Modeling Data Containing Outliers using ARIMA Additive Outlier (ARIMA-AO)

Transcription

1 Journl of Physics: Conference Series PAPER OPEN ACCESS Modeling D Conining Ouliers using ARIMA Addiive Oulier (ARIMA-AO) To cie his ricle: Ansri Sleh Ahmr e l 018 J. Phys.: Conf. Ser View he ricle online for udes nd enhncemens. This conen ws downloded from IP ddress on /0/018 07:05

2 Join Worksho of KOPI 017 & ICMSTEA 016 IOP Conf. Series: Journl of Physics: Conf. Series (018) doi : / /954/1/01010 Modeling D Conining Ouliers using ARIMA Addiive Oulier (ARIMA-AO) Ansri Sleh Ahmr 1,*, Suryo Gurino 3, Abdurkhmn 3, Abdul Rhmn 4, Awi 4, Alimuddin 4, Ilhm Minggi 4, M Arif Tiro 1, M Ksim Aidid 1, Suwrdi Anns 1, Din Umi Suiksno 5, Dewi S Ahmr, Kurniwn H Ahmr, A Abqry Ahmr, Ahmd Zki 4, Dhln Abdullh 6, Robbi Rhim 7, Heri Nurdiyno 8, Rhm Hidy 8, Drmwn Niuulu 9, Jnner Simrm 10, Nuning Kurnisih 11, Leon Andrei Abdillh 1, Andri Prnolo 13, Hviluddin 14, Whyudin Albr 15, A Nurni M Arifin 1 Dermen of Sisics, Universis Negeri Mkssr, Indonesi AHMAR Insiue, Mkssr, Indonesi 3 Dermen of Mhemics, Universis Gdh Md, Indonesi 4 Dermen of Mhemics, Universis Negeri Mkssr, Indonesi 5 Dermen of Business Adminisrion, Polieknik Negeri Ambon, Indonesi 6 Universis Mlikussleh, Aceh, Indonesi 7 School of Comuer nd Communicion Engineering, Universii Mlysi Perlis, Mlysi 8 STMIK Dhrm Wcn, Lmung, Indonesi 9Lembg Ilmu Pengehun Indonesi, Bnen, Indonesi 10 Universis Negeri Medn, Medn, Indonesi 11 Universis Pddrn, Bndung, Indonesi 1 Universis Bin Drm, Plembng, Indonesi 13 Dermen of Informics, Universis Ahmd Dhln, Yogykr, Indonesi 14 Universis Mulwrmn, Smrind, Indonesi 15 Fculy of Economics nd Bussiness, Universis Mlikussleh, Aceh, Indonesi *nsrisleh@unm.c.id Absrc. The im his sudy is discussed on he deecion nd correcion of d conining he ddiive oulier (AO) on he model ARIMA (, d, q). The rocess of deecion nd correcion of d using n ierive rocedure oulrized by Box, Jenkins, nd Reinsel (1994). By using his mehod we obined n ARIMA models were fi o he d conining AO, his model is dded o he originl model of ARIMA coefficiens obined from he ierion rocess using regression mehods. In he simulion d is obined h he d conined AO iniil models re ARIMA (,0,0) wih MSE = 36,780, fer he deecion nd correcion of d obined by he ierion of he model ARIMA (,0,0) wih he coefficiens obined from he regression Z 0,106 0, 04Z 1 0, 401Z 39X1 115X 35,9X 3 nd MSE = 19,365. This shows h here is n imrovemen of forecsing error re d. Conen from his work my be used under he erms of he Creive Commons Aribuion 3.0 licence. Any furher disribuion of his work mus minin ribuion o he uhor(s) nd he ile of he work, ournl ciion nd DOI. Published under licence by Ld 1

