Drought monitoring and forecasting play a vital role in making drought mitigation policies. In previous research, several drought monitoring tools based on the probabilistic models have been developed for precise and accurate inferences of drought severity and its effects. However, the risk of inaccurate determination of drought classes always exists in probabilistic models. The aim of this paper is to reconnaissance the advantage of the weighted Markov chain (WMC) model to accommodate the erroneous drought classes in the monthly classifications of drought. It was assumed that to increase the precision in drought prediction, the role of standardised self-correlation coefficients as weight may incorporate to establish and restructure the accurate probabilities of risk for incoming expected drought classes in the WMC framework. Consequently, the current research is based on the experimental findings of seventeen meteorological stations located in the Northern Areas of Pakistan. In this study, the standardised precipitation evapotranspiration index (SPEI) at a 1-month time scale based drought monitoring approach is applied to quantify the historical classification of drought conditions. The exploratory analysis shows that the proportion of each drought class varies from zone to zone. However, a high proportion of near-normal drought classes has been observed in all the stations. For the prediction of future drought classes, transition probability matrices have been computed using R statistical software. Our findings show that the probability of occurrences of near-normal is very high. Overall, the results associated with this study show that the WMC method for drought forecasting is sufficiently flexible to incorporate the change of drought conditions; it may change both the transition probability matrix and the autocorrelation structure.

Drought, the highest-ranked natural hazard, is the primary source of severe destructive effects on the planet (White,

However, to overcome the severe effect of the frequent occurrence of drought, forecasting plays a significant role in drought mitigation policies. In previous research, several forecasting and assessment tools for the characterisation of drought regions and the quantification of drought risk have been established. Several studies have been conducted in the field of hydrology and climatology for the assessment and modelling of drought classes for different regions in the world. Drought indices are one of the most used tools for the assessment and quantification of drought risk. A range of different drought indices involving different climatic parameters has been developed to detect dry and wet categories of a region for a specified time. Details on the list of drought indices corresponding with their variable requirement can be found in Svoboda et al. (2016).

Besides drought monitoring and assessment tools, several probabilistic and deterministic forecasting models have been developed and used to predict and forecast drought classes for various climatological regions. In recent decades, the rapid increase in the development of theories associated with the stochastic process is found for modelling many real-life uncertain phenomena. An example includes, stock market and exchange rate fluctuations; signals such as speech; audio and video; medical data such as a patient's EKG, EEG, blood pressure or temperature; and random movements, such as Brownian motion or random walks. Among several other stochastic models, the theory of Markov chains is a promising approach to dynamic model activities that have a stochastic factor (Lange,

Sen (

However, it is difficult to adjust the transition probability matrix and the precision of the forecast that is affected by objective factors. To overcome this problem, in many applications Weighted Markov Chain (WMC) method have been employed in several disciplines, including hydrology and environmental sciences (Benoit,

However, analysing droughts by using a single variable is not enough to distinguish different regions because drought hazards relate to multiple variables. A comprehensive analysis of the characterisation of drought classes is required that make a joint analysis of rainfall, runoff, and soil moisture conditions (Vicente-Serrano et al.,

The objective of this research is to handle all the difficulties in formulating a mathematical model for forecasting SPEI drought index at a one-month time scale under the WMC framework in various regions of Pakistan. We use autocorrelations from the historical series of SPEI drought index with a one-month time scale as a weight in first-order Markov chain transition probability matrices to forecast the next incidence of drought categories for seventeen meteorological stations located in the Northern Areas and KPK (Pakistan). The drought has become a recurrent phenomenon in the country. In the recent decade, due to severe drought hazards, the economic system of the country was severely disturbed. In recent decades, several authors had been working to explore the geographical and hydrological importance of this region. Awan (

In this research considered seventeen meteorological stations having different climatology and estimated historical time series data on SPEI drought index for a one-month time scale. Time series data on precipitation and temperature are used to estimate SPEI values for these stations.

The organisation of this paper is as follows. A brief description of the study area, the estimation method of SPEI, and the mathematical formulation of WMC method is presented in Section 1.

We consider seventeen meteorological stations located in different climatic regions of Northern Areas of Pakistan. These stations have high variability in rainfall throughout the season. Concerning the climatological statistics (see

Summary statistics of the selected stations.

In each season, some of the stations are continuing to bear extremely vulnerable drought conditions. In the current research, twelve meteorological stations exhibiting cold and humid climate and five meteorological stations having mild cold and arid climate are included to check the efficiency of SPEI drought index from the global warming perspective. Primary data on monthly total rainfall, mean minimum temperature, and mean maximum temperature of each station for the period 1976–2017 were collected from the Karachi data processing center through Pakistan Meteorological Department (PMD), Islamabad.

