Modeling and forecasting of rainfall data of mekele for Tigray region (Ethiopia)

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1 Statitic and pplication Volume, o. &, (ew Serie), pp Modeling and forecating of rainfall data of mekele for Tigray region (Ethiopia) maha Gerretadikan and M.K.Sharma ddi baba Univerity, ddi baba, Ethiopia btract In thi paper we have attempted to build a eaonal model of monthly rainfall data of Mekele tation of Tigray region (Ethiopia) uing Univariate Box-enkin methodology. The method of etimation and diagnotic analyi reult revealed that the model wa adequately fitted to the hitorical data. In particular, the reidual analyi, which i important for diagnotic checking confirmed that there i no violation of aumption in relation to model adequacy. Further comparion on the forecating accuracy of the model i performed by holding-out ome rainfall value. The point forecat reult howed a very cloer match with the pattern of the actual data and better forecating accuracy in validation period. Key word: Box-enkin model; Rainfall; Forecating Introduction Univariate time erie analyi and forecating ha become a major tool in hydrology, environmental management, and climatic field. Several time erie method have been ued for modeling and forecating rainfall data in literature but according to Pankratz (3) the Box and enkin method i the mot general way of approaching to forecat unlike other model, there i no need to aume initially a fixed and pecified pattern. The Univariate Box and enkin model are ueful for analyi of ingle time erie. Montgomery and ohnon (76) conidered Box and enkin methodology a probably the mot accurate method for forecating of time erie data. ccording to Caldwell (6), the Box-enkin methodology i particularly uited for development of model of proce exhibiting trong eaonal behavior. There are other forecat technique exploring the relation among obervation yield better reult; mot of thee forecat technique are baed on recent advance in time erie analyi conolidated and

2 3 MH GERRETSDIK D M.K.SHRM [Vol., o. & developed by Box and enkin (76) and further dicued in other reource uch a Chatfield (6). ail and Momani () ued Univariate Box-enkin approach and revealed that thi approach poee many appealing feature uch a the reearcher who ha a data for the pat period, for example rainfall, to forecat future rainfall without having to earch for other related time erie data. In thi paper we have alo ued Box- enkin approach to build a eaonal model of monthly rainfall data of Mekele tation in Tigray region (Ethiopia). The etimation and diagnotic analyi reult revealed that the model i well fitted to the hitorical data. The reidual analyi revealed that there wa no violation of aumption in relation to model adequacy. Further we compared the forecating accuracy of the model by holding-out ome rainfall value. The point forecat reult howed a very cloer match with the pattern of the actual data and better forecating accuracy in validation period. Material and Method. Material The ational Meteorological Service gency (MS), Ethiopia, i the reponible organization for the collection and publihing of meteorological data. The monthly rainfall data from the period anuary 75 December of Mekele tation of Tigray region were taken from MS (ppendix).. Methodology In thi article we ued Seaonal utoregreive Integrated Moving verage (SRIM) model, propoed by Box and enkin (76), for model building and forecating for rainfall data. The Box and enkin methodology i a powerful approach to the olution of many forecating problem (ohnon and Montgomery, 76) and it can provide extremely accurate forecat of time erie and offer a formal tructured approach to model building and analyi. There are many quantitative method of model building and forecating which are being ued in climatology and metrological tudie. With the development of the tatitical oftware package and it availability, thee technique have become eaier, fater and more accurate to ue. In thi tudy, we employ SS and SPSS oftware package for the tatitical data analyi. The Box- enkin methodology aume that the time erie i tationary and erially correlated. Thu, before modeling proce, it i important to check whether the data under tudy meet thee aumption or not. Let x, x, x 3,..., x t-, x t, x t+,..., x t be a

