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# SPCTR ANLYS TME SER STAT 520

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This 40 page Class Notes was uploaded by Providenci Mosciski Sr. on Wednesday September 9, 2015. The Class Notes belongs to STAT 520 at University of Washington taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/192509/stat-520-university-of-washington in Statistics at University of Washington.

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Date Created: 09/09/15

Nonparametric Spectral Estimation 0 consider stationary process Xt with spectral properties given by sdf S acvs and sdf related by 57 gt process mean denoted by u EXt sampling time given by At Nyquist frequency given by fN E 12At o observe time series that is realization of X1 X N 0 want to use N observations to estimate S o nonparametric spectral estimates based on 00 At 2 8T 7227rfTAt 7700 o idea estimate 57 somehow amp use above 0 since 57 E cov Xt7 XHT EltXt HgtltXtT W rst task is to estimate 1 The Sample Mean 1 N 0 natural estimator is sample mean X E N Z Xt t1 0 sample mean is unbiased estimator of u 1 N EfX N E Eth M t1 o X called consistent if for every 6 gt O Alim PHX m gt6O o Chebyshev7s inequality says P HX MI gt 6 S EltX 2 m2 Var 2X o consistency follows if var X gt O as N gt oo assume 2 57 lt 00 says 80 AtZsT lt oo can show N var X SOAf if 80 y 0 leads to useful approximation varX Eff i proof on page 189 uses Cesaro summability o X not consistent for all stationary processes consider Yt E Xt U Where U is an rV 2 Best Linear Unbiased Estimator 11le 11Tvrjvl1N 1N2 111T Vector ofNones 0 second estimator of u i TN is covariance matrix for X1 XN requires knowledge of 50 51 5N4 0 called best linear unbiased estimator BLUE of 1 best in sense of minimum variance 0 how do sample mean and BLUE compare O lt lt oo piecewise cont cont at f O 7 A a X E l 7 1 X said to be asymptotically e icient X ine icient for fractional difference processes SQ C sin7rf Ata C7r At a for small f Samarov and Taqqu 1988 show gtllt if 1 lt oz lt 0 then 098 lt 6Xl lt1 if OltOzlt1then OlteXllt1 if a21then 6X LO X quite ine icient if gt O rapidly as f gt O 3 Unbiased Estimation of ACVS o recall de nition of acvs 57 E GOV Xta Xt l E llXt MgtltXt7 M 0 natural estimator of 57 for 0 g 739 g N 1 AW 1 N TX X X X 57 NT g t gtlt m gt since 57 57 have for N 1 g 739 g N 1 W 1 NillX XgtltX 22gt 5 E T T N ITI t1 t tll l o replace X by u and take expectation 1 NilTl MW E X X T 57 N w lt t M W M o 57 called unbiased estimator of 57 in 39 general biased if X used in place of u Biased Estimation of ACVS o unnatural estimator of 57 p77 explained later A 1 N lTl 7 7 739 W 531 E N tg Xt XXXHM X 1 5 if replace X by u then E Tp 1 ST 571 called biased estimator of 57 but if X used 9 can be less biased than 1M standard estimator of 57 is 11 not 1 because 1 mean square error of 571 typically smaller mse 9 E Elt Tp STgt2 lt Elt Tu STgt2 E mse u mse is variance squared bias eg Emir m we E Pgtgt2ltE pgt sfgt2 Figure 192 shows variance dominates helps 11 2 var 9 var 1 510 1 liNl var 2 71 mimics 57 gt O as 739 gt oo 3 