### 1. Introduction

### 2. Data and Methods

### 2.1. Data

### 2.2. Standardized Precipitation Index

*i*and a calendar month

*j*, for a time scale

*k*requires the following steps followed by [27, 28]:

Calculation of the collective precipitation series ${X}_{ij}^{k}\hspace{0.17em}(i=1,\dots ,n)$ for a time period of interest

*j*in a year*i*, where each term is the sum of precipitation of*k*-1 past consecutive months.-
Fitting of a cumulative distribution function (usually gamma distribution function) on aggregated monthly precipitation series (e.g.,

*k*= 6 months is adopted in this study). The gamma probability density function is defined as,where*β*is a scale parameter;*α*is a shape parameter, which can be estimated using method of maximum likelihood; and Γ(*α*) is the gamma function at*α*. The cumulative probability distribution of observed precipitation event for the given month and time scale can be found with the help of estimated parameters. The cumulative distribution function (CDF) is obtained by integrating Eq. (1), i.e. -
Since two-parameter gamma function is not defined for zero values, and precipitation distribution may contain zeros, a mixed distribution function (zeros and continuous precipitation amount) is employed, and the CDF is given as follows:

where*G*(x) is the distribution function estimated for nonzero precipitation, and*q*is the zero-precipitation probability from the historical time series. -
As precipitation is not normally distributed, an equiprobability transformation (Panofsky and Brier, 1958) is carried out from the CDF of mixed distribution to the CDF of the standard normal distribution with zero mean and unit variance, which is given as follows:

*Drought severity*(

*S*) is the cumulative values of SPI within the drought duration. For suitableness, severity of drought event

*i*,

*S*

*(*

_{i}*i*= 1, 2, ...) is taken to be positive, which is given by [27]:

*SPI*

*is value of*

_{i}*i*

*period SPI for a*

^{th}*D*duration drought event.

### 2.3. Autoregressive Integrated Moving Average (ARIMA) Model

*Box-Jenkins models.*An autoregressive model of order

*p is*conventionally classified as AR (

*p*). A moving average model with

*q*terms is classified as MA (

*q*). A combined model that contains

*p*autoregressive terms and

*q*moving average terms is called ARMA (

*p, q*) [30]. If the object series is differenced

*d*times to achieve stationarity, the model is classified as ARIMA (

*p, d, q*), where the symbol “d” signifies “integrated”. A time series

*Y*

*is said to follow an ARIMA model if*

_{t}*B*is the backshift operator (i.e.

*BY*

*=*

_{t}*Y*

_{t}_{– 1}),

*ϕ*(

*B*) = (1 –

*ϕ*

_{1}

*B*–

*ϕ*

_{2}

*B*

^{2}–···–

*ϕ*

_{p}*B*

*) is the autoregressive operator,*

^{p}*θ*(

*B*) = (1 –

*θ*

_{1}

*B*–

*θ*

_{2}

*B*

^{2}–···–

*θ*

_{q}*B*

*) is the moving average operator and*

^{q}*e*

*is the random error term.*

_{t}*k*is the number of independently adjusted parameters.

### 2.4. Model accuracy Measures

*Y*

*be the observed values at time point*

_{i}*i*, for

*i = 1, 2, ..., n*and

*Ŷ*

*be the respective forecast values by the fitted model. The simple correlation coefficient between observed (*

_{ι}*Y*

*) and fitted (*

_{i}*Ŷ*

*) values is defined as:*

_{ι}##### (8)

$$\text{Cor}({Y}_{i},{\widehat{Y}}_{\iota})=r=\frac{\sum {Y}_{i}{\widehat{Y}}_{\iota}-n\overline{Y}\overline{\widehat{Y}}}{\sqrt{\sum {{Y}_{i}}^{2}-n{\overline{Y}}^{2}}\sqrt{{\sum {{\widehat{Y}}_{\iota}}^{2}-n\overline{\widehat{Y}}}^{2}}}$$*r*ranges between −1 to +1. Since here we consider the observed and predicted values of the same variable, so always we will find a non-negative (greater or equal zero) coefficient value. The decision can be made via coefficient value as: higher the value, better the model prediction.

*e*

*=*

_{i}*Y*

*–*

_{i}*Ŷ*

*. Using these notations RMSE can be expressed as follows:*

_{ι}*n*is the sample size. The MAE and ME can be defined respectively as: