뗚10뻭ꆡ뗚3웚ꆡꆡꆡꆡꆡꆡꆡ훐맺맜샭뿆톧2002쓪ꆡꆡ6퓂ꆡꆡꆡꆡCꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡ,.,ꆡ2002컄헂뇠뫅:1003-207(2002)03-0011-07훐맺막욱쫐뎡뛠훘럖탎폎ퟟ벰웤풤닢뫎붨쏴,뎣ꆡ쯉(뚫쓏듳톧뺭볃맜샭톧풺,붭쯕쓏뺩ꆡ210096)햪ꆡ튪:막욱볛룱늨뚯맦싉뗄퇐뺿쫇풤닢뗄믹뒡,뛠훘럖탎맽돌쫇웹뷱캪횹ퟮ캪럻뫏볛룱늨뚯쳘탔뗄쒣탍ꆣ놾컄퇩횤쇋훐맺막욱쫐뎡뗄뛠훘럖탎폎ퟟ,늢룹뻝뛠훘럖탎맽돌뗄뻖늿돟뛈쳘탔뫍뛠돟뛈쿠맘탔붨솢쇋킡늨뫍짱뺭췸싧쿠뷡뫏뗄막욱볛룱풤닢쒣탍ꆣ쪵횤퇐뺿뷡맻뇭쏷,놾컄쒣탍풤닢뺫뛈뷏평웤쯻쒣탍뗃떽뗄풤닢뺫뛈쏷쿔쳡룟ꆣ맘볼듊:뛠훘럖탎;킡늨;짱뺭췸싧;막욱쫐뎡;풤닢훐춼럖샠뫅:ꆡꆡꆡ컄쿗뇪쪶싫:Aꆡꆡ막욱볛룱뗄ힼ좷풤닢틢캶ퟅ춶헟룟뛮뗄쫐뎡Gh[8\]ashghaie뗈\ꆢHiroakiKatsurag\[9\]i뫍Pasquini믘놨뫍헾뢮늿쏅뛔쫐뎡뗄폐킧볠맜ꆣ캪듋ퟔ19쫀뗈\[10\뛔]늻춬뗄뷰죚쫐뎡쪵횤퇐뺿훐뗃떽퇩횤ꆣ볍막욱쫐뎡붨솢틔살,훚뛠맺췢톧헟뛔막욱볛룱늨놾컄퇩횤쇋훐맺막욱쫐뎡뗄뛠훘럖탎폎ퟟ,늢뚯맦싉벰웤풤닢쒣탍뗄퇐뺿탎돉튻룶붹뗣ꆣ퓧웚퇐룹뻝뛠훘럖탎맽돌뗄뻖늿돟뛈쳘탔뫍뛠돟뛈쿠맘탔뺿죏캪뛔쫽볛룱쫇튻늼샊퓋뚯,쫕틦쫇헽첬뗄뛀솢붨솢쇋킡늨뫍짱뺭췸싧쿠뷡뫏뗄막욱볛룱풤닢쒣춬럖늼(IndependentIdenticallyDistributed,IID)쯦탍ꆣ믺뇤솿ꆣ좻뛸뛔훚뛠막욱쫐뎡뗄퇐뺿랢쿖,막욱볛룱늨뚯훐듦퓚ퟅ폫헢튻샭싛늻럻뗄쳘탔,죧쫕틦럖1ꆡ훐맺막욱쫐뎡뛠훘럖탎폎ퟟ늼뗄엖캲뫍늨럹뗄뎤웚쿠맘탔\[1\]ꆣꆰ엖캲ꆱ쮵쏷쇋ꆡ떥럖탎맽돌볛룱뇤뚯훐놩헇뫍놩뗸랢짺뗄룅싊풶풶룟폚헽첬럖뷼쪮쓪살,럖탎샭싛,폈웤쫇떥럖탎샭싛,틑놻늼풤볆뗄쟩뿶;늨럹뗄뎤웚쿠맘탔쮵쏷쫕틦늢럇뛀튻듳엺맺췢톧헟펦폃폚뷰죚쫐뎡볛룱뗄퇐뺿훐ꆣ훷솢,뛸쫇듦퓚럇쿟탔쿠맘,듓헢튻뷇뛈뛸퇔막욱볛룱튪뗄맛뗣뻍쫇죏캪볛룱늨뚯쫇튻떥럖탎맽돌,튲벴늨뚯듦퓚튻뚨맦싉,뻟폐튻뚨뗄뿉풤닢탔ꆣ평듋맘늻춬쪱볤돟뛈짏뗄볛룱뇤뚯럖늼듦퓚춳볆짏뗄쿠쯆볼뗄컊쳢퓚폚헒떽튻훖쓜릻쏨쫶쪵볊볛룱늨뚯룷훖탔ꆣ쳘탔뗄쒣탍,늢뻝듋붨솢뫏샭뗄풤닢랽램,믱뗃ힼ좷뚨틥1:솬탸쪱볤욽컈퓶솿맽돌{X(t),tꇊT}뗄풤닢ꆣ헫뛔늼샊퓋뚯뗄좱쿝,뫳죋폖쳡돶뛠훖뗄볛룱늨뚯쒣탍볓틔룄뷸,죧L-컈뚨맽돌싺ퟣfd:X(t)=-HaX(at),tꇊT,Πa>0,0<H<1(L-Sta2퓲돆X(t)캪튻떥럖탎맽돌ꆣblePro\[2\]cess)ꆢ럖쫽늼샊퓋뚯(FractionalBrownian웤훐뗈쪽뗄몬틥쫇횸웤폐쿞캬럖늼쿠춬,H돆Motion\[3\])뫍ARCH(AutoRegressiveConditional캪HÊlder횸쫽ꆣL-컈뚨맽돌뫍럖쫽늼샊퓋뚯뻍쫇Heteroscedasticity)쒣탍\[4\뗈]뗈ꆣ떫헢킩쒣탍뚼횻뷰죚쫐뎡훐ퟮ캪훸쏻뗄솽룶떥럖탎맽돌ꆣ힡쇋볛룱늨뚯쳘헷뗄쒳튻랽쏦,ퟮ뷼평Mandelbrot평뚨틥1뿉횪,떥럖탎맽돌죏캪뇭쪾돟뛈쳘헷뗈\[5-7\쳡]돶뗄뛠훘럖탎맽돌쫇웹뷱캪횹ퟮ캪좫쏦뗄HÊlder횸쫽Ht룷룶쪱뿌쿠춬,늻쯦쪱볤뇤뮯ꆣ틲뗄쏨쫶볛룱늨뚯쳘헷뗄쒣탍,쯼늻뷶쓒삨쇋짏쫶쒣듋떥럖탎맽돌쫇튻룶늻뻟폐뻖늿돟뛈쳘헷뗄탅뫅,탍쯹쏨쫶뗄볛룱늨뚯쳘헷,룼풤볻쇋퓚웤쯻쒣탍훐믲헟쮵떥럖탎맽돌뷶뻖쿞폚떥럖탎샭싛랶돫,H닎쎻폐쳥쿖뗄볛룱늨뚯뛠훘럖탎쳘탔ꆣ헢튻쳘탔틑퓚쫽횻뇭쪾돶튻룶맽돌뗄뎤웚춳볆탐캪,늢쎻폐뾼싇쫕룥죕웚:2001퓚쒳튻쪱뿌뗄뻖늿쳘탔ꆣ-06-20ퟷ헟볲뷩:뫎붨쏴(1956-),쓐(몺ퟥ),붭쯕컞컽죋,뚫쓏듳톧뺭ꆡ뛠훘럖탎맽돌볃맜샭톧풺,뢱풺뎤,뷌쫚(늩떼),퇐뺿랽쿲:뷰죚맜막욱볛룱뗄뇤뮯럇뎣뢴퓓,벴쪹닉폃떥럖탎샭싛샭ꆢ펦벱맜샭.뷏늼샊퓋뚯뛔볛룱늨뚯폐쇋룼캪뿍맛뗄뿌뮮,떫죔풶
ꆤ12ꆤ훐맺맜샭뿆톧ꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡ2002쓪풶늻릻좫쏦ꆣ헽죧Mandelbrot횸돶\[11\],떥럖탎맽돌맽쿟탔믘맩믱뗃돟뛈몯쫽ꛓ(q)뗄맀볆횵;퓙춨맽횻쓜힡볛룱늨뚯뗄쒳튻쳘헷,죧L-컈뚨맽돌쯤좻ꛓ(q)ꆫq횮볤쫇럱듦퓚럇쿟탔맘쾵에뛏쫇럱캪뛠뿉틔쏨쫶볛룱뗄놩헇놩뗸쳘탔,좴컞램쏨쫶늨럹뗄뎤훘럖탎맽돌ꆣ웚쿠맘탔,럖쫽늼샊퓋뚯쟩뿶잡뫃쿠랴ꆣ뷸튻늽짏쫶틔뻘뗄쳘탔듓좫뻖뗄뷇뛈뚨틥뛠훘럖탎맽Mandelbrot죏캪믹폚뛠훘럖탎랶돫뗄뛠훘럖탎맽돌늢늻쫇쯼뗄튻룶횱맛쏨쫶,퇐뺿뛠훘럖탎맽돌뗄돌\[5ꆫ7,12\]뿉틔룼캪좫쏦뗘쏨쫶볛룱늨뚯뗄쳘헷ꆣ뻖늿탐캪룼캪훘튪,춬쪱튲룼쓜뷒쪾쯼뗄놾훊ꆣ뚨틥2:솬탸쪱볤맽돌{X(t),tꇊT},뻟폐욽뛠훘럖탎맽돌뻖늿탐캪뗄퇐뺿붨솢퓚뻖늿컈퓶솿쟒싺ퟣ:E[|Xq(t,ꚤt)|]=HÊlder횸쫽룅쓮믹뒡횮짏ꆣ뻖늿HÊlder횸쫽쫇쪵c(q)(ꚤꛓ(q)+1t)뇤몯쫽럖컶뗄튻룶훘튪룅쓮,쏨쫶쇋룸뚨쪱뿌뗄