3 ] MODELIG D FORECSTIG OF RIFLL DT 33 dicrete time erie meaured at equal time interval. eaonal RIM model for x t i written a [Box and enkin, 7] d D φ (B) Φ (B ){[( B) ( B ) x t ] µ} = θ (B)Θ(B ) a t Or () φ ( B ) Φ( B ) ( w µ ) = θ ( B) Θ( B ) a t t where x t i an obervation at a time t; t dicrete time; eaonal length, equal to ; µ mean level of the proce, uually taken a the average of the w t erie (if D + d > often µ ); a t normally independently ditributed white noie reidual with mean and variance σ (written a ID (, σ ) a a φ(b) = φ noneaonal autoregreive (R) operator or p Β φ Β... φ p Β polynomial of order p uch that the root of the characteritic equation φ ( B) = lie outide the unit circle for noneaonal tationarity and theφ i, i =,,..., p are the noneaonal R parameter; ( B) d d = noneaonal differencing operator of order d to produce noneaonal taionarity of the d th difference, uually d =,, or ; p Φ ( B ) = Φ B Φ B... Φ B eaonal R operator or order p uch that the p root of Φ( B ) = lie outide the unit circle for eaonal tationarity and Φ i, i =,,..., p are the eaonal R parameter; ( B) D D = eaonal differencing operator of order D to produce eaonal tationarity of the Dth differenced data, uually D =,, or ; d D w t = xt tationary erie formed by differencing x t erie ( n = d D i the number of term in the w t erie); q θ ( B) = θ B θ B... θ q B noneaonal moving average (M) operator or polynomial of order q uch that root of θ ( B) = lie outide the unit circle for invertibility andθ i, i =,,..., q; Q Θ ( B ) = Θ B Θ B... Θ B eaonal M operator of order Q uch that the q root of Θ( B ) = lie outide the unit circle for invertibility and Θ i, i =,,..., Q are the eaonal M parameter.

4 34 MH GERRETSDIK D M.K.SHRM [Vol., o.& The notation (p, d, q) (P, D Q) i ued to repreent the SRIM model (). The firt et of bracket contain the order of the noneaonal operator and econd pair of bracket ha the order of the eaonal operator. For example, a tochatic eaonal noie model of the form (,, ) (,, ) i written a ( - φ B ){[( B ) xt ] µ } = ( θb θ B ) ( Θ B ) at () If the model i non eaonal, only the notation ( p, d, q) i needed becaue the eaonal operator are not preent. When a eaonal model i tationary and require no differencing (i.e. D = and d = ), it i often referred to imply a an RM (autoregreive moving average) proce. The notation (p, q) (P, Q) i ued to repreent thi type of model. If an RM model i noneaonal, the notation (p, q) i ued to indicate order of the R and M operator, repectively. pure noneaonal R proce of order p with no differencing i often denoted by R (p). Likewie, a noneaonal M proce of order q i ometime written a M (q). Of coure an R (p) model can be repreented equivalently by the notation (p, ) or ( p,, ), while M (q) proce can alo be denoted by (, q) or (,, q)..3 Tet for Stationary Graphic Inpection: The pattern of the time erie plot (Fig.) doe not how any apparent ytematic change about the mean. The periodic peak in the plot, however, reflect the yearly regular eaonality (with eaonality interval =) of the rainfall value. The erie i, therefore, eaonal due to a large rainfall value during rainy eaon and a relatively leer peak due to mall value of rainfall in the other month. Thi indicate that the rainfall data have eaonal unit root (i.e., eaonally not tationary). The Figure exhibit the autocorrelation function plot of untranformed data in which the preence of eaonality behavior a well a eaonally non tationary of the rainfall erie i clear. Becaue there i a inuoidal wave pattern at the multiple of eaonal interval and declining lowly while non eaonal lag are relatively decaying quite lowly. It i, thu, neceary to remove the non eaonal component of the time erie correponding to the inuoidal periodic component of the autocorrelation function to make erie eaonally tationary. Dickey-Fuller Tet: The mot widely ued tet for tationary i Dickey-Fuller tet. Thi tet i baed on the etimate of the following regreion equation with no determinitic trend. x = φ x + γ x γ x + a (3) t t t p t p t

5 ] MODELIG D FORECSTIG OF RIFLL DT 35 where i the difference operator defined = t t t x x x and t x i a variable of interet. Thi model can be etimated and teted for a unit root. That i equivalent to teting φ equal to zero, γ,, γ p are p regreion coefficient and p i the number of autoregreive term. To tet the hypothei that the erie x t i tationary, we formulate the following hypothei H O : The erie i non-tationary i.e φ = H : The erie i tationary i.e φ < at α=.5. There i a need of eaonal differencing not imple differencing. Rainfall erie in mm Time Figure : Plot of monthly rainfall data The pattern of monthly rainfall erie plot and autocorrelation function ugget the need of eaonal differencing but not imple differencing.

