Multi-Input Intervention Analysis for Evaluating of the Domestic Airline Passengers in an International Airport

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1 Science Journl of Applied Mhemics nd Sisics 7; 5(3): -6 hp:// doi:.648/.sms ISSN: (Prin); ISSN: (Online) Muli-Inpu Inervenion Anlysis for Evluing of he Domesic Airline Pssengers in n Inernionl Airpor Slm Ali Wirdin,, Rdo Yendr, Suhrono 3, Moh Dnil Hendry Gml Deprmen of Mhemics, Universiy of Riu, Peknbru, Indonesi Deprmen of Mhemics, Se Islmic Universiy of Suln Syrif Ksim, Peknbru, Indonesi 3 Deprmen of Sisics, Insiu Teknologi Sepuluh Nopember, Surby, Indonesi Emil ddress: slm.li7893@unri.c.id (S. A. Wirdin), yendr_75@yhoo.com.sg (R. Yendr), suhrono@sisik.is.c.id (Suhrono), mdhgml@unri.c.id (M. D. H. Gml) Corresponding uhor To cie his ricle: Slm Ali Wirdin, Rdo Yendr, Suhrono, Moh Dnil Hendry Gml. Muli-Inpu Inervenion Anlysis for Evluing of he Domesic Airline Pssengers in n Inernionl Airpor. Science Journl of Applied Mhemics nd Sisics. Vol. 5, No. 3, 7, pp. -6. doi:.648/.sms Received: April 5, 7; Acceped: April 8, 7; Published: June 3, 7 Absrc: This ricle discusses muli-inpu inervenion nlysis o invesige he effec of inervenions which my come from inernl nd/or exernl fcors in ime series d. The obecive of his reserch is o obin muli-inpu inervenion nlysis, which cn explin he mgniude nd periodic of ech even effeced o monhly ypes of he domesic irline pssenger fligh in Peknbru irpor. The purpose of his sudy is o give heoreicl nd empiricl sudies on he muli-inpu inervenion nlysis, priculrly o develop nd consruc model procedure of muli-inpu inervenion cused by pulse nd/or sep funcion o evlue he impc of hese exernl nd/or inernl evens in ime series d. Monhly d comprising he number of he domesic irline pssenger fligh in Peknbru irpor re used s he d for his cse sudy. Generlly, he fores fires, pelnd, nd illegl burning in Riu Province give negive permnen impcs fer four monhs. This sudy focuses on he derivion of some effec shpes, i.e. he emporry, grdully or permnen monhly irline pssenger. In ddiion, he reserch lso discusses how o ssess he effec of n inervenion in rnsformion d. Keywords: Series D, Muli-inpu Inervenion Anlysis, Pulse Funcion, Sep Funcion. Inroducion Quniive models on uoregressive inegred moving verge or beer known s ARIMA were developed in 976 by Box nd Jenkins in full s sndrd procedure in he modeling of ime series nlysis. In he developmen of ime series nlysis, he vrious phenomen of ineres nd no simply hve been ofen he linkges wih he relionship beween vribles in ime series d. Sisicl modeling of ime series nlysis in is developmen presens he ARIMA model h is populr model nd widely pplied in modeling nd forecsing ime series d. Bsed on Box- Jenkins procedure which is sndrd procedure ims o obin n pproprie ARIMA model o ime series d. This procedure consiss of four sges, nmely he idenificion, prmeer esimion nd ess of significnce, nd check he dignosis nd forecsing. A he idenificion sge, his procedure requires he fulfillmen of sionry condiions on he verge vlue of he men nd vrince of ime series d. Bsed on he fc h rel ime d on he ime series d, i is ofen encounered he problem of d h re chnging he pern of he men exreme or known by he regime chnge/srucurl chnges [3, p. 74]. These chnges re usully cused by he presence of n inervenion coming from exernl fcors nd/or inernl. The pern of exreme d chnged frequenly in incorrec resuls hs implicions for he idenificion sge obining spurious or counerfei finl model for ime series d. Disser is one form of inervenion of exernl fcors which ofen impc on he chnging pern of he d in ime series. In generl, he disser h occurred in n re

