стр. 160 |

contained in the time series. We have a slight

preference for the Daniell and Parzen estimators

10

over the Bartlett estimator, for which variance

0

leakage through side lobes is more of an issue.

The rectangular spectral estimator [12.3.14] is not

-10

recommended because of the large negative side

-20

lobes in its spectral window; this estimator was 0.0 0.1 0.2 0.3 0.4 0.5

described for pedagogical reasons. Frequency

If the spectrum is suspected to contain sharp Parzen Estimate

peaks, the data should also be tapered with a

20

data taper [12.3.8] to prevent contamination of the

10

smooth part of the spectrum by variance leakage

from the spectral peak.

0

Periodogram-based estimators have a number

-10

of advantages that often make them superior

to other types of spectral estimators (see, e.g.,

-20

[12.3.21], where we discuss maximum entropy 0.0 0.1 0.2 0.3 0.4 0.5

Frequency

spectral estimators).

вЂў They are non-parametric. The only assump-

Figure 12.19: Spectral estimates (on the decibel

tions required are that the process be ergodic

scale) computed from the AR(2) time series

and weakly stationary. In addition, these es-

whose periodogram is shown in lower panel of

timators often make sense when the assump-

Figure 12.9. The horizontal bar in the upper

tions are violated, such as when the process

left hand corner indicates the bandwidth. The

has a periodic component.

vertical bar indicates the width of the asymptotic

вЂў A well-developed asymptotic theory supports conп¬Ѓdence interval. The dashed curve displays the

these estimators, and practical experience theoretical spectrum.

shows that the asymptotic results generally

hold even for time series of moderate length. shown in Figure 12.19. The parameters of each of

the estimators has been chosen so that they have

вЂў Properties of the spectral estimator, such as

bandwidth and degrees of freedom equivalent to

the bandwidth and the spectral or lag window,

that of the Daniell estimator with n = 11. The

are easily tuned to the practitionerвЂ™s own

speciп¬Ѓcs of the estimators are:

needs.

Estimator n or M EBW EDF

вЂў There are many useful extensions of this

methodology that we have not been able to Chunk 21 0.0476 22

discuss here in this short discourse. Daniell 11 0.0458 22

Bartlett 32 0.0469 22

Parzen 40 0.0465 22

12.3.20 Example. Spectral estimates, computed

from the periodogram shown in the lower The Daniell estimate is shown in the upper

panel of Figure 12.9 with the four smoothed panel of Figure 12.13 together with two other

periodogram estimators described above, are Daniell estimates that have greater bandwidth. The

12.3: Estimating the Spectrum 279

spectral estimation and auto-regressive spectral

Chunk Estimate from Tapered Chunks

estimation.

20

Auto-regressive spectral estimation (see, e.g.,

Parzen [306] or Akaike [4, 5]) is performed by:

10

вЂў assuming that the process is ergodic and

0

weakly stationary,

-10

вЂў п¬Ѓtting an AR model of some order p. The

-20

order is chosen either objectively by means

0.0 0.1 0.2 0.3 0.4 0.5

Frequency

of a criterion such as AIC [12.2.10] or BIC

[12.2.11], or subjectively using a procedure

such as the BoxвЂ“Jenkins method [12.2.1,9],

Figure 12.20: As the top panel in Figure 12.19

and

except that the chunks were tapered with the cosine

вЂў estimating the spectrum with the spectral

bell data taper.

density

ПѓZ

2

Chunk, Bartlett, and Daniell estimates are shown

(П‰) = (12.51)

p

eв€’2 ПