The Distribution of the Inverse Square Root Transformed Error Component of the Multiplicative Time Series Model

Transcription

1 Journal of Modrn Applid Saisical Mhods Volum 4 Issu Aricl Th Disribuion of h Invrs Squar Roo Transformd Error Componn of h Muliplicaiv Tim Sris Modl Brigh F. Ajibad Prolum Training Insiu, Nigria, qualrigh_brigh@yahoo.com Chinw R. Nwosu Nnamdi Azikiw Univrsiy, Nigria J. I. Mbgdu Univrsiy of Bnin, Nigria Follow his and addiional works a: hp://digialcommons.wayn.du/jmasm Par of h Applid Saisics Commons, Social and Bhavioral Scincs Commons, and h Saisical Thory Commons Rcommndd Ciaion Ajibad, Brigh F.; Nwosu, Chinw R.; and Mbgdu, J. I. (05) "Th Disribuion of h Invrs Squar Roo Transformd Error Componn of h Muliplicaiv Tim Sris Modl," Journal of Modrn Applid Saisical Mhods: Vol. 4 : Iss., Aricl 5. DOI: 0.37/jmasm/ Availabl a: hp://digialcommons.wayn.du/jmasm/vol4/iss/5 This Rgular Aricl is brough o you for fr and opn accss by h Opn Accss Journals a DigialCommons@WaynSa. I has bn accpd for inclusion in Journal of Modrn Applid Saisical Mhods by an auhorizd dior of DigialCommons@WaynSa.

2 Journal of Modrn Applid Saisical Mhods Novmbr 05, Vol. 4, No., Copyrigh 05 JMASM, Inc. ISSN Th Disribuion of h Invrs Squar Roo Transformd Error Componn of h Muliplicaiv Tim Sris Modl Brigh F. Ajibad Prolum Training Insiu Efurum, Warri, Nigria Chinw R. Nwosu Nnamdi Azikiw Univrsiy Awka, Nigria J. I. Mbgbu Univrsiy of Bnin Bnin Ciy, Nigria Th probabiliy dnsiy funcion, man and varianc of h invrs squar-roo ransformd lf-runcad N, rror componn of h muliplicaiv im sris modl wr sablishd. A comparison of ky-saisical propris of and confirmd normaliy wih man bu wih Var Var whn 0.4. Hnc 0.4 is h rquird condiion for 4 succssful ransformaion. Kywords: Muliplicaiv im sris modl, Error componn, Lf runcad normal disribuion, Invrs squar roo ransformaion, Succssful ransformaion, Momns Inroducion Th gnral muliplicaiv im sris modl for dscripiv im sris analysis is X T S C,,,... n () whr for im, X dnos h obsrvd valu of h sris, T is h rnd, S, h sasonal componn, C h cyclical rm and is h random or irrgular componn of h sris. Modl () is rgardd as adqua whn h irrgular componn is purly random. For a shor priod of im, h cyclical componn is Mr. Ajibad, B. F. is a Dpuy Chif Officr in h Dparmn of Gnral Sudis. him a: qualrigh_brigh@yahoo.com. C. R. Nwosu is an Associa Profssor in h Dparmn of Saisics. J. I. Mbgbu is a Profssor in h Dparmn of Mahs and Saisics. 7

3 AJIBADE ET AL. suprimposd ino h rnd (Chafild, 004) o yild a rnd-cycl componn dnod by M and hnc X () M S whr ar indpndn idnically disribud normal rrors wih man and varianc 0 N, According o Uch (003), h lf runcad normal disribuion N, for X is f x 0 x 0 x k 0 x (3) Using Equaion 3, Iwuz (007) obaind h lf runcad normal disribuion N, for X as f LTN x x 0 x 0 0 x (4) wih man E LTN X (5) and 73

