RELATIONSHIPS BETWEEN SPECTRAL PEAK FREQUENCIES OF A CAUSAL AR(P) PROCESS AND ARGUMENTS OF ROOTS OF THE ASSOCIATED AR POLYNOMIAL.
|
|
- Sophia Hunt
- 5 years ago
- Views:
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
Frequency Response. Response of an LTI System to Eigenfunction
Frquncy Rsons Las m w Rvsd formal dfnons of lnary and m-nvaranc Found an gnfuncon for lnar m-nvaran sysms Found h frquncy rsons of a lnar sysm o gnfuncon nu Found h frquncy rsons for cascad, fdbac, dffrnc
More informationinnovations shocks white noise
Innovaons Tm-srs modls ar consrucd as lnar funcons of fundamnal forcasng rrors, also calld nnovaons or shocks Ths basc buldng blocks sasf var σ Srall uncorrlad Ths rrors ar calld wh nos In gnral, f ou
More informationWave Superposition Principle
Physcs 36: Was Lcur 5 /7/8 Wa Suroson Prncl I s qu a common suaon for wo or mor was o arr a h sam on n sac or o xs oghr along h sam drcon. W wll consdr oday sral moran cass of h combnd ffcs of wo or mor
More informationConsider a system of 2 simultaneous first order linear equations
Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm
More informationSupplementary Figure 1. Experiment and simulation with finite qudit. anharmonicity. (a), Experimental data taken after a 60 ns three-tone pulse.
Supplmnar Fgur. Eprmn and smulaon wh fn qud anharmonc. a, Eprmnal daa akn afr a 6 ns hr-on puls. b, Smulaon usng h amlonan. Supplmnar Fgur. Phagoran dnamcs n h m doman. a, Eprmnal daa. Th hr-on puls s
More informationOscillations of Hyperbolic Systems with Functional Arguments *
Avll ://vmd/gs/9/s Vol Iss Dcmr 6 95 Prvosly Vol No Alcons nd Ald mcs AA: An Inrnonl Jornl Asrc Oscllons of Hyrolc Sysms w Fnconl Argmns * Y So Fcly of Engnrng nzw Unvrsy Isw 9-9 Jn E-ml: so@nzw-c Noro
More informationt=0 t>0: + vr - i dvc Continuation
hapr Ga Dlay and rcus onnuaon s rcu Equaon >: S S Ths dffrnal quaon, oghr wh h nal condon, fully spcfs bhaor of crcu afr swch closs Our n challng: larn how o sol such quaons TUE/EE 57 nwrk analys 4/5 NdM
More informationAdvanced Queueing Theory. M/G/1 Queueing Systems
Advand Quung Thory Ths slds ar rad by Dr. Yh Huang of Gorg Mason Unvrsy. Sudns rgsrd n Dr. Huang's ourss a GMU an ma a sngl mahn-radabl opy and prn a sngl opy of ah sld for hr own rfrn, so long as ah sld
More informationConvergence of Quintic Spline Interpolation
Inrnaonal Journal o ompur Applcaons 97 8887 Volum 7 No., Aprl onvrgnc o Qunc Spln Inrpolaon Y.P. Dub Dparmn O Mamacs, L.N..T. Jabalpur 8 Anl Sukla Dparmn O Mamacs Gan Ganga ollg O Tcnog, Jabalpur 8 ASTRAT
More informationTheoretical Seismology
Thorcal Ssmology Lcur 9 Sgnal Procssng Fourr analyss Fourr sudd a h Écol Normal n Pars, augh by Lagrang, who Fourr dscrbd as h frs among Europan mn of scnc, Laplac, who Fourr rad lss hghly, and by Mong.