Alpl is positive semide nite need not be 0 see Figure 195 for example 5 The Periodogram I o assume EXt 0 if not replace Xt with Xt X 0 base sdf estimate on 57 gt 0 estimate 57 using N77 p E Eti lXtXtlTl7 0 S 7quotl S N 1 T 0 T 0 de ne periodogram SpfgtEAt Nil p 27rfTAt 77N71 note function of f not 1fl 0 equivalent de nition for periodogram Wm proof FTX Xt FTX1t2 see also p 196 0 9 E 39 implies 4p fN Np mmm 5T fltNgtS fle all N 2 Z Xteilg 39ft At tl The Periodogram II at Fourier frequencies fk E kN At 2 A N 7239 7T Spltfkgt W ngte 2 kitN ie can compute 3pfk via DFT of X s can also use disc tinie disc freq theory to argue 9 7 N 1N and 3pfkgt1k are an FT pair Where fk E kQN At N 1N note fk7s de ne frequency grid twice as ne as fk7s inverse FT gives discrete formula for Tp7s 1 N A Ap 81 TmN 01 N 57 k7V71 7 T 7 7 7 note Riemann approximation to integral is exact Periodogram Sampling Properties 0 how well does SW20 estimate Sf 0 need to determine statistics of periodograni 1 E3pfgt 2 var 3pf 3 cows lt1 SW03 1quot f o What would be ideal l E pf for all f unbiasedness 2 var 3pf gt O as N gt oo consistency 3 cov 31 f 3pf O uncorrelatedness o What in fact holds tragedy of the periodograni 1 can be badly biased 2 inconsistent unless O 3 approximately uncorrelated if f f distinct Fourier frequencies fk kN At First Moment Properties 0 taking expectation of de nition of SW0 A N71 ESpltfgt At E89me 77N71 N71 39739 Z gt 5T 7227TfTAt 77N71 N f N m Lg 291 f f 1AtgtSltf gtdf Where DN is Dirichletls kernel 0 sketch of proof ignoring proportionality constants 13 n M mo T 1 W Dive 87 gt St 1 x m e at so ESltPgtltgt 0 alternative proof Exercise 45 uses spectral representation theoreni directly Fej r s Kernel 0 can reexpress Epf in terms of Fej r7s kernel A f ESltpgtltfgt Lg Hf f gtSltf gt df Where At sin2N7rf At 2 2 g 12V 7 27rftAt N i 7 27rft7uAt i 6 N 121 ul 0 properties assunie fN g f g m 7f gtNAtasf gtO thusf0 gt oo asN gt oo lt NAt when f y O ie peaks at f O ff gtOasN gtooforf7 0 O has nulls7 at fk kNAt amp k y 0 Width of central lobe 1N At ie grid size has unit area figigv df 1 o conclusion as N gt oo 7 acts like 6 function see Figure 200 Bias and the Periodogram 0 can argue periodogram asymptotically unbiased 39 1p i fltNgt ngnooms m e Amara fgtSltfgtdf ism 0 may may not be good approximation for nite N lim N OO Thomson quote page 199 0 dynamic range useful characterization d E 10 log10 0 three examples assume At 1 1 White noise with 02 1 1 d 0 dB p fltNgt I E8 m L Hf f gt df 1 2 AR2 process see Figure 201 d 14 dB 3 AR4 process see Figures 2023 d 65 dB 0 bias due to sidelobes of Fej rls kernel leakage can be problem for S With large d 0 two cures tapering amp preWhitening tapering replaces 7 preWhitening replaces S 11 Bias Reduction Tapering 0 given X1 XN and data taper h1hN forni tapered series thl hNXN ifEXt y 0 use h1X1 X hNXN X data Window linear taper linear Window o de nitions JO 2 Ml2 htXteWW amp Wm mm t1 SW called direct spectral estimate 0 use spectral representation for Xt N tg ht 227TftAt 7227TftAt fN 12V hteiz39Q fif At fltNgt t1 1 N WfHf fgtdzlfgt WherehtEOtlt1amptgtN N Hm E m