A Comparison of the Efficiency of Parameter Estimation Methods in the Context of Streamflow Forecasting

Transcription

1 J. Agr. Sc. Tech. (00) Vol. : A Comparson of he Effcency of Parameer Esmaon Mehods n he Conex of Sreamflow Forecasng Downloaded from jas.modares.ac.r a 8:0 IRST on Sunday Ocober s 08 L. Parvz, M. Kholgh * and A. Hoorfar ABSTRACT The forecasng of hydrologcal varables, such as sreamflow, plays an mporan role n waer resource plannng and managemen. Recenly, he developmen of sochasc models s regarded as a major sep for hs purpose. Sreamflow forecasng usng he ARIMA model can be conduced when unknown parameers are esmaed correcly because parameer esmaon s one of he crucal seps n modelng process. The man objecve of hs research s o explore he performance of parameer esmaon mehods n he ARIMA model. In hs sudy, four parameer esmaon mehods have been used: () auocorrelaon funcon based on model parameers; () condonal lkelhood; () uncondonal lkelhood; and (v) genec algorhm. Sreamflow daa of Ouromeh Rver basn suaed n orhwes Iran has been seleced as a case sudy for hs research. The resuls of hese four parameer esmaon mehods have been compared usng RMSE, RME, SE, MAE and mnmzng he sum squares of error. Ths research ndcaes ha he genec algorhm and uncondonal lkelhood mehods are, respecvely, more approprae n comparson wh oher mehods bu, due o he complexy of he model, genec algorhm has hgh convergence o a global opmum. Keywords: ARIMA model, Condonal lkelhood, Forecasng, Genec algorhm, Parameer esmaon. ITRODUCTIO Effecve plannng, managemen, and conrol of waer resources sysems requre consderable daa on numerous hydrologcal varables such as sreamflow, ranfall and emperaure. Invarably, he daa ses are recorded n me and are referred o as me seres. These seres are analyzed usng sascal mehods o evaluae he parameer of neres so as o arrve a a suable decson suppor sysem for managemen and conrol purposes [8]. Among several me seres models, he ARIMA (Auoregressve Inegraed Movng Average) model has been aracve o researchers for s power n sreamflow forecasng. Generally, n he sochasc modelng process he objecve s o develop a smple model wh he parsmony rule of he sochasc model. In order o acheve hs objecve, he model parameer esmaon ha s one of he modelng processes plays he man role for bes fng wh observed daa. Ths s because ncorrec parameer esmaon mehods lead o bas and unaccepable forecasng. Carlson e al. (970) were he frs researchers o analyses he me seres of annual sreamflow usng ARIMA []. The bass and modelng procedure of classc ARIMA models are descrbed by Box and Jenkns (976), Davs and Brockwell (978). Delleur e al. (976) and Mcleod e al. (977) have used PARIMA (Perodc Auoregressve Inegraed Movng Average) Deparmen of Irrgaon and Reclamaon Engneerng, College of Sol and Waer Engneerng, Campus of Agrculure and aural Resources, Unversy of Tehran, Karaj, Islamc Republc of Iran. * Correspondng auhor, e-mal: kholgh@u.ac.r 47