몯퓲돆X(t)캪튻뛠훘럖탎맽돌ꆣ쫽뇤뮯뗄웦틬탔ꆣ웤훐X(t,ꚤt)ꇔX(t+ꚤt)-X(t),0ꇜtꇜ뚨틥3:볙뚨몯쫽g퓚튻뚨룸뚨쪱뿌t뗄쇚폲폐T,ꛓ(q)돆캪돟뛈몯쫽ꆣ뚨틥,퓲떱ꛓ(q)=qH-1쪱뻍캪떥럖탎맽돌,웤돟뛈ꛁ(t)=Sup{ꛂꇝ0:|g(t+ꚤt)-g(t)|몯쫽ꛓ(q)쫇평H캨튻좷뚨뗄q뗄쿟탔몯쫽ꆣ떫=O(|ꚤꛂt|)asꚤtꇺ0ꛓ(q)캪럇쿟탔몯쫽쫴폚뛠훘럖탎맽돌룼캪튻냣뗄ꆡꆡ돆캪몯쫽g퓚쪱뿌t뗄HÊlder횸쫽ꆣ쟩뿶ꆣꛁ(t)튻냣뿉틔퓚(-ꇞ,+ꇞ)랶캧쓚좡횵,떫짏쫶탰쫶룸돶쇋뛠훘럖탎튻룶볲떥뗄볬퇩랽떱(쟒뷶떱)g퓚t뢽뷼폐뷧쪱ꛁ(t)럇뢺,놾컄튲횻램ꆣ뾼싇뚨틥퓚[0,T]짏볛룱탲쇐P(t),뛔쫽볛룱뾼싇헢훖쟩뿶ꆣ듓뚨틥3,늻쓑믱죧쿂뗈볛맘쾵쪽:(t)X(t)=lnP(t)-lnP(0)(1)|g(t+ꚤt)-g(t)|ꆫC(ꚤꛁ(tt)5)ꆡꆡ붫[0,T]럖룮돉N룶뎤캪ꚤt뗄쟸볤,럖룮몯쫽ꆡꆡ웤훐헽뎣솿Ct돆캪잰틲ퟓꆣ듓짏쪽뿉틔볆쯣N튻킩쳘쫢쟩뿶뗄뻖늿HÊlder횸쫽,죧늻솬탸뗣뒦캪Sq(T,ꚤt)=ꇆ-1|X(iꚤt,ꚤqt)|(2)i=00,gꆯꇙ0뗄뿉떼뗣뒦캪1ꆣ퓲평뻘뗄쳘탔벰욽컈퓶솿뿉뗃:붫짏쫶룅쓮췆맣떽솬탸쪱볤맽돌X(t)뿉뗃:lnE[Sq(T,ꚤt)]=ꛓ(q)ln(ꚤt)+c(q)lnT(3)|X(t,ꚤt)|ꆫC(ꚤꛁ(t)tt),ꆡꆡ죧맻퇹놾쫽솿뫜뛠,Sq(ꚤt)ꇖE(Sq(ꚤt),평Πtꇊ[0,T\],ꚤtꇺ0(6)듋뿉룹뻝lnSq(ꚤt)ꆫln(ꚤt)횮볤뗄맘쾵맀볆ꆡꆡꛁ(t)쪵볊쫇쏨쫶쇋X(t)뇤뮯뗄뻖늿돟뛈쳘ꛓ(q)ꆣMandelbro[7]t춨맽듳솿뗄럂헦쫽뻝퇩횤쇋헢탔,죧늼샊퓋뚯룷뗣캪1/2,럖쫽늼샊퓋뚯룷뗣캪훖볬퇩랽램뗄폐킧탔ꆣ좻뛸뛔폚컒맺탂탋뗄막욱쫐H,뿉볻헢킩맽돌뗄뻖늿뇤뮯쿠춬,뻟폐캨튻뗄뻖늿뎡,샺쪷쫽뻝뷏짙,쪮뛠쓪살닅볇슼쇋2000뛠룶죕HÊlder횸쫽ꆣ뛠훘럖탎맽돌ퟮ짮뿌뗄놾훊뻍쫇틽죫쫕엌횸쫽,짏쫶늻훘뗾뒰뿚뗄볬퇩랽램믡쯦ꚤt뗄퓶쇋뻖늿돟뛈쳘탔뗄뛠퇹탔,뻟폐솬탸뇤뮯뗄ꛁ(t),듳쪹뿉폃퇹놾쫽뻝톸쯙복짙,평듋듸살뷏듳뗄컳닮,틔듋살쏨쫶볛룱늨뚯퓚쪱볤짏뗄늻뻹퓈탔,뛸늻쫇틲듋놾컄닉폃튻훖룼캪뫏샭뗄뮬뚯뒰뿚랽램맀볆쿳틔췹뗄ARCH뗈쒣탍춨맽잰틲ퟓCt뗄뛌웚쿠맘E(S탔살쏨쫶ꆣq(ꚤt),뻟쳥죧쿂:Nꛁ(t)뗄럖늼뿉폃뛠훘럖탎웗f(ꛁ)살뿌뮮,뛠E[Sq(Tq,ꚤt)]=Eꇆ-1|X(iꚤt,ꚤt)|훘럖탎웗쫇돟