meme The Spectral Window 0 use same argument as in Exercise 45b E3dfgt 131Jfgt2 EJfgtJfgt fN gtk fN gE ch H 0 f W 0 W13 Hltf f gt dZltf gt E ll ll Hltf fquotgtHltf f EdZltf gtdZltf gt g Hltf f gtSltf gt df Where ma 2 ft Hltfgtl2 At is called spectral Window for SW N 2 Z hteilg 39ft At tl 0 note ht 1N yields periodograni 1N called default or rectangular taper becomes 7 ie spectral Window for periodograni is Fej rls kernel 0 goal select ht so H7s sidelobes less than f7s sharp discontinuities in 1N cause ringing prevent by having htls decrease to O gradually 13 Some Data TapersSpectral Windows 0 Figures 209710 show p X 100 cosine tapers p proportion of series that is downweighted7 p 0 yields default data taper p 1 yields Hanning data taper 7rt ml tradeoff between sidelobes amp central lobe Width ht hNt1oc1 cos lt 0 Figures 211712 show zeroth order dpss data tapers for speci ed W and N maximizes ratio flVW 71W df 11 Hm df WW E energy in sidelobes is as small as possible with N xed usually set W via product NW typical choices NW 1248ie W kN multiple of Width of Fej rls central lobe for NW 1 slightly better than for NW 2 similar to Hanningls 14 Examples of Leakage Suppression 0 Figures 21374 compare to Figure 202 0 illustrates tradeoff between reducing bias due to leakage introducing bias due to width of central lobe smearing loss of resolution 0 key points useful for sdfs with large dynamic range quite important for exploratory analysis not useful with white noise or ARlt2gt example failure to recognize role of dynamic range has led to controversial statements 0 if true sdf unknown can we tell if tapering needed see Figure 227 two indicators of need for taper ACVS amp Direct Spectral Estimators o recall 1 N lTl I I A N I 2 410 lt XI ZQWftAt N tg t tl7l 8739 N tg t6 o letting X htXt N yields NilTl 2 t2 htXthtlTlXtlTl Fgt At iv htXteilg 39ft At 131 o RHS is SWIM LHS is corresponding acvs estimator N7 739 d E Eti lhtXthtlTlXtlTl7 W S N 1 T O 39739 Z N 0 note that for 39739 g N 1 E Td i NilTl 5739 21 hthtlTl t o 5751 E SW yields Sdltfgt At Nil d 7 27rfTAt 77N71 T 0 yields useful equation for computations A N71 N lTl ESltdgtltfgt At sf 2 MW WW 77N71 t1 inner sum is 1 when ht1N 16 Normalization of Data Tapers I o for 739 0 have N Elt 0d So a h so require 2 h 1 for unbiased estimator t note holds for default taper 1 N 0 since ht agt Parseval 890 says h W Hm df i g Hm df 1 fltNgt note holds for Fej ns kernel 0 suppose Xt is White noise with 02 At ESltdgtltfgt ffHltf f gtSltf gtdf 02AtjivHf f df02At ie SW unbiased estimator of White noise Normalization of Data Tapers II 0 second normalization set h s such that N 1 N A d 2 A 53 Z htXtgt 2 X3 53p t1 N t1 note normalization is data dependent 0 since 31 E 3d above says 0 called restoration of power7 ie variance Tapering and Estimation of n 0 so far have assumed u 0 if unknown rede ne A N 2 Sltdgtf 2 At 2 mpg Xeme t1 ifht 1N then Sltdgt0 Wm 0 Exer 65bl arguably reasonable since 0 freq DC or mean value 0 to obtain 3d0 O for general ht can use N N H E Z htXt Z ht t1 t1 in place of X because then 2 Aw N N N EN hX 2 At hX h t1 gt 0 tgi t t 121 t