3 Join Worksho of KOPI 017 & ICMSTEA 016 IOP Conf. Series: Journl of Physics: Conf. Series (018) doi : / /954/1/ Inroducion In ime series d is someimes here's fr differen d vlues from oher d nd do no reflec he chrcerisics of se of d. The d vlue is clled he ouliers. In he nlysis of ime series d re ofen obined oulier. Oulier d is mor imc on forecsing d if we use forecsing mehods such s ARIMA, ARMA, nd ohers. Forecsing is n civiy o redic wh will hen in he fuure [1]. To forecs ime series hen we cn use forecsing mehod, e.g., ARIMA Box-Jenkins mehod []. Forecsing will be fr off from wh we redic if he d used o redic he d conined ouliers. The sudy on oulier deecion is very imorn becuse he resence of ouliers cn led o rmeer esimion or forecsing becomes inrorie. If d oulier is no enforced roerly, hen i will imc he forecs does no reflec he cul d. Reled o he inciden such s recording nd ying errors, AO is he mos common ye of ouliers re found in he ime series, so h ccording o he discussion bove, his reserch will discuss he deecion nd correcion D conining Addiive oulier (AO) in he ARIMA(,d,q) model. If he observions in ime series cn be rediced wih ceriny nd do no require furher invesigion, i is clled deerminisic ime series nd if he observions cn only show he srucure of he robbilisic se will come ime series, he ime series is clled sochsic. In he modeling of ime series nlysis ssumes h he d re sionry. Sionry ime series is sid if here is no chnge in he verge rends nd chnge in vrince. Relively sionry ime series re no incresing or decrese in he vlue of he exreme flucuions in he d, or he d is bou he verge vlue of he consn. Sionriy cn be seen by using ime series chr is scerlo beween he vlue of he vrible Z wih ime. If he digrm ime series flucues round line rllel o he ime xis (), hen he series is sid sionry in verge. When sionry condiions in he verge unme need differeniion or differencing rocess [3]. The rocess of firs-order differencing on he difference beween h d wih 1 h, i.e., Z Z Z. Forms for second order is differencing 1 Z Z Z 1 ( Z Z 1) ( Z 1 Z ) Z Z Z. If hese condiions re no me sionry in vrince, Box nd Cox (1964) 1 ( ) ( ) Z 1 inroduced he ower rnsformion Z, ( is rmeer), known s he Box-CoX rnsformion. Following re some rovisions o sbilize he vrince is [4]: The rnsformion my be mde only for he series Z re osiive. The rnsformion is done before he differencing nd modeling of ime series. The vlue seleced bsed on he sum of squres error (SSE) of he rnsformed series. The smlles vlue of SSE mos consn yield vrince. Trnsformion is no only sbilized he vrince, bu i cn lso normlize he disribuion. Inegre models Auoregressive Moving Averge (ARIMA) hs been sudied by George Box nd Gwilym Jenkins in 1976 [5], nd heir nmes re hen frequenly synonymous wih ARIMA rocesses lied o he nlysis of ime series. In generl ARIMA models is denoed by he noion ARIMA (, d, q), where denoe he order of he uoregressive (AR), d exressed differeniion (differencing), nd q exress order of he moving verge (MA). Auoregressive model (AR) ws firs inroduced by Yule on 196 [6] nd ler develoed by Wlker on 1931 [7], while model Moving Averge (MA) ws firs used by Sluzky on 1937 [8]. And 1938, Wold genere he heoreicl bsis of he combinion of ARMA nd combinions is hen ofen used [9]. Box nd Jenkins hve effecively mnged o rech n greemen on he relevn informion needed o undersnd nd use ARIMA models for ime series of he vribles [10]. In d ime series forecsing mehods ARIMA (, d, q) here is he so-clled mesures or is hses. The sges in he forecsing [11]:

