RELATIONSHIPS BETWEEN SPECTRAL PEAK FREQUENCIES OF A CAUSAL AR(P) PROCESS AND ARGUMENTS OF ROOTS OF THE ASSOCIATED AR POLYNOMIAL.

Transcription

1 RELATIONSHIPS BETWEEN SPECTRAL PEAK FREQUENCIES OF A CAUSAL AR(P) PROCESS AND ARGUMENTS OF ROOTS OF THE ASSOCIATED AR POLYNOMIAL A Wrng Proc Prsnd o T Faculy of Darmn of Mamacs San Jos Sa Unvrsy In Paral Fulfllmn of Rqurmns for Dgr Masrs of Scnc By Abra R. Jru Dcmbr 8

2 8 Abra R. Jru ALL RIGHTS RESERVED

3 APPROVED FOR THE DEPARTMENT OF MATHEMATICS Dr. Sv Crunk Dr. B Lng L Dr. Marna Brmr

4 ABSTRACT RELATIONSHIPS BETWEEN SPECTRAL PEAK FREQUENCIES OF A CAUSAL AR(P) PROCESS AND ARGUMENTS OF ROOTS OF THE ASSOCIATED AR POLYNOMIAL by Abra R. Jru Ts roc ams o assss rlaonss bwn argumns of roos of an AR olynomal and scral ak frquncs of a causal AR ( ) rocss. Frs and formos, rors of causal AR ( ) and AR ( ) rocsss ar orougly amnd. In addon, bavor of r scral dnss and frquncs a mamz scral dnss ar amnd. Any aarn rlaons a s frquncs and argumns of roos of assocad AR olynomal mg av ar also nvsgad. An am s also mad o gnralz any fndngs o gr ordr causal auorgrssv rocsss. T sudy sows a argumn of roo of an AR olynomal of a causal AR ( ) rocss r nds o or s qual o scral ak frquncy wn modulus of roo nds o on. T scral ak frquncs and argumns of roos of AR olynomal ar also found o aromaly sam wn. In addon, wn, for all < rgardlss of valu of, scral ak frquncs and argumns of roos of AR olynomal ar found o b sam.

5 DEDICATION I am ndbd o Profssor Svn Crunk for s valuabl advc, commns and suggsons a vry sag of sudy. My ds graud s also du o my wf Blm Guda for r connuous suor. v

6 Conns Inroducon Dfnons, Noaon and Fundamnal Concs 5. Tm Srs 5. Man Funcon and Auocovaranc Funcon 6. W Nos 7.4 Saonary Tm Srs 8.5 Auorgrssv (AR) Procss.6 Scral Dnsy 5 Scral Pak Frquncs and Argumns of Roos of AR Polynomal of a Causal Auorgrssv Procss 9. Causal Auorgrssv Procss of Ordr 9. Causal Auorgrssv Procss of Ordr 5.5 Causal Auorgrssv Procss of Ordr Concluson 48 Bblogray 5 v

7 Ls of Fgurs Fgur : Causal Rgon for AR() Procss Fgur : Subscon of Causal Rgon for AR() Procss Fgur : Plo of Scral Dnsy for Rgon A 5 Fgur 4: Plos of Scral Dnss of AR() Modls for Rgons B, C and D 8 Fgur 5: Plos of Scral Dnss of AR() Modls for Rgons E, F and G Fgur 6: Plos of Scral Dnss of AR() Modls for Rgon H Fgur 7: Plo of Conours of * θ, wr Arg ( z ) Fgur 8: Plo of Conours of Mnmum ( z, z ) θ r 4 Fgur 9(a): Causal Rgon of AR() w 8 Fgur 9(b): Causal Rgon of AR() w.5 9 Fgur 9(c): Causal Rgon for AR() w Fgur 9(d): Causal Rgon for AR() w. 9 Fgur 9(): Causal Rgon for AR() w -. 9 Fgur 9(f): Causal Rgon for AR() w.5 4 Fgur 9(g): Causal Rgon for AR() w Fgur 9(): Causal Rgon for AR() w.9 4 Fgur 9(): Causal Rgon for AR() w v

8 Fgur 9(): Causal Rgon for AR() w.99 4 Fgur 9(k): Causal Rgon for AR() w Fgur : Scral Dnsy of AR() w. 7,. 7,. 6 4 Fgur : Scral Dnsy of AR() w. 5,. 5,. 8 4 Fgur : Plo of Scral Dnsy of Fgur : Plo of Scral Dnsy of w w v

9 CHAPTER Inroducon T noon a a m srs s saonary s fundamnal n m srs analyss. In many lraurs rm saonary s rfrrd o as wakly saonary, saonary n a wd sns or scond ordr saonary. T saonary of m srs dnds on snc of man funcon and auocovaranc funcon of rocss. Ta s, rocss s saonary wn man funcon s a consan (ndndn of m nd) and auocovaranc funcon ss and dnds only on m lag (s Scon.4). On or and, as sad n Andrson (97) da of src saonary s a bavor of a s of random varabls a on m s robablscally sam as bavor of a s a anor m. W wll dscuss s rors orougly n lar cars. In m srs analyss, auorgrssv modl s on of mos usful m srs modls. In dalng w an auorgrssv modl of ordr, AR( ), knowng rors of modl s vry moran. Ts s bcaus orcal rors ar rfrnc fram o wc samlng rors ar comard. Sumway and Soffr (6) sad auorgrssv modl as a modl n wc currn valu of srs,, s land by as valus,,,...,, wr dnos numbr of ss no as rqurd o forcas currn valu and { } s a saonary

10 rocss. As oosd o an ordnary rgrsson modl, n an auorgrssv modl currn valu of dndn varabl s nfluncd by s own as valus, us nam auorgrssv. Ts auorgrssv modl s gvn by w, wr ar auorgrssv modl coffcns and Car for dals). w s w nos w varanc σ (s w On of ndsnsabl rors a som auorgrssv modls sasfy s causaly. T auorgrssv modl s sad o b causal f sasfs condon a roos of olynomal ( ) z z l ousd of un crcl. Ts olynomal s rfrrd o as AR olynomal of rocss. An alrnav condon for cckng causaly, wc s comlly basd on auorgrssv coffcns, s gvn n Pand and Wu (98). W wll dscuss s condon n dal n lar cars. Tr ar wo aroacs n m srs analyss: m doman analyss and frquncy doman analyss. W (6) sad a wo aroacs ar orcally quvaln. Undr assumon of absolu summabl auocovaranc funcon of a saonary m srs { }, scral dnsy, f ( ), of rocss s a funcon a caracrzs rocss n frquncy doman. T auocovaranc funcon s on of caracrscs of m srs undr m doman. Sumway and Soffr (6) sad a T auocovaranc funcon and scral dnsy conan sam nformaon, undr assumon of absolu summabl auocovaranc funcon.