2 Parvz e al. Downloaded from jas.modares.ac.r a 8:0 IRST on Sunday Ocober s 08 n he modelng of a sreamflow managemen basn. An mporan exenson of ARIMA models was nroduced by Granger and Joyeux (980), and by Hoskng (98) who proposed he FARIMA (Fraconal Auoregressve Inegraed Movng Average) model [3, 0]. Sharman and Breakenbrdge (994) revewed he form of he lkelhood funcon for ARMA sgnal models and hen hey descrbed how a genec algorhm may be employed o search he lkelhood space wh he am of fndng he maxmum pon. The use of parallel processng echnques o speed up he search procedures has been examned n hs research []. Anderson e al. (999) provded a parameer esmaon echnque ha consders wo ypes of perodc me seres model, hose wh a fne fourh momen and models wh fne varance bu an nfne fourh momen. The resuls regardng he nfne fourh momen case are of parcular neres [5]. Shn and Lee (999) have esablshed he conssency of he maxmum lkelhood esmaors for he ARIMA model wh me rends. General unform approxmaons are esablshed for he quadrac forms whch appear n he Gaussan lkelhood [7]. Lu and Chon (000) nroduced a new mehod for ARIMA parameer esmaon. Theer algorhm was based on he GMDH (Group Mehod of Daa Handlng), frs nroduced by Ivakhnenko (966 and 97). Compuer smulaons show ha n cases wh nose conamnaon and ncorrec model order assumpons, he GMDH usually performs beer han eher he FOS or he leas-squares mehods n provdng only he parameers ha are assocaed wh he rue model erms [6]. Valenzuela e al. (003) have obaned an exper sysem based on paradgms of arfcal nellgence, such as genec algorhm, so ha model can be denfed auomacally []. Wurz e al. (003) have used GARCH/APARCH errors for parameer esmaon of ARIMA models and opmzaon (maxmzaon) of he consraned log-lkelhood funcon wh he help of a SQP solver [3]. Chorng Shyong e al. (004) have provded a genec algorhms based model denfcaon o overcome he problem of local opma whch was suable for any ARIMA model. The resuls show ha he GA-based model denfcaon mehod can presen beer soluons, and s suable for any ARIMA models [4]. Jonsr e al. (006) used a parameer esmaon mehod for sochasc ranfall-runoff model. The parameer esmaon mehod was a maxmum lkelhood mehod where he maxmum lkelhood funcon s evaluaed usng he Kalman fler echnque. The maxmum lkelhood mehod esmaed he parameers n a predcon error seng; hey also esmaed he parameers by an oupu error mehod. The model performs well and parameer esmaon mehods are promsng for fuure model developmen [6]. The pon of maxmum lkelhood mehod n a falure doman yelds he hghes value of he probably densy funcon n he falure doman. Obadage and Harnporncha (006) have proposed a genec algorhm wh an adopve penaly scheme as a ool for he deermnaon of he maxmum lkelhood pon. The genec algorhm can be used as a ool for ncreasng he compuaonal effcency n he elemen and sysem relably analyss [7]. The man objecve of hs research s he comparson of four parameer esmaon mehods of he ARIMA model. Bascally, he maxmum lkelhood funcon mehods are used n hs regard. In hs sudy, hs classc parameer esmaon mehod s compared wh a new opmzaon mehod lke he genec algorhm (GA). The sreamflow me seres have been used because sreamflow forecasng has always been a challengng ask for waer resource engneers and managers and a major componen of waer resource plannng and managemen. MATERIALS AD METHODS ARIMA s he mehod frs nroduced by Box Jenkns o analyze saonary me 48

3 Parameer Esmaon n Sreamflow Downloaded from jas.modares.ac.r a 8:0 IRST on Sunday Ocober s 08 seres daa, and has snce been used n varous felds [4]. The generalzed form of ARIMA can be descrbed as: s s D d s Φ B φ B B B Z = Θ B θ B ( B ( ) ( )( ) ( ) ( ) ( ) n Z = Z ( n s Ps ( B) = Φ B Φ PB p ( B) = φb φ pb s Qs ( B) = Θ B ΘQB q ( B) θ B θ Φ φ Θ θ (3 (4 (5 = qb (6 where Z s a dscree me observaon process, s random seres wh mean zero and varance σ, B denoes he backward shf operaor, d and D denoes he nonseasonal and seasonal order of dfferences aken respecvely (ARIMA models can be fed o saonary hydrologcal seres. For he ransformaon of non saonary no saonary seres, he nonsaonary was removed by alernave mehods such as dfferencng of heorgnal seres). Φ (B), θ (B), Φ (B) and Θ (B) are polynomals n B and B s of fne order p and q, P and Q, respecvely, and usually abbrevaed as SARIMA (p, d, q)(p, D, Q) s. When here s no seasonal effec, a SARIMA (Seasonal Auoregressve Inegraed Movng Average) model reduces o pure ARIMA (p, d, q), and when he me seres daa se s saonary a pure ARIMA reduces o ARMA (p, q) [4]. The populary of he ARIMA model s due o s flexbly, and he ncluson of boh auoregressve and movng average erms. The ARIMA approach has several advanages over ohers such as a movng average, exponenal smoohng and, n parcular, s forecasng capably and s rcher nformaon on me relaed changes. I can also handle seral correlaon among observaons, whch s found n mos me seres. I also provdes sysemac searchng n each sage (of 3 sages) for an approprae model. Oher aspecs of hs model are complexy and requrng a grea deal of experence [5]. The dfferen mehods are for parameer esmaon of he ARIMA model some of whch are descrbed brefly below, namely he auocorrelaon funcon formula based on model parameers, condonal lkelhood, uncondonal lkelhood and genec algorhm (GA). ρ 0 = Auocorrelaon Funcon Formula Based on Model Parameers In hs mehod usng auocovarance and auocorrelaon funcon (ACF) afer he denfcaon of he model, some equaons can be acheved. Usng hese equaons, he parameers of model can be found. For example, usng Equaon (7-9) shows he relaonshp beween ACF (known) and he parameers (unknown) of he ARIMA (,) model and solvng hem, he parameers of ( φ θ )( φθ ) ρ = + θ φ θ (7 ρk = φρ k k (9 he model are esmaed [4]. Maxmum Lkelhood Esmaon Mehod (MLE) Maxmum lkelhood esmaon begns wh wrng a mahemacal expresson known as a lkelhood funcon of he sample daa. Loosely speakng, he lkelhood of a se of daa s he probably of obanng he parcular se of daa he chosen probably dsrbuon model (he lkelhood prncple expresses he noon ha he whole nformaon ha daa have abou parameers has been hdden n he lkelhood funcon). The dea behnd maxmum parameer esmaon s o deermne he parameers (unknown) ha maxmze he probably (lkelhood) of sample daa. From a saed (8 49

