INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES. A Simulation Study of Additive Outlier in ARMA (1, 1) Model

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1 A Simulaio Sudy of Addiive Oulier i ARMA (1, 1) Model Azami Zaharim, Rafizah Rajali, Rade Mohamad Aok, Ibrahim Mohamed ad Khamisah Jafar Absrac Abormal observaio due o a isolaed icide such as a recordig error is kow as addiive oulier ad i is ofe foud i ime series Sice ereme value of addiive ouliers may coribue o he iaccuracy of model specificaio, proper deecio procedure is sigifica o avoid such error Equaios ha eplai he aure of a addiive oulier ad he es saisics peraiig o i are discussed i his aricle his is followed by wo separae simulaio sudies ha are coduced o ivesigae he samplig behavior ad deecio performace of he es saisics i ARMA (1, 1) models Resuls for he firs simulaio sudy show ha he es saisics is a icreasig fucio of sample size Whils i he oher simulaio sudy we see ha he performace of he es saisics improves as large magiudes of oulier effec are used Keywords Addiive oulier, samplig behavior of es saisics, deecio performace of es saisics, simulaio I INRODUCION Abormal observaios i ime series ofe sigify impora eves such as a ierveio or a uepeced icide like he oubreak of war, ecoomic recessio ec hese observaios are kow as ouliers because hey are aberra from he res of he observaios o ideify a oulier based o he reasos associaed wih i, ouliers are Mauscrip received Ju, 009 his work was suppored i par by he Miisry of Sciece ad Iovaio, Malaysia uder Gra E-Sciece ( SF0407) Azami Zaharim is wih he Cere for Egieerig Educaio Research, Naioal Uiversiy of Malaysia, Bagi Selagor, Malaysia (Phoe: ; fa: ; azami@visiegukmmy) Rafizah Rajali is wih he School of Mahemaical Scieces, Naioal Uiversiy of Malaysia, Bagi Selagor, Malaysia ( rafizah_rajali@yahoocom) Rade Mohamad Aok is wih he Cere for Egieerig Educaio Research, Naioal Uiversiy of Malaysia, Bagi Selagor, MALAYSIA ( aok@vlsiegukmmy) Ibrahim Mohamed is wih he Isiue of Mahemaical Scieces, Uiversiy of Malaya Kuala Lumpur, MALAYSIA ( imohamed@umedumy) Khamisah Jafar is wih he Cere for Egieerig Educaio Research, Naioal Uiversiy of Malaysia, Bagi Selagor, MALAYSIA ( khamisah@ vlsiegukmmy) amed based o heir aribues like addiive oulier (AO), iovaioal oulier (IO), emporary chage (C) ad level shif (LS) each of which correspods o a uique icide he mos commo icide foud i a ime series is recordig error ad such eve is oed by AO hus, his sudy aims o ivesigae he samplig behavior of he es saisics used o deec a sigle AO i ARMA (1, 1) models ad is performace whe seleced crierio is applied A oulier-free ime series Z ha follows a auoregressive movig average (ARMA) process is defied as Z = Θ a Φ (1) where B is he backshif operaor such ha BZ = Z 1 Φ B) = 1 Φ B Φ p ; Θ B) = 1 Θ B Θ q B ( 1 p B are polyomials i B, { a } ( 1 q ad is a sequece of whie oise radom variables, ideically ad idepedely disribued as N(0, σ a ) From (1), ca be defied as Z Z a Θ = () Φ Fo (197) is amog he earlies o coduc a sudy o ouliers i ime series He cosidered o-seasoal AR (p) process ad wo ouliers which are AO ad IO I his work, Fo proposed a mehod which deecs ad removes he oulier effec [1] Followig ha, may sudies o ouliers i ARMA (p,q) models were carried ou like []-[4] Makig use of (1) ad (), he geeraig mechaism of a AO i ARMA process is described as Z Y = Z + ω = Z + ωi = (3) (4) Issue, Volume 3,