2 Science Journl of Applied Mhemics nd Sisics 7; 5(3): -6 cn be divided ino wo kinds, nmely nurl dissers (nurl-mde) nd mn-mde dissers (mn-mde). Some exmples of nurl dissers h hve occurred in Riu Province is csrophic smog from fores fires, soil (pe) nd/or becuse of he impc of he spred of smoke of fores fires neighboring provinces such s Jmbi nd Souh Sumr crried on he wind o Riu province, smoke ppers s resul of unconrolled burning of he coninued culivion of plnion nd hving n impc on severl res in Souhes Asi such s Mlysi nd Singpore, especilly in he Province of Riu. While exmples of mn-mde dissers re he bomb nd rw shoo in Jkr in ury 4, 6 in fron of Srinh Building, Jkr bombing in July 7, 9, Bli bombings in Ocober, nd in Ocober, 5. One of he imporn issues of he csrophic evens is how o mesure or consruc mhemicl model, he mgniude nd durion, he impc cused by he inervenion of vrious dissers. For exmple, how o mesure nd deermine he impc of dissers smoke hze from fores fires nd smoke polluion in Souhes Asi in 3 o he field of ir rnspor for domesic flighs vi Inernionl Airpor Suln Syrif Ksim II Peknbru, s mesured by he number of pssengers ypes of domesic flighs o Riu Province. The sme quesion lso ppers how he consrucion of mhemicl equion or model of forecsing re ffeced by nurl dissers o he number of pssengers ypes of domesic flighs o Riu Province. One sisicl mehod h cn ddress he concerns nd quesions is he nlysis of muli-inpu inervenion. In he ls few yers here is lo of reserch on he nlysis of inervenions h hve been conduced o evlue he effecs of n inernl nd/or exernl inervenion. Unil now, mos reserch on he nlysis of inervenions reles o he evluion of he impc of new policies on n pplied field. The subec of furher nlysis of he inervenion cn be found in severl books of ime series nlysis, mong ohers re Brockwell nd Dvis [3, p. 34], Box e l. [5, p. 48], Cryer nd Chn [, p. 49], Drkos nd Kun [], Luhkepohl [, p. 64], Enders [3, p. 8], Hmilon [6, p. 677], Mongomery e l. [6, p. 33], Tsy [35, p. 389], Wei [39, p. ], Yffee nd McGee [4, p. 65]. One of he min problems in he modeling nlysis of he inervenion is he bsence of sndrd procedure for he esblishmen of model of inervenion, eiher in single inervenion or muliple inervenions (more hn one ype of inervenion). In ddiion, mos of he discussion of inervenion nlysis on vriey of ime series nlysis reference books covers he nlysis of single inervenion. Vrious sudies on he inervenion model re no only limied o he nlysis of single inervenion (single-inpu), he sep funcion or pulse only, bu lso series of d. They cn conin more hn one ype of inervenion (muli-inpu), i.e. sep nd pulse funcion. So i is sill very in need o be furher developed he sndrd procedures in modeling he muli-inpu inervenion. These problems provide furher opporuniies o do reserch reled o he muli-inpu inervenion, nmely his reserch is pplied o model nd forecs he monhly d ypes of he number of pssengers for domesic flighs in Riu Province. Inervenion model ws firs proposed by Box nd Tio [6] who exmine he effec of he encmen of legislion on he engine design oxidn levels of polluion in he Los Angeles re. Then Ldoler nd Chn [4] use he inervenion nlysis in response o chnges in federl policy o sudy nd evlue he effec of rising he speed limi on he inerse highwy sysem counryside o 65 miles per hour (mph), higher speeds re generlly considered o genere economic benefis primrily becuse of he reducion in rvel ime. However, higher speeds re lso ssocied wih he incresed risk of ccidens. Mny differen models hve been proposed for muli-inpu inervenion ime series forecsing by reserchers. Rezeki e l. [34] use he muli-inpu inervenion nlysis model o nlyze nd evlue he impc of he Asin crisis nd he erroris cks gins ouris rrivls in Bli. The sudy concludes h in generl he Asin crisis nd he Bli bombings negive impc on he number of ouris rrivls in Bli. This is confirmed by he resuls of he sudy he number of ouriss o Bli when here re permnen evens effecs of he Asin crisis fer dely of monhs s well s he Bli bombing I nd II h hve direc impc nd emporry. Lee e l. [5] model using muli-inpu inervenions o evlue he effecs of he Asin crisis nd he erroris cks gins he number of ouris rrivls. The resuls of empiricl sudies on hese cse sudies show h he muliinpu inervenion model is proven o explin precisely he mgniude nd durion of he impc of dissers on ime series d. The fcs in hese sudies underlie he implemenion of furher sudy forecsing in he field of ir rnspor. The problems of inervenion nlysis of muli-inpu evlues he number of pssengers ypes of domesic flighs in Riu Province wih focus on he developmen of forecsing models of inervenion muli-inpu cpble of explining he period nd he mgniude of he impc hese evens. In his reserch we discuss he resuls of heoreicl nd empiricl sudies on he developmen of procedures for esblishing muli-inpu inervenion model used for he evluion of he impc of disser on ime series d. Theoreicl sudy is focused on he idenificion sge, nmely he decrese in he quniies of he sisics used s he bsis for deermining he order of he inervenion model. Furhermore, he resuls of heoreicl sudy re used s bsis o develop sndrd procedure for he esblishmen of muli-inpu inervenion model. A he end, he empiricl sudies re conduced o nlyze he issue, which is sill in quesion in evluing he impc of dissers on rel cse occurred in he province of Riu, nmely he impc of dissers smoke hze from fores fires in he ls five yers, Souhes Asi polluion smoke in 3, nd he nionl elecions s poliicl yer in July 4, o he number of domesic pssengers flying in Riu Province.

3 Slm Ali Wirdin e l.: Muli-Inpu Inervenion Anlysis for Evluing of he Domesic Airline Pssengers in n Inernionl Airpor. Generl Muli-inpu Inervenion Model A ime series d cn be ffeced by vriey of specil evens, nmely he exisence of n inervenion eiher inernl or exernl h cn cuse chnges in he pern of ime series d. In he nlysis of he inervenion, i is ssumed h he incidence of inervenion occurs ime is known s ime series. In he inervenion model, shock or series of inervenions clled discree inpu is vluble, while he ime series d o model referred o s he series oupu. Wih he inervenion model, i cn be seen how big nd long he effecs of inervenion evens. The process of esiming he effec of inervenion is clled he nlysis of he inervenion (inervenion or inerruped) ime series. series d in severl empiricl sudies ofen consis of he observions of he severl inervenion vribles. A generl formul is formed if here is more hn one ype of inervenion occurs in ime series d, hen he model suible for use is muli-inpu inervenion model whose mhemicl model is given s follows (see Abrhm [], Box e l. [5, p. 48], Box nd Tio [6], Ismil e l. [9], Mongomery nd Weherby [9], Wei [39, p. 5]) k b ω ( ) S s B B θ ( ) ( ) q B ΘQ B ζ + δ ( ) S d S D r B ϕ ( ) ( ) ( ) ( ) p B ΦP B B B Y () where polynomil form of inervenion model prmeers conined in he response is defined s s r ω B ω ω B, δ ( B) δ... δ B, while ( )... s s r r Y is response vrible ime, ζ is binry indicor vrible h shows he exisnce of n inervenion ime nd for he h wih respec o he inervenion vrible ime nd ζ is deerminisic indicor vrible, king only he vlues nd o indice nonoccurrence nd occurrence of some even respecively which indices wheher here is influence of inervenion ime, nd b is he dely ime from he effecs of n inervenion, o he declred he vlue of ω ( B) prmeer models s inervenion h he specified order-s, while he vlue δ r is sed prmeers inervenion model h he specified orderr, θ ( B ) nd ϕ ( B) re moving verge nd uoregressive q p polynomils in B of degrees q nd p, respecively. The roos of θ ( B ) lie ouside nd ϕ ( B ) lie in or ouside he uni q p circle. In order o idenify he model of inervenion b, r nd s cn be done by looking he residul plos. Residul vlue o be obined from he difference beween he observed d vlues uses noise forecsing models. Suppose residul denoe s follows: ( ) Y Y n f ζ. () Vlues s indice when he moion of response weigh begn o decline, he vlue of b is deermined by looking when he effec of he inervenion sred hppening while r is he pern of he residuls. Theoreicl nd complee sudy cn be seen in Lee e l. [5], Helfensein [8] nd Mkridkis e l [3]. Thre re wo common ypes of deerminisic inpu vrible ζ h hve been found useful o represen he impc of inervenion evens on ime series. Boh of hese re indicor vribles king only he vlues nd o denoe he nonoccurrence nd occurrence of inervenion. One ype is sep funcion ime T, given by [5, p. 53; 6, p. 46; 39, p. 3] s ( T ), < T ξ S, T, ypiclly used o represen he effecs of n inervenion h re expeced o remin permnen fer ime T o some exen. The oher ype is pulse funcion T, given by [5, p. 53; 6, p. 46; 39, p. 3] s ( T ), T ξ P, T, h represens he effecs of n inervenion h re emporry or rnsien nd will die ou fer ime T. These indicor inpu vribles re used in mny siuions where he effecs of he inervenion cnno be represened s he response o quniive vrible h does no exis or i is imprcicl or impossible o obin he mesuremens of such vrible... Cse Model Muli-inpu inervenion model wih sep funcion (b, r, s ) nd followed by pulse funcion wih (b, r, s ) is given s ( ω ω B) B + ( ω ω ) + δ B Y S B B P N, where θq ( B) N ϕ ( B)( B) ( B ) p d S D inervenion effecs h occur re k ( ) ( ) ( ) nd (3) (4) (5) δ < so h ω ω δ ω δ ω δ ω 3 δ ω δ ω k ω ω Y S + S + S S + P P, (6) which cn lso be wrien in Tble.

