Development of Time Series Autoregressive Model for prediction of rainfall and runoff in Kelo Watershed Chhattisgarh

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1 Iteratioal Joural of Advaes i Egieerig Siee ad Tehology 53 ad ISSN: 39 evelopmet of Time Series Autoregressive Model for preditio of raifall ad ruoff i Kelo Watershed Chhattisgarh Shaolee Chakraborty,.M. eis ad A. Sherrig 3 3 epartmet of Soil & Water Coservatio Egieerig M.Teh ( Soil ad Water oservatio egieerig) Professor 3 Assoiate Professor shaolee@gmail.om Abstrat: A study was oduted to develop a stohasti time series model, apable of preditio of raifall ad ruoff i Kelo watershed.the Kelo watershed of Mahaadi river basi situated i Raigarh regio of Chhattisgarh state. Raigarh is i the North Cetral part of Chhattisgarh state. The athmet area of the watershed is 75 sq km. It is situated 53 latitude ad logitude 83 4.The raifall ad ruoff data of the watershed from the year (6) was olleted ad used for the developmet of model. Autoregressive (AR) models of order, ad were tried.the goodess of fit ad adequay of models were tested by Box Piere Portmateau test, Akaike iformatio Criterio ad by ompariso of historial ad geerated data orrelogram The graphial represetatio betwee historial ad geerated orrelogram, where i both the ases of raifall ad ruoff there was a very lose agreemet betwee them. The ompariso betwee the measured ad predited raifall ad ruoff by AR() model learly shows that the developed model a be used effiietly for the future preditio o raifall ad ruoff i Kelo watershed. Keywords Stohasti time series model, Autoregressive (AR) models, Akaike iformatio Criterio, Box Piere Portmateau test I. INTROUCTION Idia is a agraria outry, due to its favourable mosoo limate ad vast area of fertile ulturable lad. The Idia has bee traditioally depedet o agriulture as 7% of its populatio is egaged i farmig. The effiiet maagemet of water ad other atural resoures will likely be a sigifiat issue. Raifall ad ruoff modelig is a importat area of hydrologial studies ad is oe i whih researh is atively arried out. Methodology for ouplig stohasti models of hydrologial proess applyig two differet time sales so that the time series geerated by differet models be osistet. (Koutosoyiais ) Speially oeptual models, whih attempt to represet the physial proess whih ours o the athmets ad mathematial models, whih oly osidered the mathematial relatioship betwee raifall ad ruoff without osiderig the physial proess.. The priipal aim of time series aalysis to desribe the history of momets i time of some variable at a partiular site. A omprehesive review o time series aalysis tehique used i limatology ad hydrology. it was suggested to use more importat powerful test for statioary ad tred detetio i time series. (Mahiwal ad Jha 6). Most hydrologi system have both determiisti as well as stohasti ompoet, but stohasti time series model suh as Autoregressive (AR)(Thomas ad Fierig,96 ;Yevjevih,963 ; Matalas, 967), movig average (MA) ad Autoregressive movig average (ARMA) (Carlso,97) are widely used to predit aual ruoff. Idetifiatio geerally depeds o the harateristis of overall water resoures system, the harateristis of time series ad the models iput. Salas ad Smith (98) demostrated these of physial osideratio of the type of model. A developed Auto regressive model is suitable for a ertai rage ad appliable for partiular zoe of limate. (Rai ad Sherrig 7). Multivariate autoregressive (MAR) ad uivariate autoregressive (AR) methods applied to regioal sale raifall ruoff modelig (Tomasz 6). Almost every model available fits i a data aalyzed oly realistially i a ertai rage. It is possible to maitai reliable flow simulatio by asadig a series of ruoff preditio regressio model that predit a dowstream flow from a upstream flow ad the iremetal raifall betwee gaugig statio. (Vogtaaboo et.al, 8). To explore the ifluee of the iflow o the outflow i a river system ad to exploit the iteral iteratio of the outflows, bivariate time series models were eeded. (Woga et al. 7). I additio to this it may be oted that models ISSN: 39 /VN: 5363 IJAEST