뛈몯쫽ꛓ(q)뗄Legendre뇤뮻:i=0ꛁ=dꛓ(q)/dq,f(ꛁ)=ꛁq-ꛓ(q)(7)=NꇁEq|X(ꚤt)|ꇖNM-ꚤt/ꛄꆡꆡ떱f=(ꛁ)뗄횧벯늻횹튻룶뗣쪱,뻍죏캪뻟폐M-ꚤt/ꛄꇁꇆ뛠훘럖탎쳘탔,틲듋듓뛠훘럖탎웗훐튲뿉틔횱맛뗘|X(ꛄj,ꚤqt)|(4)j=0에뇰뛠훘럖탎쳘탔ꆣ폐맘뛠훘럖탎웗f(ꛁ)룼캪쿪쾸웤훐ꛄ뇭쪾ퟮ킡닉퇹볤룴,M=T/ꛄ,뇭쪾ퟮ뗄쳖싛뿉볻컄쿗[6]ꆣ킡닉퇹볤룴쿂뗄쟸볤쫽ꆣ뛠훘럖탎샭싛쇭췢튻룶훘튪쳘탔뻍쫇뛠돟뛈쿠평듋룹뻝쪽(3)뫍쪽(4)헫뛔늻춬뗄q틔맘탔ꆣ뛠훘럖탎맽돌뿉평돋램벶솪(MultiplicativelnE[Sq(ꚤt)]ꆫln(ꚤt)캪ퟸ뇪ퟷ춼,죧맻Cascade)쒣탍닺짺,헢틢캶ퟅ뛠훘럖탎맽돌늻춬돟lnE[Sq(ꚤt)]ꆫln(ꚤt)횮볤듦퓚쿟탔맘쾵,퓲뿉춨뛈짏뗄늨뚯듦퓚쿠맘탔,튲벴듳돟뛈짏뗄늨뚯펰쿬
뗚3웚ꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡ뫎붨쏴뗈:훐맺막욱쫐뎡뛠훘럖탎폎ퟟ벰웤풤닢ꆤ13ꆤ킡돟뛈짏뗄늨뚯,헢퓚맺췢튻엺톧헟뛔췄쇷\[13\뫍]뷰죚쫐뎡\[14\뗄]퇐뺿훐뗃떽쇋퇩횤ꆣꆡ짏횤ퟛ횸뛠훘럖탎폎ퟟ쪵횤퇐뺿평폚짏몣막욱쫐뎡웰늽퓧,쫽뻝뛠,쟒듳엌횸쫽ퟛ뫏쇋룷룶랽쏦뗄펰쿬,뻟폐뷏잿뗄듺뇭탔,틲듋놾컄틔짏횤ퟛ횸캪퇐뺿뛔쿳,볛룱탲쇐좡1990쓪12퓂ꆫ2000쓪11퓂뎤뛈캪2448뗄죕쫕엌횸쫽,틔P(t)뇭쪾,늢내쪽(1)뚨틥X(t),뻟쳥죧춼1쯹쪾ꆣ춼2ꆡlnE\[Sq(ꚤt)]ꆫln(ꚤt)춼춼1ꆡ짏횤ퟛ횸죕쫕엌횸쫽ꆡꆡ내쪽(3)뫍쪽(4),틔lnE[Sq(ꚤt)]ꆫln(ꚤt)캪ퟸ뇪ퟷ춼,좡q=1,3,5,7,9;평폚횻뾼싇쟺쿟뗄킱싊,뷘뻠쎻폐틢틥,틲듋ퟷ춼쪱뛔쯹폐뗄q춨맽욽틆쇮lnE[Sq(1)]=0;쇭췢평폚퇹놾뎤뛈뗄쿞훆,캪놣횤볬퇩뗄뿉뾿탔,좡ln(ꚤt)=0ꆫ4,벴ꚤt=1ꆫ55ꆣ뻟쳥뷡맻죧춼2쯹쪾,듓쿂떽짏쟺쿟틀듎캪q춼3ꆡꛓ(q)ꆫq춼=1,3,5,7,9ꆣ쫗쿈듓춼2훐뿉맛닢떽뛔폚쯹폐뗄q횵듓춼3훐뿉틔뾴돶ꛓ(q)ꆫq횮볤듦퓚ퟅ쏷쿔lnE([Sq(ꚤt)]뫍ln(ꚤt)횮볤듦퓚쏷쿔뗄쿟탔맘뗄럇쿟탔맘쾵,춼4훐뿉틔뾴돶뻖늿HÊlder횸쫽퓚쾵;틔쿟탔믘맩믱뗃돟뛈몯쫽ꛓ(q)뗄맀볆횵,내뷏듳랶캧쓚뇤뮯,평듋뿉틔쮵쏷짏횤ퟛ횸뗄볛룱늨ꛓ(q)ꆫq캪ퟸ뇪ퟷ춼뇣뿉쇋뷢쯼쏇횮볤듦퓚뗄맘뚯쫇튻뛠훘럖탎쯦믺폎ퟟꆣ쾵,뻟쳥죧춼3쯹쪾ꆣ뷸튻늽틔ꛓ(q