Alcaon of ARIMA Model for Rver Dscharges Analyss Bhola NS Ghmre Journal of Neal Physcal Socey Volume 4, Issue 1, February 17 ISSN: 39-473X Edors: Dr. Go Chandra Kahle Dr. Devendra Adhkar Mr. Deeendra Parajul JNPS, 4 (1), 7-3 (17) Publshed by: Neal Physcal Socey P.O. Box : 934 Tr-Chandra Camus Kahmandu, Neal Emal: nsedor@gmal.com
JNPS 4 (1), 7-3 (17) ISSN: 39-473X Neal Physcal Socey Research Arcle Alcaon of ARIMA Model for Rver Dscharges Analyss Bhola NS Ghmre Dearmen of Cvl Engneerng, Pulchowk Camus, Insue of Engneerng, T. U., Lalur, Neal Corresondng Emal: bholag@oe.edu.n ABSTRACT Tme seres daa ofen arse when monorng hydrologcal rocesses. Mos of he hydrologcal daa are me relaed and drecly or ndrecly her analyss relaed wh me comonen. Tme seres analyss accouns for he fac ha daa ons aken over me may have an nernal srucure (such as auocorrelaon, rend or seasonal varaon) ha should be accouned for. Many mehods and aroaches for formulang me seres forecasng models are avalable n leraure. Ths sudy wll gve a bref overvew of auo-regressve negraed movng average (ARIMA) rocess and s alcaon o forecas he rver dscharges for a rver. The develoed ARIMA model s esed successfully for wo hydrologcal saons for a rver n US. Keywords: Tme Seres Analyss, ARIMA Model, Hydrologcal Process, Auocorrelaon, Seasonal Varaon. INTRODUCTION Auo-Regressve Inegraed Movng Average (ARIMA) mehod s wdely used n feld of me seres modelng and analyss. These models were descrbed by Box and Jenkns (1976) and furher dscussed by Waler (Chafeld, 1996). The Box- Jenkns aroach n hydrologcal modelng s used by several researchers. Chew e al. (1993) conduced a comarson of sx ranfall-runoff modelng aroaches o smulae daly, monhly and annual flows n egh unregulaed cachmens. Langu (1993) used me seres analyss o deec he changes n ranfall and runoff aerns. Kuo and Sun (1993) used he me seres model for en days sream flow forecas and generae synhess hydrograh caused by yhoons n Tanshu Rver n Tawan. Nall and Moman (9) used he me seres analyss for ranfall daa n Jordan. Ths aer s amed o show he usefulness of hs oular echnue ARIMA for a ycal case sudy. ARIMA MODEL Trend and redcon of me seres can be comued by usng ARIMA model. ARIMA (,d,) model s a comlex lnear model. In sascs, normally n me seres analyss, ARIMA model s generalzaon of auoregressve movng average (ARMA) models, somemes called Box-Jenkns models afer he erave Box-Jenkns mehodology. Gven a me seres of daa X, he ARMA model s a ool for undersandng and, erhas, redcng fuure values n hs seres. The model consss of wo ars, an auoregressve (AR) ar and a movng average (MA) ar. And ha of n ARIMA model he hrd ar negraed (I) ncluded. The model s usually hen referred o as he ARIMA (,d,) model where s he order of he auoregressve ar, d s he order of non seasonal dfferences and s he order of he movng average ar. The noaon AR() refers o he auoregressve model of order, whch can be wren as: (1) X c X 1 where φ 1,.., φ are he arameers of he model, c s a consan and ε s whe nose. The consan erm s omed by many auhors for smlcy. Smlarly, he noaon MA () refers o he movng average model of order. Ths can be wren as: X 1 () Where he θ 1,..,θ are he arameers of he model, μ s he execaon of X (ofen assumed eual o ), and he ε, ε -1,, ε - are he whe nose error erms. The movng average model s a fne mulse resonse fler wh some addonal nerreaon laced on. Whle combng hese wo models, he ARMA (,) s obaned. 7
