# Forecasting of SPI and SRI Using Multiplicative ARIMA under Climate Variability in a Mediterranean Region: Wadi Ouahrane Basin, Algeria

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

^{2}) using linear stochastic models known as Autoregressive Integrated Moving Average (ARIMA) and multiplicative Seasonal Autoregressive Integrated Moving Average (SARIMA). In particular, data from 6 precipitation stations and 1 hydrometric station for the period 1972–2018 were used to evaluate the Standardized Precipitation Index (SPI) and the Standardized Runoff Index (SRI) for 12 months. Then, the multiplicative ARIMA model was applied to forecasting drought based on SPI and SRI. As a result, the ARIMA model (1,0,1)(0,0,1)

_{12}for SPI and (1,0,1)(1,0,1)

_{12}for SRI were shown to be the best models for drought forecast. In fact, both models exhibited high quality for SPI and SRI of 0.97 and 0.51 for 1-month and 12-month lead time, respectively, based on validation R

^{2}. In general, prediction skill decreases with increase in lead time. The models can be used with reasonable accuracy to forecast droughts with up to 12 months of lead time.

## 1. Introduction

## 2. Data and Methods

#### 2.1. Study Area and Data

^{2}with a maximum altitude of 991 m and a minimum altitude of 165 m (Figure 1). The Wadi Ouahrane basin is limited to the east by the basin of Wadi Fodda, to the west by the Wadi Ras basin, to the north by the Wadi Allala basin, and to the south by the Wadi Sly basin. The basin is monitored by six pluviometric stations and one hydrometric station (Figure 1). It is influenced by the Mediterranean climate, with an interannual average precipitation of 333 mm during the period 1972–2018. The mean annual temperature is 18° C. The Ouled Farès sector receives a rainfall that varies from 207 to 628 mm, which is located below 200 m altitude. It occupies nearly 40% of the basin’s area. The Benairia sector, located at more than 350 m altitude, receives an average annual rainfall that varies between 234 and 749 mm. This sector covers about 60% of the basin. The Wadi Ouahrane basin is defined by impermeable marl bedrock that covers 80% of the basin’s surface. In contrast to the formations in the southern part of the basin, which are composed of conglomerate and red sand and have an average permeability, these soft lithological strata in the northern half of the basin are continually subjected to significant water erosion. The primary agricultural activities in the wadi Ouahrane watershed, in terms of land usage, are mixed farming and cereal cultivation [30]. The Köppen–Geiger classification [31] identifies the climate of the basin as a hot-summer Mediterranean climate, thus presenting relatively mild winters (with rain) and very hot summers (often very dry). With this climate, the coldest month generally averages above 0 °C, at least 1 month’s average temperature reaches values higher than 22 °C, and at least 4 months average above 10 °C. Concerning rainfall, in the hot-summer Mediterranean climate, rainfall in the wettest month of winter averages over three times that in the driest month of summer, which receives less than 30 mm.

#### 2.2. Standardized Precipitation (Runoff) Index

_{t}can be used [32]:

_{t}is the observed series, φ is the polynomial of order p, and θ is the polynomial of order q.

_{t}.

_{w}are created. A combination of seasonal and nonseasonal models forms the so-called multiplicative ARIMA models. The basic form of this model follows [33]:

_{t}) = Z

_{t−1}, (1 − B)

^{d}is equal to d of the second nonseasonal difference and (1 − B

^{w})

^{D}is equal to D of the second seasonal difference of size w [33].

#### 2.3. Model Development

_{tab}, the null hypothesis related to normality is rejected for the chosen level of significance.

#### 2.4. Kappa (κ)

#### 2.5. Model Validation

^{2}), Mean Absolute Error (MAE), and Root Mean Square Error (RMSE) [20,39].

## 3. Results

#### 3.1. Assessment of Drought Based on SPI and SRI

#### 3.2. Stochastic Model Development

_{12}and ARIMA (2,0,1)(1,0,1)

_{12}, respectively. According to the figures, it can be seen that most of the ACF and PACF values in both indices are within the confidence limit. Therefore, there is no significant correlation between the residuals; the residuals are white noise.

#### 3.3. Drought Forecasting Using Selected Models

^{2}, RMSE, and MAE values between observed data and predicted data using the selected best model for all time series. Results show that with a longer lead time, the coefficient of correlation decreases and error coefficient increases between observed and predicted data. The best models selected from the multiplicative ARIMA approach using a time series data of SPI-12 and SRI-12 series can be used for drought forecasting at whatever lag is desired.

## 4. Discussion

## 5. Conclusions

^{2}, both models had good quality; for the SPI and the SRI, R

^{2}was equal to 0.96 and 0.97, respectively, at 1-month lag. Finally, we conclude that the seasonal ARIMA model can be used to forecast meteorological and hydrological drought indices in arid regions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Location, topographic characteristics, pluviometric, and hydrometric network of the study area.

