A Study on the Nature of an Additive Outlier in ARMA(1,1) Models

Transcription

1 Europn Journl of Scinific Rsrch SSN 45-6X Vol3 No3 9, pp EuroJournls Publishing, nc 9 hp://wwwuroournlscom/srhm A Sudy on h Nur of n Addiiv Oulir in ARMA, Modls Azmi Zhrim Cnr for Enginring Rsrch Educion, Nionl Univrsiy of Mlysi E-mil: zmi@vlsingukmmy Rfizh Rli Dprmn of Sisics, School of Mhmicl Scincs Nionl Univrsiy of Mlysi E-mil: rfizh_rli@yhoocom Rdn Mohmd Aok Cnr for Enginring Rsrch Educion, Nionl Univrsiy of Mlysi E-mil:ok@vlsingukmmy Kmrulzmn brhim Dprmn of Sisics, School of Mhmicl Scincs Nionl Univrsiy of Mlysi E-mil: kmrulz@ukmmy Ahmd Mhir Rzli Dprmn of Sisics, School of Mhmicl Scincs Nionl Univrsiy of Mlysi E-mil: mhir@ukmmy Absrc Oulirs r ofn ncounrd in im sris nlysis yps of oulirs h r usully dl wih r ddiiv oulir, innovionl oulir, mporry chng nd lvl shif his sudy focuss solly on h ddiiv oulir s i is h mos common yp found in im sris du o is ssociion wih humn rror such s yping nd rcording misks o undrsnd h nur of n ddiiv oulir, simulions of ARMA, im sris r conmind wih n ddiiv oulir hn, w xmin h ffc of h rspcd ddiiv oulir on obsrvions nd rsiduls is found h n ddiiv oulir dos no ffc obsrvions prior o is xisnc nd h ons subsqun o i Howvr, givn h occurrnc of h ddiiv oulir, h ddiiv oulir is sn o hv noicbl ffc on h rsidul of nd

2 A Sudy on h Nur of n Addiiv Oulir in ARMA, Modls 363 Kywords: ARMA, Modls, Simulion Sudy, h Nur of Addiiv Oulir, nroducion Abnorml obsrvions which r ihr oo lrg or oo smll s comprd o h rs of h obsrvion r ofn ncounrd in im sris hs brrn obsrvions r rgrdd o s oulirs As nuisnc hy r in im sris nlysis, priculrly in modl idnificion, oulirs cn provid usful informion h is oo vlubl o b ignord s hir prsnc signifis imporn vns for xmpl n inrvnion or n unxpcd incidn n h ffor o unfold h undrlying fcors o h xisnc of n oulir, oulirs r chrcrizd nd nmd bsd on hir ribus such s h ddiiv oulir AO, innovionl oulir O, mporry chng C nd lvl shif LS ch of which corrsponds o spcific vn Associd wih isold incidns lik rcording nd yping rror, AO is h mos common yp of oulir found in im sris hus, his sudy focuss solly on h nur of n AO highlighing on is ffcs on obsrvions nd rsiduls Among h rlis sudy on AO is conducd by Fox 97 n his work, Fox nlyzd h AO nd O of non-ssonl AR p procsss whr h uorgrssiv prmrs r o b known H hn proposd mhod which dcs nd rmovs h oulir ffc [] An xnsion of h AR p modl is h ARMA p,q modl hroughou h yrs, mny sudis on h oulirs concrning ARMA p,q modls hv bn crrid ou for insnc [4], [8], [9] nd [] An oulir-fr im sris Z h follows n uorgrssiv moving vrg ARMA procss is dfind s Φ Z Θ p whr Φ Φ Φ q p nd Θ Θ Θ q r polynomils in, is h bckshif opror such h Z Z nd { } is squnc of whi nois rndom vribls, idniclly nd indpndnly disribud s N, σ Following Z cn lso b wrin s Θ Z Φ Modifid modl of h dscrib h gnring mchnism of AO, for h simpl cs of singl oulir, cn b dfin s Z Y Z 3 Z 4 From quion 5 whr Y is h obsrvd sris, Z is h unobsrvbl oulir fr sris in, rprsns h mgniud of h oulir, nd is im indicor vribl signifying h occurrnc of n oulir whn n oulir is spod nd ohrwis hrfor h AO modl cn b lookd upon s h gross rror modl s i only involvs h h obsrvion nd i is drminisic in nur Formulion for h Esimion of Rsiduls Esimd rsiduls { ê } which my or my no b conmind wih oulirs cn b dfind s,

3 364 Azmi Zhrim, Rfizh Rli, Rdn Mohmd Aok, Kmrulzmn brhim nd Ahmd Mhir Rzli Y 6 whr is dscribd s 7 wih bing h wighs for byond modrly lrg vlu J h ssnilly qul o whn h roos of li ousid of h uni circl W hv from 5-7 h Y 8 Following in quion 8, rsiduls bfor h occurrnc of n AO nd h rsidul prining o i r givn rspcivly s: < 9 o obin h rsiduls for n,, w xpnd quion 8 s follows: u, hrfor, im, For n numbr of obsrvions, quions 8 cn b summrizd s follows []: n n n

