Three State Markov Chain Approach On the Behavior Of Rainfall

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1 New York Sciece Joural ;3() Three State Markov Chai Approach O the Behavior Of Raifall Vivek Kumar Garg ad Jai Bhagwa Sigh School of Basic ad Applied Scieces, Shobhit Uiversity, Meerut, Uttar Pradesh, 5, Idia vg84@yahoo.co.i Abstract: A aalytical procedure for testig the idepedece of behavior of raifall usig three-state Markov Chai approach has bee used i the preset study. The developed aalytical procedure was applied for daily raifall data of 4 years (96 ) observed from IMD approved meteorological observatory, Patagar, Idia. The whole year was divided ito three differet periods viz. Pre-mosoo (Ja -May 3), Mosoo (Jue -Sep 3) ad Post-mosoo (Oct -Dec 3) for the aalysis of daily ad weekly raifall data. A day/week was take as dry if the raifall was below.5mm/7.5mm ad day/week was take as wet if the raifall was betwee (.5mm to 5mm)/(7.5mm to 35mm) respectively, otherwise it was take as a raiy day/week. Based o three coditios of raifall, durig each period, it was cocluded that cosecutive day/week are ot idepedet ad expected legth of dry, wet, raiy spells, ad weather cycles of all the three periods has bee computed. [Vivek Kumar Garg ad Jai Bhagwa Sigh. Three State Markov Chai Approach O the Behavior Of Raifall. New York Sciece Joural ;3():76-8]. (ISSN: 554-). Key words: Three -State Markov Chai, Beroulli trials, Raifall occurrece patter.. Itroductio Raifall aalysis is ot oly importat for agricultural productio but also for other admiistrative purposes. The distributio patter of raifall rather tha the total raifall durig the etire period of time is more importat for studyig the patter of raifall occurrece. The occurrece of a sequece of wet ad dry spells ca be regarded as a series of Beroulli trials. Fisher (94) studied the ifluece of raifall o the yield of wheat ad showed that durig a seaso, the distributio of raifall rather tha total amout ifluece the crop yield. Gabriel ad Neuma (96); Gree (965, 97); Weiss (964); Wiser (965); Feyerham ad Bark, (967); Ster et. al. (984) developed raifall models. Aeja & Srivastava(986,999) developed two-state with two parameters ad three-state with five idepedet parameters Markov Chai model to study the patter of raifall occurrece. Srikatha et. at. () preset a review o stochastic geeratio of aual, mothly ad daily climate data. I this review firstly they studied traditioal time series ad after that more complex models usig pseudo-cycles i the data. Purohit et.al. (8) used two state Markov chai model to fid the probabilities of occurrece of dry &wet weeks ad also did weekly aalysis of raifall at Bagalore. Garg & Sigh () studied the patter of raifall occurrece at Patagar usig two State Markov chai approach. I the preset paper, the two state Markov chai approach is beig tried to exted further for three states Markov chai approach dividig a day/week ito dry, wet ad raiy (day/week).. Aalytical Procedure Markov assumptio is that future evaluatio oly depeds o the curret state. A three state Markov chai approach is applied o the patter of occurrece of raifall. I this preset study, we cosider three types of raifall coditio viz: a day is a dry day if raifall is less tha.5mm, wet day if raifall lies betwee.5mm ad less tha 5mm ad raiy day if raifall is 5mm or more. Based o the daily raifall observatio accordigly for weekly aalysis, if total raifall of seve days is less tha 7.5mm, it is classified as a dry week, if lies betwee 7.5mm ad less tha 35mm, wet week ad raiy week if raifall i 35mm or more. So three possible states for each (day/week) are dry, wet, raiy ad three state Markov Chai approach has bee applied to study the patter of occurrece of dry, wet ad raiy (days/weeks) at Patagar durig Pre-mosoo, Mosoo ad Postmosoo. The expected legths of dry, wet, rai spells ad weather cycle have bee derived for each period of daily ad weekly raifall data. The data observed over the sequece of time (days/ weeks) ca be regarded as a three-state Markov chai with state space S = {d, w, r} as the curret (day s/week s) raifall was supposed to deped oly o the precedig (days/weeks) raifall ad the trasitio probability matrix is defied as 76 ewyorksci@gmail.com