)맀볆횵춨맽쪽2ꆡ풤닢쒣탍(7)맀볆뛠훘럖탎웗f(ꛁ),뻟쳥뷡맻죧춼4쯹쪾ꆣꆡꆡ탨튪쮵쏷뗄짏뷚뛔컒맺막욱볛룱뛠훘럖탎쳘탔뗄퇐뺿뇭쫇q<0쪱죎뫎뷏킡뗄늨뚯뚼폐뿉쓜쪹E[Sq(ꚤt)]쏷,막욱볛룱늨뚯듦퓚뻖늿돟뛈쳘헷ꛁ(t),늢평듋횵럇뎣횮듳,퓲죎뫎컳닮,쳘뇰쫇쪵볊돌탲볆쯣맽돌틽웰볛룱늨뚯퓚쪱볤짏뗄늻뻹퓈탔ꆣ뛔럖쫽늼샊퓋훐듸살뗄짡죫컳닮,붫벫듳펰쿬E[Sq(ꚤt)]뗄횵,틲뚯뛸퇔ꛁ(t)=H,H뇭쏷쇋볛룱뇤뚯퓚헻룶쪱폲듋놾컄볆쯣훐쎻폐뾼싇q<0뗄쟩뿶ꆣ춬샭,놾컄횻볆짏뗄춳볆쿠맘탔,틔듋샠췆ꛁ(t)튲뻍뇭쏷쇋퓚t쪱쯣쇋뛠훘럖탎웗뗄ퟳ냫늿럖,폒냫늿럖뛠훘럖탎웗뗄뿌뢽뷼볛룱뇤뚯뗄쿠맘탔,뛔폚뷏캪뛌웚풤닢뛸퇔,볆쯣탨폃떽q<0뗄쟩뿶ꆣ헢훖뻖늿뗄쿠맘탔뇈퓚헻룶쪱폲짏뗄춳볆쿠맘탔룼
ꆤ14ꆤ훐맺맜샭뿆톧ꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡ2002쓪폐볛횵ꆣ볛룱늨뚯뗄뻖늿돟뛈쳘헷ꛁ(t)뿉틔춨맽킡늨뇤뮻퓚킡늨폲뗄ꆰ쪱볤-돟뛈ꆱ뿕볤훐킡늨쾵쫽(1)틔쪱볤캪믹ힼ,틔쒳튻쪱뿌뗄룷돟뛈킡늨쾵횮볤뗄맘쾵믱좡[15],평듋뷼쓪살튻킩맺췢톧헟룷훖쫽ퟷ캪쳘헷솿쫤죫짱뺭췸싧믲쒣뫽쾵춳풤닢;킡늨뇤뮻쾵쫽ퟷ캪쳘헷솿쫤죫짱뺭췸싧[16]믲쒣뫽(2)틔돟뛈캪믹ힼ,퓚쎿룶돟뛈춨맽늻춬쪱뿌횮췆샭쾵춳[17]뷸탐막욱볛룱뗄풤닢ꆣ쿖폐컄쿗훐뗄헢볤뗄쿠맘탔믱뗃룷돟뛈짏풤닢횵,좻뫳춨맽킡늨랴킩풤닢랽램뿉맩뷡캪솽샠:뇤뮻훘릹볛룱늨뚯뗄풤닢ꆣ틲듋헢킩풤닢랽램횻욬쏦뗘뾼싇볛룱늨뚯훐뗄쿠맘탔,죧랽램(1)횻뾼싇쇋춬튻쪱뿌늻춬돟뛈짏볛룱늨뚯뗄쿠맘탔;랽램(2)횻뾼싇쇋춬튻돟뛈짏늻춬쪱뿌볛룱늨뚯횮볤뗄쿠맘탔ꆣ죧1뷚훐쯹쫶,뛠훘럖탎맽돌듦퓚뛠돟뛈쿠맘탔,튲벴늻뷶냼삨춬튻돟뛈늻춬쪱뿌횮볤뗄쿠맘탔,춬쪱튲냼삨늻춬돟뛈횮볤뗄쿠맘탔ꆣ평폚퓚킡늨폲쪱볤ꆪꆪꆪ돟뛈뿕볤훐뗄킡늨뇤뮻쾵쫽뿉틔쳡좡돶뛠훘럖탎맽돌뗄뻖늿돟뛈쳘탔,춬쪱뿉틔쳥쿖늻춬돟뛈짏뗄볛룱늨뚯,틲듋틔킡늨뇤뮻쾵쫽컞틉쫇쳥쿖볛룱늨뚯ퟮ뫃뗄쳘헷솿,맘볼퓚폚죧뫎ퟮ뫏샭뗘쪹폃헢킩쳘헷솿폃폚풤닢ꆣ캪듋뾼싇떽짏쫶뻖늿돟뛈쳘탔뫍뛠돟뛈쿠맘탔,놾컄붨솢쇋튻룶믹폚킡늨뇤뮻뗄틅뒫쯣램뫍짱뺭췸싧쿠뷡뫏뗄풤닢쒣탍,뻟쳥춼4ꆡ뛠훘럖탎웗f(ꛁ)춼쒣탍죧춼5쯹쪾ꆣ춼5ꆡ풤닢쒣탍ꆡꆡ짱뺭췸싧뻟폐볇틤뫍랺뮯럇쿟탔쿠맘뗄쓜솦,(2)춨맽틅뒫쯣램믱뗃룷돟뛈짏뗄ퟮ폅쪱볤퇓퓲퓚쒳튻돟뛈짏풤닢쪱,놾컄쒣탍늻뷶뾼싇쇋춬튻동;돟뛈짏볛룱늨뚯뗄쿠맘탔,춬쪱튲뾼싇쇋늻춬돟뛈(3)붫룷돟뛈짏킡늨뇤뮻쾵쫽벰웤쪱볤퇓동뗄볤볛룱늨뚯뗄쿠맘탔,뻟쳥맽돌죧쿂:킡늨뇤뮻쾵쫽쫤죫짱뺭췸싧,믱뗃풤닢뷡맻ꆣ(1)춨맽쒳튻킡늨뇤뮻믱뗃N룶돟뛈짏뗄킡늨놾컄틅뒫쯣램폃폚ퟮ폅쪱볤퇓동뫍ퟮ폅췸싧뷡뇤뮻쾵쫽;릹,뻟쳥펦폃늽훨죧쿂:
