Rainy and Dry Days as a Stochastic Process (Albaha City)

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1 IOSR Joural of Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume, Issue Ver. III (Ja - Feb. 25), PP Raiy ad Dry Days as a Stochastic Process (Albaha City) Dr. Najeeb Al-Matar., 2 Dr. El-Rayah Mohammed Ishag Osma. Assistat Professor Maagemet Iformatio Systems Departmet Albaha Uiversity Kigdom of Saudi Arabia 2 Assistat Professor Maagemet Iformatio Systems Departmet Albaha Uiversity Kigdom of Saudi Arabia Abstract: I last years, the rai i Albaha is differ from year to year, this leads to the stadard that the rai falls i this city. I this study we try to fid a mathematical model represet this observatios. The we try to cotact the plai of this study; the problem, goals, the methodology i aalysis also the cocepts that related to Markov-Chai that applied i the study. Markov-Chai helps for obtaiig the model that represet raiy, dry days for the years (43-434) for AlBaha (Rai falls). Lastly we test the model by obtaiig the matrix for raiy ad dry days. by comparig the case i each day ad the previous by four days. Keywords: Markov Chais - Raiy Days - Dry Days - Stochastic Process- Rai falls. I. The problem: Kowig of raiy ad dry days helps for forecastig, that leads for obtaiig a mathematical model represet these days. II. The Goal: To build a mathematical model represets raiy ad dry days for E-Baha ad to fid probability distributio for the first recurrece time for each day ad the average ad also to determie the duratio of weather. III. The Hypothesis: Raiy ad dry days ca be represeted by Markov-Chai. IV. The Methodology: We use the aalytical methodology ad usig Markov-Chai for represetig the data. Previous Studies: - Gambil (23), Aalyze the behavior of rai for the period (975-2) i Kuala Lumpur Usig Markov-Chai. 2- Mirato (2) Applied Markov-Chai for forecastig the duratio of dry days i Asmara (Jue September). 3- Abdo El-Malik Applied Markov-Chai for raiy ad dry days i Juba (South Suda). Markov-Chais: There is a Markov property ad it meas (the probability of trasitio is the process from the case () to case (+) depeds oly i the case ad ot depeds o cases before. If i Markov-Chai the time space ad case space is discrete the we call the process Markov-Chai for radom variable x, x2,... The here the coditioal probability depeds o x that meas p x k x k, x k..., x k ) P( x k k ) ( 2 2 That mea the probability that the process i time i case k depeds oly i case that i time is k. We ca write p x j x p ( ) ij The trasitio probability p ij did ot depeds o time ( ). That meas Markov-Chai is homogeeous (Tow State Markov-Chai). Here we suppose we have two cases (), () ad the process ca be see i time,, 2,3,... if the process i case () i sometime, the DOI:.979/ Page

2 Raiy ad Dry Days as a Stochastic Process (Albaha City) the probability of stayig i case () for comig time is p ( ) ad the trasitio probability for case () is p If i case () this trasitio probability to case () i the comig time is p ad the probability that will be i case () is p ( ) oe step The above is called oe slap trasitio probability, it ca be preseted by a matrix called trasitio probability matrix as below p p : The probability process i case () i time. p : The probability process i case () i time. or i a vector as p p p If vector of iitial probability distributio whe = the () () p( ) p p The we ca rewrite the trasitio probability matrix as ( ) ( ) p p p ( ) ( ) ( ) p p That meas as a example elemet( ij ) i matrix p gives the probability for process i case ( j ) i time. V. Probability distributio at equilibrium: After a period the Markov-Chai will be statioary, (probability will be statioary) ad if the process i two steps we ca write as a vector: ad whe statioary p ( ) p( ) where p Ad this equatio ca be solved where If the distributio i equilibrium the is the iitial distributio ( ) () p p p () p p ( ) ( ) DOI:.979/ Page

3 Raiy ad Dry Days as a Stochastic Process (Albaha City) Ad by the same method () p () (2) That mea p, p, p Ad ot depeds o () (3),... VI. Distributio of first recurrece time: Supposig the process i time () ad case () ad it retur to case () i time () the we say T (as a radom () (2) (3) variable) is a time for that. So the probability distributio for t ( f, f, f,... Where ( i) f is the probability of returig to case () for first time i time (i) ad that meas ( i) p( T i) f Say R represet the sequece till of raiy days i the process i time () the the probability of cotiuous rai till day () the it will be dry i day (+) is the p ( R ) ( x, x..., x /x ) ( 2 ) ad this is a geometric distributio by mea p ( D ) ( ) with mea so the probability distributio for dry sequece days is The applicatio of the model: Raiy Dry Raiy Dry The data represet the amout of rai by day for raiy moths ad it represets the coditio of day ad the comig day. This table represets the sequece of raiy ad dry days. The dry (), raiy by () the the trasitio probability is p The we ca calculate the trasitio probability by p This meas if the day is dry, the the probability of it will be for the comig day raiy is.4 From matrix p we ca obtai the step matrix p This meas if the day is dry, the the probability after two days will be raiy is.69 From above we ca calculate 4 steps matrix or 8 steps matrix, at steps trasitio matrix the the system will be statioary p p p The we ca obtai the probability of the process i case (), () as ( ) , DOI:.979/ Page