6 36 MH GERRETSDIK D M.K.SHRM [Vol., o.& utocorelation Lag Figure : utocorrelation plot for the untranformed monthly rainfall erie. Uually the order of p in the regreion equation i et to three. Then if the etimate of φ i nearly zero in the fitted regreion equation (), the original erie x t need firt differencing and if the etimate of φ < then the original erie i already tationary (Makridaki et al., ). It wa found that the etimated value for φ = -.4 which confirm that original time erie plot i without obviou trend at 5% ignificance level. The autocorrelation function in Fig. exhibit non - eaonally rapidly decaying trend. a reult, both tet appear to agree to avoid firt non eaonal imple differencing. Variance Comparion: The behavior of variance aociated with different order of differencing can provide a ueful mean of deciding the appropriate order of differencing (Mill, ). The rule i that the when the ample variance doe not decreae further then a tationary erie i found. If the increae in the differencing order increae the variance, it i an indication of over differencing. To examine our erie that whether it i a candidate of nondifferencing, imple differencing, eaonal differencing or double eaonal differencing for non eaonally and eaonally tationarity, we computed the ample variance for each of x t,, erie, repectively. We got the following reult:

7 ] MODELIG D FORECSTIG OF RIFLL DT 37 Var( )=64., Var(x t )=76.5, Var( )=745., and ( x t ) =75 value; Var( ) > Var(x t ); Var ( > Var (x t ) > Var ( ). Thee reult ugget that non-eaonal firt differencing ( ) ha been overdifferenced and hence the original erie i non-eaonally tationary.the firt eaonal differencing would rather be important, becaue the Var ( ) i greater than Var( ). Thee tet for tationarity eem to agree and ugget that the firt eaonal differencing in the erie can achieve tationarity around a contant mean, which i approximately zero and it tandard deviation i 5.4 mm (Figure 3). Moreover, the CF and PCF (Figure 4(a) and 4(b)) alo tell that the monthly rainfall erie i tationary in both mean and variance after firt eaonal difference. Figure 3: Plot for Firt eaonal differenced monthly rainfall erie

8 3 MH GERRETSDIK D M.K.SHRM [Vol., o.& (a) (b) Figure 4: (a): utocorrelation Function (CF) (b): Partial utocorrelation Function (PCF) for the firt eaonal differenced monthly rainfall..4 Tet for randomne ccording to Harvey (3) the implet time erie i a random model, in which the obervation vary around a contant mean, have a contant variance, and are probabilitically independent. In other word, a random time erie doe not have time erie pattern, meaning that there i no point in attempting to fit a time erie model to uch type of data. Therefore, it i important to perform tet of randomne before any attempt to modeling proce to our erie. Therefore we check our time erie through the following tet to invetigate the hypothei that the firt-eaonally differenced monthly rainfall erie are erially uncorrelated.

9 ] MODELIG D FORECSTIG OF RIFLL DT 3 Graphic Inpection: The viual inpection of the autocorrelation function plot provide ueful information to identify the type of time erie (Chatfield, 6). For example, if a time erie i a completely random erie, then for large n, r k for every k. Thi can be examined after the array of autocorrelation coefficient r k, plotted with k a abcia and r k a ordinate. Figure 4(a) exhibit the graph of ample autocorrelation againt different lag from which we can oberve viually that the autocorrelation are not all inignificant. Thi indicate that there i ome ort of dependence between value of x t erie. The randomne can alo be checked uing Bartlett Band Tet and Box-Ljung Tet Statitic. Here we ued Box-Ljung Tet Statitic. Box-Ljung Tet Statitic: Thi tatitic i ued for collectively teting the magnitude of the autocorrelation of tationary time erie for ignificance. For thi tet, we ued the ample autocorrelation coefficient of the firt eaonally differenced monthly rainfall a well. The hypothei to be teted i Ho : ll autocorrelation up to lag are zero Veru H : ot all up to lag are zero at α=.5. The tatitic for thi teting hypothei i a Q=n(n+ (4) Thi tatitic ha a chi-quare ditribution with degree of ditribution. Q- Statitic i uually computed for = 6, 4, 36 and 4 by mot of the tatitical package. However, = or 4 will prove to be atifactory (Patricia, E. G., 4). In thi regard, we compute the tet tatitic above for the firt = lag autocorrelation value and n=4 obervation. The value of the calculated Q- Statitic i found to be 43.7 and the tabulated value for chi-ditribution with degree of freedom at.5 ignificance level i.. The deciion to reject H o i baed on whether the value of Q-Statitic >,; if that doe not hold we do not reject Ho. Since Q-tatitic=43.7> =., we reject H o. we conclude that the eaonally firt differenced monthly rainfall erie are erially correlated. ow we can ay that the monthly rainfall data are tationary and erially correlated.