Study area: meteorological stations of Northern Area and KPK (Pakistan).

There are several procedures to report drought severity using a multiscalar drought index (Ali et al.,

However, this method underestimates PET values at arid regions in cold climatic regions, whereas overestimates PET values in humid regions (Ali et al.,

A stochastic process, or random process {_{t}_{t}_{−1} at time

A discrete Markov chain is a random process that describes a sequence of events from a set of finite possible states. In contrast, the current event depends only on the preceding event. It has been commonly used to model uncertain events in various disciplines. Each discrete Markov chain is characterised by a transition probability matrix that represents the probability of transition from one state to another. Shatanawi et al. (

Consequently, a historical series of drought classification states for a specified station can be embodied as a discrete Markov chain process. Here, we assume that any single class of drought in time series of SPEI depends on its previous class and then proceed to the construction of the transition probability matrix. It is just statistical compliance that allows us to consider each drought class as a first-order Markov chain. However, one can also use a second-order Markov chain, where if each class is assumed to depend on its previous two classes.

In this scenario, time-series data on drought classes determined by SPEI drought index can be assumed as a series of correlated random variables. Various empirical studies show that self-correlation coefficients in the historical data of drought classes for all the study regions have significant importance for the prediction of future drought classes. This confirms that previous drought classes (on a monthly or yearly basis) can be considered in advance to predict the present drought class. So, in our case, the basic idea behind using WMC is that weighted averages can be made according to the incidence behaviour in the past month. Hence, the probability of present or next drought classes can be inferred and predicted in advance by appropriate configuration of weights to each drought class in WMC framework.

The fundamental steps involved in the proposed method for prediction of drought classes using SPEI drought index under the WMC model are given below.

Let _{1}, _{2}, …, _{n}

Drought Classes
_{1}, _{2},…, _{n}

Classification criteria SPEI.

In this step, we classify SPEI drought index estimated with a one-month time scale according to the classification criteria provided in

Transition probability matrices for each station are computed using the

Let _{i}_{j}_{k}

Transition probability matrices for Astore and Balakot stations from the time series data on drought classes determined by SPEI drought index are shown in

One step transition probability matrix.

^{1}

_{ij}

^{1}

_{ij}

The weights (_{.}

In this step, we assume the occurrence of drought classification states in the very last month as an initial drought class Ψ_{i}_{i}

In the above equation _{i}, i ε S

For exploratory analysis, the percentage of occurrences of various drought classes in the historical time series data on SPEI-1 with a one-month time scale of the selected study regions are presented in

Percentage frequencies of drought class.

Additionally, multiple comparison tests show that the near-normal drought class has a significantly high proportion as compared to other drought classes. However, the severely dry drought class has a significant difference from the severely wet class. Besides, the proportion of the moderate dry class is significantly greater than for moderate wet.

Here, transition probability matrices are used for further prediction of future drought classes using

Standardised weights of transition from one state to another stat.

_{1}

_{2}

_{3}

_{4}

_{5}

Predicted probabilities for the month of December 2017 at Astore.

^{(5)}

^{(4)}

^{(3)}

^{(2)}

^{(1)}

_{i}

One month ahead forecast probabilities of various drought classes.

Results associated with this test show that there is a significant difference in the proportion of each drought class. These results are consistent from the ecological and climatological point of view since, in these areas, most of the stations bear longer Moonsoon and precipitation periods. In most stations, near-normal drought class found a high probability of occurrences. However, the moderate wet drought class has a low probability as compared to a severely dry drought class. However, the near-normal drought class has a high probability in most of the study region. However, predicted probabilities for each drought class in Peshawar, Bunji, and Drosh have somehow different behaviour. In Peshawar, drought classes have almost equal probabilities except for the Severely dry class. Bunji shows a 0.533 probability of occurrence for the severely dry class. In Drosh station, almost equivalent probabilities are found for near normal and moderate wet.

Prediction and forecasting play a vital role, especially in early warning situations. Consequently, accurate and precise techniques of drought forecasting may reduce their severe effect by making effective drought mitigation policies. In this article, the SPEI-1 drought index being a more comprehensive drought monitoring procedure is used to classify historical monthly drought profile for seventeen meteorological stations of Pakistan. To predict the future drought classes, the standardised self-correlation coefficient in the time series data of SPEI-1 index is used as weights in WMC method (see

The limitation of the study includes:

The current study is based on SPEI index. Other indices such as SPI and SPEI can be incorporated. Further, their comparative assessment should be made for better understanding.

In this manuscript, we have used quantitative time-series data. Consequently, our weights are based on autocorrelation. In the future, the qualitative time series of drought can be used with our novel weighting scheme for WMC (see Ali et al.