More informationA THESIS SUBMITTED TO THE GRADUATE SCHOOL OF APPLIED AND NATURAL SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY SERDAR ASLAN
NONLINEAR ESTIMATION TECHNIQUES APPLIED TO ECONOMETRIC PROBLEMS A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF APPLIED AND NATURAL SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY BY SERDAR ASLAN IN PARTIAL
More informationState Observer Design
Sa Obsrvr Dsgn A. Khak Sdgh Conrol Sysms Group Faculy of Elcrcal and Compur Engnrng K. N. Toos Unvrsy of Tchnology Fbruary 2009 1 Problm Formulaon A ky assumpon n gnvalu assgnmn and sablzng sysms usng
More informationSIMEON BALL AND AART BLOKHUIS
A BOUND FOR THE MAXIMUM WEIGHT OF A LINEAR CODE SIMEON BALL AND AART BLOKHUIS Absrac. I s shown ha h paramrs of a lnar cod ovr F q of lngh n, dmnson k, mnmum wgh d and maxmum wgh m sasfy a cran congrunc
More informationCIVL 8/ D Boundary Value Problems - Triangular Elements (T6) 1/8
CIVL 8/7 -D Boundar Valu Problm - rangular Elmn () /8 SI-ODE RIAGULAR ELEMES () A quadracall nrpolad rangular lmn dfnd b nod, hr a h vrc and hr a h mddl a ach d. h mddl nod, dpndng on locaon, ma dfn a
More informationChapter 9 Transient Response
har 9 Transn sons har 9: Ouln N F n F Frs-Ordr Transns Frs-Ordr rcus Frs ordr crcus: rcus conan onl on nducor or on caacor gornd b frs-ordr dffrnal quaons. Zro-nu rsons: h crcu has no ald sourc afr a cran
More informationSafety and Reliability of Embedded Systems. (Sicherheit und Zuverlässigkeit eingebetteter Systeme) Stochastic Reliability Analysis
Safy and Rlably of Embddd Sysms (Schrh und Zuvrlässgk ngbr Sysm) Sochasc Rlably Analyss Safy and Rlably of Embddd Sysms Conn Dfnon of Rlably Hardwar- vs. Sofwar Rlably Tool Asssd Rlably Modlng Dscrpons
More informationSafety and Reliability of Embedded Systems. (Sicherheit und Zuverlässigkeit eingebetteter Systeme) Stochastic Reliability Analysis
(Schrh und Zuvrlässgk ngbr Sysm) Sochasc Rlably Analyss Conn Dfnon of Rlably Hardwar- vs. Sofwar Rlably Tool Asssd Rlably Modlng Dscrpons of Falurs ovr Tm Rlably Modlng Exampls of Dsrbuon Funcons Th xponnal
More informationThe Variance-Covariance Matrix
Th Varanc-Covaranc Marx Our bggs a so-ar has bn ng a lnar uncon o a s o daa by mnmzng h las squars drncs rom h o h daa wh mnsarch. Whn analyzng non-lnar daa you hav o us a program l Malab as many yps o
More informationOne dimensional steady state heat transfer of composite slabs
BUILDING PHYSICS On dmnsonal sady sa a ransfr of compos slas Par 2 Ass. Prof. Dr. Norr Harmay Budaps Unvrsy of Tcnology and Economcs Dparmn of Buldng Enrgcs and Buldng Srvc Engnrng Inroducon - Buldng Pyscs
More information9. Simple Rules for Monetary Policy
9. Smpl Ruls for Monar Polc John B. Talor, Ma 0, 03 Woodford, AR 00 ovrvw papr Purpos s o consdr o wha xn hs prscrpon rsmbls h sor of polc ha conomc hor would rcommnd Bu frs, l s rvw how hs sor of polc
More informationFAULT TOLERANT SYSTEMS
FAULT TOLERANT SYSTEMS hp://www.cs.umass.du/c/orn/faultolransysms ar 4 Analyss Mhods Chapr HW Faul Tolranc ar.4.1 Duplx Sysms Boh procssors xcu h sam as If oupus ar n agrmn - rsul s assumd o b corrc If
More information167 T componnt oftforc on atom B can b drvd as: F B =, E =,K (, ) (.2) wr w av usd 2 = ( ) =2 (.3) T scond drvatv: 2 E = K (, ) = K (1, ) + 3 (.4).2.2
166 ppnd Valnc Forc Flds.1 Introducton Valnc forc lds ar usd to dscrb ntra-molcular ntractons n trms of 2-body, 3-body, and 4-body (and gr) ntractons. W mplmntd many popular functonal forms n our program..2
More informationLecture 1: Numerical Integration The Trapezoidal and Simpson s Rule
Lcur : Numrical ngraion Th Trapzoidal and Simpson s Rul A problm Th probabiliy of a normally disribud (man µ and sandard dviaion σ ) vn occurring bwn h valus a and b is B A P( a x b) d () π whr a µ b -
More informationSummary: Solving a Homogeneous System of Two Linear First Order Equations in Two Unknowns
Summary: Solvng a Homognous Sysm of Two Lnar Frs Ordr Equaons n Two Unknowns Gvn: A Frs fnd h wo gnvalus, r, and hr rspcv corrspondng gnvcors, k, of h coffcn mar A Dpndng on h gnvalus and gnvcors, h gnral
More informationUNIT #5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Answr Ky Nam: Da: UNIT # EXPONENTIAL AND LOGARITHMIC FUNCTIONS Par I Qusions. Th prssion is quivaln o () () 6 6 6. Th ponnial funcion y 6 could rwrin as y () y y 6 () y y (). Th prssion a is quivaln o
More informationEE243 Advanced Electromagnetic Theory Lec # 10: Poynting s Theorem, Time- Harmonic EM Fields
Appl M Fall 6 Nuruhr Lcur # r 9/6/6 4 Avanc lcromagnc Thory Lc # : Poynng s Thorm Tm- armonc M Fls Poynng s Thorm Consrvaon o nrgy an momnum Poynng s Thorm or Lnar sprsv Ma Poynng s Thorm or Tm-armonc
More informationChapter 13 Laplace Transform Analysis
Chapr aplac Tranorm naly Chapr : Ouln aplac ranorm aplac Tranorm -doman phaor analy: x X σ m co ω φ x X X m φ x aplac ranorm: [ o ] d o d < aplac Tranorm Thr condon Unlaral on-dd aplac ranorm: aplac ranorm
More informationChapter 7 Stead St y- ate Errors
Char 7 Say-Sa rror Inroucon Conrol ym analy an gn cfcaon a. rann ron b. Sably c. Say-a rror fnon of ay-a rror : u c a whr u : nu, c: ouu Val only for abl ym chck ym ably fr! nu for ay-a a nu analy U o
More informationProblem 1: Consider the following stationary data generation process for a random variable y t. e t ~ N(0,1) i.i.d.