4 Parvz e al. Downloaded from jas.modares.ac.r a 8:0 IRST on Sunday Ocober s 08 pon of vew, he mehod of maxmum parameer esmaon s consdered o be more robus and o yeld esmaes wh good srand properes. If he number of random samples from a socey s aken no accoun, he value of mulplyng he probably densy funcon for random quanes x, x..., x s nroduced as he lkelhood funcon for hese quanes. ( ) σ ( e f ) = L = = ( L = = π f ( x πσ e πσ πσ ), α) σ e... = σ = σ πσ e πσ e σ LL = Ln( πσ ) σ = (0 Where α s unknown parameers, L s he lkelhood funcon and f(x,α) he dsrbuon densy funcon. The normal dsrbuon s he sandard dsrbuon n hs regard. Ths mehod s based on maxmzng he lkelhood funcon based on he selecve parameers. For ease of calculaon, maxmzaon s conduced on a funcon logarhm, because he maxmzng funcon logarhm s equal o he maxmzng lkelhood funcon. Ths mehod can be used from he error frequency funcon whch follows he normal dsrbuon wh mean zero and varance σ. The errors probably densy funcon s obaned from Equaon () and, afer applyng he lkelhood funcon [Equaon ()] and hen logarhm operaon, wll ( ( (3 resul n relaonshp 3. The frs erm of Equaon (3) s consan and so maxmzaon of log-lkelhood led o he reducon of he sum squares of error. For mnmzng he sum squares of error, he hree mehods condonal lkelhood, uncondonal lkelhood and genec algorhm were nroduced n he nex secon. ~ ~ ~ = W φw... φ θ... θ and E W Condonal Lkelhood In hs mehod for calculang he sum squares of error, can be derved from he ARIMA model lke Equaon (4) (afer ransformaon of nonsaonary seres no saonary ones). pw p ~ µ (4 d [ ] =, W = W µ, W = Z q q Usng equaon 4 n he explc form s hard o calculae. One of he soluons s he deermnaon p number of W and q number of. In hs regard, he calculaon of,,..., n condonal on nal values subsequenly s necessary. For each parameer group ha mnmzed he sum squares of errors, hose parameer groups are chosen as selecve parameers. In compung nal values, uncondonal excepon of W and can be used. Uncondonal excepon of errors s zero; where he uncondonal excepon of W s zero, hen he nal values of W are equal o zero, oherwse he mean of seres are used nsead of each of he componens W. Uncondonal Lkelhood or Calculang of on Condonal Sum Squares Ths mehod used precse nal values of W and error. In hs regard, wo backward [Equaon (5)] and forward [Equaon (6)] equaons have been used, respecvely. 50