4 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Var LTN X Pr (6) Iwuz (007) also showd ha LTN f x > 0 providd < Daa ransformaions ar h applicaion of mahmaical modificaions o valus of a variabl. Thr ar a gra variy of possibl daa ransformaions, including log X, X,,, X, and. In pracic many X X X muliplicaiv im sris daa do no m h assumpions of a paramric saisical analysis; hy ar no normally disribud, h variancs ar no homognous or boh. In analyzing such daa, hr ar wo choics: i. Adjusing h daa o fi h assumpions by making a ransformaion, or ii. Dvloping nw mhods of analysis wih assumpions which fi h daa in is original form. If a saisfacory ransformaion can b found, i will almos always b asir and simplr o us i rahr han dvloping nw mhods of analysis (Turky, 957). Hnc h nd for his work which aims a finding condiions for saisfacory invrs squar roo ransformaion wih rspc o h rror componn of h muliplicaiv im sris modl from a sudy of is disribuion. A ransformaion is considrd saisfacory or succssful, if h basic assumpions of h modl ar no violad afr ransformaion. (Iwuz al., 008)Th basic assumpions of a muliplicaiv im sris modl placd on h rror componn ar: (i) uni man (ii) consan varianc (iii) Normaliy. According o Robrs (008), ransforming daa mad i much asir o work wih - I was lik sharpning a knif. For mor informaion on choic of appropria ransformaions s Osborn (00), Osborn (00) and Wahanachwakul (0). 74

5 AJIBADE ET AL. Daa Classificaion For a im sris daa o b classifid appropria for invrs squar roo ransformaion, i. h daa mus b amnabl o h muliplicaiv im sris modl. Th approprianss of h muliplicaiv modl is accssd by (a) displaying h daa in h Buy s-ballo Tabl. (b) Ploing h priodic (yarly) mans (μ i ) and sandard dviaions σ i agains h priod (yar) i. If hr is a dpndncy rlaionship bwn μ i and σ i, hn h muliplicaiv modl is appropria. ii. h varianc mus b unsabl. Th sabiliy of h varianc of h im sris is ascraind by obsrving boh h row and column mans and sandard dviaions. If h varianc is no sabl h appropria ransformaion is drmind using Barl (947) as was applid by Akpana and Iwuz (009); Y log X, X, (7) Th linar rlaionship bwn h naural log of priodic sandard dviaions (log σ i ) and naural log of h priodic mans (log μ i ) is givn as log log (8) i i Th valu of slop β according o Barl (947) should b approximaly.5 for h invrs squar roo ransformaion (s Tabl ). Tabl. Barl s ransformaions for som valus of β 0 Transformaion No ransformaion X log X 3 X 3 - X X X 75

6 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Background of h Sudy Sinc Iwuz (007) invsigad h ffc of h logarihmic ransformaion on h rror componn, ( ~ N (, σ )) of h muliplicaiv im sris modl, a numbr of sudis invsigaing h ffcs of daa ransformaion on h various componns of h muliplicaiv im sris modl hav bn carrid ou. (S Iwuz al., 008; Iwu al., 009; Ouony al., 0; Nwosu al., 03; and Ohakw al., 03). Th ovrall aim of such sudis is o drmin h condiions for succssful ransformaion. Tha is, o sablish h condiions whr: a. h rquird basic assumpions of h modl ar no violad afr ransformaion, wih rspc o (i) h rror rm (ii) h sasonal componn. b. wih rspc o h rnd componn, hr is no alraion in h form of h rnd curv. In ohr words h form of h rnd curv in h original sris is mainaind in h ransformd sris. componn varianc Iwuz (007) found ha h logarihmic ransformaion of h rror, providd 0. assumpion for h rror rm N o log is normal wih man 0 and, in which cas. I was sablishd ha h, for h addiiv modl obaind afr h logarihmic ransformaion, is valid if and only if σ < 0.0. Obsrv from Tabl ha β for a im sris daa o b classifid fi for logarihmic ransformaion. Ouony al. (0) invsigad h disribuion and propris of h rror componn of h muliplicaiv im sris modl undr squar roo ransformaion, and found ha h squar roo ransformd rror componn is normally disribud wih man and varianc ims ha of 4 h unransformd rror componn. Tha is Var Var 4 whn 0 < σ 0.3. Thus 0 < σ 0.3 is h rcommndd condiion for succssful squar roo ransformaion. Only im sris daa wih ar classifid fi for squar roo ransformaion. Similarly, Nwosu al. (03), whil invsigaing h disribuion of h invrs ransformd rror componn of h muliplicaiv im 76