2 New York Sciece Joural ;3() P (P ) P j i Where i, j S ad is give i Table. Table. 3x3 trasitio probability matrix P= dry (d) Where P dd = P(d d) : Probability of a dry (day/week) preceded by a dry (day/week). P dw = P(w d) : Probability of a wet (day/week) preceded by a dry (day/week) P dr = P(r d) : Probability of a raiy (day/week) preceded by a dry (day/week). ad so o. Subject to the coditio that the sum of probabilities of each row is oe i.e. P P P dd dw dr Pwd Pwr Prd Prw Prr Curret (Day/week) dry (d) wet (w) raiy (r) Pdd Pdw Pdr Pwd Pwr For three state Markov chai approach, we required the values of P (i,j d,w,r) ad these values ca be estimated from the 3x3 order observed frequecy table give below, Table. 3x 3 order observed frequecy table Previous (Day/ week) Previous (Day/ week) dry (d) wet (w) raiy (r) wet (w) raiy (r) Prd Curret (Day/week) dry (d) dd wd rd wet (w) Prw Prr Where (i,j d,w,r) is observed frequecy i.e. dd : Number of dry (days/weeks) preceded by dry (days/weeks) dw: Number of wet (days/weeks) preceded by dry (days/weeks) dr : Number of raiy (days/weeks) preceded by dry (days/weeks) ad so o. dw ww rw raiy (r) dr wr rr Total d w r d dd dw i.e. Total umber of dry (days/weeks) dr w wd ww wr i.e. Total umber of wet (days/weeks) r rd rw rr i.e. Total umber of raiy (days/weeks) The Maximum likelihood estimators of P (i, j d,w,r) are give by P where i,j d,w,r i g 3. Test of Goodess of fit: I this sectio, we test the validity of the three-state Markov chai approach i.e. oe-day depedece could be cosidered as explaiig the behavior of raifall. This ca be achieved by applyig traditioal Chi-square test ad the test suggested by Wag ad Maritz i 99. For applyig these tests, we have H Raifall o cosecutive day/week is idepedet Agaist the alterative hypothesis H Raifall o cosecutive day/week is ot idepedet. The traditioal Chi-square test is Oi Ei i Ei Where O i (i,,...) are observed frequecies, E i(i,,...) are correspodig expected frequecies. i umber of observatios (,,...) The critical regio for testig ull hypothesis is where c, c is the calculated value of Chi-square ad is the tabulated value of Chisquare for ( ) degree of freedom ad level of, sigificace. I 99, Wag ad Maritz observed that WS test for idepedece of three state Markov Chai is better tha the traditioal -test method. Accordig to Wag ad Maritz, the test, for testig the above ull hypothesis, is $ A $ B V $ A $ B D WS N, The variace of maximum likelihood estimator is give by 77 ewyorksci@gmail.com

3 New York Sciece Joural ;3() A $ $ B 3 V d w w r r d where $ A P P P $ B P P P P dd ww rr rd dr wr rw PdwPwd Pdd PddPrr Prr,, 3 are statioary probabilities which are calculated as p sp/q rps/q 3 p/q P P wr dd Pdr Pwd rr p P P dw P P wr rd r q P P P ww wd Prw s rr Critical regio is WS at ' ' c Z level of sigificace i.e. we reject ull hypothesis, if WS Z, where Z is the ( ) lower percetage poit of a stadard ormal distributio. 4. Expected legth of differet spells: (a) A dry spell of legth d is defied as sequece of cosecutive dry (days/week) preceded ad followed by wet or raiy (days/week) ad the probability of a sequece of d dry (days/week) is d Pd Pdd Pdd ad expected legth of dry spell is ED Pdd where P dd is the probability of a day beig wet or raiy day. (b) A wet spell of legth w is defied as sequece of cosecutive wet (days/week) preceded ad followed by dry or raiy (days/week) ad the probability of a sequece of w wet (days/week) is w Pw ad expected legth of dry spell is EW where P ww is the probability of a (days/week) beig dry or raiy (days/week). (c) Similarly, for raiy spell of legth r, we have r P r Prr Prr ad expected legth of rai spell is ER Prr where P rr is the probability of a (days/week) beig dry or wet day ad (d) Weather cycle (WC) is E WC ED EW ER (e) The umber of days/weeks (N) after which the equilibrium state is achieved is equal to the umber of times the matrix P is powered till the elemets of a (N) colum of the matrix P become equal i.e. for 3x3 matrix P 3 (N) P 3 3 where : Probability of a day/week beig a dry day/dry week : Probability of a day/week beig a wet day/wet week 3 : Probability of a day/week beig a raiy day/raiy week 5. Results: I the preset study, the daily raifall data for 4 years from Ja., 96 to Dec3,, was collected from the meteorological observatory, Patagar, Idia located at 9N latitude, 79.3E logitude ad altitude 43.84m above mea sea level. O a average the regio has a humid subtropical climate havig hot summers (4-4 C) ad cold witers (-4 C) ad rai is received from south-west mosoo durig Jue to September. The raifall occurrece, based o daily raifall ad weekly raifall, was studied for three differet periods, viz. Pre-mosoo, Mosoo ad Post-mosoo. (i) The yearly maximum daily raifall based o 4 years is preseted i Figure ad the weekly maximum raifall based o 4 years data for 5 weeks is also preseted i Figure ewyorksci@gmail.com