11 On of uss of scral analyss s o drmn ngborood of frquncy a wc valus of f ar larg (Nwon and Pagano, 98). Ty also concludd a suc frquncy comonns ar moran n lanng varably n m srs{ }. W (6) also sad a A ak n scrum ndcas moran conrbuon o varanc from comonns a frquncs n corrsondng nrval. Ts mans a scral ak frquncs av a sgnfcan rol n scral analyss. By scral ak frquncy w man frquncy a wc scral dnsy of rocss s mamzd. Suc a frquncy s also calld ak frquncy (Ensor and Nwon, 988 and Mullr and Prw, 99). T ak of scral dnsy could b r a local or an absolu mamum on. T ara undr scral dnsy wn a gvn frquncy nrval s rooron of varanc of rocss land by frquncs n nrval. T oal ara undr scral dnsy ovr nrval [-, ] s varanc of rocss. Now suos z θ r s on of roos of AR olynomal ( z) of a causal AR( ) rocss, wr r s modulus and θ s argumn of roo. Any roo of AR olynomal of rocss s andld undr m doman aroac, so do modulus and argumn of roo. Tr ar cass n wc modulus of a roo nds on (or quvalnly roo s nar un crcl) and argumn of roo r nd o or qual o on of scral ak frquncs. Trfor, rasonabl o nvsga condons undr wc θ s r nar or qual o on of scral ak frquncs by way of lorng lowr ordr auorgrssv modls frs and

12 n ry o gnralz fndngs o ur ordr auorgrssv modls. Ts s obcv of s Wrng Proc. Imoran dfnons, noaon and on orm w s roof ar rsnd n Car. Dscrons of fundamnal concs a ar lful n lar cars ar also gvn n s car. Car focuss on lorng rlaonss bwn argumns of roos of AR olynomal and scral ak frquncs for auorgrssv modl of ordr for. T bavors of scral dnss of auorgrssv rocsss ar also dscussd n s car. W also rovd wo moran orms. In Car 4, som gnralzaons ar drawn basd on rsuls n Car. Alcaon of rsuls s also gvn. 4

13 CHAPTER Dfnons, Noaon and Fundamnal Concs In s car w dfn som basc rms and dscrb noaon and fundamnal concs a w ar gong o us lar. In addon, w sa som moran assumons undr wc dfnons ar vald. W wll also rov on orm.. Tm srs Tngs a cang w m ar of scal nrs o many rsarcrs. Masurmns, ascs, faurs or caracrscs of ngs, obcs, rsons, c., a flucua w m or ovr a rod of m consu a m srs. T avrag monly mraur of San Jos Cy collcd for as undrd yars, avrag annual amoun of ranfall of San Jos Cy collcd for las ffy yars, and Dow Jons Indusral Avrag (DJIA) collcd ovr som rod of m ar amls of m srs. If obsrvaons ar collcd only a m ons ±, ±, ±,, n m srs s rfrrd o as dscr m srs wl f obsrvaons ar rcordd connuously ovr som m nrval, n srs s calld connuous m srs. W masz dscr m srs n s sudy. Hncfor by m srs w man dscr m srs n s Wrng Proc. T sascal mod a dals w analyss of m srs daa s calld m srs analyss. In or sascal analyss, varabls ar usually assumd o b 5

14 ndndn and ordr n wc obsrvaons ar collcd s no of muc moranc. Unlk or sascal analyss, n m srs analyss obsrvaons ar dndn on on anor and also collcd n an ordrly mannr. T dndnc of succssv obsrvaons s wa maks m srs daa dffrn from or sascal ss of daa. Trougou s car and succssv cars w nk of as r a random varabl or an obsrvaon dndng on con. W dfn a m srs as follows. Dfnon L b a random varabl ndd by a m on, wr, ±, ±, ±,. Tn a squnc,{ }, s dfnd as a m srs.. Man Funcon and Auocovaranc Funcon Gvn a m srs{ }, man and auocovaranc funcons ar moran caracrscs of rocss and lay a sgnfcan rol n m srs analyss. Tr dfnons and noaon ar gvn as follows. Dfnon T man funcon, µ, of m srs ( or rocss) { } s dfnd as ( ) µ, rovdd a cd valu ss. E Dfnon T auocovaranc funcon, γ ( s ), of m srs (or rocss) { } dfnd as a funcon, (, ) E (( µ )( )), s s γ ( s, ) Cov µ,, () s s s rovdd a caon ss. 6

15 Eaml Consdr β w, wr β and β β ar known consans and w ar uncorrlad random varabls w zro man and varancσ. Tn man w and auocovaranc funcons ar calculad as ( ) E( β β w ) β µ E β ( ) E (( µ )( )) γ ( s, ) Cov, µ s s s ( s ) µ s E µ E (( β β s w )( β β w )) ( β β s)( β ) E( w s w ) s β σ w f s f s. W Nos Procss Som m srs av scal caracrscs suc as r man funcon valus vans for all m ons ; y av consan varancs; and valus of r auocovaranc funcons vans a dsnc m ons. A m srs avng suc caracrscs s rfrrd o as a w nos rocss. In Eaml, { w } s a w nos rocss. W gv a formal dfnon of a w nos rocss as follows. 7

16 Dfnon 4 Suos a { w } s a rocss suc a followng condons old. ) γ ( s, ), s ; w ) γ (, ) σ <, ; and () w w ) µ,. w Tn rocss s rfrrd o as a w nos rocss. T random varabl rfrrd o as w nos. w s T dfnon ndcas a a w nos rocss s a squnc of uncorrlad random varabls w zro man and consan varanc. In mos alcaons and smulaon roblms w nos rocss s assumd o b Gaussan..4 Saonary Tm Srs Som m srs ar sml n r srucurs wl ors ar coml. An moran rory of a m srs s on n wc bavor of m srs a on m on s robablscally sam as s bavor a anor m on. Ta s, on dsrbuon of vry collcon {,..., } of {,... },, k, k s qual o on dsrbuon, k,,...,,,..., k, and, ±, ±,... Ts rory s rfrrd o as srcly saonary. Du o fac a s rory s srongr an ncssary for many alcaons, a mldr vrson of s rory, wc s consdrd as rgulary condon and ofn ms assumd n analyss of m srs, 8