5 Parameer Esmaon n Sreamflow Downloaded from jas.modares.ac.r a 8:0 IRST on Sunday Ocober s 08 ~ φ ( B) W = θ ( B) (5 B ~ φ ( F ) W = θ ( F ) (6 F For calculaon of he prevous me seres, he backward equaon has been used and for he saonary characerscs of he AR model, W esmaons for lm values of W n = -Q (Q s he me whch W s abou zero) s zero. Then, he forward equaon s used for esmaons on he bass of precse me seres. For each parameer group ha has lower sum squares, ha parameers group s seleced. I should be noed ha usng backward and forward equaons s possble n respec o her expecaons. For he reverson calculaon, Equaons (7 and 8) have been used []: φ, θ, W = j = 0,,, (7 [ j ] 0 [, φ, W ] = 0 e j θ j>q-, (8 Genec Algorhm (GA) The mos popular echnque n evoluonary compuaon research has been he genec algorhm ha was nroduced by Hollend (975). Evoluonary compuaon echnques absrac hese evoluonary prncples no algorhms ha may be used o search for he opmal soluon o a problem. In a ypcal evoluonary algorhm, a genec represenaon scheme s chosen by he researcher o defne he se of soluon ha forms he search space for he algorhm. Any ndvdual soluon n he space has a specfc represenaon. A number of ndvdual soluons are creaed o form an nal populaon. The followng seps are hen repeaed eravely unl a soluon has been found whch sasfes a pre-deermned ermnaon creron (achevng a soppng creron such as he me lm, he number of generaon). Each ndvdual s evaluaed usng a fness funcon ha s specfed o he problem beng solved. Based upon her fness values, a number of ndvduals are chosen o be parens (selecon). ew ndvduals or offsprng are produced from hose parens usng crossover operaors. The crossover operaors ac upon he nformaon avalable n he represenaons of he parens o produc new ndvdual conssen wh he represenaon scheme. These new ndvduals may be radcally dfferen from, slghly dfferen from, or even he same as her parens. A crossover operaon s conduced probably and wh regard o crossover probably. The fness values of he offsprng are deermned. To preven opmzed resuls convergng wh local opmums, a muaon operaor has been used. Some of he chromosome (ndvdual) genes afer he crossover process have been randomly alered; muaon apples o genes, whch form he chromosomes. In bnary genec algorhms, he gene whch s seleced for muaon s changed from o 0 and vce versa. The funcon of he wo las operaors whch mae bologcal processes sar producng he second generaon. Fnally, survvors are seleced from he old populaon and he new offsprng o form he new populaon of he nex generaon. The mechansms deermnng whch and how many parens o selec, how many offsprng o creae, and whch ndvduals wll move no he nex generaon ogeher represen a selecon mehod. The key aspec dsngushng an evoluonary from a radonal search algorhm s ha s populaon-based. Raher han movng from one pon n he search space o anoher durng each phase of he search, as s done n erave mprovemen algorhm, a populaon-based search moves from one se of pons o anoher se of pons. A any gven me, he pons n he se may be sampled from dfferen areas of he search space. The operaon process of GA has been represened n Fgure [,4]. Genec algorhms are parcularly effcen n opmzaon problems, especally when he respecve objecve funcons exhb many local opma or dsconnuous dervaves. In hs research 5

6 Parvz e al. Sar Downloaded from jas.modares.ac.r a 8:0 IRST on Sunday Ocober s 08 Inpu nformaon of operaors: P mu P Generae nal populaon & codng Evaluae objecve funcon Soppng crera? cross Selecon Operaon for new generaon Crossover& Muaon Operaon Yes Fgure. Represenaon of genec algorhm (GA) A process [] Selecon of he beer chromosome End Gene (φ) Chromosome Gene (θ) Fgure. Elemen of each chromosome. 5

7 Parameer Esmaon n Sreamflow Downloaded from jas.modares.ac.r a 8:0 IRST on Sunday Ocober s 08 decson varables are he parameer of ARIMA model (φ, θ) and objecve funcon s he bass of mnmzng he sum squares of error. The crera for geng opmum resuls are he number of generaons. Sudy Waershed Ouromeh Lake waershed s one of he sxh major basns n Iran. I s locaed n he orh Wes of Iran and covers an area of 5,866 Km. The coordnaes of basn are beween 35, 39 and 38, 30 44, 33 and 47, 53 E. The annual mean precpaon s varable from 03 o 688 mm and annual evaporaon s,499 mm. The annual mean emperaure of Ouromeh saon (wh,33 m hegh) s In recen years accordng o a decrease n precpaon, drough hreaens he Ouromeh Lake waershed. The mos mporan problem of hs waershed s relaed o he lack of observed sreamflow daa. Rver waer s mosly used for rrgaon n addon o drnkng and fshng. Exsence of mporan dams such as Shahd Madan on Ajcha Rver wh 7,700 km and ahand on ahandcha Rver, and envronmenal condons lke ncreasng he Tabrz cy Ouromeh Lake level of salny and polluon by decreasng he flow of he rvers are he reasons for selecon of hs basn for hs sudy. Saons chosen o be used n hs research are Vanyar saon on Ajcha Rver wh years sreamflow daa from 980 ll 000 and ahand dam enrance saon on ahandcha Rver wh 6 years daa from 98 ll 996. In Fgure 3, he hydromercal saon and Ouromeh Lake basn have been llusraed on a scale of :00,000. RESULTS AD DISCUSIO When any ype of sochasc model s beng developed o model a gven me seres s recommended o follow he denfcaon, esmaon and dagnosc check sages of model consrucon. Fgure 4 shows hs erave process o creae he ARIMA model, and hs algorhm wll be connued unl valdaon of model s deermned. Usng hs modelng process, he frs sep s o check he normaly of he me seres. For checkng he normaly of he Vanyar and ahan me seres, a probably plo of sreamflow was used. A probably plo Iran Fgure 3. Suaon of waershed and saons. 53