**Figure 5.**ACF and PACF correlograms and 95% confidence limits for stochastic monthly series for the Wadi Ouahrane basin (

**a**–

**d**).

**Figure 6.**Diagnostic check of the best-fitted multiplicative ARIMA model for SPI-12 (

**a**,

**b**) and SRI-12 (

**c**,

**d**) time series for the Wadi Ouahrane watershed.

**Figure 8.**Comparison of SPI-12 (

**a**) and SRI-12 (

**b**) series observed and predicted in the validation phase for 1-month lead time.

Stations | ID | Name | Geographical Coordinates | Elevation | |
---|---|---|---|---|---|

Longitude | Latitude | ||||

(° ′ ″) | (° ′ ″) | (m) | |||

Precipitation Stations | |||||

S1 | 012201 | LARBAT OULED FARES | 01°09′18″ | 36°16′20″ | 116 |

S2 | 012224 | BOUZGHAIA | 01°14′27″ | 36°20′15″ | 217 |

S3 | 012205 | BENAIRIA | 01°22′28″ | 36°21′04″ | 320 |

S4 | 012221 | MEDJAJA | 01°20′53″ | 36°16′39″ | 487 |

S5 | 012209 | CHETIA | 01°15′53″ | 36°12′56″ | 108 |

S6 | NMO | Airport, Chlef | 01°19′28″ | 36°13′31″ | 158 |

Hydrometric Station | |||||

HS1 | 012201 | LARBAT OULED FARES | 01°13′56″ | 36°14′14″ | 173 |

Index | Model | AIC | BIC |
---|---|---|---|

SPI-12 | ARIMA(0,1,0)(0,1,1)_{12} | 605.25 | 613.74 |

ARIMA(0,1,1)(0,0,1)_{12} | 321.57 | 334.38 | |

ARIMA(1,0,1)(1,1,0)_{12} | 754.55 | 771.54 | |

ARIMA(1,0,1)(0,1,1)_{12} | 587.61 | 604.61 | |

ARIMA(1,0,0)(2,0,1)_{12} | 320.11 | 345.74 | |

SRI-12 | ARIMA(1,0,1)(1,0,0)_{12} | −1277.37 | −1255.90 |

ARIMA(2,0,0)(2,0,0)_{12} | −1355.60 | −1329.84 | |

ARIMA(2,0,1)(1,0,0)_{12} | −1305.33 | −1279.57 | |

ARIMA(2,1,0)(2,0,0)_{12} | −1360.34 | −1338.88 | |

ARIMA(2,1,2)(1,0,1)_{12} | −1481.87 | −1451.83 |

Index | Variables in the Model | |||||
---|---|---|---|---|---|---|

Model | Parameter | Value of Parameters | Standard Error | t-Ratio | p | |

SPI | ARIMA (0,1,0)(0,1,1)_{12} | Θ1 | −1.00 | 0.02 | −57.66 | 0 |

ARIMA(0,1,1)(0,0,1)_{12} | θ1 | −0.01 | 0.04 | −0.29 | 0.77 | |

Θ1 | −0.71 | 0.03 | −24.54 | 0 | ||

ARIMA(1,0,1)(1,1,0)_{12} | ϕ1 | 0.87 | 0.02 | 34.87 | 0 | |

θ1 | 0.11 | 0.05 | 2.09 | 0.04 | ||

Φ1 | −0.71 | 0.03 | −22.22 | 0 | ||

ARIMA(1,0,1)(0,1,1)_{12} | ϕ1 | 0.91 | 0.02 | 45.49 | 0 | |

θ1 | 0.01 | 0.05 | 0.25 | 0.81 | ||

Θ1 | −1.00 | 0.02 | −53.63 | 0 | ||

ARIMA(1,0,0)(2,0,1)_{12} | ϕ1 | 0.98 | 0.01 | 121.44 | 0 | |

Φ1 | −0.15 | 0.08 | −1.93 | 0.05 | ||

Φ2 | −0.01 | 0.06 | −0.23 | 0.82 | ||

Θ1 | −0.62 | 0.07 | −9.21 | 0 | ||

SRI | ARIMA(1,0,1)(1,0,1)_{12} | ϕ1 | 0.98 | 0.01 | 118.45 | 0 |

θ1 | 0.35 | 0.03 | 10.22 | 0 | ||

Φ1 | −0.46 | 0.04 | −11.66 | 0 | ||

ARIMA(2,0,0)(2,0,0)_{12} | ϕ1 | 1.42 | 0.04 | 36.79 | 0 | |

ϕ2 | −0.44 | 0.04 | −11.22 | 0 | ||

Φ1 | −0.59 | 0.04 | −13.88 | 0 | ||

Φ2 | −0.29 | 0.04 | −6.82 | 0 | ||

ARIMA(1,0,1)(1,0,0)_{12} | ϕ1 | 1.37 | 0.11 | 12.91 | 0 | |