2 evelopmet of Time Series Autoregressive Model for preditio of raifall ad ruoff i Kelo Watershed Chhattisgarh 54 foud appliable i a partiular zoe eg. the temperate zoe ot to be appliable i other zoe suh as tropi. (Iyegar,98). The preset study develop a stohasti model of aual raifall ad ruoff time series appliable for Kelo watershed i Chhattisgarh state has bee plaed with followig objetives : () To geerate or develop stohasti time series model for preditio of raifall ad ruoff for Kelo watershed. () To estimate parameters of Autoregressive model. (3) To test the validity of the predited raifall ad ruoff with measured ad evaluated the performae of the developed model. II. MATERIALS AN METHOS The study area is Kelo watershed of Mahaadi river basi situated i Raigarh regio of Chhattisgarh state as depited i figure Fig Loatio of study area A. Stohasti Time Series Model A mathematial model represetig stohasti proess is alled stohasti time series model. It has a ertai mathematial form or struture ad a set of parameters. A simple time series model ould be represeted by sigle probability distributio futio f(x:ө) with parameters Ө = (Ө, Ө,. ) valid for all positios t =,,.N ad without ay depedee betwee X, X, X. A time series model with depedee struture a be formed as: ε t = ф ε t + η t () where, η t = A idepedet series with mea zero ad variae ( ф ) ε t = epedet series ф = Parameter of the model. (a) Autoregressive (AR) Model ISSN: 39 /VN: 5363 IJAEST

3 IJAEST, Volume, Number Shaolee Chakraborty et al. I the Autoregressive model, the urret value of a variable is equated to the weighted sum of a pre assiged o. of part values ad a variate that is ompletely radom of previous value of proess ad shok. The p th order autoregressive model AR (p), represetig the variable Y t is geerally writte as Y t ε t p Yt = Y + Φ j = j ( Y t j Y ) Where, = The time depedet series (variable) = The time idepedet series whih is idepedet of Y t ad is ormally distributed with mea zero ad variae σ Y = Mea of aual raifall ad ruoff data Φ,Φ,.. Φ p = Autoregressive parameter. + ε B. Estimatio of Autoregressiveparameter (Φ) maximum likelihoodestimate t () For estimatio of the model parameter method of maximum likelihood will be used (Box ad Jekis, 97). Cosider the sum of rossproduts, z i z j + z i+ z j+ + + z N+J z N+I ad defie where, = differee operator N = sample size i,j = maximum possible order i partiular, ij =ji = N / ( N+ij)(3), AR () : Φ = (4), AR, 3,3,3,3 () : Φ = (5), 3,3,3,3,,,3 Φ = (6), 3,3 (a) Autoorrelatio futio The autoorrelatio futio r k of the variable Y t of equatio (3.) is obtaied by multiplyig both sides of the equatio (3.) by Y t+k ad takig expetatio term by term. The relatioship proposed by Kottegoda ad Horder (98) for the omputatio of autoorrelatio futio of lag K was used whih is expressed as:,3 ISSN: 39 /VN: 5363 IJAEST

4 evelopmet of Time Series Autoregressive Model for preditio of raifall ad ruoff i Kelo Watershed Chhattisgarh 56 N K t k = t= N r ( Y Y)( Y t= t t+ k ( Y Y) Where, r k = Autoorrelatio futio of time series Y t at lag k Y t = Stream flow series (historial data) Y = Mea of time series Y t k = Lag of K time uit N = Total umber of disrete values of time series Y t Y) (7) The followig equatio was used to determie the 95 per et probability levels (Aderso, 94). r k ±.645 N K 95%) = N K ( (8) where, N = Sample size. (b) Partial Autoorrelatio futio The followig equatio was used to alulate the partial autoorrelatio futio of lag K. (urbi, 96). r k k j= = k where, PC kk = Partial autoorrelatio futio at lag K r k = Autoorrelatio futio at lag K j= PC k, j k, j k j PC k, k (9) PC r The 95 peret probability limit for partial autoorrelatio futio was alulated usig the followig equatio.96 PC k,k (95%) = () N ()Parameter estimatio of AR (p) models The average of sequee Y t was omputed by followig equatio: N Y = Y t () N t= The average σ ε of Y t was omputed by the followig equatio: N σ ε = ( Yt Y ) ( N ) () t= After omputatio of Y ad σ ε, the remaiig parameters Φ, Φ,. Φ p of the AR models were estimated by usig the sequee: Z t = Y t Y, (3) r j ISSN: 39 /VN: 5363 IJAEST

