The Comparison of Outlier Detection in Multiple Linear Regression

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1 Poceedngs of he Wold Congess on Engneeng 2012 Vol I, July 4-6, 2012, London, U.K. The Compason of Oule Deecon n Mulple Lnea Regesson Pmpan Amphanhong Absac Fou Oule deecon appoaches n mulple lnea egessons ae evewed, nvesgaed and compaed. The Mone Calo smulaon s based on he medan absolue devaon and he obusness sandad devaon ceon. The esuls of hee and fve egessos show ha he MAD s bee han he RSD fo all suaons. The dsance and he Mahalanobs dsance ( ) ae bee han he ohes fo all sample szes wh dffeen pecenages oules n case of he X s oules. Fo he Y s oules, PRESS esdual ( () ) and R-suden ( ) appoaches ae pefomed bee han he ohes. Fo he boh of X s and Y s oules, he PRESS esdual ( () ) and he Mahalanobs dsance ( ) ae bee han he ohes fo all sample szes wh dffeen pecenages oule. Index Tems Mean absolue devaon, Mean squae eo, Robusness sandad devaon, Mone Calo smulaon, Mulple lnea egesson, Resduals. T I. INTRODUCTION he mulple egesson models ae wdely used o sudy he elaonshp beween he esponse vaable and ndependen vaables. A geneal mulple lnea egesson model s y X whee y s an n 1 veco of obseved values of he dependen vaable, X x 1, x2,..., x p, x j s an n 1 veco of he values of x, o egessos, 1, 2,..., n., s a p 1 veco of unknown paamees, and s an n 1 veco of eos wh a adonal assumpon 2 of Gauss-Makov heoem s N0, I. Vaous appoaches o esmae unknown paamees of he model whch have popey as he bes lnea unbased esmao (BLUE), fo example, he odnay leas squaes (OLS) and he maxmum lkelhood esmao (MLE). Howeve, n pacce, ae no always belonged he assumpon, hen he OLS and MLE may be abaly bad. Fuhemoe, f oules ae exss n he model, hen alenave appoaches ae needed. Thee ae many auhos have been suded and analyzed he mulple lnea egesson model when daa has oules (see [1], [2], [3] and [4]). Accodng o he leaues (see[10], [11], [12], [13], [14], [15], [17], [18] and [19]), s vey mpoan o know how o deec he oules n mulple lnea egesson model and should be suded moe caefully. Manuscp eceved Apl 15, Ths wok was suppoed n pa by he Rajamankala Unvesy of Technology Suwanabhum. P. Amphanhong s wh he Rajamankala Unvesy of Technology Suwanabhum, Faculy of Scence and Techonology, Depamen of Mahemacs, Suphanbu Campus, Thaland, e-mal: pm_pmpan@homal.com. ISSN: (Pn); ISSN: (Onlne) Ths pesen pape, he auho evews fou oule deecon appoaches n mulple lnea egesson model and hen compaes hes esuls by usng he ceon whch ae called he medan absolue devaon and he obusness sandad devaon. II. ANALYTICAL APPROACHES In hs pape, we evew fou oule deecon appoaches n mulple lnea egesson model as many leaues used o denfy he exsence of oules. A. PRESS Resduals The obsevaons X j, 1, 2,, n fo each j1, 2,, p ae compued he pedcon eo whee he fed of he h ae compued based on n 1 obsevaons and deleed he h obseved values. The PRESS esduals may be compued fom he ha max and he esdual as () e /(1 h), (1) h 1, 2,, nwhee h s he dagonal elemen of 1 H X( X ' X) X '. If () 3 hen he h obsevaon s denfed as oules (see [4]). B. R-Suden A fomal esng pocedue fo oules deecon based