17 s usually usd. T followng dfnon gvs dald dscron of s mldr vrson of rory. Dfnon 5 Suos a { } s a m srs a sasfs condons ) µ µ, ; ) ( s, ) γ ( ), s γ ; and () ) (, ) Var( ) <, γ. Tn rocss { } s sad o b wakly saonary. Hncfor, rm saonary wll b akn o man wakly saonary. Suos a { } s a saonary rocss. Tn auocovaranc funcon of rocss sasfs followng rors (s Brockwll and Davs, 99 for roofs). ) ) γ γ ( ) ; ( ) γ ( ), ; and ) γ ( ) γ ( ),. A srcly saonary rocss w fn varanc s always saonary. And also f rocss s Gaussan, n srcly saonary and wakly saonary ar quvaln. Eaml T rocss n Eaml s no saonary bcaus man funcon of rocss, µ β β, dnds on m nd. 9

18 Eaml Consdr rocss.5 w, wr w s w nos w varanc on. Tn rang rocss backwards r ms, w g (. w ) w.5 w w w r r r (.5) (.5) w. r r r r r If lm (. 5) w lm(. 5) E( ) E, n (.5) w convrgs o n man squar sns and so rocss can also b wrn as (.5) from wc w can asly calcula rocss s saonary f ( ) E s fn. r ( ) µ, γ ( ) and γ ( ) 4 w,. Hnc Dfnon 6 A m srs{ } s a lnar rocss f s a lnar combnaon of w nos, w, and s wrn as µ ψ w, (4) wr ψ < and µ s a consan.

19 For rocss n (4) can b sown a ( ) E, µ, and ( ) ( )( ) ( ) µ µ γ E k k k w w E ψ ψ w ψ ψ σ. Clarly auocovaranc funcon of rocss s a funcon of m lag only. Howvr, for rocss o b saonary auocovaranc funcon mus av fn valu a ac lag as sown blow. ( ) ( )( ) ( ) ( ) ( ) ( ). snc nqualy Scwarz Caucy by Var Var w - - < < w k k w w E ψ ψ σ ψ ψ ψ σ ψ σ ψ σ µ µ γ

20 If ψ for all <, n rocss n (4) s sad o b causal (s Dfnon 8 for dal)..5 Auorgrssv (AR) Procss Sumway and Soffr (6) dfn m srs analyss as sysmac aroac by wc on gos abou answrng mamacal and sascal qusons osd by corrlaons nroducd by samlng of obsrvaons a adacn ons n m. Hr by corrlaon w man dndnc of m srs obsrvaons on on anor. Ts dndnc bwn obsrvaons s mos rlvan faur a caracrzs dynamcs of undrlyng sysm. In m srs analyss, r ar usful ways of rssng s dndnc sng bwn m srs obsrvaons. On of s ways s by usng som mamacal modl a rsss a valu of a m as a lnar combnaon of s own as valus and w nos. Ts way of rrsnng a m srs s rfrrd o as auorgrssv form. Dfnon 7 A zro man saonary m srs { } s an auorgrssv rocss of ordr, AR( ), f w can wr n form... w, (5) wr,,..., ar ral numbrs w and w s w nos. E, n s y µ, so a If ( ) µ y w y.

21 W can also wr (5) n form ( B ) w, wr ( B) B B... B and B s a backsf oraor dfnd as B,,. An AR( ) rocss s also calld an AR( ) modl. Rlacng B by z n ( B) gvs a olynomal n (6); s s rfrrd o as an AR olynomal. ( ) (6) z z Dfnon 8 An AR( ) modl ( ) B w s causal f w can wr as a lnar rocss ψ w ψ ( B) w, (7) ψ ψ, ψ <, ψ and ψ, <. B wr ( ) B Torm An AR( ) modl s causal f and only f ( z) Prory P., Sumway and Soffr, 6). for z (Scal cas of Proof Suos a ( z) for z. Tn roos z, z,... z of ( z) l ousd of un crcl. Assum w ou loss of gnraly a < z z... z. L z ε for som ε >. Tn ( z) for all z suc a z < ε. Trfor, ( z) ss z < ε and as a owr srs anson z

22 ( z) ψ z, z < ε. Coos a δ suc a < δ < ε and l z δ. Tn follows a ( z) ( δ ) ψ ( δ ) <. Hnc squnc { ( ) ψ δ } s boundd. Ta s, r ss a k > suc a ψ ( δ ) k, wc mls a < k( δ ) < ψ ( ) <. Trfor, ( B) k δ ψ. From s rsul, follows a ss. Prmullyng bo sds of AR modl ( B ) w by ( B) and sng ( B) qual o ψ ( ) ψ B B gvs ψ ( B) w ψ ( B) w ψ B w w. Trfor, by dfnon 8, AR() modl s causal. Suos a AR( ) rocss s causal. Ta s, suos as rrsnaon ψ w ψ ( B) w, wr ( ) ψ ψ, ψ < and ψ. Mullyng B B bo sds of ψ ( B) w by ( B) gvs ( B ) ( ) ( ) B ψ B w. Comarng s o AR( ) modl ( B ) w, w av w ( B) ψ ( B) w, and nc ( B) ψ ( B) s rsul w conclud a ( z) ψ ( z), z or quvalnly ( z). From ψ, for z. ( z) 4

23 ψ for z and nc, Howvr, w av ( z) ψ z ψ z < ψ < ( z) <, for z. Ts mans ( z) for all z..6 Scral Dnsy In m srs analyss rors of a nomnon ar sudd n rms of s bavor n r m or frquncy doman. In m doman, fuur valus of a m srs ar modld as funcon of rsn and as valus of m srs. On or and, n frquncy doman, flucuaon of m srs s ofn modld n rms of rgonomrc sn and cosn funcons. Indd, s rgonomrc funcons ar n urn funcons of Fourr frquncs. Ts scon focuss on dscron of funcon a caracrzs m srs n frquncy doman. Suos { } s a zro man saonary m srs. Tn w say a auocovaranc funcon of rocss, γ ( ), s absoluly summabl f ( ) γ ( ) γ ( ) < γ. (8) Ts condon guarans convrgnc of funcon n (9). Ta s, funcon n (9) convrgs unformly (s Körnr, 4) f (8) olds. f ( ) γ ( ),. (9) 5

24 6 Snc ( ) ( ), γ γ and ( ) ( ) sn cos w can also drv anor form of funcon as follows. ( ) ( ) ( ) ( ) ( ) f γ γ γ γ ( ) ( ) ( ) γ γ γ ( ) ( )( ) γ γ ( ) ( ) ( ) cos γ γ ( ) ( ) ( ) ( ) ( ) cos cos γ γ γ ( ) ( ) γ cos Suos a condon (8) olds. Tn w av ( ) ( ) ( ) ( ) ( ) γ γ d d d f. () Summaon and ngraon ar nrcangd bcaus of absolu summably of auocovaranc funcon. T ngral n las rsson s smlfd as follows.