8 Parvz e al. Prelmnary analyss Downloaded from jas.modares.ac.r a 8:0 IRST on Sunday Ocober s 08 Checkng he normaly of me seres Independence of resduals o Idenfcaon of model Parameer esmaon Goodness of f Is model accepable? Invesgaon of saonary & non saonary of me seres ormaly of resduals Yes Generaon & Forecasng Fgure 4. Ierave process of auo-regressve negraed movng average (ARIMA) modellng. dsplays he percenles (95% confdence). Usng he probably plo s possble o assess wheher a parcular dsrbuon fs he me seres. In general, he closer he pons fall o he fed lne, he beer he f. Fgures 5-7 ndcaed ha he Vanyar and ahand me seres follow normal and lognormal dsrbuon. For an ARIMA (p, d, q) model, s necessary o oban he order of he model. The nex sep s o denfy he order of dfferencng (d) needed o make he seres saonary. In hs regard wo mehods can be used. () From a plo of he normalzed seres we can observe wheher here s any non saonary elemen n he level or boh n he level and slope. The frs case may ndcae he need for frs dfferencng, he second for dfferencng wce. () Based upon he nformaon gven by he coeffcens of Auocorrelaon Funcon (ACF) and Paral Auocorrelaon Funcon (PACF), he followng seps can be defned o deermne he saonary of he me 54

9 Parameer Esmaon n Sreamflow Downloaded from jas.modares.ac.r a 8:0 IRST on Sunday Ocober s 08 Fgure 5. Probably plo of ahand sreamflow. Fgure 6. Probably plo of Vanyar sreamflow. seres. (-) If he me seres has many hgh posve auocorrelaon coeffcens, hen probably needs a hgher order of dfferenaon. (-) Sarng from an orgnal me seres wh posve auocorrelaon, f afer dfferenang he frs auocorrelaon coeffcen s close o zero or negave, hen he seres does no need a hgher order of dfferenaon. (-3) The opmum order of dfferenaon s frequenly he one n whch he sandard devaon of he seres s smaller [8]. From he plong of wo normalzed me seres s obvous ha frs dfferencng s adequae for a saonary seres. The resuls Fgure 7. Probably plo of ahand sreamflow (log-normal). of he second mehod verfed hs maer and Tables and show he resuls of mehod. Afer hs sep he Correlogram mehod, he sample PACF and he sample ACF are used n approprae dfferenced seres for denfyng he orders p and q of he ARIMA (p, q) model. However hs s complcaed = AIC ( p, q) Ln( σ ) + ( p + q) (9 = SBC ( p, q) Ln( σ ) + ( p + q) Ln( ) (0 and no easly conduced (when he me seres daa ses have a mxed ARIMA effec, he plo canno provde clear lags o denfy. In addon, he lags of a mxed ARIMA model usually nvolve subjecve judgmen whch makes he resuls unsable). Varous mnmzng AIC, SBC. Snce SBC had been proved o be srongly conssen, deermnes he rue model asympocally, and s preferred o AIC for comparng dfferen models [4]. In he case of Vanar saon daa, hs es for he ARIMA (,, ) model has had he Table. Varaon of auo-correlaon funcon (ACF) wh dfference (d) of, and 3. Saon ρ(d= ) Ρ(d= ) ρ(d= 3) ahand (Dfferenced seres) Vanyar (Dfferenced seres)