10 4 MH GERRETSDIK D M.K.SHRM [Vol., o.&.5 Model Identification Having etablihed that the monthly rainfall data are erially correlated and tationary, the next tep in the identification proce i to find the initial value for the order of non-eaonal and eaonal parameter p, q, P, and Q,, repectively. The firt tep in thi direction i to identify the ignificant autocorrelation and partial autocorrelation from the CF and PCF plot of the underlying tationary erie (Hipel et al., 77). Hence for the (- ), where B i the backward hift operator and i defined Bx t = x t- and B d i the backward hift operator of order d, we find ignificant CF at lag k=, and k=4, ee Figure 4(a). Hence, baed on the CF behavior, we gue Seaonal utoregreive Integrated Moving verage (SRIM) model (,, ) (,, 4) of the following form. (- ) (- ) x t = (- )(- (5) nother alternative model eem to be appropriate tentatively at thi tage i baed on the principle that when the proce i a purely SRIM (p, d, ) (P, D, ) model, r kk cut off and i not ignificantly from zero after lagp+sp. If r kk damp out at lag that are multiple of, thi ugget the incorporation of a eaonal moving average (M) component into the model. The failure of the PCF to truncate at other lag may ugget that a non-eaonal M term i required (Hipel et al., 77). ccordingly, we gue SRIM (,, ) (4,, ) model..6 Model Etimation and Diagnotic checking on-linear Etimation of the parameter for Box-enkin model i a quite complicated. Parameter etimate are uually obtained by maximum likelihood method, which i aymptotically correct for time erie (Brockwell and Davi, 6). pplying maximum likelihood method of etimation, we got the following etimated value of the parameter of SRIM (,,) (,,4), and (,, ) (4,,) a given in Table 3. Table 3: Parameter etimate for uggeted SRIM model. (a):(- )(- )x t = (- )(- or (,,) (,,4) (b):(- ) )x t = ( θ B 4 4 ) (- or,) (4,,) (c): (- ) ) x t = ( θ B 4 4 ) (-

11 ] MODELIG D FORECSTIG OF RIFLL DT 4 Model Parameter Etimate Standard Error φ -..5 θ (a) t-value P-value Fit tatitic <. <. IC=4.4 RMSE.=37.53 =. (b) θ <. <. IC=4. RMSE.= 3.5 =.7 (c) θ 4 θ θ <. <.. IC=4.3 RMSE.= 3.4 =. fter we have derived model and we hould allow for additional parameter in the fitted model, and determine whether or not their etimate are tatitically ignificantly different from zero. If they are, then there i caue for concern that we have not identified the model correctly. For example, we tart with over fitting by including one more eaonal moving average parameter (which meaure the error dependency effect at lag 4 and denoted by ) to the SRIM model (b) to examine whether thi model with more parameter would adequately be fitted to the eaonally firt differenced monthly rainfall data. The incluion of thi parameter can be determined by teting it ignificance and the improvement in the meaure of goodne of fit of the model. ll ubtantial parameter in all the model in Table 4 howed tatitically ignificance except the SRIM model (c) in which we have added one more parameter. One etimated parameter in (c) ( =.6, P-value=. >.5) which i inignificant. a reult, incluion of thi parameter ( ha no viible contribution in the model (c). It mean model (a) and (b) have correctly identified.

12 4 MH GERRETSDIK D M.K.SHRM [Vol., o.& Table 4: Correlation of Parameter Etimate for the two model Model (a): SRIM (,,) (,,4) Model b:srim (,,) (4,,) Paramete r θ 4 θ Paramete 4 r θ 4 θ Θ Outlier, level hift, and variance change are common place in applied time erie. The preence of thee could eaily miled the conventional time erie analyi procedure reulting erroneou concluion. In the etimation procedure, two type of outlier (5 additive and hift outlier) were detected and adjuted in the fitted model by SS oftware. IC value have been calculated by the following formula. IC=- ln (maximum likelihood) +m (6) Where m i the number of eaonal and non-eaonal autoregreive and moving average parameter to be etimated. ow we proceed to check the adequacy of thee two model uing reidual analyi. The reidual analyi i a part of diagnotic checking and tet for white noie and normality of reidual. In thi checking the utocorrelation Function (CF) and Partial utocorrelation Function (PCF) of the reidual reulted from the fitted model hould not how any pattern (trend or eaonality pattern). nd alo for a correctly fitted model the reidual correlation coefficient hould not lie outide the two tandard error at a given ignificant level. It i clear, from Figure 5 (a and b) and Figure 6(a and b) that there i no pattern in reidual CF and PCF plot for model (a) and (b), repectively. o CF or PCF coefficient lie outide the two tandard error at 5% level of ignificance for both fitted model. The graphical analyi alo how that the reidual in the model appeared to fluctuate randomly around zero with no apparent pattern (Figure 7). The figure 5(c) exhibit the reidual hitogram (normal curve) and we find that there i no violation of the model aumption i.e. the reidual hould normally ditributed with mean zero and contant variance.