25 ( ) d f f () Usng () n () and smlfyng gvs f ( ) d γ ( ). () From () w av γ ( ) f ( ) d, wc mans a ara undr funcon n nrval [-, ] gvs varanc of rocss. No a as sad n W (6) T rm f ( ) d s conrbuon of varanc arbuabl o comonn of rocss w frquncs n nrval ( d) T funcons γ ( ) and f ( ),. form Fourr ransform ar (s W, 6 for dal). T funcon f s connuous, non-ngav and rodc (w rod ). In addon, snc cosn s an vn funcon, f s also an vn funcon and nc s symmrc around zro. Consqunly, valus of f ar usually drmnd only a valus of n nrval [, ]. Tus, suffcs o consdr gra of suc a funcon only ovr nrval [, ]. T condon n (8) olds for any AR() rocss (s Sumway and Soffr, 6 for dal). Furrmor, quaon () rsns anor way of rssng funcon f for AR() rocss (dald dscron of s quaon s gvn n Andrson, 97 and also s Lysn and Tøsm, 987). 7

26 f ( ) σ, wr ( ) ( ),. () Dfnon 9 Suos a { } s a zro man saonary m srs and condon (8) olds. Tn funcon n (9) s rfrrd o as scral dnsy of m srs{ }. Eaml 4 Consdr AR() modl.5. 5 w, wr w s Gaussan w nos w varanc on. Tn AR olynomal of rocss as roos,.5 ±.5 7, ac w modulus. Hnc rocss s causal. T scral dnsy of rocss s f.5 ( ).5, [ cos( ) cos( )]. 8

27 CHAPTER Scral Pak Frquncs and Argumns of Roos of AR Polynomal of a Causal AR() Procss In s car w ry o nvsga condons undr wc scral ak frquncs of a causal AR() rocss ar r qual or aromaly qual o argumns of roos of assocad AR olynomal for. Mor mass s gvn o lowr ordr causal auorgrssv rocsss, say AR() and AR(), o s som arn so as o draw som gnralzaon for ur ordr causal auorgrssv rocsss. All los n s car ar basd on assumon of Gaussan w nos w varanc on.. Causal Auorgrssv Procss of Ordr In s scon w amn rors of roos of AR olynomal of a causal AR() rocss. W also closly amn naur of scral dnsy of rocss and nvsga rlaonss bwn scral ak frquncs and argumns of roos of AR olynomal of rocss. T AR() modl s gvn by, wr { } s a zro man saonary rocss, and ar ral numbrs w and s w nos w varanc σ. W can also wr s modl as ( B), wr ( B) B B w 9

28 and B s a backsf oraor suc a m, uns back n m. B,,. Ta s B mas a valu a z. Solvng T AR olynomal for abov rocss s ( z) z quadrac quaon z z for z gvs z ( 4 ) and z ( 4 ). (4) T wo rssons n (4) ar calld roos of AR olynomal of rocss. T causaly of rocss mls a bo of s roos of AR olynomal l ousd of un crcl. Consqunly, z ( 4 ) and ( 4 ) z (5) s, from wc w oban z z and zz. In addon, bcaus rocss s causal can b sown a < and <. T sgn of quany 4 n (4) drmns wr roos ar ral or coml. Ta s, f s quany s ngav, n bo roos ar coml. Orws, bo roos ar ral. If bo roos ar ral, n can b sown a - < z z <, from wc w oban followng nquals: < < Trfor, followng r nquals ar causal condons for AR() rocss n rms of auorgrssv coffcns.

29 < < < T r nquals form a rangular rgon gvn n Fgur, wc s rfrrd o as a causal rgon for AR() rocss. All ons nsd rangular rgon, c ons on orzonal as ( ), ar ars of valus of and for wc rocss s causal. Furrmor, f on ( ), s blow curv drmnd by quaon. 5, n bo roos of AR olynomal ar coml. Ts roos form coml conuga ar. On or and, f on ( ), s on or abov curv. 5, n bo roos ar ral. T wo ral roos ar qual f and only f on ( ), ls on curv. 5. Fgur : Causal Rgon for AR() Procss

30 No a w can also wr roos, z, n olar coordnas form, r θ, wr ( ) z Arg θ s drcd angl masurd from osv -as o z on y coml lan (Pand and Wu, 98) and z r s modulus of z. Now l us urn our anon o scral dnsy of rocss. T scral dnsy of a causal AR() rocss s ( ) σ, w f. T form of scral dnsy a wll b usd ncfor s gvn as ( ) f ( ) ) cos( ) cos( ) cos( σ w ( )( ) ) ( ) cos( 4 σ w. (6) T valus of frquncy,, a mamz scral dnsy ar ( ) ( ) ( ) < < ± ± 4 -, 4 arc cos *

31 No a f ( ) 4, n arc ( ( ) 4 ) or ± cos f ( ) 4, n arc cos( ( ) ) ls n nrval (, ) < 4. On or and,. Snc and ar rsrcd wn rangular rgon gvn n Fgur, w now amn wr nquals ( ) 4 and ( ) 4 rgon. < old wn rangular > ( ) 4 < ( ) 4 < < ( 4) < < and ( 4) < < OR < < ( 4) and - < < ( 4) Ts mans ( ) 4 olds n rgons markd by A and H n Fgur. < Wl nqualy ( ) 4 olds n rgons B, C, D, E, F and G. > Fgur : Subscon of Causal Rgon of AR() Procss