Alcaon of ARIMA Model for Rver Dscharges Analyss X c X 1 1 (3) The error erm ε are generally assumed o be ndeenden dencally-dsrbued random varables samled from a normal dsrbuon wh zero mean: ε ~ N(,σ ) where, σ s he varance. Some researchers have used he euaon n a lag oeraor form. In lag oeraor form, AR () model s gven by- 1 L X X (4) 1 And MA() model s gven by- X 1 L (5) 1 Where, φ and θ are defned by he arameers conanng nsde he arenhess of each model. Combnng hese models we can manulae and wre n he followng form. 1 L X 1 L (6) 1 1 Assume ha he olynomal of frs erm of above euaon has a unary roo of mullcy d. Then hs euaon can be udaed ncludng he dfference erm, whch can be exressed as, 1 d d L ) 1 L (1 L) (7) 1 1 An ARIMA (, d, ) rocess exresses hs olynomal facorzaon roery, and s fnally wren as: d 1 L ) (1 L) 1 L (8) 1 1 More recsely, he ARIMA (, d, ) model can be wren as: d ( B)(1 B) y ( B) (9) The model whle used wh seasonal flucuaon, wh seasonal lengh s, he rocess s called SARIMA (, d, )(P, D, Q) s, where, d, reresens he order of rocess AR, order of dfference (I) and order of rocess (MA) for non seasonal ar and P, D, Q, reresens he order of seasonal rocess AR, order of seasonal dfference and order of seasonal MA and s s he lengh of seasonal erod. The general euaon of SARIMA model s: d s s D s ( B)(1 B) P( B )(1 B ) y ( B) Q( B ) (1) Where, φ (B) s auo regressve oeraor, θ(b) s he oeraor of movng average; ΦP(Bs) s seasonal auoregressve oeraor, ΘQ(Bs) s seasonal oeraor of movng averages, ε s whe nose. STATISTICAL TESTS FOR MODEL PERFORMANCE There are several sascal ess for model erformance. In hs sudy some of hese ess are used whch are easy o undersand and use. Coeffcen of Correlaon A very moran ar of sascs s descrbng he relaonsh beween wo (or more) varables. One of he mos fundamenal conces n research s he conce of correlaon. If wo varables are correlaed, hs means ha can use nformaon abou one varable o redc he values of he oher varable. The coeffcen of correlaon s gven by followng euaon. r ( ( x x) ( y y) (11) x x) ( y y) Where, r s correlaon coeffcen; x and y are ndeenden (observed) and deenden (redced) varables and x and y are her corresondng means. Roo Mean Suare Error The roo mean suare error (RMSE)) (also roo mean suare devaon (RMSD)) s a freuenlyused measure of he dfferences beween values redced by a model or an esmaor and he values acually observed from he hng beng modeled or esmaed. RMSE s a good measure of accuracy. These ndvdual dfferences are also called resduals, and he RMSE serves o aggregae hem no a sngle measure of redcve ower. The mahemacal form of RMSE s gven by: x y RMSE (1) N Where N s oal number of daa se and oher varables are same as earler euaon. These wo 8