2 From equaio () θ = φ a + ωi he observed oulier-free series of (1) ad uobservable series are deoed as Y ad Z respecively Magiude of AO is represeed by ω ad I is a ime idicaor variable used o idicae he occurrece of a AO herefore, I =1 whe a AO is spoed ad I =0 oherwise Followig (4), AO is said o be deermiisic i aure ie o affecig observaios subseque o i [1] II ES SAISICS FOR AO DEECION Despie oly affecig he observaio a =, a AO is kow o affec up o p subseque residuals followig = [6] herefore, residual esimaes are used i he esimaio of ω which laer forms he basis of AO deecio I his secio, discussios o he esimaio of residuals ad AO effecs are show i (6) ad (7) respecively A Esimaio of residuals o faciliae udersadig of how residual esimaes are used i he AO deecio procedure, cosider a simple case whe ad all parameers i (1) are kow [1] Le π φ = θ = 1 π B 1 1 π B be he dyamic sysem where π j deoe he weighs for j beyod a moderaely large value J ha esseially equal o 0 whe he roos of θ lie ouside of he ui circle Esimaed residuals ca he be described as = π Y Usig (5), (6) ad (7), esimaed residuals of a AO coamiaed series i a ARMA process ca be wrie as = π Y θ ( ) = π B a + ( ) ωi φ B φ θ = a + ( ) ( ) ωi θ B φ B φ = a + ωi ( ) θ B = a + π ω I = + ωπ a I (8) (5) (6) (7) ( ) ( ) Noig ha I = 1 whe = ad I = 0 oherwise; residuals for < ad = are obaied as follows: a = a + ωπ < = O he corary, residual esimaes for = + j( j = 1, j) are o as sraigh forward as (9) o obai he respeced residual esimaes, we epad equaio (8) I ( ) = a + ωπ 1 3 = a ( ) ( ) + ω 1 π 1B π B π 3 B I ( ) ( ) ( ) ( = a ) + ω I π 1I 1 π I π 3 I 3 ( ) Hece, for = + j( j = 1,, ) ( ) ( ) ( ˆ ( ) ( ) 1 1 e + j = a + j + ω I + j π I π I π j I j ) ( ) ( ) ( ( ) ( ) 1 1 = a + j + ω I + j π I π I π j I ) = a + j + ω[ 0 π 1( 0) π ( 0) π j ( 1 )] = a + + ω π (10) j ( ) j For umber of observaios, equaios (8)-(10) ca be summarized as he followig [5]: a 1 0 a 1 = + ω a + 1 π 1 a + π a a 1 0 π (9) (11) B Esimaio of AO effec Le ωˆ be he esimaor of ω i (3), ωˆ is kow as he leas squares esimae of AO effec because { a } are obaied from he leas squares heory [5] From (8), le π I be represeed by, we have ha Issue, Volume 3,

3 = ˆω = (1) = ad variace of he esimaor give as ( ˆ ) = = a ˆ σ Var ω (13) hus, he sadardized versio of ωˆ is τ = ˆ ω Var ω 1 = ˆ σ a ( ˆ ) AO = = = (14) Havig saed equaios (6) o (14), he es saisics of ieres deoed by η is he absolue maima of (14) as described i (15) = 1,,, { } η ma τ = (15) (ii) Coefficies chose for ARMA (1, 1) i able 1 ABLE I ARMA (1, 1) MODELS Model AR MA o achieve his, oulier-free ime series of size 60,100 ad 00 for each of he model i able 1 are geeraed 500 imes For isace, give = 100 ad model =1, 500 es saisics for he addiive oulier η peraiig o he respeced crierio are aaied Ne, he upper perceiles of η a 1%, 5% ad 10% level are obaied for compariso Repeaig he same procedure for all possible combiaios of (i) ad (ii), he resuls are he ploed i Figure 1-3he plos show similar paers as esimaes of η a give perceiles are icreasig fucios of sample size However magiude for he icrease varies for each model ake model 3 as a eample, a 5% upper perceiles, esimaes of η correspodig wih sample size 60,100 ad 00 are 55, 70 ad 76 respecively as compared o he esimaes of Model which are 347, 371 ad 403 III ILLUSRAIONS For he illusraio of he samplig behavior ad deecio performace of he es saisics η, we cosider a simple case whe ad all parameers of ARMA (1, 1) are kow o allow a more comprehesive aalysis o be coduced, he parameers are carefully chose o form uique combiaios of ARMA (1, 1) models as show i able 1 Assumig he residuals, e follow a ormal disribuio of N (0, 1), oulier free ime series for each of he model i able 1 are geeraed usig he arimasim procedure i R package I secio 31, oulier free ime series of size are used o sudy he samplig behavior of η he, i secio 3, AO coamiaed series are geeraed by creaig a AO a =/ i he oulier free ime series hese coamiaed series are used o es he performace of η i he deecio of he arificial simulaed AO A Samplig Behavior I his secio, we ivesigae he samplig properies of η i relaio o (i) Sample size 60, 100 ad 00 Issue, Volume 3,