Eight is an unbiased estimator of u if ht 1N reduces to sample mean X Bias Reduction Prewhitening I o leakage in 31 due to interaction between sidelobes of 7 SX with large dynamic range dX E 10 log10mafoXfminfSXf 0 goal of prewhitening replace S by self with smaller dynamic range suppose Yi E 25LguXiiK7u 1 s t s M with MEN KL 0 linear ltering theory says K I 2 Syfgt EL gue m At Sxfgt o idea Choose gu so that dy ltlt dX Bias Reduction Prewhitening II 0 use 11YM to estimate Sy A M 2 89m 2 At hthe ZQWftAt t1 Where ht1N or is dpss with NW 1 or 2 ie avoids use of with large central lobe 0 estimate SX using A d 4106 Slllf X f 257L Que z27rqut called postcoloring 0 small dy implies Syltfgt 0 yields 121sz SW S 25L guei wfu At Q XU 0 two di iculties may need large K L to get small dy ie M lt N need to know S to properly design gul can sometimes guess or estimate 7 see Chap 9 21 Bias Reduction Rejection Filtration o tapering amp prewhitening all of S of interest 0 suppose need S only over ML fH 0 design band pass lter to attenuate Z ML f 111 o prevents leakage from frequencies outside of ML f 111 called rejection ltration 0 example ARlt6gt sdf near f 04 Statistical Properties of Sam I 0 let Xt be Gaussian White noise with EXt O amp var Xt 02 so 02 At 0 let A zcmmmampwWWA wm tl Af E At121htXtcos27rfmt 3m 2 At121htthin27rfmt WUUW3UFW B 39EUU MiEMU EWU O 0 can write varAf 02 At h cos227rftAt tEl varBf 02 AtahgsiHQQW At cov AfAf 02 Atihg cos 27rftAt cos 27rftAt covBfBf 02 Atihgsin27rftAtsin27rftAt covAfBf 02 Atihcos27rftAtsin27rftAt 23 Statistical Properties of SW II 0 special case ht 1N ie periodograrn and f fk kN At ie Fourier frequencies Val lfllfkll UQAlQ SUM2 0 lt fk lt fN Vamp1quotBltfkgt 02 AtQ 0 lt fk lt fN var Aloll var Afzvgt 02 All Sltfkgt 30 Bf1vgt0 COV lAlfjlaAlfkll 0 for all fj 7t fk COV lBlfjlaBlfkll 0 for all fj 7t fk COVAltfjgtBltfkgt O for all f7 and fk o Afkls Bltfk7s are Gaussian amp independent Why 0 recall de nition of Chi square distribution suppose Y1 Y2 YV are iid NO1 rvls ie independent mean 0 amp variance 1 Gaussian rvls then xi E 312 322 YV2 has Chi square distribution with V degrees of freedom dof note V and var 2V Statistical Properties of Saw III 0 for O lt fk lt fN can argue that Altfkgt Altfkgt and Bltfkgt BOW 02 Alt2W2 50902 02 Alt2W2 I 50302 are iid NO l 0 implies that Altfkgt Bun JQSWMXQ win2 Min2 80602 2 Where 27 means equal in distribution 0 equivalently 3pfk g Sltfkxg2 0 can argue 31fk g SOC x if fk O or fN 0 using V amp var 2V yields ESpltfkgt 80 0 S fk S fN varpfk 820k O lt fk lt fUV varSpfkgt 282ltfkgt7 fk07fN 0 also have Cov3pltfjgt Sltpgtltfkgt 0 12 fk 0 3 1m 3 m 25 Statistical Properties of SW IV 0 now suppose Xt not necessarily Gaussian but regular7 S not necessarily White but continuous all f f not necessarily a Fourier frequency 0 statistical theory says as N gt oo Sltpgtltfgt SltfgtXg27 0ltfltfN A ASWU SltfgtXia f07fN 07 7g fka O S S holds ok if N large enough so E pfl 801 HCL HCL 0 implies 31 approximately uncorrelated over fk7s ie grid of Fourier frequencies 0 ngkz gfm fork0N2 so there are N 2 1 uncorrelated rv7s Statistical Properties of