American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 © Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ Stochastic Model for Rainfall Occurrence Using Markov Chain Model in Kurdufan State, Sudan Rahmtalla Yousif Adam* Assistant Professor, Department of Statistics, University of Tabuk, K S A Email: rahmtallay@yahoo.com Abstract This paper will attempt to demonstrate the potential benefits of using Stochastic Processes for modeling and interpreting historical rainfall records by the examination of weekly rainfall occurrence using Markov Chains as the driving mechanism. The weekly occurrence of rainfall was modeled by two-state first and second order Markov chain. While the amount of rainfall of a rainy week was approximated by taking the maximum likelihood estimation method to predict transition probability matrices of rainfall sequences during the rainy season. Daily rainfall data for 21 years was collected from two meteorological stations located in Kurdufan State (Sudan). The data indicated that the season starts effectively, on 8th week of June at El-Obied station and sixth week of June at Kadugli station. The transition probability matrix of Markov chain model found to be homogeneous and remained constant over the period study. Accordingly, the Index of Drought-proneness degree (ID) was found to be higher in Elobied than Kadugli Station and the hypothesis is accepted at 5% level of significant with P-value (0.151). Keywords: Kordofan; Markov Chains; rainfall; stochastic process; Transition matrix; (;). 1. Introduction (Sudan enjoys an extremely diversified ecological system that provides immense fertile land of about (80) million hectares. A large number of livestock (about 121 million heads of sheep, goats cattle and camels), natural pastures of about 24 million hectares, forest area of about (64) million hectares in addition to considerable water resources from rivers, seasonal streams and rain with annual amount of (109) billions cubic of water [1]. ------------------------------------------------------------------------ * Corresponding author. 272 http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 272-286 The paper problem that, existing rainfall data is generally available for most areas on monthly basis or means. There is a need to know the probability of having a dry or wet period having a consecutive period of 2 or 3 weeks during the rainy season. Such knowledge will enable us to propose calendar for farmers and irrigation engineers suggesting the start and end of rainy season. The main objectives of this paper is to increases the understanding of the agricultural planners and irrigation engineers to identifying the areas where agricultural development should be focused as a long term drought mitigation strategy. In addition, this study will contribute toward a better understanding of the climatology of drought in a major drought-prone region of the Sudan. In addition, this study may help the agronomists and agricultural scientists to decide the timing of cultivation and introduce new crops. Markov chains (MC) have been widely used with daily rainfall models. The first stochastic model of the temporal precipitation with Markov chain (two –state first order) introduced by Gabrial and Neuman in 1962 [2]. Richardson in 1981 used first order Markov chain along with an exponential distribution to describe the daily rainfall distribution in the (USA) [3]. Akaike in 1974 used similar Markov chain to simulate the daily rainfall occurrence [4]. James. Reference [5] also used "statistical Modeling of Daily Rainfall Occurrence. All these studies has revealed that the generated data using Markov chain along with suitable probability distribution preserve the seasonal and statistical characteristics of historical rainfall data. The rest of the paper is structured as follows: In Section two, we outline the Markov modeling estimation of transition probabilities. In section three, we present the source of data. Section four, presents and discusses the results. In section, five summary was provided including conclusions and recommendations. 