Figure 18: Top row: example of a purely continuous spectrum (left) and one realization

Transcription

1 1..5 S() requency time time Lag 1..5 S() requency time time Lag Figure 18: Top row: example o a purely continuous spectrum (let) and one realization o length 128 (right). Bottom row: example o a purely discrete spectrum (let) and one realization o length 128 (right). Figure 19: Top row: realization o process with purely continuous spectrum (let) and sample autocorrelation (right). Bottom row: realization o process with purely discrete spectrum (let) and sample autocorrelation (right). 19 2

2 sd () acvs Lag sd () acvs Lag t Figure 2: Two spectral density unctions (let) and their corresponding autocovariance sequences (right). Figure 21: Illustration o the aliasing eect. The dotted curves above show cos(t) versus t. The solid curves show cos([1 + 2kπ]t) versus t or (rom top to bottom) k = 1, 2 and 3. The solid black squares show the common value o all our sinusoids when sampled at t =, 1,...,

3 Fig 21a: The concept o aliasing illustrated or two dierent sample intervals. In each picture the thin black line is the continuous spectrum, and the thick black line is the resulting spectrum or the discrete process (which is periodic, as shown by its continuation with the thick grey line).!"#$..25.5!"#%..25.5! "#$..25.5! "#% Figure 22: Examples o MA(1) spectra when θ 1,1 is positive we have a high requency spectrum and when θ 1,1 is negative we have a low requency spectrum 23

4 !"#$! "#$!"#$%&#$%' (#$%( (#$%)!" #$% #$%' (" #$%)( (" #$%)) !"#%! "#%..25.5!"#$% #$*' ("#$%)( (" #$*)!" #$%&#$*' (" #$%)( ("#$*)) Figure 23: Examples o AR(1) spectra when φ 1,1 is positive we have a low requency spectrum and when φ 1,1 is negative we have a high requency spectrum Figure 24: Examples o AR(2) spectra with real characteristic reciprocal roots, a = r 1 and b = r 2, giving AR parameter values o: φ 1,2 = r 1 + r 2 and φ 2,2 = r 1 r

5 !"#$%&'(#)# '%*+'$#,!"#$%$--#)# '%.*#, !"# $%&'(#)# '%*+'$#,!"# $%$--#)# '%.*#, Figure 25: Examples o AR(2) spectra with complex characteristic reciprocal roots, re ±i2π, with r =.99 or the plots in the let column and r =.7 or the plots in the right column, and =.1 or the plots in the irst row, and =.4 or the plots in the second row, the AR parameter values (as shown in the titles) can be calculated rom φ 1,2 = 2r cos(2π) and φ 2,2 = r 2 Figure 26: Inconsistency o the periodogram. The plots show the periodogram (on a decibel scale) o a unit variance white noise process o length (rom top to bottom) N = 128, 256 and 124. The horizontal dashed line indicates the true sd

6 Figure 27: Fejér s kernel or sample sizes N = 8, 32 and 128 Figure 28: Bias properties o the periodogram or an AR(2) process with low dynamic range. The thick curves are the true sd S(), while the thin curves are E{Ŝ(p) ()} or sample sizes (rom top to bottom) N = 16 and

7 h h h Figure 29: Bias properties o the periodogram or an AR(4) process with high dynamic range. The thick curves are the true sd S(), while the thin curves are E{Ŝ(p) ()} or sample sizes (rom top to bottom) N = 16, 64, 256 and 124. Figure 3: Dierent data tapers (let column) and associated spectral windows H() (right column), or N = 64. The tapers are a rectangular taper (top), a 2% (middle) and 5% (bottom) split cosine bell taper. 3 31

8 6 6 N=64, original data N=64, tapered data N=256, original data N=256, tapered data N=124, original data N=124, tapered data Figure 31: Bias properties o direct spectral estimators or an AR(4) process with high dynamic range, using a 2% (let column) and 5% (right column) split cosine bell taper. The thick curves are the true sd S(), while the thin curves are E{Ŝ(p) ()} or sample sizes (rom top to bottom) N = 16, 64 and 256. Figure 32: The let column shows simulations rom the AR(4) model: X t = 2.767X t X t X t X t 4 + t For (rom top to bottom) N = 64, 256 and 124. The right column shows {X t h t } where {h t } is the appropriate length 5% split cosine bell taper

9 6 N=64, Yule-Walker AR(4) 6 N=64, Yule-Walker AR(8) N=256, Yule-Walker AR(4) N=124, Yule-Walker AR(4) N=256, Yule-Walker AR(8) N=124, Yule-Walker AR(8) Figure 33: The thick line shows the spectrum o the AR(4) process associated with the Yule-Walker estimates o φ 1,4,..., φ 4,4, or the sequences shown in the let column o Figure 32 (i.e. untapered). The thin line shows the true spectrum. Figure 34: The thick line shows the spectrum o the AR(8) process associated with the Yule-Walker estimates o φ 1,8,..., φ 8,8, or the sequences shown in the let column o Figure 32 (i.e. untapered). The thin line shows the true spectrum

10 6 N=64, Tapered Yule-Walker AR(4) N=256, Tapered Yule-Walker AR(4) N=124, Tapered Yule-Walker AR(4) Figure 35: The thick line shows the spectrum o the AR(4) process associated with the Yule-Walker estimates o φ 1,4,..., φ 4,4, or the sequences shown in the right column o Figure 32 (i.e. tapered). The thin line shows the true spectrum. 36