32 4 Now, w comar scral ak frquncs and argumns of roos of AR olynomal n ac rgon. Evaluang scond drvav of scral dnsy a valus of * gvs ( ) ( ) ( ) '' 4 σ f (7) ( ) ( ) ( ) '' 4 σ ± f (8) ( ) ( ) 4 '' cos σ ± arc f (9) W drmn wr s valus of scral dnsy ar mamum or mnmum dndng on sgns of (7), (8) and (9). For nsanc, usng scond drvav s, f (7) as ngav sgn, n ( ) f s a (local) mamum and f (7) as osv sgn, n ( ) f s a (local) mnmum. T scral ak frquncy s valu of * a wc f as (local) mamum valu. Rgon A: Ts rgon s oband from nrscon of followng wo nquals. 4 < < and 4 < <

33 and 4 > 4 > '' '' f ( ) < and f ( ± ) < Furrmor, snc > '' follows a f cos ± arc > 4. Trfor, and ± ar scral ak frquncs. T frquncs and ± ar absolu and rlav scral ak frquncs, rscvly, wn >, and rlav and absolu scral ak frquncs, rscvly, wn <. On or and, bo roos z and z ar ral w z > and z < n s rgon. Trfor, θ Arg( z ) and θ Arg ( ) ± z. Ts sows a scral ak frquncs ar sam as argumns of roos of AR olynomal n rgon. Fgur s lo of scral dnsy for. and. 66 n [, ]. AR() w. and.66 Scrum Frquncy Fgur : Plo of Scral Dnsy for Rgon A 5

34 In lo blu dod vrcal ln ndcas locaon of argumn of z wl rd dod vrcal ln ndcas locaon of cos( ( ) ) mnmum of scrum occurs. arc a wc 4 Rgons B, C and D: * or ± In s rgons noc a arc cos( ( ) ) 4. In addon, nqauly ( ) < < ( 4) ( ) < < ( 4) 4 4 dos no s and nc olds. and 4 > 4 < '' '' f ( ) > and ( ± ) < f. Trfor, scral ak frquncs ar ±. Morovr, can b sown a z z ( 4 ) ) ( 4 ) ) > < θ Arg θ Arg ( z ) ( z ) ± n rgon B and z z ( 4 ) ) ( 4 ) ) < < θ Arg θ Arg ( z ) ( z ) ± ± n rgon C. Trfor, scral ak frquncs ar qual o r Arg( z ) or ( z ) Arg n rgon B and C. In addon, snc roos ar coml conuga ar n rgon D, w can wr 6

35 s roos n olar coordnas as θ z r and θ z r, wr r z z, θ Arg( z ) and θ Arg( z ). Hnc can b sown a cos( θ ) r and r, from wc w av θ arc cos(.5 ). Howvr, as nds o.5 wn rgon θ nds o. Consqunly, condon for scral ak frquncs o nd o argumns of roos s a nds o s, closr on (, ) s o curv scral ak frquncs ar o argumns of roos..5. Ta.5 n rgon, closr Fgur 4 rrsns los of scral dnss of AR() modls for rgons (B, C and D) w av dscussd abov. Eac lo s rsrcd ovr nrval [, ]. T rd vrcal lns ndca locaon of argumns of roos of AR olynomals of rocsss. In (B) and (C), argumn of z concds w scral ak frquncy. For rgon D, wo los ar consdrd. In frs lo argumn of z s sgnfcanly smallr an scral ak frquncy wl n scond lo argumn of z s clos o scral ak frquncy. In lar s vry clos o curv formd by quaon. 5 n on (, ) causal rangular rgon (Fgur ) wl n formr s on s rlavly far from curv. 7

36 (B) AR(): -.5 and.7 (C) AR(): -.45 and -.4 Scrum..6. Scrum Frquncy Frquncy (D) AR(): -.5 and -.5 (D) AR(): - and -.5 Scrum Scrum Frquncy Frquncy Fgur 4: Plos of Scral Dnss of AR() Modls for Rgons B, C and D Rgons E, F and G: In s rgons, * or ± snc arc cos( ( ) ) 4 dos no s n s rgons. In addon, nqualy ( ) < < ( 4) olds. Howvr, 4 ( ) < < ( 4) [ ( )] > and [ ( 4)] 4 4 < and 4 < 4 > '' '' f ( ) < and ( ± ) > f. 8

37 Trfor, scral ak of rocss occurs a * n s rgons. I s also * ru a mnmum valu of scral dnsy occurs a ±. W now amn wa argumns of roos of AR olynomal looks lk n s rgons and n comar m w scral frquncs as follows. In rgon E, roos ar coml conuga ar. Wrng roos n olar coordnas lk as n rgon D, w can sow a ( z ) θ arc cos(.5 ) Arg. As nds o.5, ± θ nds o ( scral ak frquncy). Hnc n s rgon, farr on ( ), s from curv drmnd by. 5, largr s dffrnc bwn scral ak frquncy and argumns of roos, and closr on s o curv smallr s dffrnc. In rgon F: z z ( 4 ) ) ( 4 ) ) > > θ Arg θ Arg ( z ) ( z ). In rgon G: z z ( 4 ) ) ( 4 ) ) > < θ Arg θ Arg ( z ) ( z ) ±. Trfor, scral ak frquncy s qual o r Arg( z ) or ( z ) and G. Arg n rgons F T los of scral dnss of AR() modls for cosn valus of and from rgons E, F and G ar gvn n Fgur 5. In los vrcal lns ar drawn 9

38 roug argumns of roos of AR olynomals of rocsss. In (E) wn s nar.5 wn rgon (s scond lo), argumns of roos of AR olynomal of rocss ar nar, scral ak frquncy. In F and G argumn of z and scral ak frquncy ar sam. (E) AR():.67 and -.9 (E) AR(): and -.5 Scrum...5 Scrum Frquncy Frquncy (F) AR():.75 and -.4 (G) AR():.6 and. Scrum Scrum Frquncy Frquncy Fgur 5: Plos of Scral Dnss of AR() modls for Rgons E, F and G Rgon H: Ts rgon s oband from nrscon of rgons oband by nquals < < ( 4) and < < ( 4) * scral dnsy occur a, ±, ± cos( ( ) 4 ). T rm valus of arc n rgon. Howvr,