5 IJAEST, Volume, Number Shaolee Chakraborty et al. t =,,. N The parameters Φ, Φ,. Φ p were estimated by solvig the p system of followig liear equatios (Yule ad Walker equatio): r k = Φ r k + Φ r k + + Φ p r kp K> Where, r, r were omputed from equatio 7 orr k = p Φ j= r j k j K> (4) C.Statistial harateristis (a) Mea Foreast Error Mea foreast error was alulated to evaluate the performae of auto regressive models fitted to time series of raifall ad ruoff. The mea foreast error (MFE) was omputed for the aual stream flow series by the followig equatio: (Raghuwashi et al., ). MFE = where, χ (t) = Computed stream flow value χ (t) = Observed stream flow value η = Number of observatios (b)mea Absolute Error MAE= χ η η χ ( t χ χ ( t ) ) (5) (6) () Mea Relative Error (d)mea Square Error (e)root Mea Square Error (f)itegral Square Error MRE = MSE = RMSE = χ χ χ η [ ] χ ( t χ η ) [ χ χ ] η (7) (8) / (9) ISSN: 39 /VN: 5363 IJAEST

6 evelopmet of Time Series Autoregressive Model for preditio of raifall ad ruoff i Kelo Watershed Chhattisgarh 58 ISE = [ χ χ ] χ (). Goodess of fit of autoregressive (AR) models The followig tests were performed to test the goodess of fit of autoregressive (AR) models. (a)boxpiere Portmoteau lak of fit test Q = N k = () Where, N = Number of observatios r k = Serial orrelatio or autoorrelatio of series Y t The statisti Q follows χ distributio with r = Kp degree of ompared with tabulated values of χ. (b)akaike Iformatio Criterio L r k The Akaike Iformatio Criterio (Akaike, 974) was used for hekig whether the order of the fitted model is adequate ompared with the order of depedee model. Akaike Iformatio Criterio for AR(p) models, was omputed usig the followig equatio. Where, N = Number of observatios σ ε = Residual variae AIC (P) = N σ ε + ( P) () A ompariso was made betwee the AIC (p) ad the AIC (p) ad AIC (p+). If the AIC (p) is less tha both AIC (p) ad AIC (p+), the the AR (p) model is best otherwise, the model whih gives miimum AIC value was the oe to be seleted model. III.RESULT AN ISCUSSION The stadardize aual raifall ad ruoff series were modeled through the autoregressive model.the modelig proedure of the data series ivolved various steps like prelimiary aalysis ad idetifiatio, estimatio of model parameters ad diagosti hekig for the adequay of the model.(salas ad Smith 98 b) Autoorrelatio ad partial autoorrelatio are used for idetifiatio of the proper type ad order of the model.the idetifiatio geerally depeds o the harateristis of overall water resoures system, the harateristis of time series ad the models iput.( Salas ad Smith 98) demostrated these of physial osideratio of the type of model. The autoorrelatio futios ad partial autoorrelatio futios were determied for the 95% probability limits. The autoorrelatio futio ad partial autoorrelatio futios with 95% probability limits upto 4 lag of the series (lag k) were omputed ad the autoregressive model of first order AR() was seleted for further aalysis. Models of Autoregressive (AR) Family The parameters of AR model were omputed for aual raifall ad ruoff (Tables ;figs 3). The predited values of aual raifall ad ruoff were ompared with the observed values. It was observed that AR(p) model upto order has show the good fit ad orrelatio betwee the observed ad predited values ad give i figs ad 3. AR (p) models for preditio of aual raifall ISSN: 39 /VN: 5363 IJAEST

