Log-periodogram regression with odd Fourier frequencies

Transcription

1 Log-priodogram rgrssio wih odd Fourir frqucis Erhard Rschhofr Dparm of Saisics ad Opraios Rsarch, Uivrsiy of Via, Ausria Uivrsiässr. 5, Via, Ausria Absrac I his papr, a modificaio of h log-priodogram rgrssio approach for simaig h fracioal diffrcig paramr is proposd. I is basd o h iclusio of addiioal frqucis which ar obaid from h Fourir frqucis by subracig h cosa π/. Th rsuls of a xsiv simulaio sudy show ha his modificaio sigificaly rducs h roo ma squard rror. Ky Words: Fracioally igrad procss, frqucy-domai simaio, priodogram. Iroducio Th shor-ru as wll as h log-ru bhavior of a saioary sris of ma-corrcd obsrvaios y may b dscribd by a fracioally igrad ARMA modl of h form p d ( φ L... φ p L ) ( L) ( + θl θq y L ) u, whr L dos h lag opraor, ad h rrors u com from a whi ois procss wih ma ad variac σ (Gragr ad Joyux, 98, Hosig, 98). Idally, h log-ru bhaviour of a ARFIMA modl is drmid by h fracioal diffrcig paramr d ad h shor-ru bhavior by h ARMA paramrs φ,,φ p,θ,,θ q. Howvr, i pracic h divisio of ass is o so clar. For xampl, Rschhofr () fid ARFIMA modls wih p, q ad p, q, rspcivly, o a log surfac-mpraur sris ad obaid i h firs cas a sima of d clos o.5 ad i h scod cas a sima of d clos o ad a ar ui roo i h auorgrssiv par. This illusras ha a covioal paramric approach may b complly iadqua o drmi h siz of d. q

2 Alraivly, h siz of h fracioal diffrcig paramr d ca also b simad wih h hlp of a log-priodogram rgrssio (Gw ad Porr-Huda, 98). This oparamric approach is basd o h fac ha h spcral dsiy f d iω iωq... ( ) σ iω + θ + + θq ω π iω iωp φ... φ p implid by a ARFIMA modl ca i a ighborhood of frqucy zro b approximad by f ( ad log f ( ω) hrfor by d iω ) C d ω C( si( ) ω ( ) d ) C cos( ω ) log f ( ω) ~ logc + d ( log( cos( ω) )). Of cours, h prcisio of h sima of d obaid by rgrssig h log of h priodogram π I ( λ ) y a h smalls K Fourir frqucis λ π /,,..., K o log( cos( λ )) criically dpds o h assumpio ha hr ar o shor-ru ffcs i h frqucy rag bw ad πk/. Thr is a rad off bw bias ad variac. I gral, h bias icrass ad h variac dcrass as K icrass. Gw ad Porr-Huda (98) usd α K wih α.5,.6,. 7 i hir simulaio xprims ad cocludd ha K should b p small o avoid h iclusio of priodogram ordias which may b corrupd by shorru ffcs. I coras, Hurvich al. (998) argud ha udr crai codiios h opimal.8.5 K is of ordr O ( ). Howvr, Figur shows ha v K is alrady much oo larg i a ypical siuaio wih oly modra shor-rm auocorrlaio (φ.5, d., σ, ). I his xampl, h us of K priodogram ordias would obviously caus a sigifica bias (s Figur.c). Of cours, hr is v mor caus for cocr abou h bias i h cas of highr posiiv auocorrlaio (s Agialogou al., 99). Th asympoic disribuio of h ormalizd priodogram ordias I(λ )/f(λ ) occurrig i log( I ( λ )) ~ I( λ ) log f ( λ ).5 + log + d ( log( cos( ω) )) C. dpds o boh d ad if d (Küsch, 986; Hurvich ad Blrao, 99; Robiso, 995), which ivalidas h aïv assumpio ha h ormalizd priodogram ordias ar asympoically i.i.d. xpoial wih ma uiy ad hir logs ar asympoically i.i.d. wih ma γ (whr γ is Eulr s cosa) ad variac π / 6. Howvr, simulaios idica firsly ha cofidc irvals basd o his aïv asympoic variac ar mor rliabl ha hos basd o h rgrssio rsidual variac (Gw ad Porr-Huda, 98) ad

