Fusion of Markov Chain and SAX Method for Drought Probability Analysis (Case Study: Eastern District of Isfahan, Iran)

Drought is one of the most powerful natural disasters, which are affected on different aspects of the environment. Most of the time this phenomenon is immense in the arid and semi-arid area. Monitoring and prediction the severity of the drought can be useful in the management of the natural disaster caused by drought. Many indices were used in predicting droughts such as SPI, VCI, and TVX. In this paper, based on three data sets (rainfall, NDVI, and land surface temperature) which are acquired from MODIS satellite imagery, time series of SPI, VCI, and TVX in time-limited between winters 2000 to summer 2015 for the east region of Isfahan province were created. Using these indices and fusion of symbolic aggregation approximation and hidden Markov chain drought was predicted for fall 2015. For this purpose, at first, each time series was transformed into the set of quality data based on the state of drought (5 group) by using SAX. Algorithm then the probability matrix for the future state was created by using Markov hidden chain. The fall drought severity was predicted by fusion the probability matrix and state of drought severity in summer 2015. The prediction based on the likelihood for each state of drought includes severe drought, middle drought, normal drought, severe wet and middle wet. The analysis and experimental result from proposed algorithm show that the product of this algorithm is acceptable and the proposed algorithm is appropriate and efficient for predicting drought using remote sensor data.
Introduction
Drought is such natural disasters that usually covers a large area and have long-term effects. Due to the impact of this phenomenon on weather, agriculture, water and socio-economic issues, it can have an infrastructural and destructive effect on the environment. In general, due to drought dependence on multiple parameters and its complexity, a definition for this phenomenon is no easy task [1]. Drought forecasting can have a useful role in mitigation of this phenomenon's damages, which depends on the exact definition of drought and linking drought with a series of associated indices. Several parameters have been defined on this basis to be modeled during the period of drought forecasting.
Based on studies in the field, these indicators can be divided into two general categories meteorological indicators and satellite remote sensing indicators [6]. The most common weather indices are Standardized Precipitation Index (SPI) and the Palmer Drought Severity Index (PDSI).Generally, satellite indices are vegetation index (VI) and land surface temperature (LST) and its derivatives [3]. Currently, the analysis of time series of drought indicators used to predict drought which is forecast the absolute numerical value based on an extrapolation of the function fitted to the time series.
Firstly, if the drought is a phenomenon with qualitative nature, so even if we express this phenomenon numerically, ultimate results must be expressed qualitatively. Secondly, the nature of the predictions is always probabilistic thus providing a fixed amount is not meaningful. Another problem of existing methods is in determining   the   communicational   interval   of any data   with previous data. Due to the uncertainty in determining these ranges (delay), an error entered into the prediction process. In this study, prediction carried out in a way that the preceding be considered in it.
Material and Methods:
This research study area is eastern Isfahan Province where has five sub-regional. The study area has semi-desert climate and is located in the range of latitude N "40 '29 ° 32 and N" 47 '45 ° 32 and longitude E "29 '42 ° 51 to" E52 '59 ° 51. Figure 1 shows the study area. The data used in this research is land surface temperature (LST), and normal differential vegetation index (NDVI) from MODIS satellite products that are free and downloaded from the NASA Earth   Observations   (NEO)   Other   data   were   also   used   is precipitation data from TRMM. The data for a period of 16 years from winter 2000 to summer 2015 were downloaded.
A. Symbolic Aggregate Approximation method
Symbolic   Aggregate   approximation   method   is   one   of   the approaches to show time series offered by Lin et al. in 2003. This process took a time series as input and turned it into a set of strings as output [15]. By the use of Symbolic Aggregate Approximation method, a time series of arbitrary length n can be converted to an arbitrary string with length w (w <n and commonly w << n). This method is based on the fact that normalized time series follow Gaussian distribution (Larsen and Marx 1986).
Symbolic Aggregate approximation method consists of two main stages. First, convert the Piecewise Aggregation Approximation (PAA) to reduce the time series dimension and second discrete time series obtained from the previous step to convert it to the string.
B. Markov Chain
A Markov chain is kind of modeling in which the current state of the system depends on its previous state. Determining the state of the system (projected) by using Markov model needs previous state of the system and the possibility of changing in system state to other possible states, the so-called transition probabilities to be known [16].
According  to  the  current  state  of  a  system,  a  square  matrix  P formed and matrix elements Pij has represented the transition probability.
In this matrix, the likelihood of early states in the left column and the possibility of cases where the system passes them along the lines of the matrix are shown.