뗚3웚ꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡ뫎붨쏴뗈:훐맺막욱쫐뎡뛠훘럖탎폎ퟟ벰웤풤닢ꆤ15ꆤ(1)좷뚨쫊펦몯쫽ꆣ쫊펦몯쫽횸쏷쇋좺쳥뷸뮯뗄x(t)캪탨튪풤닢뗄횵,컞듓믱뗃ꆣ쿖폐뗄컄쿗훐,뗄랽쿲,쫇쯹룸폅뮯컊쳢뗄쒿뇪몯쫽,뻟폐벫웤훘튪내뇪ힼ뚨틥뗄킡늨몯쫽ꛕ(t)퓚t>0쪱뚼늻캪쇣,뗄ퟷ폃,튲쫇틅뒫쯣램펦폃쫗튪뾼싇뗄컊쳢ꆣ놾컄쯆뫵쎻폐냬램뿋럾헢룶컊쳢ꆣ쒿뇪쫇쪹쒣탍뗄욽뻹풤닢뺫뛈듯떽ퟮ룟,틲듋놾컄퓚탅뫅뒦샭뗈쇬폲닉폃뗄튻룶늻뗃틔뛸캪횮뗄톡좡통솷벯쫽뻝욽뻹풤닢뺫뛈ퟶ캪쫊펦몯쫽ꆣ풤닢랽램쫇뛔폐쿞뎤뗄탅뫅뷸탐뺵쿱퇓췘,벴볙짨x(N뺫뛈톡좡MSE(MeanSquaredError)뗄떹쫽,+k)=x(N-k),N캪탲쇐뗄뎤뛈ꆣ놾컄죧퓚듋뒦MSE뚨틥캪:닉폃헢훖평듋믡듸살죧쿂솽룶컊쳢:(M1)틔캴살쪱뿌볛룱뇤뮯랴펳떱잰쪱뿌뗄볛룱SE=1NꇆN2[x(k)-
x(k)](8)k=1늨뚯쳘헷컞싛죧뫎컞램쇮죋탅럾,폐잣잿횮틢ꆣꆡꆡx(k)캪쪵볊횵;x(k)캪풤닢횵;N캪퇹놾뎤(2)듸살뗄볆쯣솿듳ꆣ뛈ꆣ놾컄퓚듋쳡돶쇋튻룶뷏캪쟉쏮뗄욽틆킡늨룅(2)좾즫쳥뇠싫:좺쳥훐뗄쎿튻룶쳥돆캪튻좾쓮,컊쳢펭죐뛸뷢ꆣ즫쳥,좾즫쳥훐뗄쎿튻캻돆캪튻룶믹틲,쎿튻룶믹틲뾼싇죎튻샠탍킡늨뗄뇪ힼ뗄킡늨쒸몯쫽ꛕ(t),듺뇭튻룶펰쿬좾즫쳥쫊펦몯쫽횵뗄닎쫽,틲듋탨튪평킡늨뗄뚨틥뿉횪ꛕ(t)퓚쪱폲뚼뻟폐뷴횧독,벴붫룸뚨뗄폅뮯컊쳢폃좾즫쳥뇠싫,뮯캪틅뒫쯣램ꛕ(t)퓚쪱폲짏뗄럇쇣뚨틥폲쫇폐쿞뗄ꆣ늻쪧튻냣뿉틔쟳뷢뗄쫽뻝룱쪽ꆣ놾컄훐탨튪좷뚨뗄닎쫽냼삨탔,볙뚨suppꛕ(t)<[ꛁ,ꛂ\],웤훐ꛂ>0ꆣ죧ퟮ캪볲N룶돟뛈짏쪱볤퇓동닎쫽뫍냼삨틾몬닣뷚뗣쫽ꆢ톧떥뗄Haar킡늨뗄킡늨몯쫽ꛁ=0,ꛂ=1ꆣ쾰싊뫍뚯솿쾵쫽퓚쓚뗄3룶췸싧뷡릹닎쫽ꆣ뾼싇ꛕ(t)퓚쪱폲짏뗄욽틆ꛕꆯ(t)=ꛕ(t+(3)좷뚨뿘훆닎쫽ꆣ뿘훆닎쫽뗄좷뚨튲쫇튻룶tꆯ),웤뷴횧독suppꛕꆯ(t)<[ꛁꆯ,ꛂꆯ\],웤훐ꛁꆯ=ꛁ훘튪컊쳢,펰쿬떽틅뒫쯣램뗄볆쯣킧싊ꆣ내헕틔췹뗄-tꆯ,ꛂꆯ=ꛂ-tꆯ,떱tꆯꇝꛂ쪱,ꛂꆯꇜ0ꆣ듋쪱뺭킡뺭퇩,놾컄좷뚨룷뿘훆닎쫽죧쿂:좺쳥룶쫽=20;뗼늨몯쫽ꛕ(t)욽틆뛸뗃뗄ꛕꆯ(t)몯쫽죔좻싺ퟣ킡늨듺듎쫽=8000;퓓붻싊=ꆫ;뇤틬싊=0.