4 New York Sciece Joural ;3() Maximum raifall (mm) Yearly maximum daily raifall years Figure. Yearly maximum daily raifall Weekly maximum raifall (96-) Table 4. Calculated values of Chi-square ad WS test Chi-square test Postmosoo Pre-mosoo Mosoo Premosoo Post- Mosoo mosoo Postmosoo WS test Pre-mosoo Mosoo (iv) The estimated trasitio probability matrices for daily raifall data are reported i table 5a (Pre mosoo), table 5b (Mosoo) ad table 5c (Post mosoo) respectively Table 5a. Pre mosoo based o daily raifall maximum raifall (mm) week umber Figure. Weekly maximum raifall (ii) Based o daily raifall data, the calculated values of traditioal Chi-square ad WS test, for three differet periods are reported i table 3. Table 3. Calculated values of Chi-square ad WS test Chi-square Premosoo Post- test Mosoo mosoo WS test (iii) Based o weekly raifall data, the calculated values of traditioal Chi-square ad WS test, for three differet periods are reported i table 4. Trasitio probability matrix d w.78.. r Table 5b. Mosoo based o daily raifall Trasitio probability matrix d w r Table 5c. Post mosoo based o daily raifall Trasitio probability matrix d w r (v) The estimated trasitio probability matrices for weekly raifall data are reported i table 6a (Pre mosoo), table 6b (Mosoo) ad table 6c (Post mosoo) respectively. Table 6a. Pre mosoo based o weekly raifall Trasitio probability matrix d w r ewyorksci@gmail.com

5 New York Sciece Joural ;3() Table 6b. Mosoo based o weekly raifall Trasitio probability matrix d w r Table 6c. Post mosoo based o weekly raifall Trasitio probability matrix d w.7.. r (vi) Based o daily raifall data, equilibrium state probabilities, expected legth of differet spells, weather cycles ad umber of days required for the system to achieve the equilibrium state (N), are reported i table 7. (vii) Based o weekly raifall data, equilibrium state probabilities, expected legth of differet spells, weather cycles ad umber of weeks required for the system to achieve the equilibrium state (N), are reported i table Coclusios: (i) The data uder study was collected for 4 years (96-) from IMD approved meteorological observatory, Patagar. From figure, it ca be see that yearly maximum daily raifall is lowest (5mm) i year 979 ad highest (75mm) i year over 4 years. Similarly, over the same period it ca be see from figure that weekly maximum raifall is lowest ( mm) i 46 th week (Nov, -8) ad highest (443mm) i 36 th week (Sep, 3-9). These values idicatig very large fluctuatio durig the period of study. (ii) Before applyig Three-state Markov Chai approach, we tested H agaist H by applyig traditioal Chi-square test ad WS test, for all three periods (based o daily ad weekly raifall data). From table 3 ad table 4, it is clear that, accordig to traditioal Chi-square ad WS test the ull hypothesis of idepedece of raifall o cosecutive day/week is rejected at 5% level of sigificace. This idicates that curret day/week raifall depeds o precedig day/week raifall. (iii) Because curret day/week raifall depeds o precedig day/week raifall so three-state Markov Chai approach is used to study the behavior of raifall. Estimated trasitio probability matrices are reported i table 5a (Pre mosoo), table 5b (Mosoo) ad table 5c (Post mosoo) for daily raifall data ad i table 6a (Pre mosoo), table 6b (Mosoo) ad table 6c (Post mosoo) for weekly raifall data respectively. (iv) Based o daily raifall data, equilibrium state probabilities, ad 3of a day beig dry, wet or raiy are give i table 7 ad o the basis of these values it ca be said that. (a) O the average for every wet day there are about 47, 3 ad 97 dry days ad about 3, 6 ad raiy days for all three periods. (b) Similarly, o the average for every raiy day there are about 9, ad 49 dry days ad about, ad wet days for all three periods. (v) Based o weekly raifall data, equilibrium state probabilities, ad 3of a week beig dry, wet or raiy are give i table 8 ad o the basis of these values it ca be said that (a) O the average for every wet week there are about 3, ad 3 dry weeks ad about, 4 ad raiy weeks for all three periods. (b) Similarly, o the average for every raiy week there are about 8, ad 3 dry weeks ad about, ad wet weeks for all three periods. (vi) The umber of days required for the system to achieve the equilibrium state (N) are 7, ad 9 for daily raifall data which idicates that after 7 days from Ja, days from Jue ad 9 days from Oct, the probability of the day beig dry, wet or raiy is idepedet of the iitial weather coditio durig the pre-mosoo, mosoo ad post-mosoo respectively. Similarly, for weekly data aalysis, the umber of weeks required for the system to achieve the equilibrium state (N) are 7, 7 ad 8 which idicates that after 7 weeks from Ja, 7 weeks from Jue ad 8 weeks from Oct, the probability of the week beig dry, wet or raiy is idepedet of the iitial weather coditio durig the pre-mosoo, mosoo ad post-mosoo respectively. 8 ewyorksci@gmail.com