(1)


           j=1,2,3…r





Discussion and Results
First, by using data from TRMM sensor and the standardized precipitation index SPI and via Equation 2 for monthly time series of the winter of 2000 until the end of 2015 summer was calculated. By using time series data, land surface temperature (LST) and the normalized  vegetation  index  (NDVI)  for  the  same  period  with precipitation  data, two indices VCI TVX were calculated  using Equations 3 and 4.





(4)

 



(5)

 



(6)

 




In the second stage, the normal time series were used as input of SAX methods. First, since environmental changes are more noticeable in seasons the amount of W considered 63 for the PAA convert to reduce seasonally adjusted time series Figure 4 Showing PPA time series of indices used in this research. The next step, assuming a Gaussian distribution for each indicator, and the values of δ and δ3 as breakpoints for SAX method were selected.
These values were based on probability levels. Moreover, strings intended to convert SAX have been chosen as follows: 1- SW: extreme wet 2- MW: wet 3- N: Normal 4- MD: drought 5- SD: severe drought.
According to these rules SAX conversion implemented and time series  converted  to  the  set  of  strings  which  are  indicators  of drought. Figure 1 showing the transformation for each series of the time.




 



Fig. 1. SAX presentation for drought index





The horizontal axis represents time, the vertical axis represents the amount of PAA for each indicator, and the number written on each of intervals represents the new time series value. In other words, the input time series after SAX conversion converted to a set of strings of qualitative drought values. After conversion of each time series into qualitative data through the SAX method, in the next stage, the collection of qualitative data by using Markov chain method were prepared to predict the probability of the next state. In other words, the transition state matrix for each index was determined. Figure 2 shows probabilistic values for each index through Markov chain.
 





(1)







SW


MW


N


MD


SD


 




0.16


0.17


0


0


0.67


SD




0.1


0.05


0.01


0.81


0.03


MD




0.05


0


0.82


0.10


0.02


N




0


0.66


0.14


0.19


0


MW




0


0.66


0.23


0.03


0


SW





 




(2)







SW


MW


N


MD


SD


 




0


0.22


0.12


0


0.66


SD




0.01


0.1


0.08


0.79


0.02


MD




0.03


0.09


0.68


0.14


0.04


N




0.18


0.73


0.12


0.07


0


MW




0.67


0


0.11


0.22


0


SW





 




(3)







SW


MW


N


MD


SD


 




0


0


0.33


0


0.66


SD




0.01


0.06


0.10


0.79


0.04


MD




0.23


0.05


0.66


0.05


0


N




0.22


0.66


0


0.12


0


MW




0.66


0


0.06


0.28


0


SW





 





Fig. 2. Probability model for drought based on (1) VCI  (2) SPI (3) TVX
The correlation coefficient between the values of the likelihood of each index with other indices to evaluate accurately calculated as follows: 1 .The correlation coefficient between SPI and TVX equal to 89. 2. The correlation coefficient between SPI and VCI equal to 95. 3. The correlation coefficient between VCI and TVX index equal to 89.
High amounts of correlation between the probability values indicate that resulted probability values are acceptable.
In the final step, after calculating the probability values by using the last status of the three indicators, drought conditions for the next month predicted. The forecast shows that in the fall of 2016 in the Eastern region of Isfahan Province by the possibility of 2% severe drought, 12% drought, 50% typical situation (normal), 25% wet and 10 % extreme wet will occur. For validation of this probability model the dataset of October, November and December for 2015 are analyzed the result shows the probability model is matched with this dataset.
Conclusion
This study was conducted to investigate, modeling and forecast drought, one of the world’s natural hazards and controversial issue. For this purpose, time series of three indices SPI, VCI, and TVX between winter 2000 and summer 2015 were designed and used, and in this study, it was done by incorporation of SAX and Markov chain. The advantage of using a combination of these methods compared to other methods is it provide a probabilistic qualitative model of drought.
In fact, due to the nature of drought that is qualitative and on the other hand, fuzzy and probabilistic nature of the predictions, this method seems more reasonable than other modeling methods. On the contrary,  due  to  the  high  correlation  between  probabilistic models obtained as well as implementing the method for certain modes and the logical outcome of this case, the accuracy of the proposed   method  was  acceptable.
The authors examining the impact of changes in the value of break points of SAX method on the result of probabilistic model and also determining these values on the basis of conditions of each climate left for the future works. In addition, using an analysis of other rules  in  order  to  qualify  the  indices  in  SAX  method  or  using another method to display time series can be helpful to advance this research.