2. Markov chains modeling: The theory of stochastic process deals with system, developing in time or space in accordance with probabilistic laws. Its concept is based on expanding the random variable concept to include time. The function ),( stX is called a stochastic process, when X random variable, a function of S possible outcomes of an experiment ( state space ), t is the parameter set of process ( time ), so that the set of possible values of an individual random variables, )( xtnX ,of a stochastic process TtXnX tn ∈≥ ),(, ,1 is known as it’s state space. Markov Chains are the simplest mathematical models for the random phenomena evolving time [6]. The stochastic process with discrete parameter space ,nX (n = 0,1,2,….) or the stochastic process with continuous parameter 0, ≥nX n is called the Markov chain . ijnnn PXiXniXiXjXP ==−=== −+ ],........1,,/[ 011,1 ……..(1) For all states jiii n ,,....., ,11 − and 1≥n . 273 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 272-286 We refer to this fundamental equation as the Markov property, the future depend on the past through the present. The random variables 110 ;.......,; −nXXX , · · · are dependent. Since the probability, Pij is non-negative process then: ∑ = = k j Pij 0 1 0,10 ≥∀≤≤ jiPij ……………….. (2) 2.1 Probability Transition Matrix If there is a limit, state n elements then transition probability for i and j values can be organized in a matrix called probability transition matrix [7]. iXjXP nnij === + /Pr( 1 Satisfy 0>ijP , 1=∑ ijP for all j These probabilities can be written in the following matrix form:             = nnnn n n ppp ppp ppp P ..... :::: ....... ....... 21 22221 11211 This matrix is called the probability transition matrix of Markov chain. 2.2 One step transition probability The Markov chain { },2,1,0, =nX n with state space { },2,1,0=S is probability transition for random process from i to j. ( ) ( ) niXjXPiXjXP nn ∀=====+ 011 || The probability of making a transition from state i to state j in one step is denoted pij For a Markov chain with 2 states, the matrix is called the one-step transition matrix [8].       2221 1211 pp pp For a Markov chain with 3 states, the one-step transition matrix is 274 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 272-286           333231 232221 131211 ppp ppp ppp 2.3 The n- step transition probability The probability transition of random processes { },2,1,0: =nX n from state i to state j after n step is called the n-step transition probability defined as: ( ) ...,2,1,0,,0,|)( =≥=== + jiniXjXPp mmn n ij This indicates the probability transition of random processes from state i to state j after n step [9]. We can write it in term of matrix )(nP as follow:               =     )( 22 )( 21 )( 20 )( 12 )( 11 )( 10 )( 02 )( 01 )( 00 )( nnn nnn nnn n ppp ppp ppp P The interpretation of matrix is: 1- If n =1 then )(n ijp becomes the probability transition of random processes, from state i to state j , with one step denoted by ijp . 2- If n = 0 then ( )    ≠ = ==== .,0 ,,1 | 00 )0( ji ji iXjXPpij 3- For all, n = 0,1,.. the matrix )(nP became a random process satisfying the two above properties. 2.4 Chapman – Kolmogorov equation Chapman – Kolmogorov equation is my help us to predict and forecast for several steps or several years in the future. The probability transition of random processes from state to state after n + m step [6]. If { },2,1,0: =nX n Markov chain with limit m states and transition probability matrix (TPM) ( )ijpP = then: 275 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 272-286 .1...,,2,1, 1 )()()( −=∀= ∑ = − nrppp m k rn kj r ik n ij ………………(3) 2.5 Maximum Likelihood Function When have a Markov Chain with state 0, 1 , 2 , 3 , ….. with unknown transition matrix P, the likelihood function is: nij ij sji PL . , ∈ ∏= ……………. (4) ijn = the number of times has state j following state i, to maximize the function: 1=∑ ∈si ijp Indicates that each row of transition matrix is equal to 1 and then: ∑∑ ∑= ∈ .i ij n n si ijp …………… (5) Where ijn the transition count for thji ),( cell and .in is the thi , row total transition count. Therefore, the random variable ijn depends on the parameter ijP ∑= ijij LPnLln …………………….. (6) 2.6 Notations For the purpose of this paper, some notions are explained as follows [10]. 