7 AJIBADE ET AL. sris modl, obaind ha h dsirabl saisical propris of and wr found o b approximaly h sam and normally disribud wih uni man for σ 0.0. Hnc, σ 0.0 is h rcommndd condiion for succssful invrs ransformaion of h muliplicaiv im sris modl. Tim sris daa classifid fi for invrs ransformaion mus hav β. Also, Ohakw al. ha N, in (03) found ha for h squar ransformaion h inrval 0 < σ Hnc, 0 < σ 0.07is h condiion for succssful squar ransformaion. Obsrv ha a im sris daa is classifid fi for squar ransformaion whn β -. No ha h ovrall aim of hs works is o sablish condiions for succssful ransformaion, hnc provid br choic of righ ransformaion. According o Robrs (008), choosing a good ransformaion improvd his analyss in hr ways: (i) incras in visual clariy as graphs wr mad mor informaiv (ii). Rducion or liminaion of oulirs (iii). Incras in saisical clariy; his saisical s bcam mor snsiiv, F and valus incrasd making i mor likly o dc diffrncs whn hy xis. Jusificaion for his Sudy Th valu of h slop, cagorizd im sris daa ino muually xclusiv groups, in h sns ha any im sris daa blongs xclusivly o on and only on group hnc can only b approprialy ransformd by only on of h six ransformaions lisd in Tabl. Thus dspi h fac ha Iwuz (007), Ouony al., (0), Nwosu al. (03), and Ohakw al. (03) carrid ou similar sudis wih rspc o h logarihmic, squar roo, invrs and squar ransformaions rspcivly, his work on invrs squar roo ransformaion is sill vry ncssary sinc rsuls sablishd for h abov lisd four ransformaions canno b applid in h analysis of im sris daa rquiring invrs squar roo ransformaion. Invrs Squar Roo Transformaion 3 Whn, adop invrs squar roo ransformaion on h muliplicaiv im sris modl givn in Equaion o obain 77

8 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Y X M S M S (9) whr M, S and, 0 M S Bcaus dos no admi ngaiv or zro valus, h us of h lf runcad normal disribuion as h pdf of shall b xploid. Thus, i will b of inrs o find wha h disribuion of is. Is iid N,. Wha is h rlaionship bwn and? Aim and Objcivs Th aim of his work is o obain h disribuion of h invrs squar roo ransformd rror componn of h muliplicaiv im sris modl and h objcivs ar: i. o xamin h naur of h disribuion. ii. o vrify h saisfacion of h assumpion on h man of h rror rms; μ =. iii. o drmin h rlaionship bwn and. Mhodology To achiv h abov sad objcivs h following wr conducd: L X = and Y = = X. Obain h pdf of, g(y).. Plo h curvs of h wo pdfs, g(y) and f LTN (x) for various valus of. 3. Obain h rgion whr g(i) saisfis h following normaliy condiions (Bll-shapd condiions). i. Mod Man. ii. Mdian Man. 78

9 AJIBADE ET AL. iii. iv. Approvd normaliy s, Andrson Darling s s saisic (AD) was usd o confirm h normaliy of h simulad rror rms and h invrs squar roo ransformd rror rm. Y = = X for som valus of σ Obain and us h funcional xprssions for h man and varianc of o valida som of h rsuls obaind using simulad daa. Th probabiliy dnsiy funcion of Y, g( y) x Givn h pdf of X in Equaion 4 and h ransformaion Y x hn X dx and y dy y 3 using h ransformaion of variabl chniqu dx g y f x LTN dy (s Frund & Walpol, 986). Hnc y,0 y g y 3 y 0 y 0 (0) 79

10 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Plo of h Probabiliy dnsiy curvs f x f x LTN and g(y) Using h pdf of h wo variabls givn in Equaion 4 and Equaion 0, h curvs f x and g(y) wr plod for som valus of (0, 0.4]. For wan of spac only fiv ar shown in Figurs o 5. Figur. Curv Shaps for σ = 0.06 Figur. Curv Shaps for σ =