4 Science Journl of Applied Mhemics nd Sisics 7; 5(3): -6 3 Tble. Mgniude of he Effecs Inervenion. Response Effec s Mgniude ω Y ( ) k + ω δ ω δ ω ( ) k + + ω δ ω δ ω ω ( ) k + + ω δ ω δ ω ω ω wih k, 3,... T T + T +k T +k T T +3 T + T + The illusrion of Eq. (6) nd is impc re represened in Figure. For illusrions, consider muli-inpu inervenion wih wo evens, nmely sep funcion occurring T where ω 5 nd ω 5, ω 3, δ,8, ω, ω wih (b, s, r ) which is 4 followed by pulse funcion T 7 wih (b, r, s ). The firs inervenion h ffeced he d, wih mgniude of 5. The pulse funcion inervenion hd n effec h lsed for 3 periods beyond T wih mgniude effec of 35 nd 4 on he second nd hird fer he inervenion, respecively. Simulion of he Inervenion Model T T7 Inervenion Effec T T7 4-8 Y() 6 Y() () (b) Figure. () Plo of Simulion he Inervenion Model, (b) Inervenion Effec of he Muli Inpu Inervenion where he Sep Funcion (b, s, r ) Occurred T nd Followed by he Pulse Funcion (b, r, s ) T 7. Figure shows he simulion sudy used o show h he response funcion o esime he order of muli-inpu inervenion model o do excly he kind of d of he number of pssengers for domesic flighs in Peknbru irpor. In Figure () i cn be seen h he d re sble from o rising 5, 8 nd o 3 nd rising gin o 4 3 nd slowly or grdully flling bck o he vlue of. While Y shows d of simulion where δ <, so h inervenion effecs h occur re scenrios wih wo inervenion orders nd coefficiens h re defined or ssumed s in he previous descripion... Cse Model Muli-inpu inervenion model wih pulse funcion (b, s, r ) nd followed by sep funcion wih (b, r, s ) is given s ( ω ω ) B B ( ω ω ω ), +, + δ B Y B B B P S N, ( ) ( ) ω, ω, ω, 3 ω, ω δ ω, ω δ ω δ, 3... Y P P P + S + S + S + (8) (7) which cn lso be wrien in Tble. Y Tble. Mgniude of he Effecs Inervenion. Response Effec s Mgniude ω T ω ω m m i i i i ω δ ω δ wih k, 3,... T + T + T + 3 T + k T, k 4 T + m, m Figure shows he simulion considering muli-inpu inervenion wih wo evens, nmely sep funcion occurring T ssuming sep funcion (b, s, r ) he iniil vlue for ω, ω 6, δ,5, ω 4, ω 5 nd ω 4, which is followed by pulse funcion (b, r, s ) occurred T 7.

5 4 Slm Ali Wirdin e l.: Muli-Inpu Inervenion Anlysis for Evluing of he Domesic Airline Pssengers in n Inernionl Airpor Simulion of he Inervenion Model Inervenion Effec 4 T T7 T T7 3-5 Y() Y() () (b) Figure. () Plo of Simulion he Inervenion model, (b) Inervenion Effec of he Muli-Inpu Inervenion where he Sep Funcion (b, s, r ) Occurred T nd Followed by he Pulse Funcion (b, r, s ) T 7. From Figure () i is shown h he d is sble from unil incresing, 3 nd 4 sill 6 nd sble in vlue 3 hen rising gin o 38 7 nd grdully or permnen effec Prmeer Esimion ( ) ( ) ( ) d d ( ) ( ) ( ) ( ) ( ) r p s p b r q The inervenion model is defined s [33, p. 33] ( B) ( ) ( ) ( B) ωs θq Y ξ + δ B ϕ B ( B) ( B ), b d S D r p Eq.(9) cn be rewrien s δ B ϕ B B Y ω B ϕ B B ξ + δ B θ B nd/or (9) ( ) ( ) ( ), ξ b c B Y d B + e B () where ( ) δ ( ) ϕ ( ) d S D p+ r r p p+ r c B B B ( B) ( B ) c B... c B, ( ) ω ( ) ϕ ( ) d S D p+ s s p p+ s d B B B ( B) ( B ) d... d B, r+ q ( ) δ ( ) θ ( )..., e B B B e B r q r+ q nd hus, nd cn be represened s: under he ssumpion h he Y c Y... c Y d ξ... d ξ + e... e, () p+ r p r b p+ s b p s r+ q r q re (, ) ( ) d ( B) e( B) c B Y ξ b, () N σ whie noise, condiionl likelihood funcion n n L( δ, ω, ϕ, θ, σ b, ξ, Y, ξ, Y, ) ( πσ ) exp. σ (3) In generl, he esimion mehods inroduced in [39, p. 45] cn lso be used o esime he prmeers δ, ω, ϕ, θ nd σ. For exmple, by seing he unknown s equl o heir condiionl expeced vlues of zero, he nonliner les squres esime of hese prmeers is obined by minimizing ( δ ω ϕ θ ) n L,,, b, (4)

