SPCTR ANLYS TME SER
SPCTR ANLYS TME SER STAT 520
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This 28 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 29 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
What is Spectral Analysis 0 one of most widely used 85 lucrative methods in data analysis 0 can be regarded as analysis of variance of time series using cosines 85 sines cosines 85 sines statistics or Fourier theory statistics 0 today7s lecture introduction to spectral analysis notion of a time7 series 025 introduction to time series analysis with some basic notions from time domain7 analysis subject of Stat 519 de nition of simpli ed version of spectrum and two methods for estimating nonparametric and parametric see Chapter 1 for details Time Series 0 what is a time series one damned thing after another7 R A Fisher denotebyt7t1N four examples7 each with N 128 Figs 2 85 3 in textbook 1es Wind Speed Time Ser First Example 120 100 40 20 Second Example Atomic Clock Time Series 120 100 80 60 40 20 20 1es iver Time Ser 80 100 1se Time Series 60 Ocean N0 Willamette R 40 I 20 Fourth Example Third Example 0 0 cV0 oo o O 0 120 100 80 60 40 20 Time Series Analysis 0 goal of time series analysis quantify characteristics of time series 0 sample mean 85 variance two well know statistics 1 N 1 N NE sct and z mt 38 t1 t1 capture univariate properties but do not capture bivariate prop erties7 ie7 do not tell us how sct and tk are related 56 Lagged Scatter Plots I o tellt us about bivariate distribution of separated pairs 0 t1 versus 33157 t 1 N 1 lag 1 scatter plot 0 four examples Fig 4 Lag 1 Scatter Plot for Wind Speed Series 0 s o a 8970 Ol 141 0 S39ll O39ll 9390 O39OI 9396 0396 9398 Lag 1 Scatter Plot for Willamette River Series 141 2 Lagged Scatter Plots II o tk versus 33157 t 1 N k lag k scatter plot 0 summarize scatter plots using linear model tk 0 k it 6m not always reasonable see Fig 9 0 Pearson product moment correlation coef cient let yl yN 85 21 zN be 2 collections of ordered values let 1 85 2 be sample means thus 3 E ZytN sample correlation coef cient 2 WW Z 2 22 22 Z 1 A t ltp gt 1 27 2 measures strength of linearity 1 1413 Sample Autocorrelation Sequence olet yttkt1N k andzttt1N k o for each lag k7 plug these into 3 3Zt 2 2a m2 m a2 12 and to get after a little tweaking N k 4 4 A t1 lttk gtltt 55 pk N 4 Ztiltt 33V 0 mm k O N 17 called sample acs 0 four examples Figs 6 and 7 O39L 9390 0390 9390 039 L Sample ACS for Wind Speed Series O39L 9390 0390 9390 f 19 939 o 0 o Q OV O O o o o o 0 o O O 7tao G O O39L 9390 0390 9390 039 L Sample ACS for Willamette River Series O39L 9390 0390 9390 Modeling of Time Series 0 assume mt is realization of random variable X t 0 need to specify properties of Xt ie7 model 3 0 simplifying assumptions related to stationarity 3k estimates time independent theoretical acs pk 2 mm Jain02 2 mm mow ll02 where u E EXt and 02 E EXt m2 X s are multivariate Gaussian 0 statistics of X s completely known if M 02 and pk7s known 0 critique of time domain7 characterization M7 027 pk not easy to visualize sct from pys statistical properties of gs difficult to use 1419 I l Frequency Domain Modeling 0 idea express Xt in terms of cosines and sines ie7 sinusoids 0 consider arti cial time series cos27rft 85 sin27rft t 1 128 where f is the frequency of the sinusoid and 11 is the period 7 0 consider ten different frequencies carefully chosen f 1 3 17 19 m7 m7 39 39 39 7 m7 T28 oletfj187wherej1319 o in following twenty overheads7 top plots show sinusoidal time series whose tth elements are cos27rf1t sin27rf1t cos27rf3t sin27rf3t cos27rf19t sin27rf19t Frequency Domain Modeling II 0 bottom plots Show cumulative sums of series 00827rf1t 00827rf1t 00827rf1t cos 27rf1t 00827rf1t sinlt27rf1t 00827rf19t 00827rf1t sinlt27rf1t 00827rf19t sinlt27rf19t sinlt27rf1t sinlt27rf1t 00827rf3t sinlt27rf1t 00827rf3t sin27rf3t AAAA 142 2 Frequency Domain Modeling III 0 sum of all 20 sinusoids highly structured and nonrandom in appearance 0 let7s repeat this exercise but now multiply each sinusoid by a random amplitude A each sinusoid gets a different amplitude 0 As Chosen from a standard Gaussian normal distribution zero mean unit variance Random Amplitude Sinusoid amp Sum of Sinusoids f 19128 A 115 00 N 1 D O V D N 039 O Frequency Domain Modeling IV 0 generalize to following simple model for Xt N2 Xt u Z Aj cos 27rfjt Bj sin 27rfjt j1 holds for t 1 2 N7 where N is even fj E jN xed frequencies cyclesunit time called Fourier or standard frequencies Aj7s and Bj7s are random variables gtllt EAj EBj 0 var Aj var Bj a now allowed to depend on j cov 17 Ak cov 3 Bk O forj y k gtllt cov 1117 Bk O for all j k 142 5 The