SW V o if ht is regular7 taper can argue as N gt oo 9 SURE2 0 lt f lt fuv Sltdgtltfgt SltfgtXi7 f07fN cov3ltdgtltfgt Wm 0 g Hltf f M Hltvgt dv HCL HCL 2 Where C E SmaXAt amp hit a 0 says cov 3dfgt Sltdgtltflgt 0 if lf gt Width of central lobe of 0 SW thus approximately uncorrelated over coarser grid than fk7s 3ltdgtltogt Wm Sltdgtlt gtgt Where K is such that K Kl c NAt 3 WW lt NAt o typically C 2 Hanning or 4 dpss with NW 4 0 yields K l lt N 2 l uncorrelated rv7s Statistical Properties of SW VI Sltdgtf 2 sang2 gt var3dltfgt 820 0 holds for all N 620 inconsistent if y 0 cf var X SON At 0 Figure 225 illustration of inconsistency for White noise 0 Figure 226 effect of tapering ARlt2gt series Why does SW look smoother than 3p consider extreme taper with only thyj y O wlzw A N 2 Mg At htXte ZQWftAt AthizX t1 local7 variability constant across frequencies 0 Figure 227 effect of tapering ARM series PDF for SW f o recall glen g SltfgtXg2 o probability density function for U E S f Xg 2 e uSmSU u 2 0 mill l0 ult0 U also called exponential rv SQ 2 yields xg rv 0 pdf of xg shown on LHS of Figure 228 0 random sample of iid Xas tends to have upshootsl recall Figure 41 o Xas on dB scale have downshootsl Figure 225 o to resolve paradoxf nd pdf of V E 10 log10Xg log2i310gt10moeNUlow 00 lt v lt 00 V U shown on RHS of Figure 228 o y 2 shifts pdf to left or right 0 note 95 of deviates fall Within lle dB of mean A Test for White Noise 0 Q is time series a sample of White noise 0 null hypothesis X1 X N are Gaussian White noise 0 glpfk7s for O lt fk lt fN are iid exponential under null hypothesis 0 form normalized cumulative periodogram k E 21glpltfjgt 29151fo7 Where M E UN 12j k1M 1 o for level oz test plot 7 versus fk with lines Law 2 M and mm 2 fl M where Dltozgt E Ola M 112 012 011M 112 and eg C005 1358 0 reject null hypothesis if 7 versus fk falls outside of lines at any fk 0 see Figure 234 for example 30 Smoothing Sldl I S ldlq inconsistent estimator of SO var Sdf 82f for all N want estimator with reduced variance gives better Visual indication of sdf leads to more powerful statistical tests idea average SW across frequencies makes sense if S slowly varying SOC7M 39 quot SW SltfkMgt assume also N large enough so that SltpgtltfkeMgt7 7311307 731me are approximately uncorrelated estimators of S can then de ne 7 1 M A 5 E mjM SltPgtltfkijgt so that 7 7 Em m amp MW Ml 31 Smoothing Sldl II o rectangular smoothers can induce ringing recall Figure 86 from Chapter 3 o more generally can de ne A M A k WWmE awWWQL E j M discretely smoothed direct spectral estimator typically N Z N ie at least as ne as Fourier gj is LTl lter with gj gnj usually 0 another formulation AW fltNgt 1d sltngmmc mswma real valued symmetric 2 f N periode typical shape similar to Gaussian pdf m controls degree of smoothing ie sets Width 0 convolutions expensive7 amp multiplications cheapl so better to transfer to other domain 32 Smoothing Windows amp Lag Windows de ne vnm agt can write Why Sam At Nil UT m d 7 27rf7 At 77N71 7 T v77m7s for 39739 Z N dont matter since 9 0 then de ne lag Window 107m as w 2 11mm 39739 ltN 7 0 ITI 2 