*2.2. Data Sources and Preparation*

This study has been carried out on the basis of data collected from Pakistan Meteorological Department (PMD) Karachi. The research includes precipitation records and varying air temperatures of Mithi Meteorological station, Tharparkar. The data also include some facts obtained from close localities of Mithi such as Islamkot and Diplo by utilizing linear regression methods. With the help of nonparametric tests, the parameters of the acquired data values were thoroughly checked. In this research, data has been obtained from the Mithi weather station for 2005–2019 period, whose altitude is 42 m (138 feet), longitude is 69.800430 m and latitude is 24.740065 m [38].

The future climate data will be generated through ARIMA model using R-programming language. From the collected data, a long-term dataset for SPEI series of 3, 6, 9, 12- and 24-month time scales will be established. Time-series plots provide long-term and concrete information regarding drought situations at the regional scale with a 0.5-degree geographic resolution and monthly time resolution. It also has a multi-scale nature that gives timescales for the SPEI between 1 and 8 months. The potential evapotranspiration (PET) equation to calculate the SPEI indices is given as:

$$\text{PET} = \frac{0.408\Delta (R\_n - G) + \gamma \left(\frac{900}{T + 273}\right) \text{LI}\_2 (\varepsilon\_s - \varepsilon\_a)}{\Delta + (1 + 0.34 \text{LI}\_2)} \tag{1}$$

where Δ is the slope of vapor pressure curve (kPa ◦C−1), *Rn* the surface net radian (MJ.m−<sup>2</sup> day−1), *G* the soil heat flux density (MJ.m−<sup>2</sup> day−1), *γ* the psychometric constant (kPa ◦C<sup>−</sup>1), *T* the mean daily air temperature (◦C−1), *U*<sup>2</sup> the wind speed (m.s−1), es the saturated vapor pressure and *ea* the actual vapor pressure [39]. Table 2 illustrates the standard precipitation evaporation and transpiration (SPEI) value of the dry/wet classification value [33,40,41].

#### *2.3. Autoregressive and Moving Average Model (ARMA)*

From the statistical analysis of time series, ARMA model gives a wretched representation of a static stochastic technique as far as 2 polynomials, the 1st for the A.R and the 2nd for the M.A. The A.R part contains reverting the variable all alone slacked values. The M.A part includes modeling the error term as a linear combination of error terms arising

contemporaneously and at changed time in earlier. The model is typically mentioned to as the ARMA (*p*, *q*) display where *p* is the order of the A.R part and *q* is the order of the M.A [42].

$$Z\_t = \omega + \eta\_t + \sum\_{i=1}^p \beta\_i Z\_{t-i} + \sum\_{i=1}^q a\_i \eta\_{t-i} \tag{2}$$

where *ω* is mean of the series, *α<sup>i</sup>* is the parameter of the moving average model, and *β<sup>i</sup>* is the parameter of the autoregressive model, whereas *ηt*, *ηt*−*i*, and *Zt*−*<sup>i</sup>* are error terms known as white noise and *Zt* is the time series.

#### **Table 2.** SPEI scale [33,40,41].


#### *2.4. Autoregressive Integrated Moving Average Model (ARIMA)*

In econometrics and statistics and specifically in time series analysis, the ARIMA model is speculation of ARMA model. Together the formulations are fixed to time series data either to all the more likely comprehend the data or to forecast future outcomes in row. ARIMA models are related sometimes where data shows non-stationarity proofs, where an underlying differencing step can be connected at least multiple times to wipe out the non-stationarity [43].

#### 2.4.1. Non-Seasonal Model

In general the non-seasonal ARIMA model is A.R having order *p* and M.A of order *q* and operate on the time series differences *zt*; thus ARIMA family formulation is categorized by three parameter (*p*, *d*, *q*) that can either have 0 or positive integral values.

Generally non-Seasonal ARIMA model is written as:

$$
\beta(\mathcal{C})\nabla^d z\_t = a(\mathcal{C})a\_t \tag{3}
$$

where *β*(*C*) and *α*(*C*) are polynomials of order *p* and *q*, ∇ shows the order of difference [44].