6 Science Journl of Applied Mhemics nd Sisics 7; 5(3): -6 5 mx + +, nd re he residuls under he whie noise ssumpion nd norml disribuion. where { p r b p s } 4. Reserch Mehodology The inervenion response or Y is esily formuled using he response vlues chr for deermining he order of he inervenion model using he orders b, s nd r. The inervenion response denoed s Y is bsiclly residul or error which is he difference beween he cul d nd he ARIMA model forecss bsed on he d before he inervenion. A complee procedure of he inervenion model building cn be used o evlue hese k inervenion funcions ime T, T,, T k s ccording o he following procedures. This procedure is described in Ismil e l. [9], Lee e l. [5], Mongomery e l. [6, p. 39], Yffee nd McGee [4, p. 8]. The Firs Procedure: Dividing he d se of ime series ino ( k + ) prs Pr : The firs pr is he d before he firs inervenion, s mny s n ime periods, i.e.,,, T. Denoed he d s Y. b Pr : The second pr is he d from he firs inervenion unil us before he second inervenion, s mny s n ime periods, i.e. T, T +, T +,, T. Denoed he d s Y. c Pr ( k + ) : The ( k + ) s pr is he d from he kh inervenion unil he end of d nlysis bsed on s mny s n k ime periods, i.e. Tk, Tk +, Tk +,, n. Denoe hese d s Y k. The Second Procedure: Modeling of he firs inervenion Sep : () Apply Box-Jenkins procedure o ge review ARIMA model building for ime series d before he firs inervenion Y occurs. I cn be simply wrien s follow: Y θq ( B) ΘQ ( B ) ϕ ( B) Φ ( B )( B) ( B ) p P S S d S D. (5) () Forecs Pr dse Y using he bes of ARIMA model. In his sep, he forecsed d re Y ˆ, Y ˆ,..., Y ˆ T T + T + n. b Sep : () Clcule he response vlues of he firs inervenion or Y. These re he residuls of he d for ime periods T, T +, T +,..., T, bsed on he forecsing of he ARIMA model in he firs sep. This sep produces response vlues of he firs inervenion, T T + T i.e. Y, Y,..., Y. () Deerminion of se order b, s, r from he firs inervenion by using he plo of response vlues Y, Y,..., Y nd confidence inervl of widh, T T + T i.e. 3 ˆ σ, ± where σˆ is Roo Men Squre Error (RMSE) of he previous ARIMA model. (3) This inervl is bsed on he deerminion of conrol chr bounds during sisicl quliy conrol for deecing oulier observions. c Sep 3: () Esime he prmeer nd es he significnce for he firs inervenion model. () Conduc dignosic check o exmine he residul ssumpion, i.e. whie noise nd normliy disribuion. In his sep, he firs inpu inervenion model is b S ωs ( B) B θ ( ) ( ) q B ΘQ B ξ, + S d S D δr ( B) ϕ ( ) ( ) ( ) ( ) p B ΦP B B B Y. (6) The Third Prosedure: Modeling of he mh Inervenion Model, where m, 3,..., k Sep : m + m s Forecs d ( ) Y bsed on he ( ) m inervenion model. In his sep, we obin he forecsed m s inervenion model, i.e. vlues from he ( ) Yˆ, Yˆ,..., Yˆ. T T + T + n m m m m b Sep : () Clcule he mh inervenion responses Y m, which is he residul of he d periods for Tm, Tm +,, T m +, This is bsed on he s inervenion model. These forecsing of he ( m ) response vlues re denoed s T T + T Y, Y,..., Y. m m m + b s r from he ( ) () Idenify presume orders m, m, m m h inervenion model from he plo of response vlues YT, Y,..., m Tm + Y Tm +, nd he confidence inervl of widh ± 3 ˆ σ. m c Sep 3: () Perform prmeer esimes s mesure seeking he bes or mos efficien esimes for he prmeers in he model nd es of significnce s in he comprive evluion of he mh inervenion models. () Checks dignosic o evlue nd/or exmine he suibiliy of residul ssumpion h if residul eligible whie noise nd normlly disribued. This sep

7 6 Slm Ali Wirdin e l.: Muli-Inpu Inervenion Anlysis for Evluing of he Domesic Airline Pssengers in n Inernionl Airpor cn lso be esed gins he residul is whie noise process cn be done individully or oinly. Tesing cn be done individully if known disribuion of he esimed residul, which is generlly ner norml wih men zero. While esing he model ll ogeher is by Lung-Box. The resul of his sep is Y m b ωs ( B) B ξ + N δ ( ) r B (7) This procedure is done ierively unil he ls kh inervenion. As resul of hese seps, evenully we obin he following muli-inpu inervenion model s k b ω ( ) S s B B θ ( ) ( ) q B ΘQ B ξ, + S d S D δ ( ) r B ϕ ( ) ( )( ) ( ) p B ΦP B B B Y (8) Bsed on he procedures for esblishing muli-inpu inervenion model in Secion 4, he number of pssengers crried d modeling ypes of domesic flighs in Peknbru Airpor wih he following seps. 5. Resuls nd Discussion 5.. Pre-inervenion Model Resuls This secion presens he resuls of he Box-Jenkins procedure [5, p. 93; 4, p. ; 6, p. 3] uilized for his reserch which includes he idenificion, prmeer esimion, dignosic checking nd forecsing o find he bes ARIMA model before he firs inervenion, i.e. he smog disser since June 3. The idenificion sep shows h he d re no sionry boh in vrince nd men. In his cse, homogeneous nonsionry ime series cn be reduced o sionry ime series by king proper degree of differencing [39, p. 7]. Sionry d on he men obined hrough differencing before he firs inervenion fer regulr nd sesonl differencing (d, D nd S ). Bsed on his Box-Cox rnsformion [39, p. 85], nurl log is employed o cuse he vrince d o be sionry s shown in Figure 3. Series Plo of Number of he Domesic Airline Pssengers Fligh in Peknbru Airpor,6 S S P3 S4 S5 Monhly Pssenger Tols (Ln Trnsformion),4,,,8,6,4, Monh Yer Figure 3. Nurl log (ln) Trnsformion of he Monhly Types of he Domesic Airline Pssenger in Peknbru Airpor from ury 4 - December 6. Auocorrelion Funcion for Pre-Inervenion (wih 5% significnce limis for he uocorrelions) Auocorrelion,,8,6,4,, -, -,4 -,6 -,8 -, Lg ()

8 Science Journl of Applied Mhemics nd Sisics 7; 5(3): -6 7 Pril Auocorrelion Funcion for Pre-Inervenion (wih 5% significnce limis for he pril uocorrelions) Pril Auocorrelion,,8,6,4,, -, -,4 -,6 -,8 -, Lg (b) Figure 4. () Plo of ACF nd (b) PACF of he Sionry D before he Firs Inervenion fer Regulr nd Sesonl Differencing (d, D nd S ). A relizion of his model wih is smple uocorrelion funcion is given in Figure 4. A visul inspecion revels h he men nd vrince remins sble while here re some shor runs where successive observions end o follow ech oher for very brief durions, suggesing h here is indeed some negive uocorrelion s confirmed by he smple ACF plo. The ACF plo of before he firs inervenion d show h ACF lg nd lg re significnly differen wih zero or hey re greer hn he confidence inervl of ACF. There re severl non sesonl lgs (lg,,..., 8) nd he ACF ends o be cu off fer lg wheres PACF Finl Esimes of Prmeers diminishes dies down. On he oher hnd, ACF nd PACF sesonl lgs (lg, 4,...) end o cu off fer lg. Hence, here re possible pproprie orders of his ARIMA model, i.e. (,, )(,, ) nd (,, )(,, ). Tble 3 shows he resuls of he prmeer esimion, prmeer significnce es, nd dignosic checking. From his ble, we know h boh models re pproprie s mens for forecsing he monhly ypes of he domesic irline pssenger in Peknbru irpor before he firs inervenion. Tble 3. Resuls of Prmeer Esimion, Prmeer Significnce Tes nd Dignosic Checking. Type Coef SE Coef T P MA,67,86 7,35, SMA,7583,937 8,9, Consn 83, 5,6,,3 Differencing regulr sesonl of order Number of observions Originl series 9 fer differencing 96 Residuls SS (bckforecss excluded) MS DF 93 Modified Box-Pierce (Lung-Box) Chi-Squre sisic Lg Chi-Squre 4,6 4, 7, 85, DF P-Vlue,4,4,, Compleely, he bes ARIMA of pre-inervenion model for his cse cn be wrien s ln Y (,67 B)(, 7583 B ) ( B)( B ) 5.. The Firs Inervenion Model Resuls (9) This secion presens he resuls of he inervenion model by illusring he impc of he firs sep funcion, nmely he smog disser from Februry 3 unil December 3 or he ime,,...,. Mhemiclly, he firs inervenion ype of sep funcion of deerminisic (dummy) inervenion indicor is wrien s ( ), 9 ξ S,,,,,. () The firs sep in his model is o deermine he order b, r nd s for he firs sep funcion inervenion model [39, p. 34; 4, p. 36]. In his work, Figure 5 illusres chr of he residuls o deermine he pern of orders for he firs inervenion.

9 8 Slm Ali Wirdin e l.: Muli-Inpu Inervenion Anlysis for Evluing of he Domesic Airline Pssengers in n Inernionl Airpor Chr of Response Vlues of he Number of Domesic Airline Pssengger Fligh in Peknbru Airpor 6 S Residul fer he Firs Inervenion T- T- T- T-9 T-8 T-7 T-6 T-5 T-4 T-3 T- T- T T+ T+ T+3 T+4 T+5 T+6 T+7 T+8 T+9 T+ T+ Figure 5. Response vlues of he Domesic Airline Pssenger Fligh in Peknbru Airpor. Bsed on he Figure 5, we cn see h he response vlues ime T, T +,..., T +5 hve less bsolue vlues hn he confidence inervl. This grph lso illusres h only he vlues T +6 is close o he lower confidence inervl. Thus, here is no possible se of order for he firs sep funcion inervenion model. Prmeer esimion nd significnce ess show h his model order yields significn prmeers from his firs inervenion. Tble 4 presens he Minib oupu bsed on he firs inervenion model. Tble 4. Resuls of Prmeer Esimion, Prmeer Significnce Tes nd Dignosic Checking. Finl Esimes of Prmeers Type Coef SE Coef T P MA,65,83 7,37, SMA,7964,96 8,7, Consn 45,3 34,8,8,84 Differencing regulr, sesonl of order Number of observions Originl series 9 fer differencing 96 Residuls SS (bckforecss excluded) MS DF 94 Modified Box-Pierce (Lung-Box) Chi-Squre sisic Lg Chi-Squre 4,4 4,5 69,5 87,6 DF P-Vlue,8,4,, Bsed on he resul prmeer esimion nd significnce ess show h boh ses of he model orders yield significn prmeers nd residuls h sisfy he whie noise nd norml disribuion ssumpions. An inervenion model for he number of fer he firs sep funcion inervenion nd prior o he second sep funcion cn be wrien s ( ) (,65 B)(, 7964 B ) ln Y ( B)( B ) 5.3. Resuls from he Second Inervenion Model () Afer modeling he firs inervenion bsed on he inervenion model due o he disser smog, furhermore, noher nlysis of he second sep funcion inervenion is conduced. This is bsed on Februry 4 which is equed wih T. Mhemiclly, he inervenion ype of sep funcion is wrien s ( T ) ( ), ξ S,,, 3,, 6. () As explined bove he forming of inervenion model. The firs sep in his modeling is o deermine he order b, s nd r for he second sep funcion inervenion model. This is done o deermine he order of he inervenion model nd o explin he decrese in he number ypes of he domesic irline pssenger fligh in Peknbru irpor.

10 Science Journl of Applied Mhemics nd Sisics 7; 5(3): -6 9 Chr of Response Vlues of he Number of Domesic Airline Pssenger Fligh in Peknbru Airpor S Residul fer he Second Inervenion T-3 T- T- T- T-9 T-8 T-7 T-6 T-5 T-4 T-3 T- T- T T+ T+ T+3 T+4 Figure 6. Response vlues of he Domesic Airline Pssenger Fligh in Peknbru Airpor. The oupu in Figure 6, we cn see h he response vlues ime T, T +, T +3 nd T +4 hve less bsolue vlue hn he confidence inervl. This grph lso illusres h only he response vlues ime T + (Mrch 4) is close o he confidence inervls. Thus, here is pproprie presume order of he second sep funcion inervenion model, i.e. b, s nd r. Prmeer esimion nd significnce ess show h his model order yields significn prmeers from his second inervenion. Tble 5 presens he SAS oupu bsed on he second sep funcion inervenion. Tble 5. Resuls of Prmeer Esimion, Prmeer Significnce Tes nd Dignosic Checking. Esimsi Prmeer hiung p vlue Type of significn ω 39.3,8. significn Tble 6. Resuls of Prmeer Esimion, Prmeer Significnce Tes nd Dignosic Checking. ω -548.,3.93 significn -5,76. significn Finl Esimes of Prmeers Type Coef SE Coef T P MA,746,69,3, SMA,8456,843,3, Consn 95,97 9,7,6,33 Differencing regulr, sesonl of order 9 Number of observions Originl series fer differencing 96 Residuls SS (bckforecss excluded) MS DF 6 Modified Box-Pierce (Lung-Box) Chi-Squre sisic Lg Chi-Squre 6, 4, 7,4 85,8 DF P-Vlue,6,4,, ω Bsed on he resuls lised in Tble 5, n inervenion model for he number ypes of he domesic irline pssenger fligh in Peknbru irpor fer he firs sep funcion inervenion nd prior o he hird sep funcion inervenion cn be wrien s ( ) ln Y ( 39.3S 548.S 5.76S 3 ) + N (3) where (,746 B)(,8456 B ) N ( B)( B ) Resuls from he Third Inervenion Model The finl nlysis of he hird pulse inervenion bsed on nionl elecion even. The bes of inervenion model for he nionl elecion s poliicl yer, which ook even on July 4 is equed wih T 3 7. Mhemiclly, he pulse funcion in his inervenion could be wrien s ( T ) ( 7), 7 ξ P3,, 7. (4) As described in he previous secion, he firs sep in his modeling is o deermine he order b 3, r 3 nd s 3 for he hird

11 Slm Ali Wirdin e l.: Muli-Inpu Inervenion Anlysis for Evluing of he Domesic Airline Pssengers in n Inernionl Airpor pulse funcion inervenion model. This is done o deermine he order of he inervenion model nd o explin he decrese in he number monhly ypes of he domesic irline pssenger in Peknbru irpor due o he nionl elecion nd o deermine he order of inervenion. A residul chr is s in Figure 7 is used o show his sep. Chr of Response Vlues of he Number he Domesic Airline Pssenger Fligh in Peknbru Airpor P3 7 Residul fer he Third Inervenion T3- T3-9 T3-8 T3-7 T3-6 T3-5 T3-4 T3-3 T3- T3- T3 T3+ T3+ T3+3 T3+4 T3+5 Figure 7. Response vlues of he Domesic Airline Pssenger Fligh in Peknbru Airpor. From Figure 7 shows chr of he residuls o deermine he order of b 3, s 3, nd r 3 for he hird inervenion model bsed on he nionl elecion nd we cn see he response vlues ime T 3, T 3 +, T 3 +,..., T 3 +5 hve less bsolue vlues hn he confidence inervl. This grph lso illusres h here is no vlues is close o he lower confidence inervl. Thus, here is no Finl Esimes of Prmeers possible se of order for he hird pulse funcion inervenion model. Prmeer esimion nd significnce ess show h his model ARIMA yields significn prmeers from his hird inervenion. Afer we esimed he prmeers of ech idenified model, he esimed model should be esed o verify he ssumpion of residul. Tble 7. Resuls of Prmeer Esimion, Prmeer Significnce Tes nd Dignosic Checking. Type Coef SE Coef T P MA,694,7 9,89, SMA,7976,55 7,56, Consn 58,5 34,,44,663 Differencing regulr, sesonl of order 9 Number of observions Originl series fer differencing 96 Residuls SS (bckforecss excluded) MS DF Modified Box-Pierce (Lung-Box) Chi-Squre sisic Lg Chi-Squre 6,4 7,4 4,8 5, DF P-Vlue,59,59,8,76 The SAS oupu shown in Tble 6 shows h he finl muli-inpu inervenion model for he number of he domesic irline pssengers in Peknbru irpor fer he hird pulse inervenion funcion cn be wrien s ˆ( 3) (,694 B)(,7976 B ) ln Y. ( B)( B ) (5) 5.5. Resuls from he Fourh Inervenion Model Afer modeling he hird inervenion bsed on he inervenion model due o he disser smog, noher nlysis of he fourh sep funcion inervenion ws conduced. This ws bsed on ury 5 which is equed wih T Mhemiclly, he inervenion ype of sep funcion is wrien s

12 Science Journl of Applied Mhemics nd Sisics 7; 5(3): -6 ( T ) ( 33), 3 ξ S4,, 33, 34,, 4. (6) The firs sep in his modeling is o deermine he order b 4, s 4 nd r 4 for he hird sep funcion inervenion model. This is done o deermine he order of he inervenion model nd o explin he decrese in he number of he number ypes of he domesic irline pssenger fligh on Riu Province vi Suln Syrif Ksim II Inernionl irpor due o he disser smog. Chr of Response Vlues of he Number of Domesic Airline Pssengger Fligh in Peknbru Airpor S Residul fer he Fourh Inervenion T4- T4-9 T4-8 T4-7 T4-6 T4-5 T4-4 T4-3 T4- T4- T4 T4+ T4+ T4+3 T4+4 T4+5 T4+6 T4+7 Finl Esimes of Prmeers Figure 8. Response vlues of he Domesic Airline Pssenger Fligh in Peknbru Airpor. Tble 8. Resuls of Prmeer Esimion, Prmeer Significnce Tes nd Dignosic Checking. Type Coef SE Coef T P MA,5788,797 7,6, SMA,4,53,9,58 Consn -53,7 5,4 -,5,6 Differencing regulr sesonl of order Number of observions Originl series 33 fer differencing Residuls SS (bckforecss excluded) MS DF 7 Modified Box-Pierce (Lung-Box) Chi-Squre sisic Lg Chi-Squre 7,9,4 3, 37,9 DF P-Vlue,54,43,57,766 Figure 8 shows h he response vlues ime T 4 + nd T 4 +6 hve greer bsolue vlues hn he confidence inervls. This mens h here re possible ses of orders for he fourh sep funcion inervenion model, i.e. he firs se order is b 4, s 4, r 4 nd he second is b 4, s 4 3, r 4. Prmeer esimion nd significnce ess show h boh ses of he model orders yield significn prmeers nd residuls h sisfy he whie noise nd Norml disribuion ssumpions. Tble 9. Resuls of Prmeer Esimion, Prmeer Significnce Tes nd Dignosic Checking. Prmeer Esimion vlue p vlue Type of significn θ,5788 7,6, Significn Θ,4,9,58 Significn ω -.9, Significn 4 ω -58.7, <. Significn ω 68.4,3 4 4,44 <. Significn ω , Significn