4 Join Worksho of KOPI 017 & ICMSTEA 016 IOP Conf. Series: Journl of Physics: Conf. Series (018) doi : / /954/1/ Idenificion Model Idenificion of he model ws done o see he significnce nd sionry of uocorrelion d, so wheher or no o do rnsformion or differencing rocess (differeniion). From his sge, he model is obined while esing he model will be mde wheher or no he corresonding d. b. Assessmen nd Tesing Models Afer idenifying he model, he nex se is he ssessmen nd esing of he model. A his sge, divided ino wo rs, nmely rmeer esimion nd dignosic model. c. Prmeer Esimion Afer obining one or more models while he nex se is o find esimes for he rmeers in he model. d. Dignosic Model Dignosic checking o do check wheher he model esimed sufficien or deque fi wih he d. Dignosic checking is bsed on he nlysis of residuls. The bsic ssumion is h he residuls of ARIMA models re indeenden normlly disribued rndom vribles wih zero men consn vrince. The oulier is roblem ofen encounered in finncil d. Wih he d ouliers re he ries o redicion or forecs of he d will hve roblems such s he roblem of lck of ccure in forecsing. Gounder, e l [1] describes severl yes of ouliers re: (1) Addiive oulier (AO). Ouliers of he simles nd mos frequenly sudied in he nlysis of ime series re n ddiive oulier, lso known s Tye I ouliers. An AO only ffecs single observion, which could be worh smll or lrger vlue comred wih he execed vlue in he d. Afer his effec, he d series bck o he norml rck s if nohing hened. Effecs of n AO is indeenden of ARIMA models nd consrined. Oulier AO is he mos roblemic becuse i conins wo consecuive residues, one before nd he oher fer AO. An AO cn hve serious effecs on he observed roeries. This will ffec he residul susicion nd suosiion rmeers. This cn be roved in generl h he AO lrge would encourge ll uocorrelion coefficiens owrds zero. The influence of ouliers is reduced o lrge smle size. The formul of he AO re s follows [11]: Z, T Y Z, T Z where: Y = Series observions Z = rry oulier-free observions = mgniude oulier T I = Indicor vrible h indices he ime oulier I () innovionl oulier (IO), in conrs o AO, n innovionl oulier is lso known s Tye II ouliers (Fox, 197) h ffec some observions. An AO only ffecs only one residue, he ime of he occurrence of ouliers. Effec of IO on he observed series consiss of n iniil shock h sred in subsequen observions by he weigh of he moving verge reresenion (MA) of he ARIMA models. So fr he weigh is ofen exlosive, my influence he IO, in some cses, he incidence hs incresed nd coninues o increse o more nd more vlue from h ime unil he evens of he s, s well s in he fuure more unwelcome. Effec of IO deends on he riculr model of he series, nd series wih sionry rnsformion effec indefiniely. For series conining rend nd sesonl, he IO will ffec boh. IO resens some serious drwbcks h should be voided. The roblem hen is h he oher hree yes of ouliers h cn no exlin he chnges in he sesonl comonen. T (1) 3

5 Join Worksho of KOPI 017 & ICMSTEA 016 IOP Conf. Series: Journl of Physics: Conf. Series (018) doi : / /954/1/ Mehod The mehod used in his reserch is he sudy of lierure by nlyzing he deecion nd correcion of d AO on he ARIMA(,d q) model. The iniil se in he nlysis is o sge in forecsing ARIMA (,d,q) in order o obin he corresonding llegions of ARIMA model. Furher, esing of sionry residul. If he residul of sionry is no obined hen lleged h occurs becuse of he d h i conins ouliers. Then do he deecion nd correcion of d conining AO wih n ierive rocedure o obin he residul of sionry nd hs he smlles MSE. 3. Resul nd Discussion The oulier is roblem ofen encounered in finncil d. Wih he d ouliers re he ries o redicion or forecs of he d will hve roblems such s he roblem of lck of ccure in forecsing. For sionry rocess, sy Z s he observed series nd X s n oulier-free series. Assume h X follows he generl model ARMA (, q): B Z B, ~ WN 0,,, R, R () q i i wih B 1 1B... B dn 1 B 1 B... B is sionry nd inverible oerors who do no shre he sme fcor. Addiive ouliers (AO) is defined s: Z, T Y Z, T (3) where: Y Z I T B T Y I, ~ WN 0,, i, i R, R B T 1, T, I 0, T, (4) (5) (6) is n indicor vrible reresening wheher or no n oulier ime T. In generl, if he ime series d re k ddiive oulier (AO), he generl model of n oulier his: T B B k, ~ 0,,, i, i, 1 Y I WN R R (7) 3.1. Esimed Effec Oulier when Oulier Time Unknown As mer of moivion o deec AO rocedure, we consider he simle cse when T nd ll he rmeers in (3) known. For exmle: nd defined: B B 1 1B B... (8) B e B Y (9) 4