on R-suden s gven by 2 e / ˆ( ) (1 h), (2) 1, 2,, nwhee ( /2 n), n ( p 1) ndcaes he exsence oules. (see [4]). C. Dsance The s 1/2 ( h e)/( (1 h)), (3) 1, 2,, n. Fo each obsevaon compue o ( he )/(1 h) whch ells how much he pedced value y ˆ, a he desgn pon x would be affeced f he h case wee deleed. Belsley, Kuh and Welsch [5] suggesed ha any obsevaon fo whch 2 p / n waans aenon fo oules. D. Mahalanobs Dsance The measue of he leveage by means fo (Mahalanobs dsance) s ( ) ( )' ( n 1)[ h 1/ n] 1 2 2, (4) 1, 2,, n whee 1/ n( ) and 2 1/( n 1)* n 1

2 Poceedngs of he Wold Congess on Engneeng 2012 Vol I, July 4-6, 2012, London, U.K. n 2 2 ( )'( ). If p 1 whee 2 p 1,0.95 s he 95 h 1,0.95 pecenle of a ch-squae dsbuon wh p 1 degees of feedom hen hee s an oule (see [6]). A. Ceons III. MAIN RESULTS Thee ae many sascal values compued fom he sample daa ha can be used o denfy he exsence of oules. Mos eque dffeen sascal ceon of he sandad devaon (S.D.) of he esduals, e1, e2,, en, bu he measue based on he mean squaed eo (MSE) s no obus, snce may be hghly nfluenced by evens of small pobably. Ths pape, auho uses he medan absolue devaon whch s denoed MAD (see [7], [16]) and he obusness sandad devaon whch s denoed RSD (see [8]), ae defned as and MAD( e ) Med e Med( e) , (5) RSD( e ) 2.1Med e, (6) P. Amphanhong and P. Suwaee [1] suded he exsences of oule s deecon n sascs and hen compason pocedues n he mulple lnea egesson. They showed ha Mahalanobs dsance denfes he pesence of oules moe ofen han he ohes fo small, medum and lage sample szes wh dffeen pecenages oules n he X-oules and n boh he X-Y oules. The nex bes sascs fo he deecon ae R-suden and dsance. As fo he Y- oules, R-suden and PRESS esdual pefom bee han he ohe appoach. B. Numecal Resuls One housand of daa ses ae geneaed fom he model y 0 1x 1 e, 1, 2,, n whee all egesson coeffcens ae fxed j 1, fo each 1, 2,, n and j1, 2,, p and he eos ae assumed o be ndependen. n p The explanaoy vaables xj R ae sampled ndependenly fom a N (0,1). The sample daa ses ae geneaed unde (p=3, p=5) egessos and he sample szes ae small szes (n=10), medum szes (n=20 and n=30), and lage szes (n=50 and n=100), wh dffeen pecenage of oules (10%, 20% and 30%). The vaaon of fou oule deecon appoaches povde an ndcaon of he sensvy of hem, hen compason of hes esuls by counng he numbe of mes ha each appoaches can be denfy oules. The compuaons gve he bes of oule deecon appoaches fo dffeen sample szes and pecenages of oule wh 1,000 eplcaons. The esuls of fou oule deecon appoaches ae as followng; Table 1: Compasons of sascs value of oule deecon by pecenage of X s Oules wh hee egessos. () () Table 2: Compasons of sascs value of oule deecon by pecenage of X s Oules wh hee egessos (con.) I can be seen fom able 1 o 2, he bes ceon s medan absolue devaon, and he bes of X s oule deecon appoaches ae and. The pefomances ae hghes values of oule deecon (1.000) fo all sample szes and pecenage of oules. Fuhemoe, he pefomance of () and ae hgh fo lage sample szes and all pecenage of oules [Fg. 1(a)]. ISSN: (Pn); ISSN: (Onlne)