13 Slm Ali Wirdin e l.: Muli-Inpu Inervenion Anlysis for Evluing of he Domesic Airline Pssengers in n Inernionl Airpor Bsed on he resuls in Tble 7, n inervenion model for he number ypes of he domesic irline pssenger fligh in Peknbru irpor fer he fourh sep funcion inervenion nd prior o he fifh sep funcion inervenion cn be wrien s ( 4) ( 4, 9 4, 4, 4, ) lny.9s S 68.44S S + N where (,5788 B)(,4 B ) N ( )( ) B B (7) The clculions show h he disser smog resuled in decrese he number of domesic pssengers flying ir pssing hrough he Suln Syrif Ksim II Peknbru irpor per monh of 93 9 pssengers. This decline hs occurred since he incidence of fire disser unil prior o he fifh inervenion, which begn in Februry 5 nd flucued unil Augus Resuls from he Fifh Inervenion Model The finl nlysis of he fifh sep inervenion funcion bsed on he disser smog 5 which ook plce on Sepember 5 or he ime T 5 4 unil T 5 5. Mhemiclly, he inervenion ype of sep funcion is wrien s ( T ) ( 4), 4 ξ S5,, 4, 4,, 5. (8) The firs sep in his nlysis is o deermine he order of he 5h inervenion model. Figure 9 shows chr of he residuls o deermine he order of b 5, s 5 nd r 5 used in he inervenion model. The residuls will be used o model he decrese of he number ypes of he domesic irline pssenger fligh in Peknbru irpor vi Suln Syrif Ksim II Inernionl irpor due o he disser smog. Chr of Response vlues of he Number he Domesic Airline Pssenger Fligh in Peknbru Airpor 5 S5 4 Residul fer he Fifh Inervenion T5- T5-9 T5-8 T5-7 T5-6 T5-5 T5-4 T5-3 T5- T5- T5 T5+ T5+ T5+3 T5+4 T5+5 T5+6 T5+7 T5+8 T5+9 Figure 9. Response vlues of he Domesic Airline Pssenger Fligh in Peknbru Airpor. Bsed on he resuls chr of he residuls in Figure 9, his mens h here re possible ses of order for he fifh sep funcion inervenion model, he firs se order is b 5, s 5 (, ), r 5 nd he second is b 5, s 5, r 5 due o decrese in residul perns end o form srigh line, which indices consn effec, prmeer esimion nd significnce ess show h boh se of he model orders yield significn prmeers nd residuls h sisfy he whie noise nd norml disribuion ssumpions. The comprison of SBC crieri shows h he second model yields beer resul hn he firs. The resuls in Tble 8 re shown using he SAS oupu.

14 Science Journl of Applied Mhemics nd Sisics 7; 5(3): -6 3 Tble. Resuls of Prmeer Esimion, Prmeer Significnce nd Dignosic Checking. Finl Esimes of Prmeers Type Coef SE Coef T P MA,764,66,58, SMA,848,937 8,7, Consn -,6 4,7 -,98,38 Differencing: regulr sesonl of order Number of observions Originl series4 fer differencing8 Residuls: SS (bckforecss excluded) MS DF5 Modified Box-Pierce (Lung-Box) Chi-Squre sisic Lg Chi-Squre 5,3 4, 44, 5,6 DF P-Vlue,8,88,9,3 Tble. Resuls of Prmeer Esimion, Prmeer Significnce es nd Dignosic Checking. Prmeer Esimion vlue p vlue Type of significn θ,764,58, Significn Θ,848 8,7, Significn ω -6.55,4-4.7 <. Significn In his cse, he inervenion model for he number ypes of he domesic irline pssenger fligh in Peknbru irpor vi Suln Syrif Ksim II Inernionl irpor fer he s, nd, 3rd, 4h nd 5h sep inervenion funcion cn be wrien s ( 5) (,764 B)(,848 B ) lny -6.55,4 S5, 9 + (8) ( B)( B ) The effec of he reconsrucion nd he forecs of he finl inervenion model s he for rnsformion d (nurl log) re presened in Figure. Ln Trnsformion D,6,4,,,8,6 Number Types of he Domesic Airline Pssengger Fligh in Peknbru Airpor Vrible s Inervenion nd Inervenion 3rd Inervenion 4h Inervenion 5h Inervenion Ln Akul Finl Model S S P3 S4 S5,4, Monh Yer Figure. Effec on he Reconsrucion nd Forecs vlues of he Domesic Airline Pssenger Fligh in Peknbru Airpor. 6. Evluing nd Monioring Forecsing Model Performnce This pper considers how o evlue he performnce of forecsing echnique for priculr ime series or pplicion. I is imporn o crefully define he mening of performnce. I is emping o evlue performnce on he bsis of he fi of he forecsing or ime series model o hisoricl d. An evluion of he impc for ech inervenion could no be done direcly bsed on he model of equion (8). This hs cused he d no in origin scle, so he effec of ech inervenion could no be direcly used s he esimed prmeers. The rionl for his semen is bsed on he ssumpion h he inervenion model h we wn o

15 4 Slm Ali Wirdin e l.: Muli-Inpu Inervenion Anlysis for Evluing of he Domesic Airline Pssengers in n Inernionl Airpor evlue is s follows (see Lee e l. [5], nd Zhng e l. [4]) where Y The cul d Y Y + n (9) Y The inervenion effec n Noise model follows o ARIMA(p, d, q)(p, D, Q) S for error (d wihou inervenion effec) Besides h, he nex ssumpion is h he inervenion effec follows he simples model, i.e. Y ( T ) ω ξ (3) where ( T ) ( T ) ξ S The sep funcion cerin T. In his cse, he effec of he inervenion T is T T T Y Y n ω. Thus, we could direcly use he esimed prmeers o mesure he impc of n inervenion. On he oher, we cn ssume h he vrince d is no sionry nd we mus rnsform his d by using nurl log. So he process cn be wrien s Yɶ ln Y (3) If i is defined s Yɶ Yɶ n. (3) + ( T ) Yɶ ω S (33) hen he effec of his inervenion T on he rnsformion d is T T T Yɶ Yɶ n ω (34) Hence, he impc of his inervenion on he originl d is: T. Yɶ n n Yɶ T + T T T Y e e e. (35) This resul shows h he esimed prmeers of his inervenion model rnsformion d could no be inerpreed direcly due o he mgniude of he effecs of n inervenion. Therefore, he effec of he inervenion on he rnsformion d cerin ime mus be clculed by using he difference beween he forecs of his inervenion nd he pre-inervenion models. Following his, we cn rnsform he d o he originl scle o exmine he impc nd he resuls of ech inervenion s shown in Figure. Bsed on his conversion o he originl d scle, he impc of he firs, second, hird, fourh nd fifh inervenions re summrized in he following secions. And he inervenion model is Series Plo for Effecs of he Reconsrucion nd Forecss of he Inervenion Models S S P37 S433 S54 D of he Domesic Airline Pssenger Fligh in Peknbru Airpor Vrible s Inervenion nd Inervenion 3rd Inervenion 4h Inervenion 5h Inervenion Finl Model Y() 5 Monh Yer Figure. Effecs of he Reconsrucion nd Forecss of he Firs, Second, Third, Fourh nd Fifh Inervenions Models (Originl D). 7. Conclusions In his sudy, he heoreicl nd empiricl sudies on he inervenion model were crried ou o deermine he order of he inervenion model. This model includes he derivion of some effec shpes cegorized s emporry, grdully or