6 Join Worksho of KOPI 017 & ICMSTEA 016 IOP Conf. Series: Journl of Physics: Conf. Series (018) doi : / /954/1/01010 from (6) cn be esblished: T AO : e B I, ~ WN 0,,, R, R (10) i i For n observions mde, AO models bove cn be wrien s: Le AT e1 0 1 e T 1 0 T 1 et 1 T e T 1 1 T 1 et T en nt n s he les squres esimor of o model AO. Becuse i is whie noise, hen he heory of les squres, found h: AO : wih * nt F 1 1F F... nt F AT e * T 1 T nt 0, F is he forwrd shif oeror so h Fe e 1 nt dn 0. The vrince of he esimor is: Vr AT 1 T F e T e * Vr * Vr F e Procedure Using Ierive Oulier Deecion If T is no known, bu he rmeers of ime series re known, hen we cn clcule for ech 1, 1,,..., n, nd hen mke decision bsed on he resuls of hese smling. However, in rcice ofen be found h rmeer ime series,, nd usully unknown nd mus be redicble. Wih he oulier hen mke redicions rmeer would be bised. In riculr, will end o be exggered, s indiced erlier. Therefore i is he Chng nd Tio (1983) roosed n ierive rocedure o deec nd del wih he siuion when he number of AO is no known o exis. In n oulier deecion nd correcion of he d i needs is clled rocedure. The rocedure o be used in he deecion rocedure is ierions. The rocedure is s follows (Box, Jenkins, nd Reinsel, 1994): F e (11) (1) (13) 5

7 Join Worksho of KOPI 017 & ICMSTEA 016 IOP Conf. Series: Journl of Physics: Conf. Series (018) doi : / /954/1/01010 Se 1: Downlod he d Se : Esimed rmeers of ARIMA (usully ssumed h he d conins no ouliers) Se 3: Clcule he residuls of he ARIMA models re: B e Y BY B wih : nd le : B 1 1B... B nd 1 (14) B 1 B... B (15) 1 n e (16) n 1 ) Se 4: Clcule he es sisic resence of AO ( A, T wih : 1, T AT (17) AT e T e T i T i i1 nt i io (18) Jik 1,T c hen here is AO on he ime. nt i i0 (19) c is he criicl vlue h is clculed using he formul resened by Lung in he SAS / ETS User's Guide. AO effecs my remove of residul by defining: T e e AT B I e AT T, T (0) In oher cses, new esimes clculed from he modified residul e. If here re ny ouliers re idenified he modified residul e nd modificion esimes, bu B B used o clcule he sic of 1,T. Ses re reeed unil B wih he sme rmeers ll ouliers re idenified. Le oulier idenificion rocedure k oin in ime, T 1, T,..., T k. So he overll oulier correcion model of i: 6

8 Join Worksho of KOPI 017 & ICMSTEA 016 IOP Conf. Series: Journl of Physics: Conf. Series (018) doi : / /954/1/01010 k T B B Y I 1 The formul bove is esimed for he observion sequence Z for AO. And i wih revised se of residuls: k B T e Y I () B 1 nd obined from his model fi. To modify he observion ime series conining he effecs of ouliers cn remove nd correcion of he d done by consrucing he generlized esiming equions s follows: k T 1 z Z I (1) (3) 4. Simulion The d used in his simulion is he d obined from he ARIMA simulion resuls using he R licion. The ses ken in he simulion forecs bove is s follows: 4.1. Phse Idenificion The firs se in he modeling of he d s shown below. ime series on he idenificion is checked sionry d by loing () (b) (c) Figure 1. () Plo Time Series D, (b) ACF Plo Time Series D, (c) PACF Plo Time Series D From he icure looks sionry in he vrince of he d nd hen do he ACF nd PACF los her. Bsed on he d digrm, he vlue of ACF nd PACF, he d hs been sionry in he 7