A/CN C m Sr Anal Profor Òcar Jordà Wnr conomc.c. Dav POBLM S SOLIONS Par I Analcal Quon Problm : Condr h followng aonar daa gnraon proc for a random varabl - N..d. wh < and N -. a Oban h populaon man varanc
More informationApplying Software Reliability Techniques to Low Retail Demand Estimation
Applyng Sofwar Rlably Tchnqus o Low Ral Dmand Esmaon Ma Lndsy Unvrsy of Norh Txas ITDS Dp P.O. Box 30549 Dnon, TX 7603-549 940 565 3174 lndsym@un.du Robr Pavur Unvrsy of Norh Txas ITDS Dp P.O. Box 30549
More informationBoosting and Ensemble Methods
Boosng and Ensmbl Mhods PAC Larnng modl Som dsrbuon D ovr doman X Eampls: c* s h arg funcon Goal: Wh hgh probably -d fnd h n H such ha rrorh,c* < d and ar arbrarly small. Inro o ML 2 Wak Larnng
More informationHomework: Introduction to Motion
Homwork: Inroducon o Moon Dsanc vs. Tm Graphs Nam Prod Drcons: Answr h foowng qusons n h spacs provdd. 1. Wha do you do o cra a horzona n on a dsancm graph? 2. How do you wak o cra a sragh n ha sops up?
More information1) They represent a continuum of energies (there is no energy quantization). where all values of p are allowed so there is a continuum of energies.
Unbound Stats OK, u untl now, w a dalt solly wt stats tat ar bound nsd a otntal wll. [Wll, ct for our tratnt of t fr artcl and w want to tat n nd r.] W want to now consdr wat ans f t artcl s unbound. Rbr
More informationarxiv: v1 [math.ap] 16 Apr 2016
Th Cauchy problm for a combuson modl n porous mda J. C. da Moa M. M. Sanos. A. Sanos arxv:64.4798v [mah.ap] 6 Apr 6 Absrac W prov h xsnc of a global soluon o h Cauchy problm for a nonlnar racon-dffuson
More information( r) E (r) Phasor. Function of space only. Fourier series Synthesis equations. Sinusoidal EM Waves. For complex periodic signals
Inoducon Snusodal M Was.MB D Yan Pllo Snusodal M.3MB 3. Snusodal M.3MB 3. Inoducon Inoducon o o dsgn h communcaons sd of a sall? Fqunc? Oms oagaon? Oms daa a? Annnas? Dc? Gan? Wa quaons Sgnal analss Wa
More informationFI 3103 Quantum Physics
/9/4 FI 33 Quanum Physcs Aleander A. Iskandar Physcs of Magnesm and Phooncs Research Grou Insu Teknolog Bandung Basc Conces n Quanum Physcs Probably and Eecaon Value Hesenberg Uncerany Prncle Wave Funcon
More informationThe Fourier Transform
/9/ Th ourr Transform Jan Baptst Josph ourr 768-83 Effcnt Data Rprsntaton Data can b rprsntd n many ways. Advantag usng an approprat rprsntaton. Eampls: osy ponts along a ln Color spac rd/grn/blu v.s.
More informationNDC Dynamic Equilibrium model with financial and
9 July 009 NDC Dynamc Equlbrum modl wh fnancal and dmograhc rsks rr DEVOLDER, Inmaculada DOMÍNGUEZ-FABIÁN, Aurél MILLER ABSTRACT Classcal socal scury nson schms, combnng a dfnd bnf hlosohy and a ay as
More informationdy 1. If fx ( ) is continuous at x = 3, then 13. If y x ) for x 0, then f (g(x)) = g (f (x)) when x = a. ½ b. ½ c. 1 b. 4x a. 3 b. 3 c.