7 IJAEST, Volume, Number Shaolee Chakraborty et al. AR () : Yt = (Yt 5.8) AR () : Yt = (Yt 5.8) AR (p) models for preditio of aual ruoff AR () : Yt = (Yt 69.7) AR () : Yt = (Yt 69.7) Table Statistial parameters of autoregressive (AR) model for raifall Model AR() AR () AR () Autoregressive parameter Φ=.7876 Φ=.839 Φ =.4546 White oise variae Akaike Iformatio Criteria Value of port moteaue statistis egree of freedom upto 4 lags Table value of χ at 5% level of sigifiae Table Statistial parameters of autoregressive (AR) model for ruoff Model AR() AR () AR () Autoregressive parameter Φ=.3 Φ=.546 Φ =.58 White oise variae Akaike Iformatio Criteria Value of port moteaue statistis egree of freedom upto 4lags 4 3. Table value of χ at 5% level of sigifiae ISSN: 39 /VN: 5363 IJAEST

8 evelopmet of Time Series Autoregressive Model for preditio of raifall ad ruoff i Kelo Watershed Chhattisgarh 6 Raifall, mm Measured Predited Years Fig Compariso of orrelogram of measured ad predited series for Raifall 7 R =.99 5 Predited raifall, mm Measured raifall, mm Fig Compariso betwee measured ad predited aual raifall of Kelowatershed Box Piere Portmoteau ad Akaike Iformatio riterio test for AR models Box Piere Portmoteau lak of fit test ad Akaike Iformatio riterio (AIC) for all three models to hek the adequay for both aual raifall ad ruoff for AR(), AR() ad AR() models were estimated. The result of Box Piere Portmoteau lak of fit test reveled that all three models viz., AR(), AR() ad AR() were givig good fit ad were aeptable but AIC values of AR() i both raifall ad ruoff are lyig betwee AR() ad AR(), therefore AR() was osidered for further preditio of aual raifall ad ruoff. Compariso of the observed ad predited aual raifall ad ruoff The orrelogram of observed ad predited series for aual raifall ad ruoff were developed by plottig autoorrelatio futio agaist lag k. A graphial ompariso of observed ad predited aual raifall ad ruoff with the seleted model are show i fig ad 4. The graphial represetatio of the data shows a loser agreemet betwee observed ad predited aual raifall ad ruoff seleted model.it reveals that developed model for raifall ad ruoff a be utilized for the preditio of future treds with the miimum hae of error. ISSN: 39 /VN: 5363 IJAEST

9 IJAEST, Volume, Number Shaolee Chakraborty et al. Measured Predited 5 Ru o ff Years Fig 3 Compariso of orrelogram of measured ad predited series for ruoff P redited Fig 4 Compariso betwee measured ad predited aual ruoff of Kelo watershed Statistial harateristis of data The mea, stadard deviatio ad skewess of historial ad geerated data was alulated to evaluate the fittig of the model i momet preservatio. The results are tabulated i Table3. The results learly shows that the skewess of geerated data by AR() model ad historial data is lyig betwee to + ad therefore AR() model preserved the mea ad skewess better. R = Measured Table 3 Statistial harateristis of observed ad predited raifall ad ruoff for Kelo watershed Statistial harateristis Raifall Observed Raifall Predited Ruoff Observed Ruoff Predited Mea Stadard deviatio Skewess Mea foreast error Mea absolute error Mea relative error ISSN: 39 /VN: 5363 IJAEST

