Ratio Mathematica Volume 44, 2022 Markov Chain Model and its Application Yearly Rainfall Data in Nagapattinam District Dr. S. Santha1 T. Subasini2 Abstract A stochastic model expresses a sequence of possible events in which the possible event of each event depends on the previous event and is called a Markov chain. This paper has analyzed yearly rainfall in the Nagapattinam district and formulated three-state models. The first-order Markov chain to determine the long-term probability of rainfall in the following years and the steady-state. It can be used to make a forecast of the annual rainfall pattern. This model can give some information about rainfall to farmers and the government to plan strategies for high crop production in the Nagapattinam district. Keywords: Markov chain, Yearly Rainfall, Transition probability AMS Classification 2010: 37A30, 47D073. 1 Assistant professor and Head, Department of Mathematics, Government Arts and Science College, Nagercoil, Affiliated by Manonmaniam Sundaranar University, Tirunelveli Tamil Nadu, India. santhawilliam14@gmail.com. 2 Reg No.18121172092018, Research Scholar, Rani Anna Government Arts and Science College for Women, Tirunelveli, Affiliated by Manonmaniam Sundaranar University, Tirunelveli Tamil Nadu, India. subasinit@gmail.com. 3 Received on June 18th 20th, 2022. Accepted on Sep 1st, 2022. Published on Nov 30th, 2022. doi: 10.23755/rm.v44i0.887. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY license agreement. 36 mailto:santhawilliam14@gmail.com mailto:subasinit@gmail.com Dr. S. Santha and T. Subasini 1. Introduction Forecasting is a science of the future. Here based on the knowledge of the previous, predictions are made on the future. This involves a knowledge of all forecasting methods. The importance of forecasting is becoming necessary for prosperity. Preparation of a forecast needs mathematical formula and historical data into the future. The most effective forecasting methods is to use mathematical techniques to routinely forecast demands. By combining mathematical techniques with informed judgement, they can serve checks on each other and tend to eliminate gross errors. Agriculture largely depends on water resources. The variation and quantity of rainfall have two extreme impacts – bumper harvest or lean year. Rainfall modelling plays a prominent role for rainfall prediction. Apart from data generation, the application of rainfall modelling is vital for water resource management and in the field of hydrology and agriculture. Data relating to climate and rainfall needs a wide range of models in which time and spatial scales are involved. Rainfall, its arrival, intensity, duration, was generally decided by the atmospheric factors which are available just before the onset. Hence, for forecasting the future rainfall it is difficult to estimate the probable environmental factors which are the causes of the rainfall. In view of this the researchers are left with the option of predicting it with the previous data set. Abubaker et al [10] have formulated a, four- state Markov model of annual rainfall in Minna with respect to crop production in the region. The present study emphasizes analyzing the annual rainfall data in a three-state model. In this paper the forecasting of annual rainfall pattern for the period of 20 years (2001 to 2020) in Nagapattinam District. 2. Methodology Nagapattinam district is one of the 38 districts (a coastal district) of Tamilnadu state in southern India. The district lies between northern latitude 10.7906 degrees and 79.8428 degrees Eastern longitude. Since the study was forecasting of rainfall, the annual rainfall data was collected from the statistical department in Nagapattinam from the year 2001 to 2020. Figure: 1 Annual rainfall in Nagapattinam district from 2001-2020. 