AP CALCULUS BC SUMMER ASSIGNMENT DO NOT SHOW YOUR WORK ON THIS! Complt ts problms during t last two wks of August. SHOW ALL WORK. Know ow to do ALL of ts problms, so do tm wll. Itms markd wit a * dnot
More informationELEN E4830 Digital Image Processing
ELEN E48 Dgal Imag Procssng Mrm Eamnaon Sprng Soluon Problm Quanzaon and Human Encodng r k u P u P u r r 6 6 6 6 5 6 4 8 8 4 P r 6 6 P r 4 8 8 6 8 4 r 8 4 8 4 7 8 r 6 6 6 6 P r 8 4 8 P r 6 6 8 5 P r /
More informationAnalysis of decentralized potential field based multi-agent navigation via primal-dual Lyapunov theory
Analyss of dcnralzd ponal fld basd mul-agn navgaon va prmal-dual Lyapunov hory Th MIT Faculy has mad hs arcl opnly avalabl. Plas shar how hs accss bnfs you. Your sory mars. Caon As Publshd Publshr Dmarogonas,
More informationReview - Probabilistic Classification
Mmoral Unvrsty of wfoundland Pattrn Rcognton Lctur 8 May 5, 6 http://www.ngr.mun.ca/~charlsr Offc Hours: Tusdays Thursdays 8:3-9:3 PM E- (untl furthr notc) Gvn lablld sampls { ɛc,,,..., } {. Estmat Rvw
More informationanger: Blood Circle Knife Safety Check list:
m a so k o lo ly. v a ll a nf saf Kn w k n a Knf Safy Cck ls: ng cum s do ays of us In w rn 1, Frs Ad k dff 2. Trand Frs Adr 3. Ladr can s vryon (scous s n a sm-crcl) 4. Evryon sad as n pcur o lf. 5. Us
More informationGaussian Random Process and Its Application for Detecting the Ionospheric Disturbances Using GPS
Journal of Global Posonng Sysms (005) Vol. 4, No. 1-: 76-81 Gaussan Random Procss and Is Applcaon for Dcng h Ionosphrc Dsurbancs Usng GPS H.. Zhang 1,, J. Wang 3, W. Y. Zhu 1, C. Huang 1 (1) Shangha Asronomcal
More informationLecture 4 : Backpropagation Algorithm. Prof. Seul Jung ( Intelligent Systems and Emotional Engineering Laboratory) Chungnam National University
Lcur 4 : Bacpropagaon Algorhm Pro. Sul Jung Inllgn Sm and moonal ngnrng Laboraor Chungnam Naonal Unvr Inroducon o Bacpropagaon algorhm 969 Mn and Papr aac. 980 Parr and Wrbo dcovrd bac propagaon algorhm.
More informationNAME: ANSWER KEY DATE: PERIOD. DIRECTIONS: MULTIPLE CHOICE. Choose the letter of the correct answer.
R A T T L E R S S L U G S NAME: ANSWER KEY DATE: PERIOD PREAP PHYSICS REIEW TWO KINEMATICS / GRAPHING FORM A DIRECTIONS: MULTIPLE CHOICE. Chs h r f h rr answr. Us h fgur bw answr qusns 1 and 2. 0 10 20
More informationLectures 9-11: Fourier Transforms
Lcurs 9-: ourr Transforms Rfrncs Jordan & Smh Ch7, Boas Ch5 scon 4, Kryszg Ch Wb s hp://wwwjhudu/sgnals/: go o Connuous Tm ourr Transform Proprs PHY6 Inroducon o ourr Transforms W hav sn ha any prodc funcon
More information8. Queueing systems. Contents. Simple teletraffic model. Pure queueing system
8. Quug sysms Cos 8. Quug sysms Rfrshr: Sml lraffc modl Quug dscl M/M/ srvr wag lacs Alcao o ack lvl modllg of daa raffc M/M/ srvrs wag lacs lc8. S-38.45 Iroduco o Tlraffc Thory Srg 5 8. Quug sysms 8.