1- )/( 1−iim WWP = conditional probability of a wet week on week )(i given a wet week on week ( 1−i ) in a certain period m. 2- )/( 1−iim WDP = conditional probability of a dry week on week )(i given a wet week on week ( 1−i ) in a certain period m. 3- )/( 1−ii DW = conditional probability of a wet week on week )(i given a dry week on week ( 1−i ) in a certain period m. 4- )/( 1−ii DD = conditional probability of a dry week on week )(i given a dry week on week ( 1−i ) in 276 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 272-286 a certain period m. Thus, for each week, four elements in the transition matrix were to be determined in first order Markov chain. For a second order chain, eight elements of the transitional probability matrix were to be determined. 1- )/( 21 −− iiim WWWP = conditional probabilities of a wet week following two a wet week in certain period m. 2- )/( 21 −− iiim DWWP 3- )/( 21 −− iiim WDWP 4- )/( 21 −− iiim DDWP 5- )/( 21 −− iiim DDDP 6- )/( 21 −− iiim DWDP 7- )/( 21 −− iiim WDDP 8- )/( 21 −− iiim WWWP 2.6.1 Initial Probability n FP D D = ……………………….. (7) n FP W W = ……………………….. (8) 2.6.2 Conditional Probabilities D DD DD F FP = ……………………….. (9) W WW WW F FP = ……………………….. (10) DDWD PP −=1 ……………………….. (11) WWDW PP −= 1 ……………………….. (12) 2.5.3. Consecutive dry and wet week probabilities: )13......(...............................2 11 DDwDw PPD = )14........(.....................2 21 WWWwWw PPW = 277 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 272-286 )15...(......................3 321 DDwDDwDw PPPD = )16..(......................3 321 WWwWWwWw PPPW = Where, Wet week: A week with rainfall of 7mm or more. Dry week: A week with rainfall of less than 7mm. =DP Probability of the dry week =WP Probability of the wet week =DF Number of dry weeks =WF Number of wet weeks n - Number of years of data =DDP Probability (conditional) of a dry week preceded by a dry week WWP = Probability (conditional) of a wet week preceded by a wet week WDP = Probability (conditional) of a wet week preceded by a dry week =DWP Probability (conditional) of a dry week preceded by a wet week =DDF Number of dry weeks preceded by another dry week =WWF Number of wet weeks preceded by another wet week 2D = Probability of two consecutive dry weeks. 2W = Probability of two consecutive wet weeks. 3D = Probability of three consecutive dry weeks. 278 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 272-286 3W = Probability of three consecutive wet weeks. =1DwP Probability of the dry week (first week) =2DDwP Probability of the second dry week, given the preceding week dry =3DDwP Probability of the third dry week, given the preceding week dry =1WwP Probability of the wet week (first week) =2WWwP Probability of the second wet week, given the preceding week wet =3WWwP Probability of the third wet week, given the preceding week wet 3. Source of data The present work is based on data related to the autumn season (May-November) daily rainfall reported by two stations in Sudan. Elobied in North Kurdofan state (longitude 12:30° and 14:30 North, and 29° and 32° East) and Kadougli in south Kurdofan (latitudes 90° 45′ and 12° 45′ N, and longitudes 29° 15′ and 32° 30′ E), over priod of 20 years (1990-2009) from Kadougli station, and 21 years (1990-2010) from Elobied station [1]. We transfer the original daily data to weekly data by dividing the month into four classes. The autumn season (1- Mayto 30-November), have (28 Standard Metrological Weeks (SMW). 