Bhola NS Ghmre euaons are used for he comarson of observed and redced values. AIC and BIC crera The wo mos commonly used enalzed model selecon crera, he Akake s nformaon creron (AIC) and he Bayesan nformaon creron (BIC), are examned and comared for ARIMA model selecon. AIC - In general case, AIC k nln( SSE / n) (13) Where k s he number of arameers n he sascal model, n s he number of observaons and SSE s suare sum of error gven by - SSE n (14) 1 BIC - In general BIC s gven by- BIC k ln( n) nln( SSE / n) (15) The mnmum values of hese AIC and BIC crera gve he beer model erformance. CASE STUDY DESCRIPTION Dscharge Daa For he alcaon demonsraon of ARIMA model, he me seres daly dscharge daa wo saons n Schuylkll Rver a Berne (Saon no: 1475, La. 4 º 31'1'' and Long. 75 º 59'55'') and Phladelha (Saon no: 14745, La. 39 º 58'4'' and Long. 75 º 11'''), USA are aken. The cachmens area of Berne saon s abou 919.45 km and ha of Phladelha saon s 49.85 km. Ths nformaon was obaned from USGS webse. The daa from he erod Ocober 1, o Seember 3, 6 were aken for boh of he saons. Inal all sx years daa were aken for ARIMA model develomen and fnally usng model las one year daa (Ocober 1, 6 o Seember 3, 7) were redced for boh he saons. Some of he sascal arameers for hese ses are shown n Table 1 of he dscharge daa. The arameers μ, σ, σ/μ, C sk, C kr, X max, X mn are mean, sandard devaon, varance, skew-ness, kuross, maxmum and mnmum values resecvely. The dscharge lms of Berne saon are.13 o 97.1 m 3 /s and ha of Phladelha saon are.4 o 1484.94 m 3 /s. Table 1. The daly sascal arameers for Schuylkll Rver. Saon Basn Area (Km ) μ σ σ/μ C sk C kr X max X mn Berne 1475 Phladelha 14745 919.45 49.85.9 99. 33.9 118.55 1.53 1.195 1.18 4.54 7.59 33.7 97.1 1484.94.13.4 Develomen of ARIMA Models From he me seres lo for he gven daa (fgure 1), can be observed ha here s no seasonaly for daly daa. In fac, s very dffcul o fx he seasonaly for daly daa and due o he large san of me (365 days), s unrelable oo. So, he model formulaon has done whou seasonaly. The ARIMA models for he boh saons are develoed by usng SPSS. Inally, he several models were esed based on he AIC and BIC creron. The AIC and BIC values for few models for hese are gven n he followng Table. Table. The AIC and BIC values for some esng ARIMA models for Berne daa and Phladelha daa. ARIMA - Berne Daa (1,,) (1,1,) (,,) (1,,1) (1,1,1) (1,1,) (1,,1) (1,,) (,,) AIC 345 377 345 344 3413 3411 3798 381 3939 BIC 3437 3784 344 344 3431 3438 385 3813 3968 ARIMA Phladelha Daa (,1,) (,,) (1,,) (1,1,) (1,1,1) (,11) (,,1) AIC 9897 31744 9568 9896 9544 95 977 BIC 993 3175 958 998 9561 955 9796 9
Alcaon of ARIMA Model for Rver Dscharges Analyss From Table, can be judged ha on he rncle of AIC and BIC es, ARIMA (1, 1, ) s suable for Berne saon and ARIMA (1, 1, ) s suable for Phladelha saon. However whle we observed he correlaon marx of he ARIMA arameers for Berne Saon from Table 3, he arameers MA(1) and MA() has very hgh correlaon aroachng o uny. So ha her effecs n ARIMA model are neglgble and we can reduce hs MA arameer. Then he ARIMA (1, 1, 1) s roosed for he furher analyss even hough ARIMA (1, 1, ) has farly less AIC and BIC values. The correlaon marx and arameers for fnal model for he Berne saon are gven n he followng Table 4. Table 3. The correlaon marx for arameers of Berne daa. ARIMA (1,1,) Non-Seasonal φ 1 θ 1 θ Consan Non-Seasonal φ 1 1..737 -.716 θ 1.737 1. -.993 θ -.716 -.993 1. Consan 1. Table 4. The correlaon marx and fnal arameers for Berne Saon. a. The correlaon marx b. The fnal arameers ARIMA (1,1,1) Non-Seasonal Non-Seasonal φ 1 θ 1 φ 1 1..3 