10 Parvz e al. Table. Varaon of sandard devaon wh dfference (d) of, and 3. Saon d= d= d= 3 ahand Vanyar Downloaded from jas.modares.ac.r a 8:0 IRST on Sunday Ocober s 08 lowes quany among models wh varous orders ( AIC= 4.46, SBC= 4.7 ) and n he case of he ahand dam enrance saon seres, hs es has also had he lowes quany for he ARIMA(,, ) model (AIC=.4, SBC=.48 ). Afer prmary analyss of me seres and denfcaon, he nex sep s parameer esmaon. For he model parameer esmaon a compuer program n Mahemacal Laboraory (MATLAB 7) was wren wh he objecve of mnmzng he sum squares of error usng he maxmzaon lkelhood and genec algorhm mehods. Genec algorhm has ournamen for selecon funcon, unform for muaon funcon and sngle pon for crossover funcon. The opmzed paramers of he genec algorhm wh regard o mmmzaon of he objecve funcon has been goen afer several runs. These resuls are gven n Table 3. The resuls of parameer esmaon for he model usng four mehods are gven n Table 4. For he comparson among he menoned mehods of parameer esmaon, wo Table 3. Opmal values of genec algorhm (GA) parameers. Parameer value Probably of muaon.00 Probably of crossover.8 Inal populaon 0 umber of generaons 500 approaches have been used. () Forecasng me seres usng esmaed parameers and comparson of hese seres wh some crera. () Deermnaon of he parameers se whch has he lowes he sum squares of n error ( ) = For evaluang he performance of forecased values, s common o use 5% of daa for hs purpose. In hs research forecasng of me seres usng emaed parameers wh several parameer esmaon mehods was conduced by ITSM sofware. Forecased values n he Vanyar saon seres are relaed o he years from 00 up o 004 and, n he case of he ahand dam enrance saon seres, are relaed o he years from 997 up o 999. Resuls have been shown graphcally n Fgures 8 and 9. By observng he plos can be undersood ha wh a noceable decreasng or ncreasng (rend) n he forecased years, dfferences beween observed and smulaed flows wll be ncreased. Because hese models use pervous daa for parameer esmaon and such rends have no been noed n observed daa lke he flow of 997 and 998 a he ahand saon ha has a noceable ncrease n observed flows n forecased years han n he pervous me seres (Mean= 5.58). Model Performance Indcaors Table 4. The resuls of parameer esmaon usng dfferen mehods. The mehods of parameer esmaon Vanyar saon ahand saon Φ, θ Φ, θ Auocorrelaon funcon formula based on model parameers -0.33, , 0.96 Condonal lkelhood -0.4, , 0.6 Uncondonal lkelhood -0.3, , 0.8 Genec algorhm - 0.6, ,

11 Parameer Esmaon n Sreamflow Downloaded from jas.modares.ac.r a 8:0 IRST on Sunday Ocober s 08 Flow (m 3 s - ) observed Condonal lkelhood Genec algorhm Auocorrelaon funcon Uncondonal lkelhood Tme (year) Fgure 8. Forecased and observed values of dscharge (Ajcha Rver, Vanar saon) Flow(m 3 s - ) Observed Condonal lkelhood Genec algorhm Auocorrelaon funcon Uncondonal lkelhood Tme(year) Fgure 9. Forecased and observed values of dscharge (Ajcha Rver, ahand saon) To evaluae he adequacy of he model wh he proposed parameer esmaon mehods, he performance of he models should be analycally measured. Such crera for he goodness of f are obaned as: Where Qˆ s he forecased sreamflow (n hs case forecasng usng he menoned CE SE = = = = = Qˆ ( Q ) Q MAE = n RME ( Qˆ Q ) ( Q Q ) [ Q Qˆ ] = = = Q Q Qˆ ( ( (3 (4 Mean(m 3 s - ) Vanyar ahand Fgure 0. Comparson beween sascs descrpons, mean, (m 3 s - ). mehods of parameer esmaon n modelng process), Q he observed sreamflow, Q he mean observed sreamflow and he number of observed daa ems. The smalles RMSE (Roo Mean Square Error) deermnes he mehod havng he mos accurae local or small-scale esmaes. The smalles MAE (Mean Absolue Error) s ndcave of he mos accurae global esmaes [9]. An RME (Relave Mean Error) value near zero mples ha he model s provdng a good esmae of observed values. The mnmum and maxmum of hese crera are relaed o he genec algorhm and condonal lkelhood mehods. The resuls n Table 5 llusraed ha hese crera decreased from mehod one o four excep n he second mehod, and s an ndcaon of he accuracy of parameer esmaon mehods from he frs mehod o he fourh. In he second approach, he sum squares of error were calculaed for each group of parameers usng he above menoned mehods. The mnmum of he sum squares of error for example n Vanyar saon accordng o Table 6 was relaed o he genec algorhm mehod. Anoher comparson s beween sascal properes (mean and sandard devaon) beween observed and smulaed seres. Fgure 0 shows ha sascal properes lke he mean wh an ndcaon ha genec algorhm has a lower dfference beween observed and forecased sreamflow usng 57