Spectrum I 0 properties of simple model Exercise 11 EiXt M a 7s decompose population variance N2 02 EXt m2 Z a j1 a 7s determine acs 4 j1 pkg NQ 2 Zj1 039 0 de ne spectrum as Sj E 0 1 g j g N2 ZN2 0 cos 27rfjk 142 6 The Spectrum II o fundamental relationship 2 5339 02 jl decomposes 02 into components related to fj Sj7s equivalent to acs and 02 Exercise 15 0 easy to simulate t7s from simple model 0 four examples of spectra versus fj acs7s versus k t7s versus t 9390 9390 170 8390 80 L0 00 Theoretical Spectrum for Wind Speed Series l i 00 01 02 03 04 05 O39L 9390 0390 9390 039 L Theoretical and Sample ACSs for Wind Speed Q39L 9390 00 Theoretical Spectrum for Atomic Clock Series O ooO oooooooooooooooooO 00 01 05 039L 9390 0390 90 Theoretical and Sample ACSs for Atomic Clock Actual and Simulated Atomic Clock Series O O I 128 o o o 96 o 64 1433 Theoretical Spectrum for Willamette River Series O 04 02 1434 Theoretical and Sample ACSs for Willamette River Ol 96 128 L 64 L Actual and Simulated Willamette River Series 32 ZL39O OL39O 800 900 1700 800 000 Theoretical Spectrum for Ocean Noise Series 00 01 04 05 O39L 9390 0390 90 Theoretical and Sample ACSs for Ocean Noise Actual and Simulated Ocean Noise Series 00 N 1 D O V D N 039 O Nonparametric Estimation of Sj I 0 problem estimate spectrum Sj from X1 XN o mine out Afs 85 st since Sj var Aj var Bj 0 could use linear algebra N knowns and N unknowns 0 can get Afs Via discrete Fourier cosine transform since N N47 2 Xt cos 27rfjt T tgl N 2 0 yields for 1 g j lt N2 Aj NZXt cos 27rfjt tl Nonparametric Estimation of Sj II N 2 o Bj7s from sine transform B39 N 1 Xt sin 27rfjt 0 since 839 var Aj var ByL can estimate Sj using 2 2 9 2 N 2 N 2 Z Xt cos 27rfjt Z Xt sin 27rfjt tgl tgl 2 N2 0 examples Figs 20 and 21 8390 9390 170 8390 0390 TheoreticalEstimated Spectra for Wind Speed 04 05 9398 0398 9398 0398 939L 039L 9390 0390 TheoreticalEstimated Spectra for Atomic Clock 00 05 8390 8390 L390 0390 TheoreticalEstimated Spectra for Willamette River o 000 v 00 04 05 9390 170 8390 80 L0 00 TheoreticalEstimated Spectra for Ocean Noise ml AM I vwlvv VV W I 00 01 03 04 05 Nonparametric Estimation of Sj III 0 points about 7 uncorrelatedness of Afs and st implies st approximately uncorrelated exact under Gaussian assumption easy to test hypothesis using st dif cult for sample acs 7 is 2 degrees of freedom7 estimate if Sj7s slowly varying can average Sj7s locally Parametric Estimation of Sj I o assume st depend on small number of parameters 0 simple model S 39 oz jl 10z2 20zcos27rfj related to rst order autoregressive process 0 estimate Sj7s by estimating a7 A A A B S 39 oz A g 1d2 2dcos27rfj Parametric Estimation of Sj II 0 can show that p1 a7 so let a m o requiring A2 A j11 oz 204 cos 2m 0 examples theoretical7 spectra for wind speed atomic clock and ocean noise doesnt work well Willamette River series which points out need to be careful about parameterization 1448 8390 9390 170 8390 0390 ParametricNonparametric Estimated Spectra for Wind Speed 04 05 9398 0398 9398 0398 939L 039L 9390 0390 ParametricNonparametric Estimated Spectra for Atomic Clock 00 9390 170 8390 80 L0 00 ParametricNonparametric Estimated Spectra for Ocean Noise AMA W 00 Viva w I 01 03 04 05 Industrial Strength Theory I 0 simple model not adequate in practice frequencies in model tied to sample size N time series treated as if it were circular7 ie7 XkXk1 XN1XNX1X2 XN o assume stationarity which means that EXt a var Xt 02 and cov XE Xtk pk02 Xk i has same spectrum as X1 X2 Industrial Strength Theory II 0 under stationarity7 simple model extends to become 12 u eWt dZltfgt 12 H 2 AQ cos27rft BU Si lt27rftgtl7 f Xt 22 where dZQ yields AQ and BQ 7 and we now use claw E cos27rft i sin27rft i E 1 o analogous to simple model7 we use vardZf 50 df to de ne a spectral density function SQ Industrial Strength Theory III 0 fundamental relationship now becomes 12 Sm df 02 12 0 SQ and pk02 related Via 12 0 pkaQ Sltfe 2 fk df and so 0 2 Z pke ZQWfk 12 k oo 0 basic estimator of SQ is periodogram N I 2 1 N Xt Xe 27rfIt where X E N tigXt SW06 E 1 N Industrial Strength Theory IV 0 ideally it would be nice if 1 ESltPgtltfgt Sc 2 var 8pf a O as N a 00 but alas 1 periodogram can be badly biased for nite N can correct using data tapers 2 var 3pf 820 as N a 00 ifO lt f lt can correct using smoothing Windows Uses of Spectral Analysis 0 analysis of variance technique for time series 0 some uses testing theories eg7 wind data exploratory data analysis eg7 rainfall data discriminating data eg7 neonates diagnostic tests eg7 ARIMA modeling assessing predictability eg7 atomic clocks o applications tout le monde
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