Parametric Methods. Autoregressive (AR) Moving Average (MA) Autoregressive - Moving Average (ARMA) LO-2.5, P-13.3 to 13.4 (skip

Transcription

1 Pmeti Methods Autoegessive AR) Movig Avege MA) Autoegessive - Movig Avege ARMA) LO-.5, P-3.3 to 3.4 si ) /

2 Time Seies Modes Time Seies DT Rdom Sig /

3 Motivtio fo Time Seies Modes Re the esut we hd tht eted outut PSD to iut PSD fo ie, time-ivit system: h Sig Beig Modeed Iut RP WSS w/ S ω) Outut RP LTI System WSS w/ S ω) Imuse Resose ht) Feuey Resose Hω) F{ht)} S ω) H ω) S ω) If the iut is white with owe the: S ω) H ω) The She of outut PSD is ometey set y Hω)!!! 3/

4 Time Seies Modes Pmeti Modes) Thus, ude this mode owig the LTI system s tsfe futio o feuey esose) tes eveythig out the PSD. The tsfe futio of LTI system is ometey detemied y set of metes { } d { }: H ) B ) A ) If if, if, if!!!) we ssue ouseves tht the dom oesses we e to oess e modeed s the outut of LTI system dive y white oise, the. Estimtig Pmetes Estimtig PSD ote: We Limit Disussio to Re-Vued Poesses 4/

5 5/ Pmeti PSD Modes ) j j e e S ω ω ω The most gee meti PSD mode is the: } {, } {, Mode Pmetes The outut of the LTI system gives time-domi mode fo the oess: ) Thee e thee sei ses tht e osideed fo these modes: Autoegessive AR) Movig Avege MA) Autoegessive Movig Avege ARMA)

6 6/ Autoegessive AR) PSD Modes If the LTI system s mode is ostied to hve oy oes, the: A H ) ) ) Outut deeds egessivey o itsef Ode of the mode is : ed AR) mode ) j AR e S ω ω TF hs oy Poes Poes Give Rise to PSD Sies Emes: LO Fig.. & Fig..

7 Movig Avege MA) PSD Modes If the LTI system s mode is ostied to hve oy eos, the: H ) B ), TF hs oy Zeos Ode of the mode is : ed MA) mode Outut is vege of vues iside movig widow S MA ω) e jω Zeos Give Rise to PSD us Emes: LO Fig..3 & Fig..4 7/

8 8/ Autoegessive Movig Avege ARMA) If the LTI system s mode is owed to hve Poes & Zeos, the: A B H ) ) ) Ode of the mode is, : ed ARMA,) mode Poes & Zeos Give Rise to PSD Sies & us ) ) j j e e S ω ω ω

9 9/ ACF Mode of Poess So f we ve see etioshis etwee: PSD Mode Time-Domi Mode These modes imt oesodig mode to the ACF: Let the oess oey ARMA,) mode To get ACF: mutiy oth sides of this y - & te E{}: E E E } { } { } { eed This!

10 / ACF Mode of Poess ot.) To evute this wite s outut of fite with iut : } { } { δ h h E h h E E We hve ssumed us fite fo mode: >

11 /,,, h ACF Mode of Poess ot.) Usig this esut gives the Yue-We Eutios fo ARMA: ARMA) These eutios e the ey to estimtig the mode metes!!! We ow oo t simifitios of these fo the AR & MA ses.

12 / ACF Mode fo AR Poess Yue-We Eutios AR) Seiiig to the AR se, we set d get: h ow, we see tht im ) im H h Iiti Vue Theoem fo Z-Tsfom

13 3/ ACF Mode fo AR Poess ot.) If we oo t,, fo these AR Yue-We eutios, we get simuteous eutios tht e soved fo the mode metes of { i } i,, d : If we ow the AC Mti, the we sove these eutios fo the mode metes!!! Yue-We Eutios AR)

14 4/ ACF Mode fo MA Poess,,, h Seiiig to the MA se, we set d get: But fo the MA se the system is FIR fite d we hve othewise h,,,,,,,, Yue-We Eutios MA)

15 Pmeti PSD Estimtio As metioed ove, the ide hee is to fid good estimte of the mode metes d the use those to get estimte of the PSD. The si ide hods egdess if it is ARMA, AR, o MA. Howeve, the deivtio of the mete estimtes is uite hd fo the ARMA d MA ses. So we oside oy the AR se ut eve thee we ey o ituitio to some degee. Thee hs ee HUGE mout of eseh o how to estimte the AR mode metes. EE5 disusses this to some etet; hee we simy stte few tiu methods. 5/

16 6/ Pmeti PSD Estimtio ot.) Hee is the gee AR method: Give dt {, -}. Estimte the AC Mti fom the dt:. Sove the AR Yue-We Eutios fo the AR Mode }, { }, { 3. Comute the PSD estimte fom the mode ) j AR e S ω ω

17 7/ Pmeti PSD Estimtio AR Cse ot.) i i i, Autooetio Method Estimte the ACF usig:,, j j j Covie Method Estimte usig: Sove Usig: Two ommo methods ut thee e my othes):

18 Lest Sues Method & Lie Peditio Thee is othe method tht is ofte used tht omes t the oem fom itte diffeet dietio. Re: The ove ide ws sed o the Yue-We eutios, whih e i tems of the ACF whih is uow i tie!!) Thus we eed to estimte the ACF to use this view Lest Sues ovides diffeet wy to estimte the AR metes. Re: The outut of AR mode is give y 8/

19 LS Method & Lie Peditio ot.) If we e-ge this outut eutio we get: Peditio Eo Peditio of Thee e ots of itios whee ie editio is used: Dt Comessio Tget Tig oise Cetio Et. Go: Fid set of editio oeffiiets { } suh tht the sum of sues of the editio eo is miimied Lest Sues!!! miimie V - 9/

20 / LS Method & Lie Peditio ot.) To hoose the { } to miimie V we diffeetite d set V ; ow we use: V ;

21 LS Method & Lie Peditio ot.) So to sove the LS Lie Peditio oem we eed: ; ) Defie:. Mti Γ with eemets λ. Veto λ with eemets λ 3. Veto with eemets,, whee λ ;, The ) e witte s eoitig tht ): Γ λ Γ λ /