3 scodly ha ay bias rducio rsulig from h omissio of h mos qusioabl (i.., h vry firs) priodogram ordias is mor ha offs by h associad icras i variac (Hurvich ad Blrao, 99). Assumig ha K ad K log K /, Hurvich al. (998) sablishd h cosiscy of h origial log-priodogram rgrssio simaor. Hasslr (99), Piris ad Cour (99), ad Ris (99) proposd modificaios which ar basd o smoohd vrsios of h priodogram. Th rsuls of a simulaio sudy (Ris al., ) idica ha Ris s modificaio ad h origial simaor ouprform Robiso s modificaio (omissio of h vry firs priodogram ordias) as wll as h paramric simaor basd o h maximizaio of h frqucy-domai lilihood. Th fac ha Ris s modificaio also appars o b suprior o h origial simaor wih rspc o h ma squar rror is slighly discrdid by is dpdc o a largr umbr of uig paramrs. Idd, is prformac dpds o oly o h choic of K bu also o h choic of h lag widow ad, v mor imporaly, h rucaio poi of h lag widow. Figur. Ralizaio of ARFIMA(,d,) procss wih φ.5, d., ad σ (a) Tim sris plo () (b) Priodogram ad spcral dsiy a Fourir frqucis λ,,,[ / ] υ log cos (h dashd lis rprs h ru slop, i.., d.) (c) Log priodogram ad log spcral dsiy plod agais ( ( )) λ (a) (b) (c) -6-5

4 I h x scio, a furhr modificaio of h log-priodogram rgrssio simaor is iroducd which is basd o h iclusio of K addiioal frqucis. Bcaus his modificaio sill dpds o oly o uig paramr (jus as h origial simaor), a fair compariso is possibl which is o compromisd by poial daa-soopig ffcs. Scio prss h rsuls of a simulaio sudy. Scio cocluds.. Esimaors basd o odd Fourir frqucis Fourir frqucis corrspod o priods of dividd by,,,, m / ad ar hrfor paricularly usful for h dscripio of saioary im sris. Th d poi is always jus idical o h sarig poi. I coras, h bs way o dscrib a icrasig (or dcrasig) rd wih a priodic fucio li a siusoid is o sar a h miimum (maximum) ad d a h maximum (miimum), which ca b accomplishd by usig hos frqucis which corrspod o priods of dividd by.5,.5,.5,, m-.5 < /. Ths frqucis ar rlad o h Fourir frqucis via λ λ λ /,,,m, ad will b calld odd Fourir frqucis. Thy shar a ic propry wih h Fourir frqucis, amly ha cos ( λ ) ( i λ + ) i λ + + i λ - ad aalogously which implis ha whr π si ( λ ), I ( λ ) y 8π ( Â + Bˆ ), A ˆ y cos( λ ), Bˆ y si( λ ar jus h las squars simas of h paramrs A ad B i h rgrssios y A cos(ω )+u ad y B si(ω )+u, rspcivly. Bu hr is also a impora diffrc. Whil ) {cos( λ ) + isi( λ )} implis ha cos( λ ) si( λ ),