39 < < and 4 < < 4 > 4 and > 4 and 4 < 4 < '' '' f ( ) > and f ( ± ) > '' n rgon w also av f ( ± arc ( ( ) 4 )) < Snc < * scral aks occur a ± cos( ( ) 4 ) * scral dnsy occur a, ±. cos. Hnc arc. T mnmum valus of On or and, roos ar coml conuga ar n s rgon. Hnc, lk n rgon D, wrng roos z and z n olar coordnas can b sown a Arg ( z ) θ arc cos(.5 ) and ( z ) θ arc cos(.5 ) Arg. In addon, argumns of roos of AR olynomal ar sam as scral ak frquncs wn. In s cas, argumns of roos and scral ak frquncs ar ±. Howvr, wn s no dffcul o sow a as nds o -, scral ak frquncs nd o argumns of roos. Fgur 6 rrsns los of scral dnss of AR() modls wn auorgrssv coffcns ar cosn from rgon H. T los ndca a closr

40 s o - closr ar argumns of roos of AR olynomal o scral ak frquncs of rocss. (H) AR():. and -.9 Scrum Frquncy (H) AR():.4 and -. Scrum Frquncy Fgur 6: Plos of Scral Dnss of AR() Modls for Rgon H W av sown a mamum on on scral dnsy occurs a a frquncy qual o on of argumns of roos of AR olynomal wn roos ar ral. In addon, wn roos ar coml w av sn a mamum ons of scral dnsy may occur a or nar argumns of roos of AR * olynomal. To mak s mor undrsandabl w gv conours of θ ovr nrval [, ] n Fgur 7, wr Arg( ) θ and bo roos ar coml. Wn roos ar ral obvously scral ak occurs a on of argumns of roos and nc s s omd from fgur. As w can s fgur, θ * > wn, z >

41 θ * < wn and * θ wn. In addon, dffrnc s clos o zro nar. < AR() : Conours of θ - * Fgur 7 : Plo of Conours of * θ, wr θ Arg( z ) I s also rmarkabl o s rlaonss bwn argumn of any roo of an AR olynomal and scral ak frquncs wn roo s nar un crcl. Any roo, say z, of an AR olynomal can b wrn n a olar coordna form wr r z and θ Arg( z). As r nds o θ, z r nds o θ r, θ. Hnc ( θ ) nds o zro snc z s roo of ( z), from wc w conclud a ( θ ) f nds o

42 scral ak. Snc scral dnsy s symmrcal abou zro follows a f ( θ ) also nds o scral ak. Trfor, θ nds o scral ak frquncy. T conours of mnmum valus of z and z ar gvn n Fgur 8. As w can s from fgur a las on of roos s vry clos o un crcl wn (, ) s nar boundars of rangular rgon. AR(): Conours of r Fgur 8: Plo of Conours of r Mnmum ( z, z ) 4

43 . Causal Auorgrssv Procss of Ordr In s scon, causal condons for AR() rocss wll b rsnd n rms of auorgrssv coffcns. For gvn valu of, w wll amn naur of rmanng AR coffcns wn rocss s causal. T rlaons bwn argumns of roos of an AR olynomal and scral ak frquncs of rocss s also nvsgad. T AR() modl s gvn by w, wr ar ral numbrs w and w s w nos w varanc σ w. T AR olynomal of rocss s ( ) z roos of ( ) z z z. Ts rocss s causal f and only f all z l ousd un crcl by Torm. Howvr, s ossbl o consruc quvaln condons basd on auorgrssv coffcns. Pand and Wu (98) consrucd causal condons for AR() rocss basd on auorgrssv coffcns of rocss, from wc w oban causal condons gvn n () for AR() rocss. < < < < < < < < () By addng nd and 4 nquals n (), w g ( ) < ( )( ) and dvdng bo sds of rsul by ( ) gvs >. Smlarly, from 5

44 s and 4 nquals w g <. From s rsulng wo nquals w oban nqualy < <. In addon, 4 nqualy n () sows a < <. Morovr, from s and nd nquals w av and <. Lng m o b mnmum of and, condons n () ar quvalnly wrn as < < < < < m () For any gvn valu of, can b sown a r s a (-, -) and b (, ) suc a (a, b). In addon, f <, n (a, ), b a 4, < a < 4 and (-, ) (a, ) (a, b). In gnral, f k ma{ a, b}, n ( ) k, w < k < 4. For a arcular valu of, causal condons ar rducd o a rangular rgon. For nsanc, rangular rgons n Fgur 9(a) roug Fgur 9(k) rrsn causal rgons for AR() modls for gvn valus of. Now l us dscrb rlaonss bwn scral ak frquncs and argumns of roos of AR olynomal for som gvn valus of. Aloug by dfnon, f w assum a, n nquals n () gvn abov ar rducd o causal condons for AR() rocss dscussd n Scon.. For comarson, causal rgon for AR() s gvn n Fgur 9(a). 6

45 If s osv, n rgon bavs lk rangular rgons n Fgur 9(b), (d), (f), (g) and (). In suc a cas closr s o, closr rangular rgon s o a ln sgmn w nd ons ( ) (-, ) and ( ),, (, -). If s ngav, n rgon bavs lk rangular rgons n Fgur 9(c), (), (g), () and (k). T closr s o -, closr s rangular rgon o a ln sgmn w nd ons ( ) (, ) and ( ), of ac roo o., (-, -). T closr s o ±, closr s modulus In Fgur 9(), as (, ) nds o ln n rangular rgon, modulus of ac of coml roos nds o and argumns for s roos also nd o frquncs a mamz scral dnsy. In Fgur 9(f), f and ar cosn nar.5, n modulus of coml roos sgnfcanly dva from wl f y ar cosn n ngborood of wn rangular rgon, n modulus of ac of coml roos nds o and also argumn of s roos nd o frquncs a mamz scral dnsy. In lar cas coml roos ar nar un crcl wl n formr cas coml roos dva from un crcl. Ta s, closr ons (, ) ar o ln.5, farr coml roos ar from un crcl. In Fgur 9(), all r roos of AR olynomal ar nar un crcl for all (, ) n rangular rgon. Ts s du o fac a closr s o 7

46 ±, closr ar roos of AR olynomal o un crcl (s Torm n Scon.). Smlarly, n Fgur 9(), all r roos ar nar un crcl for all (, ) n rangular rgon. In s cas argumns of roos nd o scral ak frquncs of rocss. In gnral, f nds o -, n modulus of a las on of roos of AR olynomal nds o. In s cas s roo s nar un crcl and argumn of roo nds o frquncy a mamzs scral dnsy of rocss. In addon, farr valus of ar from -, farr roos ar from un crcl and nc largr dffrnc bwn argumns of roos and scral ak frquncs. On or and, f bo and ar zro, n argumns of roos of AR olynomal ar acly sam as scral ak frquncs. Fgur 9(a): Causal rgon for AR() rocss 8