4 The Raiy ad Dry Days as a Stochastic Process (Albaha City) This assume that the ratio of dry days i the seaso of rai for the log ru is 88.3 The the distributio of first recurrece time for case () is 2 f p( T ) ( ), 2,3,... Ad by usig. 4 ad. 86 we get f 2,3,... ( ) ( 86).86.4(.4) (.86).98(.4) () f,.4.886, The the probability distributio for first recurrece to case () is 2.98(.4) 2,3,... f.886 The the day of first recurrece time to case () is ( ) ( ) E T f Also for case () the mea The the probability distributio for R (raiy days) is. 63 (Geometric.86 distributio) ad the probability distributio D (dry days) is That meas the circulatio of weather is days, ie are dry ad oe is raiy. Testig the model: Here we try to test the efficiecy of the model by obtaiig the (4) step trasitio probability (actual) ad comparig by the trasitio probability model that obtaied from the model. The below is (T.P.M.) with 4 steps (dry ad raiy) we compare the case i each day ad comig day for four days * p ad compared with p.883. The variatios are ot differ, this meas the model is with high efficiecy for represetig the raiy ad dry days. VII. Coclusios ad recommedatios: By applyig this model with (TPM) it appears 88% from days i the seaso of rai is raiy. We get the first recommedatio is.33 for dry days ad for raiy days also 9 days are raiy ad oe day is dry; the model compared by the actual ad it explais for comparig of high the forecastig values do ot differ from the actual ad that meas the model is efficiecy ad represets the data with high predictio so we highly recommed to apply this model for other cities. DOI:.979/ Page

5 Raiy ad Dry Days as a Stochastic Process (Albaha City) Ackowledgmets: This research is a part of a project etitled "Raiy ad Dry Days as a Stochastic Process (AlBaha City)". This project was fuded by the Deaship of Scietific Research, Albaha Uiversity, KSA. the assistace of Deaship is gratefully ackowledged. Referece: []. Baigorria G, Joes J (2). GiST, A stochastic model for geeratig spatially ad temporally correlated daily raifall data. J Clim, 23, [2]. Baily T. (967) "The Elam ails of stochastic process with applicatio to the atioal scieces" Joh Wiley ad Sos. [3]. Cappe O (25). Iferece i hidde Markov models. Spriger, New York [4]. Dauso F. (22). Climak: a stochastic model for weather data geeratio. Italy, J. Agro. 6():57-7. [5]. Dei, S. M. ad Jemai, A. A. (29). Fittig the distributio of dry ad wet spells with alterative probability models. Met. ad Atmosph. Physics,4, [6]. Edword P.C. Kao (997) " A Itroductio to stochastic processes". [7]. Ferraris L, Gabellai S, Rebora N, Provezale A (23). A compariso of stochastic models for spatial raifall dowscalig. Water Resour Res,39(2), [8]. Furrer EM, Katz RW (27). Geeralized liear modelig approach to stochastic weather geerators. Clim Res, 34(2), 29. [9]. Gabrial ad Neuma (96). " A Markov-Chai model for occurreces of raiy days i Tel Aviv". []. Hayhoe, H. N. (2). Improvemets of stochastic weather data geerators for diverse climates, Clim. Res, 4, []. Kaufma L, Rousseeuw PJ (99). Fidig groups i data: a itroductio to cluster aalysis. Wiley, New York. [2]. Sharma, A. ad Lall, U.( 999). A oparametric approach for daily raifall simulatio. Math. Comput. Simulat., 48, [3]. Thyer, M.A. ad Kuczera, G. (2). Modellig log-term persistece i hydro-climatic time series usig a hidde state Markov model. Water Resour. Res., 36, [4]. Wilks, D.S., (992). Adoptig stochastic weather geeratio algorithms for climate chage studies. Climatic Chage, 22, [5]. Wilks, D.S., (998). Multisite geeralisatio of a daily stochastic precipitatio geeratio model. J. Hydrol., 2, [6]. Zucchii, W. ad Guttorp, P. (99). A hidde Markov model for space-time precipitatio. Water Resour. Res., 27, DOI:.979/ Page