5 λ {cos( λ ) + isi( )} i λ ( λ i - ) implis ha cos( λ ) - cos( λ ) -, cos( ) λ ad cos( λ ) - si( λ ) si( λ ) λ si( λ ) cos( ) cos( ) λ λ ~, λ λ si( λ ) ~ ( )π if. Thus, i migh b mor appropria i h cas of odd Fourir frqucis o us h quaiis whr ( I λ ) 8π ( A ~ + B ~ ), ~ A y ( cos( λ ) + ), ~ B si( λ ) y si( λ) ( cos( λ ) si ( λ ( cos( λ )) ) ar jus h las squars simas of h paramrs A ad B i h rgrssios y µ+ A cos(λ )+u ad y µ+ B si(λ )+u, rspcivly. Figur illusras diffr mhods for h valuaio of h priodogram a o- Fourir frqucis. I gral, sigifica discrpacis occur oly for frqucis clos o zro. Th oly xcpio is wh a ozro ma is o a io accou (s Figur.b). No surprisigly, i his cas h mos xrm disorios ar foud jus a h odd Fourir frqucis. Figur.c shows ha ma-corrcio alo is o suffici wh dalig wih o-fourir frqucis. Th high agrm bw Figurs.d,., ad.f idicas ha i dos o ma ay sigifica diffrc whhr w us (i) y µ+a cos(λ)+ε, y µ+b si(λ)+ε, (ii) y µ+a cos(λ)+b si(λ)+ε, or (iii) y µ+r si(λ+φ)+ε 5

6 for h calculaio of h priodogram ordias a a o-fourir frqucy λ. Oc h K, λ K priodogram has b valuad a h frqucis λ, λ,, λ, h fracioal diffrcig paramr d ca asily b simad by h log-priodogram rgrssio mhod. I h x scio, h ffc of h iclusio of h odd Fourir frqucis i addiio o h sadard Fourir frqucis will b xamid by a simulaio xprim. I his ivsigaio, h rgrssio (ii) will b usd. Figur : Logarihms of ma spcral simas obaid from, sampls of siz from a ARFIMA(,d,) procss wih φ.5, d., ad σ. Th blac circls corrspod o Fourir frqucis ad h whi circls o odd Fourir frqucis. Th log spcral dsiy (blac li) is plod i h frqucy rag bw π/() ad π/(). (a) Log of ma priodogram (b) Log of ma priodogram of shifd daa (addiiv cosa: ) (c) Log of ma priodogram of dmad daa (d) Log of ma spcral simas obaid from h simpl rgrssios (i) () Log of ma spcral simas obaid from h rgrssio (ii) (f) Log of ma spcral simas obaid from h oliar rgrssio (iii) (a) (b) (c) (d) () (f)

7 . Simulaio sudy To bolsr h sigificac of our simulaio sudy, a larg umbr of ARFIMA modls, i.., 5 of ordr (,d,) ad 5 of ordr (,d,), ar usd o gra h syhic im sris. W s h paramr valus as follows: d, ±., ±., φ,θ, ±., ±.6, ±.9, σ. Fiv housad ralizaios of lgh ar grad. For ach ralizaio, h fracioal diffrcig paramr d is simad by rgrssig h log priodogram a h firs K Fourir frqucis λ ad opioally also a h firs K odd Fourir frqucis λ o log( cos( λ )) ad log( cos( λ )), rspcivly. Th simad slop is usd as sima of d. Tabls ad giv h mas ad h roo ma squar rrors for K6, 8,. Also icludd ar h simas obaid by omiig h priodogram ordias a h firs Fourir frqucy ad a h firs odd Fourir frqucy, rspcivly. Th rsuls of our simulaio sudy show ha h iclusio of h odd Fourir frqucis rducs h roo ma squar rror sigificaly bu has i gral o posiiv ffc o h bias. Th simaor basd oly o h Fourir frqucis is oly compiiv i h cas d-., which is o vry rlva i pracic. Similarly obvious is h poilssss of dcrasig K or omiig h lows frqucis. Tabl : Mas (ialic) ad roo ma squar rrors of various log-priodogram rgrssio simas of h fracioal diffrcig paramr d obaid from 5, sris of obsrvaios from a ARFIMA(,d,) procss. Th simas d, 6,8,, ar basd o h firs Fourir frqucis ad h simas d addiioally o h firs odd Fourir, xcludig h frqucis. Excludig h vry firs Fourir frqucy givs h simas firs Fourir frqucy ad h firs odd Fourir frqucy givs h simas d φ d 6 d 8 d d 6 d 8 d d 6 d 8 d d d d 6 d 8 d 7