몯쫽뗄뿉죝탭탔쳵볾,벴2201ꆫꆣCꛕ|ꛗꇤ(ꛘ)|)|ꇤ=ꇒ+ꇞ-ꇞꛘdꛘ=ꇒ+ꇞiꛘtꇤ|eꛗ(ꛘ-ꇞꛘdꛘ(4)룸뚨튻킩쯦믺뗄돵쪼닎쫽,좻뫳룹뻝쫊펦몯쫽횵춨맽뢴훆ꆢ퓓붻뫍뇤틬뗈닙ퟷ뗼듺쯑톰떽ퟮ폅=ꇒ+ꇞ|ꛗ(ꛘ2)|-ꇞꛘdꛘ<ꇞ(10)믲뷼쯆ퟮ폅뗄좾즫쳥ꆣ쇭췢탨튪뾼싇뗄튻룶훘튪컊쳢쫇톡좡뫎훖랽쪽ꆡꆡ틲듋ꛕꆯ(t)죔좻캪킡늨쒸몯쫽,폫ꛕ(t)뻟폐뗄킡늨뇤뮻싺ퟣ놾컄풤닢펦폃뗄탨쟳ꆣ솬탸킡늨뇤쿠춬뗄탎ힴ,쟒suppꛕꆯ(t)<[ꛁꆯ,ꛂꆯ\]ꛂ,ꆯꇜ0ꆣ퓙튻듎뾼싇킡늨뇤뮻뮻릫쪽죧쿂:Wt-ꛓTx(a,ꛓ)=1)dt(11)aꇒx(t)ꛕꇤ(WaTx(ꛁt-ꛓ,ꛓ)=1ꛁꇒx(t)ꛕ(ꛁ)dt(9)ꆡꆡ평폚듋쪱suppꛕꆯ(t)<[ꛁꆯ,ꛂꆯ\]ꛂ,ꆯꇜ0,틲듋ꆡꆡ떱붫돟뛈샫즢캪ꛁ=2j쪱뻍캪뛾뷸킡늨뇤뮻,퓚볆쯣킡늨쾵쫽WTx(a,ꛓ)쪱,퓚볆쯣훐쯹튪폃떽놾컄헽쫇닉폃쇋헢훖킡늨뇤뮻랽쪽,떫캪쇋쿂쏦쳖뗄x(t),t<ꛓ,헢뻍뿋럾쇋킡늨럖뷢쪱뗄풤닢뇟뷧싛랽뇣,듋뒦죔틔솬탸킡늨뇤뮻캪샽ꆣ컊쳢ꆣ춨맽퇐뺿랢쿖,죧쪽(9)쯹쪾뗄뒫춳킡늨뇤뮻퓚듋췢닉폃뫎훖샠탍뗄킡늨믡펰쿬뛔뻖늿돟뛈쳘놾컄틔풤닢캪쒿뗄뗄펦폃훐늻쓜쫊폃,믡폶떽풤닢헷뗄쳡좡쓜솦,룹뻝Mallat뗈톧헟뗄퇐뺿놾컄닉폃뇟뷧뗄컊쳢ꆣMarr킡늨(튲돆캪쒫컷룧쎱킡늨)뇤뮻,뻟쳥뿉닎쯹캽풤닢뇟뷧컊쳢뻍쫇,룸뚨튻룶폐쿞뎤뗄뷰볻컄쿗[18]ꆣ횵뗃힢틢뗄쫇Marr킡늨뻟폐컞쿞뎤죚쪱볤탲쇐,x(t),t<=Tꆣ내헕짏쪽,죧맻쒸킡늨뗄횧벯,떫웤폐킧횧벯캪[-5,5\],틲듋욽틆쪱쫇내몯쫽퓚t>0쪱ꛕ(t)ꇙ0,벴t>ꛓ쪱ꛕ((t-ꛓ)/a)헕Marr킡늨뗄폐킧횧벯,뛸늻쫇쯼뗄쪵볊횧벯뎤ꇙ0,퓲퓚볆쯣ꛓ쪱뿌뗄킡늨쾵쫽WTx(a,ꛓ)쪱,탨뛈ꆣ튪폃떽t>ꛓ뗄x(t)ꆣ떱ꛓ캪뇟뷧믲뾿뷼뇟뷧쪱,볆쯣W3ꆡ풤닢쪵횤퇐뺿Tx(a,ꛓ)뻍탨튪폃떽뇟뷧췢,벴t>T쪱뗄x(t)ꆣ뛔폚틔풤닢캪쒿뗄뗄킡늨럖뷢뛸퇔,t>T쪱춬퇹틔짏횤ퟛ횸캪퇐뺿뛔쿳,뻟쳥늽훨죧쿂쯹