12 Parvz e al. Downloaded from jas.modares.ac.r a 8:0 IRST on Sunday Ocober s 08 Table 5. Performance of parameer esmaon mehods. Vanyar saon ahand saon Mehod RMSE RME SE MAE RMSE RME SE MAE Auocorrelaon funcon() Condonal lkelhood() Uncondonal Lkelhood (3) Genec algorhm(4) Table 6. Sum squares of error (Vanar saon). Mnmum sum squares Mehod of error Condonal lkelhood Uncondonal lkelhood Genec algorhm he GA parameer esmaon mehod because of he precsely esmaed parameer. Maxmum lkelhood esmaon s a reasonably well-prncpled way o work ou wha compuaon s needed for learnng some knds of model from he daa. Some advanages of he maxmum lkelhood mehod over oher mehods are: has a lower varance han oher mehods; he mehod s sascally well founded; and uses all he sequences nformaon. In hs research usng he maxmum lkelhood esmaon mehod for parameer esmaon of he ARIMA model led o he leas squares n whch, for mnmzng he sum squares of error, he hree mehods of condon lkelhood, uncondonal lkelhood and genec algorhm have been used. The man advanages of back box models n hydrology are ha hey are no as daa demandng as physcal models. In prncples, he parameers n a physcally based model can be esmaed by feld measuremens, bu such an deal suaon requres comprehensve feld daa whch cover all he parameers. Because of he large number of parameers n a physcally based model, parameer esmaon canno be done by free opmzaon for all parameers. Ths research ndcaes ha genec algorhm and uncondonal lkelhood mehods are respecvely more approprae n comparson wh oher mehods. These resuls are gven usng some crera such as RMSE, SE and mnmzng he sum squares of error. Also he comparson beween sascal properes ndcaed he ncreasng accuracy of he frs parameer esmaon mehod unl he fourh excep he second mehod. The auocorrelaon funcon formula mehod based on model parameers gave accepable resuls bu, by ncreasng he orders of model, dervng equaons of parameers and solvng of hem s complcaed. In he condon lkelhood mehod, calculaon and wrng he program s smple bu for accurae esmaon of he sum squares of error, hs mehod dose no use precse prevous me seres. The uncondonal lkelhood mehod usng wo forward and backward equaons can overcome hs dffculy. In addon, maxmum lkelhood s very CPU nensve and hus exremely slow. Genec algorhm does no work wh parameers bu works wh he codng of parameers se. In hs regard, search space exponenally ncreased and has hgh confdence for achevng he global opmum. Genec algorhm by speedng up he search producer and geng he global opmum can conquer he dffcules of oher parameer esmaon mehods especally when he orders and he number parameers of he model are ncreased. I mus be menoned ha hese mehods for parameer esmaon have used observed daa and he perssence of rend (noceable decrease or ncrease) n he observed daa of forecased years can be effeced he forecased daa. One of he modelng seps s checkng he normaly of me seres. Then, usng approprae ransformaon for hs regard s necessary. One of he reasons of hgh dfference beween observed and forecased 58