4. Results and Discussion 4.1 Conditional Probabilities We can compute the conditional probability states of the (SMW) according to the following formulas dddw dw dw NN Np + = …(17), dwdd dd dd NN Np + = …(18), wwwd wd wt nN Np + = ….(19), wdww ww ww NN Np + = .( 20) 4.2 Initial Probability Markov chains could give probability of spell lengths within a given period as well as probability of a specified amount of rain within a given period. To compute the initial probability from data we can use the equations (7) and (8) as follow: 279 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 272-286 Table 1: Conditional probabilities of (SMW) Elobied Station Kadugli Station Class SMW ddP dwP wdP wwP ddP dwP wdP wwP 1-7may 1 0.95 0.05 0.50 0.50 0.88 0.12 0.67 0.33 8-15may 2 0.81 0.19 0.60 0.40 0.71 0.29 0.67 0.33 16-22may 3 0.82 0.18 0.75 0.25 0.82 0.18 1.00 0.00 23-31may 4 0.67 0.33 0.83 0.17 0.64 0.36 0.56 0.44 1-7jun 5 0.75 0.25 0.60 0.40 0.50 0.50 0.60 0.40 8-15jun 6 0.50 0.50 0.56 0.44 0.30 0.70 0.60 0.40 16-22jun 7 0.30 0.70 0.55 0.45 0.40 0.60 0.60 0.40 23-30jun 8 0.58 0.42 0.44 0.56 0.67 0.33 0.27 0.73 1-7jul 9 0.00 1.00 0.33 0.67 0.43 0.57 0.31 0.69 8-15jul 10 0.40 0.60 0.19 0.81 0.43 0.57 0.23 0.77 16-22jul 11 0.00 1.00 0.25 0.75 0.33 0.67 0.21 0.79 23-31jul 12 0.00 1.00 0.05 0.95 0.00 1.00 0.18 0.82 1-7aug 13 0.00 1.00 0.11 0.89 0.00 1.00 0.19 0.81 8-15aug 14 0.00 1.00 0.24 0.76 0.00 1.00 0.12 0.88 16-22aug 15 0.33 0.67 0.06 0.94 0.33 0.67 0.06 0.94 23-31aug 16 0.40 0.60 0.13 0.88 0.33 0.67 0.21 0.79 1-7sep 17 0.00 1.00 0.18 0.82 0.40 0.60 0.20 0.80 8-15sep 18 0.00 1.00 0.17 0.83 0.33 0.67 0.21 0.79 16-22sep 19 0.36 0.64 0.70 0.30 0.46 0.54 0.86 0.14 23-30sep 20 0.46 0.54 0.75 0.25 0.50 0.50 0.63 0.38 1-7oct 21 0.22 0.78 0.50 0.50 0.00 1.00 0.25 0.75 8-15oct 22 0.62 0.38 0.50 0.50 0.18 0.82 0.78 0.22 16-22oct 23 0.50 0.50 0.86 0.14 0.33 0.67 0.73 0.27 23-31oct 24 0.95 0.05 1.00 0.00 0.75 0.25 0.75 0.25 1-7nov 25 0.89 0.11 0.67 0.33 0.64 0.36 0.67 0.33 8-15nov 26 0.89 0.11 0.67 0.33 0.88 0.12 0.50 0.50 e16-22nov 27 0.89 0.11 1.00 0.00 1.00 0.00 - - 23-30nov 28 1.00 0.00 1.00 0.00 - - 4.3 Consecutive dry and wet week probabilities For a second order chain, there are eight elements of the transitional probability matrix to be determined. We can also drive a third order equation to calculate three consecutive dry or a wet days. 280 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 272-286 1- 2DDwP means that the probability of the second week being dry, given the preceding week is dry, denoted by 2D. 2- 2WWwP means that the probability of the second week being wet, given that the preceding week is wet, denoted by 2W. 3- 3DDwP means that the Probability of the third week being dry, given that the preceding week is dry, denoted by 3D. 3WWwP means that the Probability of the third week being wet, given that the preceding week is wet, denoted by 3W. By equations (13), (14), (15), and (16) we can calculate the 2D, 2W, 3D, and 3W respectively as in the table (3). The analysis of consecutive dry and wet spells (Table 3) during rainy season reveals that there is an interval limits (90-10) % chances that 2 consecutive dry weeks may occur during (1st -8th ) SMW and (19th -27th) SMW . Similarly, the probabilities of occurrence of three consecutive dry weeks are also very high with interval limits (71-14) % during the (1st -6th) SMW and (20th -27th) SMW. The probability of occurrence of two consecutive wet weeks are (10-80) % during (5th – 18th) SMW and the probability of occurrence of three consecutive wet weeks are (14-57) % during (7th – 18th ) for ElObied Station data. For Kadugli Station analysis of consecutive dry and wet spells explained that, during rainy season there is being interval limits (13-63) % chances that two consecutive dry weeks may occur during (1st -9th ) SMW and (16th - 27th) SMW . Similarly, the probabilities of occurrence of three consecutive dry weeks are low- compared with that of ElObied Station data - with interval limits (20-60) % during (1st -4th) SMW and (20th -27th) SMW. The probability of occurrence of two consecutive