N and via 107m lt gt is smoothing Window standardized now have Alw fltNgt Yd S f MN Wmf gtS lt gt dqs Nil me deiz3927rfTAt 77N71 7 T can de ne Alw 177735 607 39739 S N 57 7 0 lTl Z N and thus have 910 gt SUMO 33 Lag Window Spectral Estimators o gllw called lag Window spectral estimator lots of other names 7 see page 242 0 how are 3d5 and 3lw related can show page 238 N71 2 77N71 WSW 723927rf739 At 739 e k 7 SW01 At Where M 227139 739 N U739 Z 976 j jiM regard gj as discrete smoothing Window thus all 3d5 can be expressed as 3lw With fk E kQN At can argue N WW 97 jN1 3lwfk7s determine 3lwf g fN thus all gllw can be expressed as 3d5 3lw avoids convolution thus preferable 34 Conditions on Smoothing Window 0 assunie is real valued even and 2 f N periodic this implies wqm w m 0 conditions required on to get consistency necessary but not su icient 1 figlgv df 1 ltgt w07m 1 for all m in terms of Q weights condition is N 97 1 jN1 2 as m gt oo acts like a 6 function convention sniall m iniplies rnore smoothing 0 additional desirable condition WM 2 0 all f not required to get consistency suf cient but not necessary for 3lwf Z O 35 Smoothing Window Bandwidth 0 condition 1 and Z 0 say is pdf 0 can de ne bandwidth in terms of variance f N 2 12 W 12103 f we df i 1 1 Nil 12 i At 7r2 71 72 ml 127 gives rectangular its natural7 Width due to Grenander but With 127 omitted can be hard to compute accurately can be imaginary when Wmf lt O for some f 0 second prefered bandwidth measure 1 1 ffN W2 df At zivjm 1113M fltNgt 7 BW E equivalent Width of autocorrelation of see 85c due to Jenkins 36 First Moment Properties of 010M I o recall smoothing Window representation for 3lw 3ltzwgtltfgt W Wmltf f gt3ltdgtltf gt df fltNgt 0 taking expectations E3 wltfgt W Wmltf f gtE3dltf gtdf fltNgt 3 Mi 1quot g H0 gtSlt gt dqs df g 33 Wmltf f Wf lt26 df Slt gt dqs 371 MW gtSlt gt dqs Where is the spectral Window for Sllw 24mm 2 W Wmltf f gtHltf gt df fltNgt note amp are periodic with period 2fN 0 Exercise 617 gives 2nd expression for Wf 0 note similarity to spectral Window for 3d ESltdgtltfgt W Hltf gtSlt gt dqs fltNgt 37 First Moment Properties of 010M II 0 claim f 24mm Lg Wmc f gtHltf gt df N71 NilTl At 2 wT7m Z hthtTgt 7227rf739At 77N71 t1 o sketch of proof wnm a ht E HO M ht E Hlt39gt2 Ht h ht gtlt w m agt Wm o by construction acts like as N gt oo acts like as m gt oo o by appropriately letting m gt 00 as N gt 00 can argue Wm gtllt acts like also 0 conclusion can obtain Nggmmswn hm W Umltf gtSlt gtd 5ltfgt N7m oo fN o for nite N can set m to get E lwf 38 Smoothing Window Bias if Esltdgtf 5f then ESlwf a g Wmltf f gtSltf gt df 0 can quantify bias bW due to alone bwltfgt W Wmltf f gtSltf gt df Sltfgt fN W V W Wmltf 1quot Son Sm df 2 f f fltNgt W W lt gt W lt26 Sm dqs fN f m Way W wmlt gt Sn q Sm das fltNgt HT 0 assume S has Taylor series expansion about f 2 50 lt26 80 qss m S ltfgt on 2 bwltfgt W Wmlt gt qss m S ltfgt dqs fltNgt Why S ltfgt fN i5 ltfgt 7 TWm gt 2d i 24 5124 o interpretation large bias if bandwidth of large large bias if S rapidly varying 39

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