6 New York Sciece Joural ;3() Table 7. Equilibrium state probabilities, expected legth of differet spells, weather cycles ad umber of days for equilibrium state Expected legth of 3 Dry ru Wet ru Rai ru Cycle N Total days Pre mosoo Mosoo Post mosoo Table 8. Equilibrium state probabilities, expected legth of differet spells, weather cycles ad umber of weeks for equilibrium state Expected legth of 3 Dry ru Wet ru Rai ru Cycle N Total weeks Pre mosoo Mosoo Post mosoo Refereces:. Aeja D.R., Srivastava O.P. Markov chai model for raifall occurrece. Joural Idia Society Agriculture Statistics. 999;5(): Aeja D.R., Srivastava O.P. A study techique for raifall patter. Aligarh Joural of Statistics. 986; vol.6: Feyerherm A. M., Bark L. D. Goodess of fit of a Markov Chai Model for Sequeces of wet ad dry days. Joural of Applied Meteorology. 967; vol.6: Fisher R.A. The ifluece of the raifall o the yield of wheat at Rothamsted. Philosophical trasactio of the Royal Society of Lodo. 94; Series B, Vol.: Gabriel, K R., Neuma, J. A Markov chai Model for study for daily raifall occurrece at Tel Aviv. Quarterly Joural Roy. Met. Society. 96; 88: Garg V.K., Sigh J. B. Markov Chai Approach o the behavior of Raifall. Iteratioal Joural of Agricultural ad Statistical Scieces. ; vol.6: No.. 7. Gree J. R. Two probability models for sequeces of wet or dry days. Mothly Weather Review. 965; 93: Gree J. R. A geeralized Probability Model for Sequeces of Wet ad Dry Days. Mothly Weather Review. 97; vol.98: No Purohit R.C., Reddy G. V. S., Bhaskar S. R., Chittora A. K. Markov Chai Model Probability of Dry, Wet Weeks ad Statistical Aalysis of Weekly Raifall for Agricultural Plaig at Bagalore. Karataka Joural of Agricultural Sciece. 8; ():-6.. Srikatha, R., McMaho, T. A. Stochastic Geeratio of Aual, Mothly ad Daily Climate Data: A Review. Hydrology ad Earth System Scieces. ; 5(4): Ster R.D., Coe R. A model fittig aalysis of daily raifall data. Joural of Royal Statistical Society. 984; 47(): Wag D. Q., Martiz J. S. Note o testig a threestate Markov Chai for idepedece. Joural Statistics Computatio ad Simulatio. 99; vol. 37: Weiss L. L. Sequeces of wet or dry days described by a Markov Chai probability Model. Mothly Weather Review. 964; 9: Wiser E. H. Modified Markov probability models of sequeces of precipitatio evets. Mothly Weather Review. 965: /7/ 8 ewyorksci@gmail.com

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

Rainy and Dry Days as a Stochastic Process (Albaha City) IOSR Joural of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 239-765X. Volume, Issue Ver. III (Ja - Feb. 25), PP 86-9 www.iosrjourals.org Raiy ad Dry Days as a Stochastic Process (Albaha City) Dr. Najeeb