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9 Tabl : Mas (ialic) ad roo ma squar rrors of various log-priodogram rgrssio simas of h fracioal diffrcig paramr d obaid from 5, sris of obsrvaios from a ARFIMA(,d,) procss. Th simas d, 6,8,, ar basd o h firs Fourir frqucis ad h simas d addiioally o h firs odd Fourir, xcludig h frqucis. Excludig h vry firs Fourir frqucy givs h simas firs Fourir frqucy ad h firs odd Fourir frqucy givs h simas d θ d 6 d 8 d d 6 d 8 d d 6 d 8 d d d d 6 d 8 d 9

10 Discussio I gral, icrasig h umbr of Fourir frqucis usd i h log-priodogram rgrssio icrass h bias ad dcrass h variac of h simaor of h fracioal diffrcig paramr. Bcaus of lacig idpdc i is hard o assss h ffc of h addiioal iclusio of odd Fourir frqucis o h variac. Howvr, h rsuls of a xsiv simulaio sudy show ha i has a posiiv ffc. Th roo ma squar rror dcrass sigificaly. I coras, h omissio of h vry lows frqucis has a gaiv ffc. A alraiv approach o improv h prformac of h log-priodogram simaor is o rplac h priodogram by a smoohd vrsio (Hasslr, 99; Piris ad Cour, 99; Ris, 99). Howvr, hr is o d o l hs wo modificaios comp agais ach ohr bcaus hy ca asily b combid simply by firs valuaig h priodogram o oly a h Fourir frqucis bu also a h odd Fourir frqucis ad h usig all availabl priodogram ordias for h smoohig as wll as for h rgrssio. A daild ivsigaio of h rsulig simaor ad is dpdc o h choic of h lag widow ad h rucaio poi is byod h scop of his shor o.

11 Rfrcs Agialogou, C., Nwbold, P., Wohar, M. (99) Bias i a simaor of h fracioal diffrc paramr. Joural of Tim Sris Aalysis, 5 6. Gw, J., Porr-Huda, S. (98) Th simaio ad applicaio of log mmory im sris modls. Joural of Tim Sris Aalysis, -8. Gragr, C.W.J., Joyux, R. (98) A iroducio o log-mmory im sris modls ad fracioal diffrcig. Joural of Tim Sris Aalysis, 5-9. Hasslr, U. (99) Rgrssio of spcral simaors wih fracioally igrad im sris. Joural of Tim Sris Aalysis, Piris, M.S., Cour, J.R. (99) A o o h simaio of dgr of diffrcig i log mmory im sris aalysis. Probabiliy ad Mahmaical Saisics, -9. Hosig, J.R.M. (98) Fracioal diffrcig. Biomria 68, Hurvich, C.M., Blrao, K.I. (99) Asympoics for h low-frqucy ordias of h priodogram of a log mmory im sris. Joural of Tim Sris Aalysis, Hurvich, C.M., Blrao, K.I. (99) Auomaic smiparamric simaio of h mmory paramr of a log mmory im sris. Joural of Tim Sris Aalysis 5, 85-. Hurvich, C.M., Do, R., Brodsy, J. (998) Th ma squard rror of Gw ad Porr- Huda's simaor of h mmory paramr of a log-mmory im sris. Joural of Tim Sris Aalysis 9, 9 6. Küsch, H.R. (986) Discrimiaio bw moooic rds ad log-rag dpdc. Joural of Applid Probabiliy, 5-. Ris, V.A. (99) Esimaio of h fracioal diffrc paramr i h ARIMA(p,d,q) modl usig h smoohd priodogram. Joural of Tim Sris Aalysis 5, 5-5. Ris, V.A., Abraham, B., Lops, S.R.C. () Esimaio of h paramrs i ARFIMA procsss: a simulaio sudy. Commuicaios i Saisics: Simulaio ad Compuaio, Rschhofr, E. () Robus sig for saioariy of global surfac mpraur. To appar i Joural of Applid Saisics. Robiso, P.M. (995) Log-priodogram rgrssio of im sris wih log rag dpdc. Aals of Saisics, 8-7.