9 Join Worksho of KOPI 017 & ICMSTEA 016 IOP Conf. Series: Journl of Physics: Conf. Series (018) doi : / /954/1/01010 verge. Once he d is sionry, hen he nex se is o idenify he model emorrily. Idenificion of he model done by looking he vlues of he coefficien uocorrelion, he ACF nd PACF los of he d h hs been sionry. From he form of he ACF nd PACF re runced form fer lg of nd droed exonenilly suseced h he rorie model for hese d is ARIMA (,0,0) or AR() wih rmeers 1 0,37 nd 0, Assessmen nd dignosic checks The erly sges of inerreing he resuls of ime series nlysis re o look is significnce model rmeers h hve been modeled. The model is he firs roximion obined AR(), so we need o es he rmeers of Auoreg. gressive Esimed rmeers of he model AR() is significnly differen from zero wih 95% confidence level. This cn be seen in he -vlues. For rmeer AR(1) h 1, -vlue = nd for AR() is 1, -vlue = If he clculions re erformed wih comuer sisicl ckge, hen simly use he -vlue lredy known. Crieri conclusion h significnce esing if < α nd no significn when α. Bsed on he resulss of Minib for he d obined -vlue = < α = 0.05, which mens significn esing. Afer esing is significnce rmeers, he nex se is o es he suibiliy of he model. To suibiliy models include he dequcy of he model (es wheher he remining whie noise) nd es he ssumion of norml disribuion. In Minib resuls lso cn be seen h he vlue of Lung-Box sisics is dislyed on lg 1, lg 4 nd lg 36. Lung-Box sisic vlue lg 1, lg 4 nd lg 36 consecuive shows -vlue = 0.619, 0.50, 0.03, nd vlue is greer hn α = This mens h he remining eligible whie noise. Nex, es he remining norml disribuion. The resuls of he es chr norml disribuion for he residul shown in figure. () (b) (c) (d) 8

10 Join Worksho of KOPI 017 & ICMSTEA 016 IOP Conf. Series: Journl of Physics: Conf. Series (018) doi : / /954/1/01010 (e) (f) (g) (h) (i) () (k) (l) Figure. () norml disribuion for he remining es (Kolmogorov-Smirnov es); (b) Plo Residul Model AR(1); (c) Boxlo D Residul AR(1); (d) Plo Time Series residuess Addiion Process Deecion Ouliers 1 h ; (e) Tes he normliy of Process Residues Oulier Deecion Addiion 1 h ; (f) BoxPlo Residue from Process Enhncemens Oulier Deecion 1 h ; (g) Plo Time Series Process Residues of Oulier Deecion Addiion nd ; (h) Tes he normliy of Process Residues he ddiion of oulier deecion nd ; (i) residuess BoxPlo oulier deecion rocess Addiion nd ; () Plo Time Series Process Residues of oulier deecion Addiion 3 rd ; (k) Tesing normliy of Process Residues oulier deecion Addiion 3 rd ; (l) BoxPlo residues The ddiion of hree oulier deecion rocess. 9

11 Join Worksho of KOPI 017 & ICMSTEA 016 IOP Conf. Series: Journl of Physics: Conf. Series (018) doi : / /954/1/01010 Bsed on he Kolmogorov-Smirnov es (Figure ) obined -vlue <0.010 which is smller hn α=0.05, his shows h he res do no mee he norml disribuion ssumion. This is resumbly due o he resence of ouliers in he d. From he resuls of he inerim model idenificion, he model AR() wrien mhemiclly s follows: Z 0, 37Z 0, 48Z (4) 1 wih MSE of To deec he resence of ouliers or ouliers in he d is crried ou gins he residul lo nd boxlo of he erlier models we derive he AR() model. From he figure boxlo bove shows h here is n oulier = 98; hus obined: T 1, 98, I 0, ohers (5) By using he mehod of he les squre regression equion nd imlemenion: Z 0,0600 0, 09Z 0, 417Z (6) 1 wih MSE = Then wih he ddiion of oulier deecion ARIMA models will be: Z 0,0775 0, 1Z 0, 411Z 36 X ( ) (7) -1 1 wih MSE = This se is done coninuously in order o obin smller MSE, normliy ess re me, nd here is no longer n imge boxlo oulier d. Here re he resuls of oulier deecion ierion s indiced by imroved is MSE. The ddiion of second oulier, nmely: wih he regression obinble : wih MSE =.808. T 1, 16, I 0, ohers -1 1 (8) Z 0,09 0, 06Z 0, 410Z 37 X ( ) 114 X ( ) (9) The ddiion of hird oulier, nmely: T 1, 180, I 0, ohers (30) wih he regression obinble : Z 0,106 0, 04Z 0, 401Z 39 X ( ) 115 X ( ) 35,9 X (31) wih MSE = Wih he fulfillmen of he residul normliy es nd is boxlo lo he model fi o he d is ARIMA(,0,0) wih he coefficiens obined from he regression. One of he usefulness of his deecion mehod is h i cn reduce he error re of d redicion. This cn be seen in he decline in he vlue of MSE of ech model occurs in he firs ierion o he second ierion. The following ble of comrisons of ech deecion nd correcion nd ime series models digrm MSE imirmen. 10