0 500 1000 1500 2000 2 0 0 1 2 0 0 2 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 2 0 0 8 2 0 0 9 2 0 1 0 2 0 1 1 2 0 1 2 2 0 1 3 2 0 1 4 2 0 1 5 2 0 1 6 2 0 1 7 2 0 1 8 2 0 1 9 2 0 2 0 37 Markov Chain Model and Its Application Yearly Rainfall Data in Nagapattinam District From Figure 1 time-series analysis of yearly rainfall is converted into rainfall states prepared by a suitable frequency distribution table. The frequency distribution of each class is specified states of rainfall and is denoted by S1, S2, and S3 in table 1. Table: 1 3. Markov Chain Modeling Definition: 3.1 A Markov chain is a random sequence (Xn, n Є N) for all Xo, X1 ……. Xn-1 Є I such that P (Xn = Jn / X0 = J0, X1 = J1, X2 = J2………Xn-1 = Jn-1) = P (Xn = Jn / Xn-1 = Jn-1) (1) Definition: 3.2 If a Markov Chain (Jn, n ≥ 0) is homogeneous. We consider P (Xn =j / Xn-1 = i) = Pij and we put the matrix P i.e., P = [Pij] The Markov process X has steady state transition probabilities if for any pair of states i, j: The first step transition probabilities Pij, Pij n denote the n step transition probability. That is, Pij n = P {Xn+m = J / Xm = i} n ≥ 0 all i, j ≥ 0 We have Pij 1 = Pij. Pij n+m = ∑ 𝑃∞𝑘= 0 ik n Pkj m for all n, m ≥ 0 (2) From (2) P n+m = p(n)×p(m) P(2 )= P(1+1) = P2 Hence by induction P(n) = P(n-1+1) = P(n-1) P1 = Pn We have Pn = P0 Pn (3) Here P0 denote the initial state vector of transition matrix Pn denote limiting state probability. Now, let Pn = [Pn1, P2 n, Pn3]. Also let P0 = [ P01 P 0 2 P 0 3]. Limiting state probabilities: 3.3 The probability distribution π = [π1 π2………. πn] is called the limiting distribution of the continuous time Markov chain X (t) if π = (π1π2 π3) Howard [8] let n→ ∞ in equation (3) we get π = πP (4) also, π = ∑ 𝜋𝑖 = 13𝑖=1 Abubakar et al [10] These equations will be used to find the limiting state equilibrium probabilities. SL. No class interval Frequency distribution states 1. below 1100 8 S1 2 1100-1600 9 S2 3 above 1600 3 S3 38 Dr. S. Santha and T. Subasini 4. Result and Discussion Let the model for yearly rainfall is S1: less than 1100 S2: in between 1100-1600 S3: greater value of 1600. Therefore, the transition probability matrix P = [ 𝑃11 𝑃12 𝑃13 𝑃21 𝑃22 0 𝑃31 0 0 ] The probability of transition matrix P =| 4 3 1 4 5 0 3 0 0 | (5) i, e. Pij = 𝑓𝑖𝑗 ∑ 𝑓𝑖𝑗 i, j = 1,2,3. Tamil and Samuel (9) Where fij → transition frequency from state i to state j, 0 ≤ Pij ≤ 1. We get the probability matrix P = | 4/8 3/8 1/8 4/9 5/9 0 3/3 0 0 |, P = | 0.5 0.375 0.125 0.44 0.56 0 1 0 0 | n-step transition probability we have P2 = [ 0.5415 0.396 0.0625 0.4689 0.4756 0.0555 0.5 0.375 0.0625 ] P3 = [ 0.5091 0.4232 0.0442 0.5011 0.4402 0.0586 0.5415 0.396 0.0625 ] P4 = [ 0.5101 0.4262 0.0519 0.5046 0.4326 0.0521 0.5091 0.4232 0.0442 ] 39 Markov Chain Model and Its Application Yearly Rainfall Data in Nagapattinam District P5 = [ 0.5079 0.4282 0.0509 0.5071 0.4297 0.0519 0.5101 0.4262 0.0519 ] P6 = [ 0.5079 0.4284 0.0515 0.5074 0.4290 0.0514 0.5079 0.4282 0.0509 ] P7 = [ 0.5078 0.4286 0.0515 0.5076 0.4287 0.0515 0.5079 0.4284 0.0515 ] P8 = [ 0.5078 0.4286 0.0515 0.5077 0.4286 0.0515 0.5078 0.4286 0.0515 ] p10 = [ 0.5078 0.4286 0.0515 0.5078 0.4286 0.0515 0.5078 0.4286 0.0515 ] (6) p10 = [ 0.51 0.43 0.05 0.51 0.43 0.05 0.51 0.43 0.05 ] corrected to 2 decimal places Limiting state probabilities After n steps pn gets the fixed value (6) i, e. n ≥10 Let us take P0 = (1 0 0 ) P0. P n = (100)[ 0.507 0.4286 0.0515 0.5078 0.4286 0.0515 0.5078 0.4286 0.0515 ]= = (0.5078 0.4286 0.0515) = (0.51 0.43 0.05) From equation (4) and (6), n = nP = (0.51 0.43 0.05). This result shows that the probabilities of yearly rainfall after ten years. In the first- year probability is (0.5 0.38 0.13) respectively. By the comparison of the probabilities, state 3 dropped slowly and the probability of state 1 and 2 increased in the above ten years. This shows that in the 51% annual rainfall in Nagapattinam will be state 1, 43%will be state 2 and 5% will be state 3. 5. Conclusion This article show analyzes the yearly rainfall data used by the first-order Markov chain model. Yearly rainfall forecasting of the current year can be used to make a 40 Dr. S. Santha and T. Subasini forecast for the following year and in the long run. A yearly rainfall forecasting pattern used to give details about the production of the crops in the Nagapattinam district. References [1] Akintunde A.A. 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