θ 1.3 1. Consan 1. Consan ARIMA (1, 1, 1) Esmaes Sd Error Non-Seasonal φ 1.698.15 θ 1.991.3 Consan -.1.14 Smlarly, for Phladelha saon, AIC and BIC values for some models, correlaon marx for seleced model and fnal ARIMA model arameers are gven n Table 5. Table 5. The correlaon marx and fnal arameers for Phladelha daa. a. The correlaon marx b. The fnal arameers ARIMA (,1,1) Non- Seasonal Non-Seasonal φ 1 φ θ 1 φ 1 1. -.598.53 φ -.598 1..14 θ 1.53.14 1. Consan 1. Consan ARIMA (,1,1) Esmaes Sd Error Non-Seasonal φ 1.787. φ -.141. θ 1.968.6 Consan -.9.14 As dscussed n earler, for ARIMA (1, 1, 1), he general euaon can be reduced as: ( B)(1 B) y ( B) (16) 1 1 1 Whle subsung he model arameers n he above euaon and smlfy, we ge he fnal model for Berne saon as: y 1.698 y.698 y.991e.1 (17) 1 1 Smlarly, for ARIMA (, 1, 1) model, he general euaon s gven by: ( B)(1 B) y ( B) (18) 1 1 3
Predced Dscharge (Mm 3 ) Predced Dscharge (Mm 3 ) Dscharge (Mm 3 ) Dscharge (Mm 3 ) Bhola NS Ghmre Based on he model arameers, he fnal model for Phladelha saon s gven by: y 1.787 y.98 y.141 y.968 e.9 (19) 1 3 1 RESULT AND DISCUSSIONS The me seres lo of orgnal daa and redced daa from he fnal models for boh he saons are gven n fgure 1. Ths shows ha he general rend s followed by redced daa o ha of observed daa. In he lo, he sx years observed daa followed by las one year redced daa for he boh saons. 1 5 Observed Predced Berne 15 1 5 Observed Predced Phladelha 5 1 15 5 Tme n Days 5 1 15 5 Tme n Days Fg. 1. Tme seres lo of observed and redced daly dscharge a Berne and Phladelha Saons. The scaer los of hese wo saons are also gven n fgure. Ths shows ha hey are ue good models. The R for Berne saon and Phladelha saons are:.991 and.9688 resecvely. 7 Berne 1 Phladelha 6 R =.99 1 R =.97 5 8 4 6 3 4 1 4 6 8 1 Observed Dscharge (Mm 3 ) 5 1 15 Observed Dscharge (Mm 3 ) Fg.. Scaer lo of observed and redced daly dscharge a Berne and Phladelha Saons. CONCLUSIONS Tme seres analyss for he rver dscharge shows ha s an moran ool for modelng and forecasng. ARIMA (1, 1, 1) model s fed for Berne saon and ARIMA (, 1, 1) s fed for Phladelha saon. Boh he saons are les n he same rvers bu hey have dfferen cachmens coverage. So should be noed ha even he rver s same, deendng uon he cachmens characerscs, alcable models are dfferen 31
Alcaon of ARIMA Model for Rver Dscharges Analyss ndvdual ses. The coeffcen of deermnaons (.99 for Berne and.969 for Phladelha) shows ha he model s useful for runoff forecasng. REFERENCES: Box, G. E. P., and Jenkns, G. M. (1976). Tme seres analyss: forecasng and conrol. Revsed Edn. Holden-Day, San Francsco.... Chafeld, C. (1996). Analyss of me seres: an nroducon. 5 h ed. Chaman and hall, Boca Raon. Chew, F. H. S.; Sewardson, M. J., and McMahon, T. A. (1993). Comarson of sx ranfallrunoff modellng aroaches. J. of Hydrology, 147(1-4): 1-36. Langu, E. M. (1993). Deecon of changes n ranfall and runoff aerns. J. of Hydrology, 147 (1-4): 153-167. Kuo, J. T., and Sun, Y. H. (1993). An nervenon model for average 1 day sream flow forecas and synhess. J. of Hydrology, 151(1): 35-56. Nal, P. E., and Moman, M. (9). Tme seres analyss model for ranfall daa n Jrdan: case sudy for usng Tme seres analyss. Amercan J. of Envronmenal Scences, 5(5): 599-64. 3