4 Fig1 1% upper perceiles of η Ne, we eamie he deecio performace of η associaed wih (i) Sample size of 60, 100 ad 00 (ii) Coefficies chose for ARMA (1, 1) i able 1 (iii) AO effec ω of magiudes 5, 10 ad 15 For each possible combiaio of (i),(ii) ad (iii), 500 AO coamiaed series are geeraed by allocaig a AO of he respeced ω a = / i each of he series as suggesed i (3) For eample, give = 100,model =1 ad ω =5, we acquire he proporios of he AOs correcly deeced a 1% sigificace level from he 500 AO coamiaed series i relaio o he respeced crierio Repeaig he same procedure for all possible combiaios of (i), (ii) ad (iii), he resuls are he ploed i Figure 4-15 Overall he plos sugges ha he deecio performace of η improves whe large ω are used, his is especially evide i model 3 as ehibied i Figure 10-1 Fig 5% upper perceiles of η Fig 4 Proporio of he AO correcly deeced i Model 1 whe ω = 5 Fig3 10% upper perceiles of η B Deecio Performace Issue, Volume 3,

5 Fig 5 Proporio of he AO correcly deeced i Model 1 whe ω = 10 Fig 7 Proporio of he AO correcly deeced i Model whe ω = 5 Fig 6 Proporio of he AO correcly deeced i Model 1 whe ω = 15 Fig 8 Proporio of he AO correcly deeced i Model whe ω = 10 Issue, Volume 3,

6 Fig 9 Proporio of he AO correcly deeced i Model whe ω = 15 Fig 11 Proporio of he AO correcly deeced i Model 3 whe ω = 10 Fig 10 Proporio of he AO correcly deeced i Model 3 whe ω = 5 Fig 1 Proporio of he AO correcly deeced i Model 3 whe ω = 15 Issue, Volume 3,

7 Fig 13 Proporio of he AO correcly deeced i Model 4 whe ω = 5 Fig 15 Proporio of he AO correcly deeced i Model 4 whe ω = 15 IV CONCLUSION A AO is ofe associaed wih isolaed misake such as a recordig error herefore, i has a deermiisic aure because he AO effec ω does o affec subseque observaios as described i equaio (4) However, accordig o equaios (8) - (10), residuals ha come afer a AO may subsaially be affeced by ω I secio 31, simulaio sudy o he samplig behavior suggess ha esimaes of he es saisics η are icreasig fucios of O he oher had, he simulaio sudy i secio 3 show ha he deecio performace of η improves whe large magiudes of ω are used I his sudy, eiher he samplig behavior or he deecio performace of η idicaes ay obvious relaioship wih he coefficies chose for ARMA (1, 1) Fig 14 Proporio of he AO correcly deeced i Model 4 whe ω = 10 ACKNOWLEDGMEN he auhors hak he Miisry of Sciece ad Iovaio Malaysia, for heir geerosiy i providig fiacial suppor eeded i coducig his sudy REFERENCES [1] A Zaharim Ouliers ad Chage Pois i ime Series Daa (Upublished PhD hesis), Uiversiy of Newcasle Upo ye, 1996 [] I Chag Oulier i ime Series (Upublished PhD hesis), Deparme of Saisics, Uiversiy of Wiscosi-Madiso, 198 [3] WS Cha A Noe o ime Series Model Specificaio i he Presece of Ouliers, Joural of Applies Saisics, vol 19, o 1, pp Issue, Volume 3,

8 [4] RS say ime Series Model Specificaio i he Presece of Ouliers, Joural of America Saisical Associaio vol81, pp13-141, 1986 [5] WSWei ime Series Aalysis: Uivariae ad Mulivariae Mehods (Book syle), Boso: Pearso Addiso Wesley, 006, pp3-9 [6] I Mohamed Ouliers i Biliear ime Series Model (Upublished PhD hesis), Uiversii ekologi Mara, 005 [7] G Rischard ad G Aille, A Robus Look a he Use of Regressio Diagosics he Saisicia, vol 41, o 1, pp 41-53, 199 [8] A Zaharim, I Mohamed, I Ahmad, S Abdullah & MZ Omar Performace es Saisics for Sigle Oulier Deecio i Biliear (1, 1, 1, 1) Models, WSEAS rasacios o Mahemaics, issue 1, vol 5, pp [9] A Zaharim, I Mohamed, S Abdullah & MSYahya A Evaluaio of es Saisics for Deecig Level Chage i BL (1, 1, 1, 1) Models, WSEAS rasacios o Mahemaics, issue, vol 7, pp [10] K Ibrahim, R Rajali & A Zaharim O he Deecio of Ouliers for Waer Levels of Laga River, Proceedigs of he 1 s WSEAS Ieraioal Coferece o Mulivariae Aalysis ad is Applicaio i Sciece ad Egieerig, pp Issue, Volume 3,