More information(heat loss divided by total enthalpy flux) is of the order of 8-16 times
16.51, Rok Prolson Prof. Manl Marnz-Sanhz r 8: Convv Ha ransfr: Ohr Effs Ovrall Ha oss and Prforman Effs of Ha oss (1) Ovrall Ha oss h loal ha loss r n ara s q = ρ ( ) ngrad ha loss s a S, and sng m =
More informationA Note on Estimability in Linear Models
Intrnatonal Journal of Statstcs and Applcatons 2014, 4(4): 212-216 DOI: 10.5923/j.statstcs.20140404.06 A Not on Estmablty n Lnar Modls S. O. Adymo 1,*, F. N. Nwob 2 1 Dpartmnt of Mathmatcs and Statstcs,
More informationConventional Hot-Wire Anemometer
Convnonal Ho-Wr Anmomr cro Ho Wr Avanag much mallr prob z mm o µm br paal roluon array o h nor hghr rquncy rpon lowr co prormanc/co abrcaon roc I µm lghly op p layr 8µm havly boron op ch op layr abrcaon
More informationGrand Canonical Ensemble
Th nsmbl of systms mmrsd n a partcl-hat rsrvor at constant tmpratur T, prssur P, and chmcal potntal. Consdr an nsmbl of M dntcal systms (M =,, 3,...M).. Thy ar mutually sharng th total numbr of partcls
More informationDynamic Power Allocation in MIMO Fading Systems Without Channel Distribution Information
PROC. IEEE INFOCOM 06 Dynamc Powr Allocaon n MIMO Fadng Sysms Whou Channl Dsrbuon Informaon Hao Yu and Mchal J. Nly Unvrsy of Souhrn Calforna Absrac Ths papr consdrs dynamc powr allocaon n MIMO fadng sysms
More informationAdvanced time-series analysis (University of Lund, Economic History Department)
Advanced me-seres analss (Unvers of Lund, Economc Hsor Dearmen) 3 Jan-3 Februar and 6-3 March Lecure 4 Economerc echnues for saonar seres : Unvarae sochasc models wh Box- Jenns mehodolog, smle forecasng
More informationAR(1) Process. The first-order autoregressive process, AR(1) is. where e t is WN(0, σ 2 )
AR() Procss Th firs-ordr auorgrssiv procss, AR() is whr is WN(0, σ ) Condiional Man and Varianc of AR() Condiional man: Condiional varianc: ) ( ) ( Ω Ω E E ) var( ) ) ( var( ) var( σ Ω Ω Ω Ω E Auocovarianc
More informationDouble Slits in Space and Time
Doubl Slis in Sac an Tim Gorg Jons As has bn ror rcnly in h mia, a am l by Grhar Paulus has monsra an inrsing chniqu for ionizing argon aoms by using ulra-shor lasr ulss. Each lasr uls is ffcivly on an
More informationRobustness Experiments with Two Variance Components
Naonal Insue of Sandards and Technology (NIST) Informaon Technology Laboraory (ITL) Sascal Engneerng Dvson (SED) Robusness Expermens wh Two Varance Componens by Ana Ivelsse Avlés avles@ns.gov Conference
More informationOUTLINE FOR Chapter 2-2. Basic Laws
0//8 OUTLINE FOR Chapr - AERODYNAMIC W-- Basc Laws Analss of an problm n fld mchancs ncssarl nclds samn of h basc laws gornng h fld moon. Th basc laws, whch applcabl o an fld, ar: Consraon of mass Conn
More informationLecture 3: Phasor notation, Transfer Functions. Context
EECS 5 Fall 4, ctur 3 ctur 3: Phasor notaton, Transfr Functons EECS 5 Fall 3, ctur 3 Contxt In th last lctur, w dscussd: how to convrt a lnar crcut nto a st of dffrntal quatons, How to convrt th st of
More informationRandom Process Part 1
Random Procss Part A random procss t (, ζ is a signal or wavform in tim. t : tim ζ : outcom in th sampl spac Each tim w rapat th xprimnt, a nw wavform is gnratd. ( W will adopt t for short. Tim sampls
More informationLucas Test is based on Euler s theorem which states that if n is any integer and a is coprime to n, then a φ(n) 1modn.
