015**, 12, 1201–1210. [Google Scholar] [CrossRef] - Borji, M.; Malekian, A.; Salajegheh, A.; Ghadimi, M. Multi-time-scale analysis of hydrological drought forecasting using support vector regression (SVR) and artificial neural networks (ANN). Arab. J. Geosci.
**2016**, 9, 725. [Google Scholar] [CrossRef] - Kousari, M.R.; Hosseini, M.E.; Ahani, H.; Hakimelahi, H. Introducing an operational method to forecast long-term regional drought based on the application of artificial intelligence capabilities. Theor. Appl. Climatol.
**2017**, 127, 361–380. [Google Scholar] [CrossRef] - Sheffield, J.; Wood, E.; Chaney, N.; Guan, K.; Sadri, S.; Yuan, X.; Olang, L.; Amani, A.; Ali, A.; DeMuth, S.; et al. A drought monitoring and forecasting system for sub-Sahara African water resources and food security. Bull. Am. Meteorol. Soc.
**2014**, 95, 861–882. [Google Scholar] [CrossRef] - Dehghani, M.; Saghafian, B.; Rivaz, F.; Khodadadi, A. Evaluation of dynamic regression and artificial neural networks models for real-time hydrological drought forecasting. Arab. J. Geosci.
**2017**, 10, 266. [Google Scholar] [CrossRef] - Alsharif, M.H.; Younes, M.K.; Kim, J. Time series ARIMA model for prediction of daily and monthly average global solar radiation: The case study of Seoul, South Korea. Symmetry
**2019**, 11, 240. [Google Scholar] [CrossRef] - Mishra, A.K.; Desai, V.R. Drought forecasting using stochastic models. Stoch. Environ. Res. Risk Assess.
**2005**, 19, 326–339. [Google Scholar] [CrossRef] - McKee, T.B.; Doesken, N.J.; Kleist, J. The relationship of drought frequency and duration to time scales. In Proceedings of the Eighth Conference on Applied Climatology, Boston, MA, USA, 17–22 January 1993; Volume 17, pp. 179–183. [Google Scholar]
- WMO. Standardized Precipitation Index User Guide; Svoboda, M., Hayes, M., Wood, D.A., Eds.; WMO: Geneva, Switzerland, 2012. [Google Scholar]
- Bari, S.H.; Rahman, M.T.; Hussain, M.M.; Ray, S. Forecasting monthly precipitation in Sylhet city using ARIMA model. Civ. Environ. Res.
**2015**, 7, 69–77. [Google Scholar] - Box, G.E.P.; Jenkins, G.M. Time Series Analysis: Forecasting and Control; Holden-Day: San Francisco, CA, USA, 1976. [Google Scholar]
- Box, G.E.P.; Jenkins, G.M.; Reinsel, G.C. Time Series Analysis, Forecasting and Control; Prentice Hall: Englewood Cliffs, NJ, USA, 1994. [Google Scholar]
- Modarres, R. Streamflow drought time series forecasting. Stoch. Environ. Res. Risk Assess.
**2007**, 21, 223–233. [Google Scholar] [CrossRef] - Brockwell, P.J.; Davis, R.A. Introduction to Time Series and Forecasting; Springer Science and Business Media LLC: Berlin/Heidelberg, Germany, 2016. [Google Scholar]
- Akaike, H. A New look at the statistical model identification. Funct. Shape Data Anal.
**1974**, 19, 215–222. [Google Scholar] - Schwarz, G. Estimating the dimension of a model. Ann. Stat.
**1978**, 6, 461–464. [Google Scholar] [CrossRef] - Rahman, M.A.; Yunsheng, L.; Sultana, N. Analysis and prediction of rainfall trends over Bangladesh using Mann–Kendall, Spearman’s rho tests and ARIMA model. Meteorol. Atmos. Phys.
**2017**, 129, 409–424. [Google Scholar] [CrossRef] - Widowati; Putro, S.P.; Koshio, S.; Oktaferdian, V. Implementation of ARIMA model to asses seasonal variability macrobenthic assemblages. Aquat. Procedia
**2016**, 7, 277–284. [Google Scholar] [CrossRef] - Huang, Y.F.; Mirzaei, M.; Yap, W.K. Flood analysis in Langat river basin using stochastic model. Int. J. GEOMATE
**2016**, 11, 2796–2803. [Google Scholar] - Central Weather Bureau, Monthly Report on Climate System: Typhoon Climate Analysis; Central Weather Bureau, Ministry of Transportation and Communications: Taipei, Taiwan, 2019.
- Li, W.; Li, L.; Ting, M.; Liu, Y. Intensification of Northern Hemisphere subtropical highs in a warming climate. Nat. Geosci.
**2012**, 5, 830–834. [Google Scholar] [CrossRef] - Taiwan Climate Change Projection and Information Platform (TCCIP). Taiwan Climate Change Science Report 2017—Physical Phenomena and Mechanisms; TCCIP: Taipei, Taiwan, 2017. [Google Scholar]

**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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**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