13 Parameer Esmaon n Sreamflow Downloaded from jas.modares.ac.r a 8:0 IRST on Sunday Ocober s 08 flows n ahand may be he need for oher normal ransformaon. COCLUSIOS Parameer esmaon s one of he man seps of me seres modelng. Ths research nvesgaes usng a sochasc model by dfferen parameer esmaon mehods for forecasng annual sreamflow. Four parameer esmaon mehods - auocorrelaon funcon based on model parameers, condonal lkelhood, uncondonal lkelhood and a genec algorhm- have been used. Some crera such as RMSE, MRE, SE, MAE and he mnmum sum squares of error have been used for comparson of he performance of parameer esmaon mehods. Ths research ndcaes ha genec algorhm and uncondonal lkelhood mehods are respecvely more approprae n comparson wh oher mehods bu, due o he complexy of he model, genec algorhm (GA) has a hgh convergence speed o global opmum. I mus be menoned ha, for geng precse forecased values, approprae normal ransformaon usng parameers esmaon mehod has an mporan affec. Therefore, we can propose hs mehod for ARIMA sreamflow modelng. REFERECES. Box, E. P. and Jenkns, G. M Tme Seres Analyss: Forecasng and Conrol. Prence-Hall, Englewood Clffs, J.. Sharman, K. C. and Breakenbrdge, E Esmaon of Sgnal Parameers Usng he Maxmum Lkelhood Mehod. Mahemacal Aspecs of Dgal Sgnal Processng, IEE Colloquum, 8: Taqqu, S Fraconally Dfferenced ARIMA Models Appled o Hydrologc Tme Seres: Idenfcaon, Esmaon and Smulaon. Waer Resour. Res., 33(5): Salas, J. D., Delleur, J. W., Yevjevch, V. and Lane, W. L Appled Modelng of Hydrologyc Tme Seres. Waer Recourse Publcaon, PP Anderson, P. L., Meerschaer, M. M. and Veccha, A. V Innovaons Algorhm for Perodcally Saonary Tme Seres. Sochasc Processes and Ther Applcaons, 83: Lu, S. and Chon, K. H A ew Algorhm for ARIMA Model Parameer Esmaon Usng Groupmehod of Daa Handlng. Boengn. Con., Proc. of he IEEE 6 h Annual orheas. PP Shne, D. W. and Lee, J. H Conssency of he Maxmum Lkelhood Esmaors for onsaonary ARIMA Regressons wh Tme Trends. Journal of Sascal Plannng and Inference, 87: Khall, M., Panu, U. S. and Lennox, W. C. 00. Groups and eural eworks Based Sreamflow Daa Infllng Procedures. J. Hydrol., 4: Schloeder, C. A., Zmmerman,. E. and Jacobs, M. J. 00. Comparson of Mehods for Inerpolang Sol Properes Usng Lmed Daa. Sol Sc. Soc. Amer. J., 65: Chen, B. S., Lee, B. K. and Peng, S. P. 00. Maxmum Lkelhood Parameer Esmaon of F-ARIMA Process Usng he Genec Algorhm, In: "The Frequency Doman". IEEE Trans. on processng, 50(9): Karamouz, M. and Kerachan, R Waer Qualy Plannng and Managemen. Amrkabr Unversy of Technology Puplcaon. PP Valenzuela, O., Marquez, L., Pasadas, M. and Rojas, I Auomac Idenfcaon of ARIMA Tme Seres by Exper Sysems Usng Paradgms of Arfcal Inellgence. Mongrafas del Semnaro Garca de Galdeeano, 3: Wurz, D., Chalab, Y. and Luksan, L.003. Parameer Esmaon of ARMA Models wh GARCH/APARCH Errors an R and S Plus Sofware Implemenaon. J. Sa. Sof., 5(): Shyonong, C., Huang, J. J. and Tzeng, G. H Model Idenfcaon of ARIMA Famly Usng Genec Algorhms. Appled Mahemacs and Compuaon 64(3): Kurunce, A., Yurekl, K. and Cevl, O Performance of Two Sochasc Approaches 59

14 Parvz e al. Downloaded from jas.modares.ac.r a 8:0 IRST on Sunday Ocober s 08 for Forecasng Waer Qualy and Sreamflow Daa from Yeslrmak Rver, Turkey. Envron. Model. Sof., 0: Jonsdor, H., Madsen, H. and Palsson, O. P Parameer Esmaon n Sochasc Ranfall-runoff Models. J. Hydrol., 36: Obadage, A.S. and Harnporncha, Deermnaon of Pon of Maxmum Lkelhood n Falure Doman Usng Genec Algorhms. In. J. of Pressure Vessels and Ppng, 83: Valenzuela, O., Rojas, I., Rojas, F., Pomares, H., Herrera. L. J., Guullen, A., Marquez, L. and Pasadas, M Hybrdzaon of Inellgen Technques and ARIMA Models for Tme Seres Predcon. Fuzzy Ses and Sysems, 59: ARIMA.. :. ARIMA RMSE, RME, SE, MAE. 60