11 AJIBADE ET AL. Figur 3. Curv Shaps for σ = 0.5 Figur 4. Curv Shaps for σ = 0.3 Figur 5. Curv Shaps for σ = 0.4 8

12 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Obsrvaions: i. Th curv g(y) is posiivly skwd for σ > 0.5 (s Figurs 3-5). ii. f(x) is posiivly skwd for σ > 0.30 (s Figur 5) as rpord in Iwuz (007). Normaliy Rgion for g(y) From Figurs o 5, i is clar ha h curv g(y) has on maximum poin, y max (mod), and on maximum valu, g(y max ), for all valus of σ. To obain h valus of σ ha saisfy h symmric and bll-shapd condiion of mod = man, w invok Roll s Thorm and procd o obain h maximum poin (mod) for a givn valu of σ. Diffrniaing g(y) in Equaion 0 givs g'( y) 4 4 y 3 y 3y y 3 y y () y ( y ) y y Equaing g`(y) = 0, givs ( y ) 3 y 0 y y y 0 () Puing w = y in Equaion, givs 8

13 AJIBADE ET AL. 3 w w 0 (3) Solving Equaion 3, givs Bcaus y max is posiiv hn 6 w 3 6 w 3 hnc y 6 3 and y 6 3 max Th bll-shapd condiion would imply y max, s Tabl for h numrical compuaion of y max

14 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Tabl. Compuaion of y max 6, for [0.0, 0.3] 3 y max y max 6 3 y max 6 3 y max Thus g(y) is symmrical abou wih Mod Man corrc o wo dcimal placs whn 0 < σ < and corrc o on dcimal plac whn 0 < σ <

15 AJIBADE ET AL. Us of simulad rror rms To find h rgion whr h bll-shapd condiions (ii-iii) lisd in mhodology N, for, ar saisfid, w mad us of arificial daa gnrad from subsqunly ransformd o obain for Valus of h rquird saisical characrisics wr obaind for ach variabl and as shown in Tabls 3 o 6. For ach configuraion of (n = 00, 0.05 σ 0.5), 000 rplicaions wr prformd for valus of σ in sps of 0.0. For wan of spac h rsuls of h firs 5 rplicaions ar shown for h configuraions, (n = 00, σ = 0.06), (n = 00, σ = 0.), (n = 00, σ = 0.5), and (n = 00, σ = 0.). Funcional xprssions for h man and varianc of g(y) By dfiniion, h man of Y, E(Y) is givn by: y 0 E( Y) yg( y) dy dy 0 y (4) l u, hn y and du 3 y u dy, for u 0 u 0 u u du k 3 u 0 (5) E( Y) k u u du whr k l u z, hn z u and du dz for z z z k k E( Y) ( z ) dz ( z ) dz (6) 85

16 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Tabl 3. Simulaion Rsuls whn σ = 0.06 X N,, 0.06 Y,, N, 0.06 Man SD Varianc Mdian AD p-valu Man SD Varianc Mdian AD p-valu No. For ach row, Var Var quals 4. 86

17 AJIBADE ET AL. Tabl 4. Simulaion Rsuls whn σ = 0. X N,, 0. Y,, N, 0. Man SD Varianc Mdian AD p-valu Man SD Varianc Mdian AD p-valu No. For ach row, Var Var quals 4. 87

18 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Tabl 5. Simulaion Rsuls whn σ = 0.5 X N,, 0.5 Y,, N, 0.5 Man SD Varianc Mdian AD p-valu Man SD Varianc Mdian AD p-valu No. For ach row, Var Var quals 4 xcp whr indicad by. For hos rows, Var Var quals 3. 88

19 AJIBADE ET AL. Tabl 6. Simulaion Rsuls whn σ = 0. X N,, 0. Y,, N, 0. Man SD Varianc Mdian AD p-valu Man SD Varianc Mdian AD p-valu < < < < < < < < < < < < < < < No. For ach row, Var Var quals 3. 89