wet weeks are (13-63) % during (4th – 23rd) SMW and the probability of occurrence of three consecutive wet weeks are (20-80)% during (5th – 21st )SMW. Table (4) shows that the probability of a week being a wet after two weeks (ElObied Station). This insures that the starting point occurs with probability more than 51% and the probability of end point correspond to 5% and reaches a higher point of 94% at 12nd SMW. In addition, table (4) shows that the probability of a week being a wet after two weeks (Kadugli Station). In a station the starting point occurs with probability more than 55% and the probability of end point correspond to 13% and reaches a higher point equal to 91% at 12nd SMW. 281 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 272-286 Table 2: No of dry and wet, initial probability for SMW and percentage (Elobied Station) Elobied Station Kadugli Station No of dry and wet Initial probability for SMW Initial probability for SMW% No of dry and wet Initial probability for SMW Initial probability for SMW% Class SMW Dry Wet PD PW PD% PW% Dry wet PD PW PD% PW% 1-7may 1 19 2 0.90 0.10 90.48 9.52 17 3 0.85 0.15 85 15 8-15may 2 16 5 0.76 0.24 76.19 23.81 14 6 0.7 0.3 70 30 16-22may 3 18 3 0.86 0.14 85.71 14.29 17 3 0.85 0.15 85 15 23-31may 4 13 6 0.62 0.29 61.90 28.57 12 8 0.6 0.4 60 40 1-7jun 5 13 6 0.62 0.29 61.90 28.57 10 10 0.5 0.5 50 50 8-15jun 6 11 10 0.52 0.48 52.38 47.62 9 11 0.45 0.55 45 55 16-22jun 7 9 12 0.43 0.57 42.86 57.14 10 10 0.5 0.5 50 50 23-30jun 8 11 10 0.52 0.48 52.38 47.62 10 10 0.5 0.5 50 50 1-7jul 9 5 16 0.24 0.76 23.81 76.19 7 13 0.35 0.65 35 65 8-15jul 10 5 16 0.24 0.76 23.81 76.19 6 14 0.3 0.7 30 70 16-22jul 11 4 17 0.19 0.81 19.05 80.95 5 15 0.25 0.75 25 75 23-31jul 12 1 20 0.05 0.95 4.76 95.24 3 17 0.15 0.85 15 85 1-7aug 13 2 19 0.10 0.90 9.52 90.48 3 17 0.15 0.85 15 85 282 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 272-286 8-15aug 14 4 17 0.19 0.81 19.05 80.95 2 18 0.1 0.9 10 90 16-22aug 15 2 19 0.10 0.90 9.52 90.48 2 18 0.1 0.9 10 90 23-31aug 16 4 17 0.19 0.81 19.05 80.95 5 15 0.25 0.75 25 75 1-7sep 17 3 18 0.14 0.86 14.29 85.71 5 15 0.25 0.75 25 75 8-15sep 18 2 19 0.10 0.90 9.52 90.48 5 15 0.25 0.75 25 75 16-22sep 19 11 10 0.52 0.48 52.38 47.62 12 8 0.6 0.4 60 40 23-30sep 20 14 7 0.67 0.33 66.67 33.33 12 8 0.6 0.4 60 40 1-7oct 21 8 13 0.38 0.62 38.10 61.90 5 15 0.25 0.75 25 75 8-15oct 22 12 9 0.57 0.43 57.14 42.86 10 10 0.5 0.5 50 50 16-22oct 23 13 8 0.62 0.38 61.90 38.10 10 10 0.5 0.5 50 50 23-31oct 24 20 1 0.95 0.05 95.24 4.76 16 4 0.8 0.2 80 20 1-7nov 25 18 3 0.86 0.14 85.71 14.29 16 6 0.8 0.3 80 30 8-15nov 26 19 2 0.90 0.10 90.48 9.52 16 4 0.8 0.2 80 20 e16-22nov 27 19 2 0.90 0.10 90.48 9.52 20 0 1 0 100 0 23-30nov 28 21 0 1.00 0.00 100.00 0.00 20 0 1 0 100 0 283 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 272-286 Table 3: the probability of occurrence of two and three consecutive (wet and dry) weeks and Percentages limits Stations ElObied Kadugli 2 consecutive dry SMW limits (1st-8th )SMW and (19th-27th)SMW (1st -9th )SMW and (16th - 27th)SMW Percentages limits (90-10) % (13-63) % wet SMW limits (5th – 18th) SMW (4th – 23rd ) SMW Percentages limits (10-80) % (20-60) % 3 consecutive dry SMW limits (1st-6th) SMW and (20th -27th) SMW (1st-4th)SMW and (20th -27th) SMW Percentages limits (71-14) % (20-60) % wet SMW limits (7th – 18th ) SMW (5th – 21st )SMW Percentages limits (14-57)% (20-80)% Table 4: The probability of a week being dry or wet after two weeks Station Starting End season Higher point ElObied 8th SMW (11 – 17th June) 51% 24th SMW (23 – 31th October) 5% 94%at 12nd SMW Kadugli 7th SMW (16 – 22th June) 55% 26th SMW (8 – 15th November) 13% 91%at 15nd SMW 4.4 Drought-proneness Index (DI) frequencies Table (4-5) explain the stationary distribution and Index of Drought-proneness. The result reveal that the DI of areas as in the following table. Table 5: Drought-proneness Index frequencies Criteria Degree of drought- proneness ElObied station SMW Kadugli Station SMW 0.00