10 evelopmet of Time Series Autoregressive Model for preditio of raifall ad ruoff i Kelo Watershed Chhattisgarh 6 Root mea square error Itegralsquare error Performae Evaluatio of AR() model for raifall ad ruoff Performae of the model was also estimated by estimatig statistial harateristis suh as MFE, MAE, MRE, MSE, RMSE ad ISE were also used to prove the adequay of the model for future preditio with higher degree of orrelatio to previous measured observatios. I raifall ad ruoff, the MFE for AR() model is 4.67mm ad 6.94 mm respetively. The differet errors betwee measured ad predited values by AR () for both raifall ad ruoff were estimated ad whih represets that for preditio of aual raifall ad ruoff by AR() model is givig the best results. Sie all the errors are omparatively less ad graphial ompariso of observed ad predited values of raifall ad ruoff idiates that the AR() model a be used effiietly for preditio of aual raifall ad ruoff i Kelo watershed. IV.CONCLUSION O the basis of estimated errors, statistial harateristis ad orrelatio betwee observed ad predited values, it is oluded that the proposed autoregressive AR() model a be used to predit the aual raifall ad ruoff i Kelo watershed of Chhattisgarh state. REFERENCE [] Akaike, H, (974). A ew look at the statistial model idetifiatio. IEEE Trasatios o automati otrol, AS9, pp [] Aderso, R. L., (94): istributio of the serial orrelatio oeffiiets. Aals of Math. Statistis.3(): 3. [3] Box, G.E.P. ad Jekis, G., (97). Time series Aalysis, Foreastig ad otrol. First Ed. Holde ay I., Sa Fraiso. [4] Carlso, R.F (97). Appliatio of liear models to four aual stream flow series. Water Resoure Researh, 64:778 [5] urbi, J, (96). The fittig of time series model. Rev. It. Ist. stat., 8: 333. [6] Iyegar, R. N., (98). Stohasti modelig of mothly raifall. J.Hydrolog,57: [7] Kottegoda, N. T., Horder, M. A., (98), aily flow model raifall oueres usig pulse ad a trasfer futio. J. Hydrology,47: 534. [8] Koutosoyia,., (): Couplig stohasti models of differet time sales. Water Resoures Researh, 37():37939 [9] Mahiwal,. ad Jha, M.K.(6). Timeseries aalysis for water resoures plaig ad maagemet. Joural of Hydrology ad Hydro mehais, Vol.54(3), pp: [] Matalas, N.C Mathematial assessmet of sytheti yology. J. Water Resoure. Res., Vol3, No.4,pp: [] Raghuwashi, S, Walleder, W.S., () Foreastig aily Evapotraspiratio for a Grass Referee Crop. Agriultural Egg. Joural, 9 (): 6. [] Rai, E ad Sherrig, A., (7). evelopmet of autoregressive time series model for preditio of raifall ad ruoff for Mashara watershed of lower Gomati athmet. J. Agriultural Egieerig. Vol.44(4) pg :384 [3] Salas, J.. ad Smith, R.A.,(98b). Uertaities i hydrologi time series aalysis. Paper preseted at the ASCE sprig Meetig, Portlad, Orego, Preprit, 858. [4] Salas, J.. ad Smith, R.A.,(98). Physial basis of stohasti models of aual flows. Water Resoures Researh,7 (): [5] Thomas,H.A. ad Fierig, M.B.,96. Mathematial sythesis of streamflow sequees for the aalysis of river basis by simulatios. I esig of Water Resoure Systems, pp: , Cambridge, Massahussetts, Harvard Uiversity Press. [6] Tomasz.(6). A data based regioal Sale auto regressive raifall ruoff model. A study from Odra river.,. J. Hydrology Vol.., No.6, pp: [7] Vogtaaboo, S. Lim, H.S. ad Rihards, K. (8). ata based Mehaisti RaifallRuoff Modellig for a Large Mosoo Cathmet i Thailad,. J. Hydrology Vol. (), pp: 3. [8] Woga, G. T., (7). Improved flood routig by ARMA modelig ad the kalma filter s tehique. J. Hydrology, 93:759. [9] Yevjevih, V.M..,963. Flutuatios of wet ad dry years. Part I.Researh data assembly ad mathematial models. Hydrology Paper., Colorado State Uiversity,Fort Collis, Colorado. [] Yule,G.V., (97).O a method of ivestigatig periodiities i disturbed series with speial referee to Wolfer s suspot umber.phil. Tra.A 6:67. ISSN: 39 /VN: 5363 IJAEST

11 IJAEST, Volume, Number Shaolee Chakraborty et al. ISSN: 39 /VN: 5363 IJAEST