13 ] MODELIG D FORECSTIG OF RIFLL DT 43 From plot in Figure 5(d) and Figure.6(c), it i obviou that the et of autocorrelation for reidual are not ignificant and we cannot reject the hypothei that the autocorrelation of the reidual are zero. Thee reult are in agreement with the hypothei that the reidual reulted from each of the uggeted model do not how any correlation or pattern and thee are normally ditributed, we conclude that the two SRIM (,, ) (,, 4) and (,, ) (4,, ) model are found to be adequately fitted to the eaonally firt differenced monthly rainfall erie. (a) (b)

14 44 MH GERRETSDIK D M.K.SHRM [Vol., o.& (c) (d) Figure 5: (a): utocorrelation (CF) (b): Partial utocorrelation (PCF) (c): ormality ditribution Diagnotic plot (d): White noie tet p-value Plot for Reidual reulted from SRIM (,, ) (,, 4) model.

15 ] MODELIG D FORECSTIG OF RIFLL DT 45 (a) (b)

16 46 MH GERRETSDIK D M.K.SHRM [Vol., o.& (c) Figure 6: (a): utocorrelation (CF) (b): Partial utocorrelation (PCF) (c): White noie Tet P-value Plot for Reidual reulted from SRIM (,, ) (4,, ) model. Figure 7: Scatter plot of reidual from the fitted model. fter the diagnotic, there are further tet which are neceary to elect the better of the two model in relation to better forecating accuracy. Therefore,

17 ] MODELIG D FORECSTIG OF RIFLL DT 47 further tet hould be done baed on the forecating reliability of competing model that are adequately fitted. 3 Forecating the two 3. forecating ccuracy ement of the model We have elected two model after diagnotic checking. ow we proceed to compare their forecating performance uing the variou accuracy meaure. For thi purpoe we did not ue obervation from (Oct. to Dec. ) of monthly rainfall data for calculation of forecating error uing following equation. = - (7) Table 5: Reult of ccuracy for the two model Model(SRIM) ME MPE ME MSE THIEL S (,,) (4,, ) (,, ),, 4) To meaure the forecating ability of the two model, we have etimated within-ample and out-of-ample forecat. If the magnitude of the difference between the forecated and actual value i low then the model ha good forecating performance. In thi cae, the eaonal RIM (,, ) (,, 4) model ha hown better reult which i evident from the Table 5 except for the ME value. The value of Thiele U-Statitic are.7 and.3, repectively, for SRIM (,, ) (4,, ) and (,, ) (,, 4) model. Both reult indicate that the two model are reaonably better than the naïve forecating model. However, ince the value of the Thiele U-Statitic i.3 for the SRIM (,, ) (,, 4) which i le than the value.7 of SRIM (,, ) (4,, ) model which indicate that the SRIM (,, ) (,, 4) model perform better in forecating accuracy than the SRIM (,, ) (4,, ). It can be concluded that the forecating ability of the SRIM (,, ) (,, 4) model i better for the purpoe future monthly rainfall data forecating. Graphical analyi alo exhibit cloene of the forecated value with the holding out data. Figure (a and b) repreent the forecat for the validation period and future forecat of monthly rainfall data uing SRIM (,, ) (,, 4) model. It i noteworthy that the forecat in the validation period are reaonably cloe to the actual erie and captured the turning point pattern a well.