ꆤ16ꆤ훐맺맜샭뿆톧ꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡꆡ2002쓪쪾:뇭2ꆡ웤쯼쒣탍닎쫽(1)붫1990쓪12퓂ꆫ2000쓪11퓂뎤뛈캪2448쒣탍닎쫽뗄내쪽NNin=10,hn=16,lr=,mc=(1)뚨틥뗄뛔쫽볛룱X(t)틔Marr뛾뷸욽틆랽램(1)in=6,hn=8,lr=,mc=킡늨뇤뮻랽쪽퓚6룶돟뛈짏뗃6룶킡늨쾵쫽탲쇐,랽램(2)S1:in=10,hn=12,lr=02,mc=;뻟쳥뷡맻죧춼6쯹쪾ꆣS2:in=8,hn=8,lr=,mc=;S3:in=6,hn=6,lr=,mc=;S4:in=6,hn=4,lr=,mc=;S5:in=6,hn=4,lr=,mc=;S6:in=5,hn=4,lr=,mc=;ꆡꆡ웤훐in뇭쪾쫤죫닣뷚뗣쫽;hn뇭쪾틾몬닣뷚뗣쫽;lr뇭쪾톧쾰싊;mc뇭쪾뚯솿쾵쫽;S1ꆫS6뇭쪾돟뛈ꆣ쯄훖쒣탍풤닢뺫뛈뇈뷏죧춼7쯹쪾ꆣ춼6ꆡMarr뛾뷸욽틆킡늨뇤뮻쾵쫽춼7ꆡ룷쒣탍풤닢뺫뛈뇈뷏ꆡꆡ(2)내쪱볤쮳탲붫짏쫶룷탲쇐(냼삨X(t))럖ꆡꆡ듓춼7뿉틔뾴돶,춨맽틔킡늨뇤뮻쾵쫽캪쳘헷캪통솷벯(t=1ꆫ2000),닢쫔벯(t=2001ꆫ솿뗄죽룶풤닢쒣탍닢쫔벯풤닢뺫뛈뚼쏷쿔룟폚춬샠2248)ꆣ짱뺭췸싧쒣탍,MSE풤닢뺫뛈훁짙복짙쇋50%,쮵(3)내헕춼5쯹쪾뗄믬뫏퓓붻쒣탍뒦샭늽훨,돵쏷쇋킡늨쾵쫽ퟷ캪볛룱늨뚯풤닢쳘헷솿뗄폐킧탔ꆣ쪼뮯-쯦믺좺쳥,쫤죫통솷퇹놾,좷뚨룷돟뛈ퟮ폅쪱쳘뇰쫇놾컄쒣탍평폚늻뷶뾼싇쇋춬튻돟뛈늻춬쪱뿌볤퇓동뫍짱뺭췸싧닎쫽,뻟쳥뷡맻죧뇭1쯹쪾ꆣ웤뗄쿠맘탔춬쪱튲뾼싇쇋늻춬돟뛈볤뗄쿠맘탔,쪹풤훐hn뇭쪾틾몬닣뷚뗣쫽,lr뇭쪾톧쾰싊,mc뇭쪾닢뺫뛈뷏랽램(1)뫍랽램(2)쒣탍뚼폐쇋뫜듳뗄쳡뚯솿쾵쫽,S1ꆫS6뇭쪾돟뛈ꆣ룟,퓚쮵쏷랽램(1)뫍랽램(2)쒣탍뻖쿞탔뗄춬쪱뇭뇭1ꆡퟮ폅닎쫽쏷쇋놾컄쒣탍퓚힡쇋볛룱늨뚯뗄뻖늿돟뛈쳘헷뫍돟뛈뗄쪱볤퇓동뛠훘럖탎뗄뛠돟뛈쿠맘탔뫳믱뗃쇋벫듳돉릦ꆣꆡ뷡싛ꆡꆡ(1)놾컄쪵횤퇐뺿쇋컒맺막욱쫐뎡뗄뛠훘럖탎(4)붫닢쫔벯퇹놾쫽뻝내헕뛔펦뗄쪱볤퇓동닎쫽쫤죫쿠펦뗄짱뺭췸싧풤닢ꆣ쯦믺폎ퟟ,쮵쏷쇋웤훐듦퓚뛠훘럖탎쳘탔ꆣ놾컄샻폃뛠훘럖탎맽돌뗄뻖늿돟뛈쳘헷뫍뛠돟뛈쿠맘탔붨(5)춬쪱퇐뺿횱뷓쫤죫짱뺭췸싧풤닢쒣탍솢쇋튻룶킡늨-틅뒫쯣램-짱뺭췸싧쿠뷡뫏뗄풤닢(NN,NeuralNetwork),짏쫶뗄랽램(1)뫍랽램(2)쒣탍뗄풤닢뺫뛈,춨맽쯄헟뗄뇈뷏쇋뷢킡늨럖뷢ꆢ춬쒣탍,믱뗃쇋뇈틔췹쒣탍룼룟뗄풤닢뺫뛈ꆣ튻돟뛈늻춬쪱뿌쿠맘탔뫍늻춬돟뛈횮볤쿠맘탔뛔풤(2)놾컄쯹폃뗄킡늨뇤뮻횻퓚만뚨뗄돟뛈뿕볤닢뺫뛈뗄펰쿬,룷쒣탍닎쫽죧뇭2쯹쪾ꆣ훐뷸탐킡늨뇤뮻,폐뿉쓜늻쓜힡쒳킩훘튪돟뛈짏
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