Modul 10 Addtonal Topcs 10.1 Lctur 1 Prambl: Dtrmnng whthr a gvn ntgr s prm or compost s known as prmalty tstng. Thr ar prmalty tsts whch mrly tll us whthr a gvn ntgr s prm or not, wthout gvng us th factors
More informationTIME-DOMAIN EQUIVALENT EDGE CURRENT (EEC's) TECHNIQUE TO IMPROVE A TLM-PHYSICAL OPTICS HYBRID PROCEDURE
TIME-DOMAIN EQUIVALENT EDGE CURRENT (EEC's TECHNIQUE TO IMPROVE A TLM-PHYSICAL OPTICS HYBRID PROCEDURE J. Lanoë*, M. M. Ny*, S. L Magur* * Laboraory of Elcroncs and Sysms for Tlcommuncaons (LEST, GET-ENST
More informationPartition Functions for independent and distinguishable particles
0.0J /.77J / 5.60J hrodynacs of oolcular Syss Insrucors: Lnda G. Grffh, Kbrly Haad-Schffrl, Moung G. awnd, Robr W. Fld Lcur 5 5.60/0.0/.77 vs. q for dsngushabl vs ndsngushabl syss Drvaon of hrodynac Proprs
More information8-node quadrilateral element. Numerical integration
Fnt Elmnt Mthod lctur nots _nod quadrlatral lmnt Pag of 0 -nod quadrlatral lmnt. Numrcal ntgraton h tchnqu usd for th formulaton of th lnar trangl can b formall tndd to construct quadrlatral lmnts as wll
More informationPublished in: Proceedings of the Twenty Second Nordic Seminar on Computational Mechanics
Downloadd from vbn.aau.dk on: aprl 09, 019 Aalborg Unvrs Implmnaon of Moldng Consrans n Topology Opmzaon Marx, S.; Krsnsn, Andrs Schmd Publshd n: Procdngs of h Twny Scond Nordc Smnar on Compuaonal Mchancs
More informationCS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 4
CS434a/54a: Paern Recognon Prof. Olga Veksler Lecure 4 Oulne Normal Random Varable Properes Dscrmnan funcons Why Normal Random Varables? Analycally racable Works well when observaon comes form a corruped
More informationAn Indian Journal FULL PAPER. Trade Science Inc. The interest rate level and the loose or tight monetary policy -- based on the fisher effect ABSTRACT
[Typ x] [Typ x] [Typ x] ISSN : 0974 7435 Volum 10 Issu 18 BoTchnology 2014 An Indan Journal FULL PAPER BTAIJ, 10(18), 2014 [1042510430] Th nrs ra lvl and h loos or gh monary polcy basd on h fshr ffc Zhao
More informationEconomics 600: August, 2007 Dynamic Part: Problem Set 5. Problems on Differential Equations and Continuous Time Optimization
THE UNIVERSITY OF MARYLAND COLLEGE PARK, MARYLAND Economcs 600: August, 007 Dynamc Part: Problm St 5 Problms on Dffrntal Equatons and Contnuous Tm Optmzaton Quston Solv th followng two dffrntal quatons.
More informationSearching for pairing interactions with coherent charge fluctuations spectroscopy
Sarchng for parng nracons wh cohrn charg flucuaons spcroscopy J. Lornzana ISC-CNR, Sapnza, Unvrsy of Rom B. Mansar, A. Mann, A. Odh, M. Scaronglla, M. Chrgu, F. Carbon EPFL, Lausann Ouln Raman scarng Cohrn
More informationThermodynamic Properties of the Harmonic Oscillator and a Four Level System
www.ccsn.org/apr Appld Physcs Rsarch Vol. 3, No. ; May Thrmodynamc Proprs of h Harmonc Oscllaor and a Four Lvl Sysm Oladunjoy A. Awoga Thorcal Physcs Group, Dparmn of Physcs, Unvrsy of Uyo, Uyo, Ngra E-mal:
More informationEngineering Circuit Analysis 8th Edition Chapter Nine Exercise Solutions
Engnrng rcu naly 8h Eon hapr Nn Exrc Soluon. = KΩ, = µf, an uch ha h crcu rpon oramp. a For Sourc-fr paralll crcu: For oramp or b H 9V, V / hoo = H.7.8 ra / 5..7..9 9V 9..9..9 5.75,.5 5.75.5..9 . = nh,
More information3-2-1 ANN Architecture
ARTIFICIAL NEURAL NETWORKS (ANNs) Profssor Tom Fomby Dpartmnt of Economics Soutrn Mtodist Univrsity Marc 008 Artificial Nural Ntworks (raftr ANNs) can b usd for itr prdiction or classification problms.
More informationCONTINUOUS TIME DYNAMIC PROGRAMMING
Eon. 511b Sprng 1993 C. Sms I. Th Opmaon Problm CONTINUOUS TIME DYNAMIC PROGRAMMING W onsdr h problm of maxmng subj o and EU(C, ) d (1) j ^ d = (C, ) d + σ (C, ) dw () h(c, ), (3) whr () and (3) hold for
More informationComparative Study of Finite Element and Haar Wavelet Correlation Method for the Numerical Solution of Parabolic Type Partial Differential Equations
ISS 746-7659, England, UK Journal of Informaon and Compung Scnc Vol., o. 3, 6, pp.88-7 Comparav Sudy of Fn Elmn and Haar Wavl Corrlaon Mhod for h umrcal Soluon of Parabolc Typ Paral Dffrnal Equaons S.