18 4 MH GERRETSDIK D M.K.SHRM [Vol., o.& We are giving below the month-wie forecat and it interval of monthly rainfall erie at Mekele tation in Tigray region by uing the elected model Table (6). 3. Forecating Monthly Rainfall value ow the final model for forecating of hitorical monthly rainfall erie of Mekele tation i a given below. The SRIM model (,, ) (,, 4) can be written a: (- )(- )x t =(- )(- () Thi equation () can alo be written a given below.. x t =x t-+ x t- -x t-4 ) + a t - a t-4 - a t- + a t-4 () fter ubtituting the etimated parameter value to Eq. () above, we obtain the following difference equation which can be ued for forecating purpoe. x t =x t- -.4 x t- -x t-4 ) + a t -.3a t-4 +.a t-.a t-4 ()

19 ] MODELIG D FORECSTIG OF RIFLL DT 4 Table 6: Forecat of the Rainfall erie from the period anuary -September. Month Forecat (5 % Lower Limit) (5 % Upper Limit) an Feb Mar pr May un ul ug Sep Oct ov Dec an Feb Mar pr May un ul ug Sep Mean Standard Deviation(S. D)

20 5 MH GERRETSDIK D M.K.SHRM [Vol., o.& monthly rainfall in mm Time ctual and Forecated monthly rainfall at Mekele (a) monthly rainfall in mm Time S E P O C T O V D E C F E B M R P R M Y U U L U G S E P O C T O V D E C F E B M R P R M Y U U L U G S E P O C T O V D E C F E B M R P R M Y U U L U G S E P Where Rainfall erie from Oct-Dec are validation period ctual and Forecated monthly rainfall (b)

21 ] MODELIG D FORECSTIG OF RIFLL DT 5 Figure : (a) Plot of the model etimation from (an75- Sep) period and (an-sep); (b) Plot of the model validation period (Oct- Dec) and forecated monthly rainfall erie for the period from (an- Sep). 4 Concluion In thi paper the ue of Univariate Box- enkin methodology ha been hown in hitorical rainfall data. The etimation and diagnotic analyi reult revealed that model are adequately fitted to the hitorical data. In particular, the reidual analyi, which i important for diagnotic checking confirmed that there i no violation of aumption in relation to model adequacy. Further comparion baed on the forecating accuracy of the model i performed with the hold-out ome rainfall value. The point forecat reult howed a very cloer match with the pattern of the actual data and better forecating accuracy in validation period. cknowledgement We wih to thank the editor and referee both for their comment and uggetion. Reference Box,G. E. P.and enkin, G.M., 76. Time Serie nalyi: Forecating and Control; Holden day Brockwell, P.. and Davi, R.., 6. Introduction to Time Serie and Forecating, nd. ed., Springer, ew York. Caldwell.G., 6. The Box-enkin Forecating Technique Poted at Internet webite Chatfield, C. 6: The nalyi of Time Serie, 5th ed., Chapman & Hall, London. Montgomery, D.C. and ohnon, L., 76. Forecating and Time Serie nalyi. ew York, McGraw Hall. Harvey,.3. Time Serie Model; Harveter Wheatear, London. Hipel, K. W., McLeod,.I and Lennox W.C., 77. dvance in box and enkin modeling; Water Reource Reearch, Vol.3, o. 3, pp.

22 5 MH GERRETSDIK D M.K.SHRM [Vol., o.& Kava, M., and. Dulleur, 75. The Stochatic and Chronological Structure of Rainfall equence pplication to India; Water reource Reearch Center o. 57, Perdue Univerity Makridaki, S., Wheelwright, S.C. and Hyndman, R..,. Forecating method and pplication; ew York: ohn Wiley & Son. Mill, T.. The Economic Modeling of Financial Time erie; Cambridge Univerity Pre, Cambridge. aill, P.E and Momani M.,. Time Serie nalyi Model for Rainfall Data in ordan: Cae Study for Uing Time Serie nalyi; merican ournal of Environmental, Vol.5, 5-6 pp; eddah, Kingdom of Saudi rabia. Patricia, E. G., 4: Introduction to Modeling and Forecating in Buine and Economic; McGraw-Hill,Inc.,ew York. Pankratz,., 3. Forecating with Univariate Box-enkin: Concept and Cae; ohn Wiley & Son, Inc. ew York.

23 ] MODELIG D FORECSTIG OF RIFLL DT 53 PPEDIX(). Monthly rainfall data at Mekele tation Year an Feb Mar pr May un ul ug Sep Oct ov Dec Total Source: ational meteorological gency, ddi baba, Ethiopia. uthor for correpondence M.K.Sharma ddi baba Univerity, ddi baba, Ethiopia mk_ubah@yahoo.co.in