θ1 | 0.08 | 0.1 | 0.77 | 0.44 | ||

Φ1 | −0.47 | 0.04 | −12.03 | 0 | ||

ARIMA(2,1,0)(2,0,0)_{12} | ϕ1 | 0.38 | 0.04 | 8.83 | 0 | |

ϕ2 | 0.12 | 0.04 | 2.69 | 0.01 | ||

Φ1 | −0.59 | 0.04 | −13.98 | 0 | ||

Φ2 | −0.30 | 0.04 | −7.13 | 0 | ||

ARIMA(2,1,2)(1,0,1)_{12} | ϕ1 | −0.06 | 0.36 | −0.18 | 0.86 | |

ϕ2 | 0.45 | 0.26 | 1.74 | 0.08 | ||

θ1 | 0.45 | 0.36 | 1.25 | 0.21 | ||

θ2 | −0.14 | 0.15 | −0.93 | 0.35 | ||

Φ1 | 0.05 | 0.05 | 0.99 | 0.32 | ||

Θ1 | −0.95 | 0.03 | −36.98 | 0 |

**Table 4.**Comparison of statistical properties of the observed and predicted data in model validation.

Index | SPI | SRI |
---|---|---|

Model | ARIMA (1,0,1)(0,1,1)_{12} | ARIMA (1,0,1)(1,0,1)_{12} |

Kw | 0.79 | 0.88 |

Variance (Observed) | 0.97 | 0.17 |

Variance (Forecasted) | 0.81 | 0.16 |

F test | 1.12 | 1.06 |

Mean (Observed) | −0.187 | −0.263 |

Mean (Forecasted) | −0.174 | −0.26 |

Z | −0.013 | −0.012 |

RMSE | 0.46 | 0.067 |

MAE | 0.02 | −0.002 |

R | 0.89 | 0.98 |

**Table 5.**Coefficient of correlation, RMSE, and MAE between observed and predicted data for lead time 1 to 12.

Lead Time | SPI-12 | SRI-12 | ||||
---|---|---|---|---|---|---|

R^{2} | RMSE | MAE | R^{2} | RMSE | MAE | |

1 | 0.96 | 0.43 | −0.026 | 0.97 | 0.061 | −0.001 |

2 | 0.9 | 0.45 | −0.028 | 0.966 | 0.06 | −0.016 |

3 | 0.87 | 0.55 | −0.049 | 0.961 | 0.058 | −0.018 |

4 | 0.86 | 0.58 | −0.054 | 0.96 | 0.11 | 0.03 |

5 | 0.85 | 0.64 | −0.089 | 0.95 | 0.14 | 0.07 |

6 | 0.8 | 0.67 | −0.001 | 0.92 | 0.17 | 0.09 |

7 | 0.78 | 0.78 | −0.012 | 0.91 | 0.19 | 0.1 |

8 | 0.71 | 0.84 | −0.022 | 0.88 | 0.23 | 0.12 |

9 | 0.68 | 0.89 | −0.033 | 0.86 | 0.28 | 0.15 |

10 | 0.65 | 0.91 | −0.052 | 0.84 | 0.29 | 0.16 |

11 | 0.58 | 0.93 | −0.080 | 0.74 | 0.37 | 0.19 |

12 | 0.51 | 0.98 | −0.091 | 0.7 | 0.39 | 0.21 |

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**MDPI and ACS Style**

Achite, M.; Bazrafshan, O.; Azhdari, Z.; Wałęga, A.; Krakauer, N.; Caloiero, T.
Forecasting of SPI and SRI Using Multiplicative ARIMA under Climate Variability in a Mediterranean Region: Wadi Ouahrane Basin, Algeria. *Climate* **2022**, *10*, 36.
https://doi.org/10.3390/cli10030036

**AMA Style**

Achite M, Bazrafshan O, Azhdari Z, Wałęga A, Krakauer N, Caloiero T.
Forecasting of SPI and SRI Using Multiplicative ARIMA under Climate Variability in a Mediterranean Region: Wadi Ouahrane Basin, Algeria. *Climate*. 2022; 10(3):36.
https://doi.org/10.3390/cli10030036

**Chicago/Turabian Style**

Achite, Mohammed, Ommolbanin Bazrafshan, Zahra Azhdari, Andrzej Wałęga, Nir Krakauer, and Tommaso Caloiero.
2022. "Forecasting of SPI and SRI Using Multiplicative ARIMA under Climate Variability in a Mediterranean Region: Wadi Ouahrane Basin, Algeria" *Climate* 10, no. 3: 36.
https://doi.org/10.3390/cli10030036