16 Science Journl of Applied Mhemics nd Sisics 7; 5(3): -6 5 permnen effec. Bsed on he resuls of he discussion bove, i cn be concluded h ech model, s resul of heir inervenion hs consruced forecsing model highly influenced by he phenomenon of exernl nd/or inernl fcors. The finl resuls of his heoreicl sudy were developed procedure for esblishing n inervenion model which includes hree min sges of modeling. All he compuions involved in his reserch hve been performed by using MINITAB version 6 nd SAS. This pper lso shows h he inerpreion of n inervenion model for rnsformion d could no be done direcly bsed on esimed model prmeers. Furher reserch is needed o undersnd he precise impc of he inervenions on oher forms of d rnsformion. However, he vilbiliy of sophisiced sisicl sofwre pckges such s Minib, JMP nd SAS mkes i possible for he prciioner o consider severl differen models wih vrious orders nd compre hem bsed on he model selecion crieri such s AIC, AICC nd SIC nd residul nlysis. Acknowledgemen We would like o hnk he Suln Syrif Ksim II Inernionl Airpor for providing us wih he number of pssenger ypes domesic fligh d. These suppors re grefully cknowledged. References [] B. Abrhm, Inervenion nlysis nd muliple ime series, Biomerik, (98), [] M. S. Akins, A Cse sudy on he use of inervenion nlysis pplied o rffic ccidens, Journl of he Operionl Reserch Sociey, 7 (979), [3] P. J. Brockwell nd R. A. Dvis, Inroducion o Series nd Forecsing, Spinger-Verlg, New York, 996. [4] B. L. Bowermn, R. T. O Connell nd A. Koehler, Forecsing, Series nd Regression: An Applied Approch, 4h Ediion, Duxbury Press, Belmon, Cliforni, 4. [5] G. E. P. Box, G. M. Jenkins nd G. C. Reinsel, Series Anlysis Forecsing nd Conrol, 5h Ediion, John Wiley & Sons, Inc., Hoboken, New Jersey, 6. [6] G. E. P. Box nd G. C. Tio, Inervenion nlysis wih pplicions o economic nd environmenl problems, Journl of he Americn Sisicl Associion, 7 (975), [7] L. Binchi, W. Guschi, J. Jrre nd R. C. Hnumr, Improving forecsing for elemrkeing ceners by ARIMA modeling wih inervenion, Inernionl Journl of Forecsing, 4 (998), [8] M. N. Bhchryy nd A. P. Lyon, Effeciveness of se bel legislion on he Queenslnd rod oll n Ausrlin cse sudy in inervenion nlysis, Journl of he Americn Sisicl Associion, 74 (979), [9] N. B. Chng nd Y. T. Lin, An nlysis of recycling impcs on solid wse generion by ime series inervenion modeling, Resources, Conservion nd Recycling, 9 (997), [] C. Chfield, Series Forecsing, Chpmn Hll, London,. [] J. D. Cryer nd Kung-Sik Chn, Series Anlysis wih Applicions in R, nd ediion, Springer, New York, 8. [] K. Drkos nd A. Kun, Regionl effecs of errorism on ourism in hree Medierrnen counries. Journl of Conflic Resoluion, 47 (3), [3] W. Enders, Applied Economeric Series, John Wiley & Sons, Inc., New York, 995. [4] W. Enders, T. Sndler, nd J. Culey, Assessing he impc of erroris hwring policies: An inervenion ime series pproch. Defense nd Pece Economics, (99), - 8. [5] A. Hrvey nd J. Durbin, The effecs of se bel legislion on Briish rod csulies, Journl of he Royl Sisicl Sociey, Series A, 4 (986), [6] J. D. Hmilon, Series Anlysis, New Jersey: Princeon Universiy Press, New York, 994. [7] M. E. Hilon, The impc of recen chnges in Cliforni drinking-driving lws on fl cciden levels during he firs pos inervenion Yer: An Inerruped ime series nlysis, Lw & Sociey Review, 8 (984), [8] U. Helfensein, The use of rnsfer funcion models, inervenion nlysis nd reled ime series mehods in epidemiology, Inernionl Journl of Epidemiology, 3 (99), [9] Z. Ismil, Suhrono, A. Yhy nd R. Efendi, Inervenion model for nlyzing he impc of errorism o he ourism indusry, Journl of Mhemics nd Sisics, 5 (9), [] []H. Jorquer, W. Plm, nd J. Tpi, An inervenion nlysis of ir quliy d Snigo, Chile. Amospheric Environmen, 34 (), [] Rising Fuel Price, Minisry of Finnce of Republic of Indonesi, hp:// ccessed in My h, 6.35 PM. [] H. Luhkepohl, New Inroducion Muliple Series Anlysis, nd Ediion, Springer, New York, 5. [3] C. Y. Lm, W. H. Ip nd C. W. Lu, A business process civiy model nd performnce mesuremen using ime series ARIMA inervenion nlysis, Exper Sysems wih Applicions, 36 (9), [4] J. Ledoler nd K. S. Chn, Evluing he impc of he 65- mph mximum speed limi on Iow inerses. The Americn Sisicin, 5 (996), [5] M. H. Lee, Suhrono nd B. Snugi, Muli-inpu inervenion model for evluing he impc of he Asin crisis nd he erroris cks on ouris rrivls, Memik, 6 (), [6] D. C. Mongomery, C. L. Jennings nd M. Kulhci, Inroducion o Series Anlysis nd Forecsing, John Wiley & Sons, Inc., New Jersey, 8.

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