20 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Using h binomial xpansion, 3 n nx n( n ) x n( n )( n ) x ( x)... (7)!! 3! (Smih and Minon, 008). 3 z ( z ) ( z ) ( z )...!! 3! 3 z 3( z ) 5( z )... (8) z k z 3( z ) 5( z ) E( Y)... d 8 48 (9) 3 z z z z z 3( z ) 5( z ) dz dz dz dz z z z dz dz EY ( ) z 3 z 3( z) 5( z) dz dz

21 AJIBADE ET AL. z z z dz dz EY ( ) z 3 3 z 3z 5z dz dz Pr z Pr x () Pr () EY ( ) Pr () () EY ( ) Pr (0) 9

22 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL To find h varianc, firs obain h scond momn; ( ) ( ) E Y y g y dy 0 y 0 y dy du l u hn du dy, 3 and dy 3 for u 0 y y u 0 u ( u) du k 3 0 () E( Y ) k u u du u whr k l u z hn uz and du dz for z z k E( Y ). ( z) dz Using h binomial xpansion on (+zσ) -, givn in Equaion 6 w hav 3 ( z ) z ( z ) ( z )... z k 3 E( Y ) [ z ( z ) ( z )...] dz () 9

23 AJIBADE ET AL. EY ( ) z z dz z dz z 3 z 3 z dz z dz Pr z Pr x () Pr x () 3 () E( Y ) Pr (3) 3 Obsrv h following:. Subsqun rms in sris (0) and (3) for E(Y) and E(Y ) rspcivly all hav as a facor. 93

24 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL 0. for σ 0. corrc o 4 dcimal placs. (S Tabl 7, column 3) 3. Condiions () and () imply ha all subsqun rms for E(Y) and E(Y ) ar all zros for σ 0.. Thus, wihou loss of gnraliy 3 EY ( ) Pr () for 0. 6 (4) and EY ( ) Pr () for 0. (5) hus Var( Y) E( Y ) [ E( Y)] Var( Y ) Pr () 3 Pr () 6 94

25 AJIBADE ET AL. Pr () 8 3 Pr () 6 (6) Numrical compuaions of man and varianc of Y Now compu h valus of E(Y) and Var(Y) for σ [0.0,0.] using h funcional xprssions obaind in Equaions 4 and 6, rspcivly. Tabl 7 shows h compuaions of E(Y) and Var(Y). For hs compuaions w wri and EY 3 B 0. 8A Var Y B 3 B 8A 6A whr A and B Pr From Tabl 7, columns 4 and 5, A = and B = for <0. 3 EY 0. (7) 8 and Var Y (8) 95

26 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Equaion 7 is h rlaionship obsrvd wih simulad daa in Tabls 3-6. Rsuls Th following rsuls wr obaind from h invsigaions carrid ou on h pdf of, g( y ) whr N,, lf runcad a 0. i. Th curv shaps ar bll-shapd, wih mod man whn 0 < σ 0.45 corrc o dcimal plac. Using simulad daa, whnvr σ < 0.5 ii. Mdian Man iii. E 3 Var iv. v. Var 8 4, hus var( ) Var( ) 4 is normally disribud whn σ 0.4. I was obsrvd ha h normaliy of a pdf curv a a poin b implid normaliy a poins 0 a b. Using h funcional xprssions for man and varianc of vi. vii. viii. 3 E 0. 8 corrc o dcimal placs (dp) whn σ 0. corrc o dp whn σ 0. 3 Var( ) Var 4 Var corrc o dp whn σ 0.04 corrc o dp whn σ

27 AJIBADE ET AL. Tabl 7. Compuaions of E(Y) & Var(Y) for σ [0.0, 0.3] A B ( ) EY Var( Y ) VarX / Var Y From h probabiliy dnsiy curvs, h rsuls obaind from simulad daa and h funcional xprssions for h man and varianc, σ 0.4 (inrscing rgion) is h rcommndd condiion for succssful invrs squar roo ransformaion. Th rsuls of his invsigaion oghr wih findings from similar N, undr ohr yps of invsigaions wih rspc o h rror rm 97