47 Fgur 9(b): Causal Rgon for AR() w.5 Fgur 9(c): Causal Rgon for AR() w -.5 Fgur 9(d): Causal Rgon for AR() w. Fgur 9(): Causal Rgon for AR() w -. 9

48 Fgur 9(f): Causal Rgon for AR() w.5 Fgur 9(g): Causal Rgon for AR() w -.5 Fgur 9(): Causal Rgon for AR() w.9 Fgur 9(): Causal Rgon for AR() w -.9 4

49 Fgur 9(): Causal Rgon for AR() w.99 Fgur 9(k): Causal Rgon for AR() w -.99 Suos a AR olynomal as wo coml roos and on ral roo. Tn argumn of ral roo qual o scral ak frquncy f s modulus s smallr an a of coml roos. In s cas scral ak frquncy s r or ±. T argumns of coml roos of AR olynomal nd scral ak frquncs f s roos ar vry clos o un crcl and ral roo s far from un crcl. Eaml 5 Fgur s lo of scral dnsy of AR() w auorgrssv coffcns. 7,. 7, and. 6 ovr nrval [, ]. In fgur, dasd ln, wc concds w on of dod lns, corrsonds o scral ak frquncy (aromaly.6). On or and, AR olynomal of 4

50 rocss as coml conuga ar roos,.869 ±.498 (o r dcmal lacs), w modulus and argumns aromaly qual o and ±.6, rscvly, and a ral roo,. 766 (o r dcmal lacs), w modulus aromaly qual o.766 and argumn qual o. In fgur, dod vrcal lns corrsond o argumns of ral roo and coml roo I s no dffcul o s a scral ak frquncy s vry clos o argumn of s coml roo. Eaml 6 Fgur sows lo of scral dnsy of a causal AR() rocss w auorgrssv coffcns. 5,. 5, and. 8. T AR olynomal of rocss as roos. 9 and.456 ±.76 (o r dcmal lacs). In fgur, dod lns corrsond o argumns, and.46, of AR olynomal roos,.9 and , rscvly. T modulus of coml conuga ar roos s aromaly.9. T frquncs a mamz scral dnsy n nrval [, ] ar aromaly and. 66. T dasd ln corrsonds o frquncy.66. Obsrv a r s a nocabl dffrnc bwn.46 and.66. T sgnfcanc of s dffrnc s du o fac a closnss of modulus of ral roo o on ovrwlms ffc of coml roo a frquncy. 66. Noc a scral ak frquncy,, and argumn of ral roo ar sam. In gnral, f modulus of on of roos of AR olynomal s nar on, n argumn of s roo and scral ak frquncy ar r sam or aromaly sam. 4

51 Fgur : Plo of Scral Dnsy of AR();. 7,. 7,. 6 Fgur :Plo of Scral Dnsy of AR();. 5,. 5, 8. 4

52 . Causal Auorgrssv Procss of Ordr P for P 4 In s scon w sar our dscusson by rovng Torm. Ts orm gvs rlaons bwn and modulus of ac roo of AR olynomal of a causal AR() rocss. In wo rcdng scons w dscussd abou condons undr wc scral ak frquncs ar qual o or nd o argumns of roos of AR olynomal for causal AR() and AR() rocsss. A suffcn condon for scral ak frquncs o b qual o or aromaly qual o argumns of roos of AR olynomal was a roos ar nar un crcl. W rov s samn as Torm for any causal AR() rocss n s scon. Torm L w b a causal AR( ) rocss w AR olynomal ( ). L z,,..., z z, b roos of ( z). Tn nds f and only f z nds,. Proof : Gvn a causal AR( ) modl w, wr w s w nos w varanc σ w. Consdr AR olynomal, ( z) can facorz ( z) as ( z z )( z z ) ( z... z ) z, wr, of rocss. Tn w z ar roos of an AR olynomal for,,,. By andng s roduc w also g ( ) z z... z as a coffcn of z and comarng s o coffcn of las rm of AR 44

53 olynomal gvs ( )... z z z. Tn from causaly of rocss w g nds f and only f z, nds. T abov orm ndcas a f s nar on or ngav on, n all roos of AR olynomal ar nar un crcl and convrsly, f all roos ar nar un crcl, n far from un crcl, n s nar on or ngav on. Howvr, f a las on of roos s dvas from on or ngav on accordngly. In Scon. w av sn a f on of roos of an AR olynomal of a causal AR() rocss s nar un crcl, n argumn of a roo nds o scral ak frquncy. W now rov a rlad orm for a causal AR() rocss. Torm L z θ b a roo of AR olynomal of a causal AR( ) rocss. r * Tn r ss a scral ak frquncy suc a z * θ ± or f z s a ral roo * θ f z s a coml roo. Proof : Suos a z θ s on of roos of AR olynomal, wr r r and θ. If z θ r nds o, follows a z o nds. θ Trfor, ( ) nds o ( ), wc mls a ( ) θ f θ σ ( ) z * s n ngborood of on of aks of scral dnsy. Hnc θ s nar, w 45

54 * wr s on of scral ak frquncs. From symmrc rors of scral dnsy follows a θ nds o *, scral ak frquncy. If roo s ral, n obvously θ or ± and nc s qual o scral ak frquncy. T convrs of Torm s no ru. To dmonsra s, suos a, < and. Tn scral dnsy of rocss s funcon f ( ) σ,. w ( cos( ) ) I can b sown a Erm valus of funcon occur a frquncs, *, wr,,,...,. T scral ak frquncs can asly b dnfd from s frquncs dndng on sgn of and wr s odd or vn. If s odd, scral ak frquncs ar gvn by *, ±,..., ± ±, ±,..., ± ( ) ( ) f >, f < If s vn, scral ak frquncs ar gvn by, ±,..., ± * ±, ±,..., ± ( ) ( ), f > f < 46

55 On or and, AR olynomal of rocss s ( ) z z. If s odd, n s olynomal as coml roos and on ral roo. T argumns of s roos ar, ±,..., ± θ ±, ±,..., ± ( ) ( ) f >, f < If s vn, n olynomal as coml roos wn <, and wo ral and coml roos wn >. T argumns of s roos ar gvn by, ±,..., ± θ ±, ±,..., ± ( ) ( ), f > f < Consqunly, * θ for any causal AR() rocss w, < and. Howvr, f, < and s n ngborood of zro, n modulus of ac roo of AR olynomal s sgnfcanly largr an on, wc conradcs convrs of Torm. In fac, all roos of suc a cas av sam modulus. 47