12 Join Worksho of KOPI 017 & ICMSTEA 016 IOP Conf. Series: Journl of Physics: Conf. Series (018) doi : / /954/1/01010 Tble 1 Comrison of Model Resuls MODEL MSE Regression ARIMA (,0,0) ARIMA (,0,0) + oulier deecion nd correcion 1 h wih vlue: AT 0, Z 0, , 1Z 0, 411Z 36X 1 1 ARIMA (,0,0) + oulier deecion nd correcion nd wih vlue: AT 0,07 Z 0, 09 0, 06Z 0, 410Z 37X 144X 1 1 ARIMA (,0,0) + oulier deecion nd correcion 3 rd wih vlue: AT 0,066 Z 0,106 0, 04Z 0, 401Z 39X 115X 35,9 X Conclusion From he discussion nd simulion d on d oulier deecion nd correcion using ierive mehods on he ARIMA model (, d, q) cn be obined severl conclusions, s follows: (1) The rocess of deecion nd correcion of d conining Addiive Oulier on ARIMA (, d, q) using ierion mehod nd using he sofwre Minib 16 nd Microsof Excel 007. The generl model for equliy Addiive Oulier deecion ARIMA (, d, q): wih: k T Z I B 1 T 1, T B I nd B 0, T B () From he simulion resuls obined n erly model for he d conining ouliers i.e., Model AR() by he equion: Z 0, , 09Z 1 0, 417Z by MSE of Afer he rocess of oulier deecion nd correcion of d obined using n ierive mehod is he bes model forecsing model AR () wih coefficien obined from regression nd end of he equion: Z 0,106 0, 04Z 0, 401Z 39X 115X 35,9 X by MSE of (3) By using he deecion nd correcion of d conining Addiive Oulier on ARIMA (,d,q) using ierive mehods rovide rocess imrovemens nd decrese MSE models nd will cerinly rovide residul ends o qulify he norml disribuion. In his simulion shows h here is imrovemen MSE vlue of 47.34% of he iniil model. References [1] Ahmr, A. S., Rhmn, A., Arifin, A. N. M., & Ahmr, A. A Predicing movemen of sock of Y using sue indicor. Cogen Eco. Finnce, 5, 1 doi: /

13 Join Worksho of KOPI 017 & ICMSTEA 016 IOP Conf. Series: Journl of Physics: Conf. Series (018) doi : / /954/1/01010 [] Rhmn, A., & Ahmr, A. S Forecsing of rimry energy consumion d in he unied ses: A comrison beween ARIMA nd holer-winers models. Per resened he AIP Conference Proceedings, 1885, doi: / [3] Wei, W. W. S Time Series Anlysis : Univrie nd Mulivrie Mehods (nd ed.). New York, NY: Person Addison Wesley. [4] Box, G. E. P., & Cox, D. R An nlysis of rnsformions. J. of Royl S. Soc. Series B (Mehod.), [5] Box, G. E. P., & Jenkins, G. M Time series nlysis, conrol, nd forecsing. Sn Frncisco, CA: Holden Dy, 36(38), 10. [6] Yule, G. U Why do we someimes ge nonsense-correlions beween Time-Series: sudy in smling nd he nure of ime-series. J. of Royl S. Soc., 89(1), [7] Wlker, G On eriodiciy in series of reled erms. Proceedings of he Royl Sociey of London. Series A, Conining Pers of Mhemicl nd Physicl Chrcer, 131(818), [8] Sluzky, E The summion of rndom cuses s he source of cyclic rocesses. Economeric: J. of Eco. Soc., [9] Wold, H A Sudy in he Anlisys of Sionery: Time Series. Almqvis & Wiksells Bokryckeri. [10] Mkridkis, S., Wheelwrigh, S. C., & Hyndmn, R. J Forecsing mehods nd licions. John Wiley & Sons. [11] Wei, W.W.S., 1994 Time series Anlysis:Univrie nd Mulivrie Mehods, Addison- Wesley Publising Comny, Cliforni. [1] Gounder, M.K., Mhendrn, dn Rhmullh Deecion of Oulier in Non-liner Time series: A Review. Fesschrif in honor of Disinguished Professor Mir Msoom Ali. ges