More informationVertical Sound Waves
Vral Sond Wavs On an drv h formla for hs avs by onsdrn drly h vral omonn of momnm qaon hrmodynam qaon and h onny qaon from 5 and hn follon h rrbaon mhod and assmn h snsodal solons. Effvly h frs ro and
More informationRetarded Interaction of Electromagnetic field and Symmetry Violation of Time Reversal in Non-linear Optics
Rardd Inracon of Elcromagnc fld and Symmry Volaon of Tm Rrsal n Non-lnar Opcs M Xaocun (Insu of Torcal Pyscs n uzou, Cna, E-mal:mxc@6.com Absrac Basd on Documn (, by consdrng rardd nracon of radaon flds,
More informationInstitute of Actuaries of India
Insiu of Acuaris of India ubjc CT3 Probabiliy and Mahmaical aisics Novmbr Examinaions INDICATIVE OLUTION Pag of IAI CT3 Novmbr ol. a sampl man = 35 sampl sandard dviaion = 36.6 b for = uppr bound = 35+*36.6
More informationBethe-Salpeter Equation Green s Function and the Bethe-Salpeter Equation for Effective Interaction in the Ladder Approximation
Bh-Salp Equaon n s Funcon and h Bh-Salp Equaon fo Effcv Inacon n h Ladd Appoxmaon Csa A. Z. Vasconcllos Insuo d Físca-UFRS - upo: Físca d Hadons Sngl-Pacl Popagao. Dagam xpanson of popagao. W consd as
More informationCHAPTER 33: PARTICLE PHYSICS
Collg Physcs Studnt s Manual Chaptr 33 CHAPTER 33: PARTICLE PHYSICS 33. THE FOUR BASIC FORCES 4. (a) Fnd th rato of th strngths of th wak and lctromagntc forcs undr ordnary crcumstancs. (b) What dos that
More informationA L A BA M A L A W R E V IE W
A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N
More informationFolding of Regular CW-Complexes
Ald Mathmatcal Scncs, Vol. 6,, no. 83, 437-446 Foldng of Rgular CW-Comlxs E. M. El-Kholy and S N. Daoud,3. Dartmnt of Mathmatcs, Faculty of Scnc Tanta Unvrsty,Tanta,Egyt. Dartmnt of Mathmatcs, Faculty
More informationAlmost unbiased exponential estimator for the finite population mean
Almos ubasd poal smaor for f populao ma Rajs Sg, Pakaj aua, ad rmala Saa, Scool of Sascs, DAVV, Idor (M.P., Ida (rsgsa@aoo.com Flor Smaradac ar of Dparm of Mamacs, Uvrs of Mco, Gallup, USA (smarad@um.du
More informationCOMPLEX NUMBER PAIRWISE COMPARISON AND COMPLEX NUMBER AHP
ISAHP 00, Bal, Indonsa, August -9, 00 COMPLEX NUMBER PAIRWISE COMPARISON AND COMPLEX NUMBER AHP Chkako MIYAKE, Kkch OHSAWA, Masahro KITO, and Masaak SHINOHARA Dpartmnt of Mathmatcal Informaton Engnrng
More informationA MATHEMATICAL MODEL FOR NATURAL COOLING OF A CUP OF TEA
MTHEMTICL MODEL FOR NTURL COOLING OF CUP OF TE 1 Mrs.D.Kalpana, 2 Mr.S.Dhvarajan 1 Snior Lcurr, Dparmn of Chmisry, PSB Polychnic Collg, Chnnai, India. 2 ssisan Profssor, Dparmn of Mahmaics, Dr.M.G.R Educaional
More informationTransient Analysis of Two-dimensional State M/G/1 Queueing Model with Multiple Vacations and Bernoulli Schedule
Inrnaonal Journal of Compur Applcaons (975 8887) Volum 4 No.3, Fbruary 22 Transn Analyss of Two-dmnsonal Sa M/G/ Quung Modl wh Mulpl Vacaons and Brnoull Schdul Indra Assoca rofssor Dparmn of Sascs and
More informationCHAPTER CHAPTER14. Expectations: The Basic Tools. Prepared by: Fernando Quijano and Yvonn Quijano
Expcaions: Th Basic Prpard by: Frnando Quijano and Yvonn Quijano CHAPTER CHAPTER14 2006 Prnic Hall Businss Publishing Macroconomics, 4/ Olivir Blanchard 14-1 Today s Lcur Chapr 14:Expcaions: Th Basic Th
More informationTHE STRUCTURE OF THE COST OF CAPITAL UNDER UNCERTAINTY. Avraham Beja
THE STRUCTURE OF THE COST OF CAPITAL UNDER UNCERTAINTY Avraham Bja 1 Inroducon In h analyss of modls of compv marks undr uncrany, dffrn approachs can b dsngushd. On approach, ypcally dal wh n wlfar conomcs,