3 Poceedngs of he Wold Congess on Engneeng 2012 Vol I, July 4-6, 2012, London, U.K. Table 3: Compasons of sascs value of oule deecon by pecenage of Y s Oules wh hee egessos. () () Table 4: Compasons of sascs value of oule deecon by pecenage of Y s Oules wh hee egessos (con.) I can be seen fom able 3 o 4, he bes ceon s medan absolue devaon, and he bes of Y s oules deecon appoach s (), fo all pecenage of oules and sample szes. Fuhemoe, he pefomance of s bee han he and n medum (n=30) and lage sample szes fo all pecenage of oules [Fg. 1(b)]. Table 5: Compasons of sascs value of oule deecon by pecenage of boh X s and Y s Oules wh hee egessos. () () Table 6: Compasons of sascs value of oule deecon by pecenage of boh X s and Y s Oules wh hee egessos (con.) I can be seen fom able 5 o 6, he bes ceon s medan absolue devaon and he bes of boh X s and Y s oule deecon appoaches ae () and. The pefomances of () and appoaches ae hghes values of he deecon oules (1.000) n all sample szes and pecenage of oules. Fuhemoe, he pefomance of s bee han he fo lage sample szes and all pecenage of oules [Fg. 1(c)]. ISSN: (Pn); ISSN: (Onlne)

4 Poceedngs of he Wold Congess on Engneeng 2012 Vol I, July 4-6, 2012, London, U.K. values of oules deecon Oules n X-decon () () Table 7: Compasons of sascs value of oule deecon by pecenage of X s Oules wh fve egessos. () () () Sample szes () (a) values of oules deecon values of oules deecon Oules n Y-decon Sample szes (b) Oules n X-Y-decon () () () () () () () () Table 8: Compasons of sascs value of oule deecon by pecenage of X s Oules wh fve egessos (con.) Sample 30 szes (c) Fgue.1 A Compason of Sascs value of oule deecon by Sample Szes wh Thee Regessos.(a) X s Oules; (b) Y s Oules; (c) Boh X s and Y s Oules. I can be seen fom able 7 o 8, he bes ceon s medan absolue devaon. The pefomance of and ae hghes values of deecon oule (1.000) fo all sample szes and pecenage of oules. Fuhemoe, we compae he esuls of fve egessos. The compuaons gve he bes of oule deecon appoaches fo dffeen sample szes and pecenages of oule wh 1,000 eplcaons, he esuls ae as followng; ISSN: (Pn); ISSN: (Onlne)

5 Poceedngs of he Wold Congess on Engneeng 2012 Vol I, July 4-6, 2012, London, U.K. Table 9: Compasons of sascs value of oule deecon by pecenage of Y s oules wh fve egessos. () () Table 11: Compasons of sascs value of oule deecon by pecenage of boh X s and Y s Oules wh fve egessos. () () Table 10: Compasons of sascs value of oule deecon by pecenage of Y s Oules wh fve egessos (con.) Fom able 9 o 10, he bes ceon s medan absolue devaon. The bes of Y s oules deecon s (), s pefomance ae good fo all sample szes and pecenages of oules. Fuhemoe, he pefomance of s bee han he and fo lage sample szes and all pecenage of oules. Table 12. Compasons of sascs value of oule deecon by pecenage of boh X s and Y s Oules wh fve egessos (con.) Fom able 11 o 12, he bes ceon s medan absolue devaon. The bes oule deecon appoaches n boh X s and Y s oules ae () and. The pefomance of and () ae hghes values of deecon oule (1.000) fo all sample szes and pecenage of oules. Fuhemoe, he pefomance of s bee han he fo lage sample szes and all pecenage of oules, he pefomance of s bee han fo small and medum szes (n=30) and all pecenage of oules. ISSN: (Pn); ISSN: (Onlne)