56 CHAPTER 4 Concluson In s car som moran ons of sudy ar dscussd by way of summarzng rsuls from rvous cars. Alcaons of Torm and Torm ar rsnd. An auorgrssv rocss of ordr wo s causal wn coffcns of AR olynomal sasfy condons, and. Undr s < < condons w found a scral ak frquncs nd o or ar qual o argumns of roos of AR olynomal wn s nar. T scral ak frquncy s qual o a las on of argumns of roos of AR olynomal wn bo roos ar ral. W also found a wn s zro argumns of roos ar acly qual o scral ak frquncs of rocss, rgardlss of valu of. In addon, argumns of coml roos of AR olynomal ar found o b aromaly qual o scral ak frquncs wn s n ngborood of zro. < For an AR() rocss f D U E (s Fgur for rgons) suc a s no nar -, n scral ak frquncs sgnfcanly dva from argumns of roos. In s cas w found a lo of scral dnsy s r srcly 48

57 ncrasng or srcly dcrasng ovr nrval [, ] and nc aks of lo of scral dnsy occur a frquncs or. Howvr, argumns of roos of AR olynomal of rocss l n nrval (, ). T closr s o. 5 smallr ar dffrncs bwn argumns of roos of AR olynomal and scral ak frquncs. In addon, closr s o ( ) or ( 4) 4 largr ar dffrncs bwn argumns of roos of AR olynomal and scral ak frquncs. An AR() rocss s causal wnvr coffcns of AR olynomal of rocss sasfy condons n (). For a causal AR() rocss argumn of any roo of rocss s aromaly or acly qual o on of scral ak frquncs wn nds o -. In addon, ac scral ak frquncy s qual o on of argumns of roos of AR olynomal of rocss wn bo and ar zro. W found n Scon. a argumn of any roo of AR olynomal of a causal AR() rocss s qual o on of scral ak frquncs wn, <. In addon, w found ou a for a causal AR( ) rocss,, argumn of any roo of AR olynomal of rocss nds o or s qual o on of scral ak frquncs wn s modulus nds o or quvalnly wn nds o -. Ts as followng rmarkabl alcaon. 49

58 Suos a w ar nrsd n scrum of a causal AR() rocss a as only on ak n nrval [, ], say a a frquncy. Tn usng rsuls gvn abov w can fnd a rocss w s rory. Clarly AR olynomal of rocss as coml conuga ar roos. Trfor, ordr of rocss could b grar an or qual o wo. If s vry clos -, n rocss could b of ordr wo, n wc cas ordr of rocss s qual o wc numbr of aks n nrval. T rocss could also b of ordr r. Ts ans wn wo coml roos ar locad nar un crcl and ral roo s locad far from un crcl, n wc coml roos ovrwlm ffc of ral roo on scral dnsy. Tr could b or ossbls, wc w can nrr smlarly. Now w dmonsra cas wr rocss s of ordr wo. Torm ndcas a nds o - s quvaln o z nds o,. I follows from Torm a argumn of on of roos of AR olynomal nd o and a of or nds o. Suos a z z. o and Arg( ) z r so a Arg( ) θ nds θ nds o. In addon, wrng roos n olar coordnas and usng m n AR olynomal gvs z θ θ ( r ) ( r ) θ θ ( r ) ( r ).... 5

59 Solvng s sysm of quaons gvs Hnc w can us s valus of and o aroma causal AR() modl w gvn rors by modl w , wr { } s a zro man saonary rocss and w s w nos. T roos of AR olynomal of s rocss ar aromaly.55 ±.875 (roundd o r dcmal lacs). T argumns of s roos ar aromaly qual o ±. 47 (roundd o r dcmal lacs). W can also smula lo of scral dnsy of abov rocss. Fgur rrsns lo of scral dnsy of rocss wn w nos rocss s Gaussan. Noc a scral ak frquncy. 47 s aromaly sam as argumn of on of roos..99 and -.98 Scrum Frquncy Fgur : Plo of Scral Dnsy of w

60 5 A rocss of ordr r s also ossbl. W dmonsra s as follows. Hr w forc coml roos o b locad nar un crcl and ral roo o b locad far from un crcl so a argumns of coml roos nd o ±. To accomls s l us coos modulus of ral roo o b aromaly qual o and a of coml roos o b aromaly qual o.. Assum a ral roo s osv so a s argumn s. Tn usng s nformaon n AR olynomal of AR() rocss sows a ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) r r r r r r r r r θ θ θ θ θ θ ( ) ( ) ( ) Trfor, w can aroma causal AR() modl w gvn rors by w ,

61 wr { } s a zro man saonary rocss and w s w nos. Undr assumon a w nos rocss s Gaussan, lo of scral dnsy of rocss s aromad by smulad lo gvn n Fgur..9, -.79 and.98 Scrum Frquncy Fgur : Plo of Scral Dnsy of w

62 Bblogray Andrson, T. W. (97). T Sascal Analyss of Tm Srs. Nw York: Wly. Brockwll, P. J and Davs, R. A. (99). Tm Srs: Tory and Mods, nd d. Nw York: Srngr-Vrlag. Ensor, K. B. and Nwon, H. J. (988). T Effc of ordr Esmaon on Esmang Pak Frquncy of an Auorgrssv Scral Dnsy. Bomrka, 75, Körnr, T. W. (4). A Comanon o Analyss. Provdnc: AMS, 6. Lyns, D. and Tøsm, D. (987). Loss of Scral Paks n Auorgrssv Scral Esmaon. Bomrka, 74, -6. Mullr, H. and Prw K. (99). Wak Convrgnc and Adav Pak Esmaon for Scral Dnss. Ann. Sas.,, Nwon, H. J. and Pagano, M. (98). A Mod for Drmnng Prods n Tm Srs. J. Amr. Sas. Assoc., 78, Sumway, R. H. and Soffr, D. S. (6). Tm Srs Analyss and Is Alcaons w R Eamls, nd d. Nw York : Srngr. Sn, R. A. and Saman, P. (99). Bas of Auorgrssv Scral Esmaors. J. Amr. Sas. Assoc., 85, W, W. W. S. (6). Tm Srs Analyss: Unvara and Muvara Mods, nd d. Boson: Parson Educaon, Inc. 54