More informationFluctuation-Electromagnetic Interaction of Rotating Neutral Particle with the Surface: Relativistic Theory
Fluuaon-lroagn Inraon of Roang Nural Parl w Surfa: Rlavs or A.A. Kasov an G.V. Dov as on fluuaon-lroagn or w av alula rar for of araon fronal on an ang ra of a nural parl roang nar a polarabl surfa. parl
More informationSun and Geosphere, 2008; 3(1): ISSN
Sun Gosphr, 8; 3(): 5-56 ISSN 89-839 h Imporanc of Ha Conducon scos n Solar Corona Comparson of Magnohdrodnamc Equaons of On-Flud wo-flud Srucur n Currn Sh Um Dn Gor Asronom Spac Scncs Dparmn, Scnc Facul,
More informationAdaptive Neural Network Flight Control Using both Current and Recorded Data
AIAA Gudanc Navgaon and Conrol Confrnc and Eh - 3 Augus 7 Hlon H Souh Carolna AIAA 7-655 Adav Nural Nork Flgh Conrol Usng oh Currn and Rcordd Daa Grsh Chodhary * and Erc N. Johnson Darmn of Arosac Engnrng
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
More information1. Inverse Matrix 4[(3 7) (02)] 1[(0 7) (3 2)] Recall that the inverse of A is equal to:
Rfrncs Brnank, B. and I. Mihov (1998). Masuring monary policy, Quarrly Journal of Economics CXIII, 315-34. Blanchard, O. R. Proi (00). An mpirical characrizaion of h dynamic ffcs of changs in govrnmn spnding
More informationPhysics of Very High Frequency (VHF) Capacitively Coupled Plasma Discharges
Physcs of Vry Hgh Frquncy (VHF) Capactvly Coupld Plasma Dschargs Shahd Rauf, Kallol Bra, Stv Shannon, and Kn Collns Appld Matrals, Inc., Sunnyval, CA AVS 54 th Intrnatonal Symposum Sattl, WA Octobr 15-19,
More informationCase Study 1 PHA 5127 Fall 2006 Revised 9/19/06
Cas Study Qustion. A 3 yar old, 5 kg patint was brougt in for surgry and was givn a /kg iv bolus injction of a muscl rlaxant. T plasma concntrations wr masurd post injction and notd in t tabl blow: Tim
More information5. An object moving along an x-coordinate axis with its scale measured in meters has a velocity of 6t
AP CALCULUS FINAL UNIT WORKSHEETS ACCELERATION, VELOCTIY AND POSITION In problms -, drmin h posiion funcion, (), from h givn informaion.. v (), () = 5. v ()5, () = b g. a (), v() =, () = -. a (), v() =
More informationIntroduction to logistic regression
Itroducto to logstc rgrsso Gv: datast D { 2 2... } whr s a k-dmsoal vctor of ral-valud faturs or attrbuts ad s a bar class labl or targt. hus w ca sa that R k ad {0 }. For ampl f k 4 a datast of 3 data
More informationScattering at an Interface: Oblique Incidence
Course Insrucor Dr. Raymond C. Rumpf Offce: A 337 Phone: (915) 747 6958 E Mal: rcrumpf@uep.edu EE 4347 Appled Elecromagnecs Topc 3g Scaerng a an Inerface: Oblque Incdence Scaerng These Oblque noes may
More informationDr. Junchao Xia Center of Biophysics and Computational Biology. Fall /21/2016 1/23
BIO53 Bosascs Lcur 04: Cral Lm Thorm ad Thr Dsrbuos Drvd from h Normal Dsrbuo Dr. Juchao a Cr of Bophyscs ad Compuaoal Bology Fall 06 906 3 Iroduco I hs lcur w wll alk abou ma cocps as lsd blow, pcd valu
More informationReliability analysis of time - dependent stress - strength system when the number of cycles follows binomial distribution
raoal Joural of Sascs ad Ssms SSN 97-675 Volum, Numbr 7,. 575-58 sarch da Publcaos h://www.rublcao.com labl aalss of m - dd srss - srgh ssm wh h umbr of ccls follows bomal dsrbuo T.Sumah Umamahswar, N.Swah,
More informationCONSUMER BEHAVIOR CHANGES ACROSS INCOME LEVELS: MEAT MARKET ANALYSIS. A Dissertation. presented to. the Faculty of the Graduate School
CONSUMER BEHAVIOR CHANGES ACROSS INCOME LEVELS: MEAT MARKET ANALYSIS A Dssraon rsnd o h Faculy of h Gradua School a h Unvrsy of Mssour-Coluba In Paral Fulflln of h Rqurns for h Dgr Docor of Phlosohy by
More information