6 Poceedngs of he Wold Congess on Engneeng 2012 Vol I, July 4-6, 2012, London, U.K. IV. CONCLUSIONS The Mone Calo smulaon shows he pefomance of fou oule deecon appoaches n mulple lnea egesson. We use he MAD and RSD as he ceons, whch MAD s bee han RSD fo all suaons. We ge he same ageemen fo hee and fve egessos, he dsance and he Mahalanobs dsance ( ) ae bee han he ohes fo all sample szes wh dffeen pecenages oules n case of he X s oules. The PRESS esdual ( () ) and R-suden ( ) appoaches ae pefomed bee han he ohes n he case of Y s oules. The PRESS esdual ( () ) and he Mahalanobs dsance ( ) ae bee han he ohes fo all sample szes wh dffeen pecenages oule n he case of boh of he X s and Y s oules. Fuhemoe, we have seen ha he pefomance of s bee fo lage sample szes and all pecenage of oules n all cases of he oules. REFERENCES [1] P. Ampanhong and P. Suwaee, A Compaave Sudy of Oule Deecon Pocedues n Mulple Lnea Regesson (Peodcal syle Submed fo publcaon), IMECS 2009 submed fo publcaon. [2] A. C. Aknson, In: Plos, Tonasfomaons, and Regesson: An Inoducon o Gaphcal Mehods of Dagnosc Regesson Analyss, Oxfod : Claendon Pess, [3] A. S. Had and J. S. Smonoff, Pocedues fo he Idenfcaon of Mulple Oules n Lnea Models, J. Amme. Sas. Assoc. Vol. 88, 1993, pp [4] P. J. Hube, Robus Sasc, New Yok : John Wley & Sons, [5] D. A. Belsley and R. E. S. Welsch, Regesson Dagnoscs : Idenfyng Influenal Daa and Souce of Collneay, New Yok: John Wley & Sons, [6] P. J. Rousseeuw and A. M. Leoy, Robus Regesson and Oule Deecon, New Yok : John Wley & Sons, [7] A. S. Kosnsk, A pocedue fo he deecon of mulvaae oules, Compuaonal Sascs and Daa Analyss. Vol. 29, 1998, pp [8] J. O. Ramsay, A Compaave Sudy of Seveal Robus Esmaes of Slope, Inecep, and Scale n Lnea Regesson, Jounal of he Amecan Sascal Assocaon. Vol. 72, 1977, pp [9] Y. Jazhong, A Mone Calo Compason of Seveal Hgh Beakdown and Effcen Esmao, Compuaonal Sascs & Daa analyss. Vol. 30, 1999, pp [10] H. P. Lopuhaa and P. J. Rousseeuw, Beakdown Pon of Affne Equvaan Esmaos of Maulvaae Locaon and Covaance Mace, Techncal Repo, Faculy of Mahemacs and Infomacs, Nehelands: Delf Unvesy of Technology, [11] D. C. Mongomey, E. A. Peck and G.G. Vnng, Inoducon o Lnea Regesson Analyss, 3d ed. New Yok: John Wley & Sons, [12] V. Bane and T. Lews, Oules n Sascal Daa, 3d ed. UK : Wley, Chcese, [13] D. Bkes and Y. Dodge, Alenave Mehods of Regesson, New Yok : John Wley & Sons, [14] A. S. Had, A Modfcaon of a Mehod fo he Deecon of Oules n Mulvaae Samples, J. Roy. Sas. Soc. Se B. Vol. 56, 1994, pp [15] T. P. Ryan, Moden Regesson Mehods, New Yok: John Wley & Sons, [16] P. J. Rousseeuw, Leas Medan of Squaes Regesson, J. Amme. Sas. Assoc. Vol. 79, 1984, pp [17] P. J. Rousseeuw and K. V. Dessen, A Fas Algohm fo he Mnmum Covaance Deemnan Esmao, Technomecs, Vol. 41, 1999, pp [18] J. W. Wsnowsk, D. C. Mongomey and J. R. Smpson, A Compaave Analyss of Mulple Oule Deecon Pocedues n he Lnea Regesson Model, Compuaonal Sascs and Daa Analyss. Vol. 6, 2001, pp [19] J. You, A Mone Calo compason of seveal hgh beakdown and effcen esmaos, Compuaonal Sascs and Daa Analyss. Vol. 30, 1999, pp ISSN: (Pn); ISSN: (Onlne)