4 A Sudy on h Nur of n Addiiv Oulir in ARMA, Modls llusrions o illusr h nur of n AO in ARMA, modl, w considr simpl cs whn nd h prmrs of ARMA, r known Nx, n oulir-fr im sris of n is simuld using h rimsim procdur in R pckg wih r 7 nd m 3 hs obsrvions r dnod s Y nd plod in Figur whil h rsiduls r shown in Figur Figur : Plo for h unconmind sris Y Figur : Plo for h rsiduls of h unconmind sris Y 3 AO Effcs on Obsrvions o illusr h ffcs of n AO on obsrvions, of mgniuds 5, nd 5 r pplid o cr n rificil AO 5 s mniond in 3-4 Givn h non-conmind

5 366 Azmi Zhrim, Rfizh Rli, Rdn Mohmd Aok, Kmrulzmn brhim nd Ahmd Mhir Rzli obsrviony 5 756, w r lf wih Y 5, , Y 5, 756 nd Y 5, rspcivly For clrr picur, only Y4 o Y6 of h conmind nd unconmind sris r plod in Figur 3 As shown in h plo, h dgr of n AO crd 5 corrsponds wih h mgniud of ssocid wih i Dspi h suddn ump 5 in ll h conmind sris, h plo indics no sign of h AO ffcing obsrvions subsqun o i 3 AO Effcs on Rsiduls Using quions 8-, rsiduls of h conmind sris mniond in scion 3 r obind for nlysis Accommoding rsiduls ê4 o ê 6 of h conmind sris nd h unconmind sris in Figur 4, w find h h rsiduls prior o n AO rmin unchngd s dscribd in 9 Similr o Figur 3, suddn umps 5 suggs h h AO conmind rsidul corrsponds wih h mgniud of ssocid wih i Howvr, unlik Figur 3, h AO sms o hv subsnil ffc on h rsidul of 5 Figur 3: Plo for h conmind nd unconmind sris

6 A Sudy on h Nur of n Addiiv Oulir in ARMA, Modls 367 Figur 4: Plo for h rsiduls of conmind nd unconmind sris 4 Conclusion AO is h mos common yp of oulir found in im sris s i is usully ssocid wih humn rror such s yping misk hrfor n AO is sid o b drminisic in nur i no ffcing ohr obsrvions subsqun o i s dscribd in quion 4 On h conrry, quion suggss h n AO my hv subsnil impc on h rsidul concrning i nd h ons subsqun o i o illusr h nur of n AO in ARMA, modls, simulion sudis r crrid ou o nlyz h ffcs of n AO on obsrvions nd rsiduls in scion 3 nd 3 rspcivly is found h n AO dos no ffc obsrvions prior o is xisnc nd h ons following i Howvr, givn h occurrnc of n AO, h AO is sn o hv noicbl ffc on h rsidul nd of n ARMA, modl

7 368 Azmi Zhrim, Rfizh Rli, Rdn Mohmd Aok, Kmrulzmn brhim nd Ahmd Mhir Rzli Rfrncs [] A Zhrim 996 Oulirs nd Chng Poins in im Sris D, PhD hsis, Univrsiy of Nwcsl upon yn [] A Zhrim, Mohmd, Ahmd, S Abdullh & MZ Omr 6 Prformnc s Sisics for Singl Oulir Dcion in ilinr,,, Modls, WSEAS rnscions on Mhmics, ssu, Vol 5, pp [3] A Zhrim, Mohmd, S Abdullh & MSYhy 8 An Evluion of s Sisics for Dcing Lvl Chng in L,,, Modls, WSEAS rnscions on Mhmics, ssu, Vol 7, pp 67-7 [4] C Chn & LM Liu 993 Join simion of modl prmrs nd oulir ffcs in im sris, Journl of Amricn Sisicl Associion 88, pp [5] Chng 98 Oulir in im Sris, PhD hsis, Dprmn of Sisics, Univrsiy of Wisconsin-Mdison [6] Mohmd 5 Oulirs in ilinr im Sris Modl, PhD hsis, Mr Univrsiy of chnology [7] K brhim, R Rli & A Zhrim 8 On h Dcion of Oulirs for Wr Lvls of Lng Rivr, Procdings of h s WSEAS nrnionl Confrnc on Mulivri Anlysis nd is Applicion in Scinc nd Enginring, pp9- [8] RS sy 986 im Sris Modl Spcificion in h Prsnc of Oulirs, Journl of Amricn Sisicl Associion 8, pp3-4 [9] G Rischrd & G Anill 99 A robus look h us of rgrssion dignosics, h Sisicin 4, pp 4-53 [] WS Chn 99 A No on im Sris Modl Spcificion in h Prsnc of Oulirs, Journl of Applis Sisics, Vol 9, no, pp 7-4 [] WSWi im Sris Anlysis: Univri nd Mulivri Mhods, oson: Prson Addison Wsly, 6, pp3-9