28 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Tabl 8. Summary of his and similar findings wih rspc o h rror rm log Disribuion of 0,,,, N, undr diffrn ransformaions Condiion for succssful ransformaion N 0. N 0. Rlaionship bwn σ and σ N 0.59 N,, 0.07 N 0.4 Conclusion From h rsuls of h invsigaions of h disribuions of h rror rm of h muliplicaiv im sris modl and is invrs squar roo ransformd rror rm, i is clar ha h condiion for succssful invrs squar roo ransformaion is σ < 0.4. This is bcaus h wo sochasic procsss and ar normally disribud wih man, bu wih h varianc of invrs squar roo ransformd rror rm bing on quarr of h varianc of h unransformd rror componn whnvr σ < 0.4, ousid his rgion ransformaion is no advisabl sinc h basic assumpion on h rror rm ar violad afr h ransformaion. This rlaionship bwn h wo variancs, Var Var, agrs wih findings of Ouony al. (0) undr squar 4 roo ransformaion, howvr h rgion of succssful ransformaion obaind is closr o h rgion obaind for h logarihmic and invrs ransformaions by Iwuz (007) and Nwosu al. (03). 98

29 AJIBADE ET AL. Rfrncs Akpana, A. C. & Iwuz I. S. (009). On applying h Barl ransformaion mhod o im sris daa. Journal of Mahmaical Scincs, 0(3), Barl, M. S. (947). Th us of ransformaions. Biomrica, 3(), doi:0.307/ Chafild, C. (004). Th analysis of im sris: An inroducion (6h d.). London: Chapman & Hall/CRC. Frund, J. E & Walpol, R. E (986). Mahmaical saisics (4h d.). Uppr Saddl Rivr, NJ: Prnic-Hall, Inc. Iwu, H., Iwuz, I. S., & Nwogu, E. C. (009). Trnd analysis of ransformaions of h muliplicaiv im sris modl. Journal of Nigrian Saisical Associaion,, N, Iwuz, I. S. (007). Som implicaions of runcaing h disribuion o h lf a zro. Journal of Applid Scinc, 7(), Iwuz, I. S., Akpana, A. C., & Iwu, H. C. (008). Sasonal analysis of ransformaions of h muliplicaiv im sris modl. Asian Journal of Mahmaics and Saisics (), doi:0.393/ajms Nwosu, C.R., Iwuz, I.S., & Ohakw J. (03). Condiion for succssful invrs ransformaion of h rror componn of h muliplicaiv im sris modl. Asian Journal of Applid Scinc 6(), -5. doi:0.393/ajaps Ohakw, J., Iwuoha, O., & Ouony, E. L. (03). Condiion for succssful squar ransformaion in im sris modling. Applid Mahmaics, 4(4), doi:0.436/am Osborn, J. (00). Nos on h us of daa ransformaions. Pracical Assssmn Rsarch and Evaluaion, 8(6). Availabl onlin: hp://pareonlin.n/gvn.asp?v=8&n=6 Osborn, J. W. (00). Improving your daa ransformaion. Pracical Assssmn Rsarch & Evaluaion, 5(). Availabl onlin: hp://paronlin.n/gvn.asp?v=5&n= Ouony, E. L., Iwuz, I. S., & Ohakw, J. (0). Th ffc of squar roo ransformaion on h rror componn of h muliplicaiv im sris modl. Inrnaional Journal of Saisics and Sysms, 6(4), Robrs, S. (008). Transform your daa. Nuriion, 4, doi:0.06/j.nu

30 ERROR COMPONENT DISTRIBUTION OF THE TIME SERIES MODEL Smih, R. T., & Minon, R. B. (008) Calculus (3rd d.). NY: McGraw Hill. Turky, J. W. (957). On h comparaiv anaomy of ransformaions. Th Annals of Mahmaical Saisics, 8(3), doi:0.4/aoms/ Uch, P. I (003). Probabiliy: Thory and applicaions. Ikga, Lagos: Longman Nigria PLC. Wahanachwakul, L. (0). Transformaion wih righ skw daa. In S. I. Ao, L. Glmn, D. W. L. Hukins, A. Hunr and A. M. Korsunsky, Eds. Procdings of h World Congrss on Enginring Vol.. London, UK: Nwswood Limid. 00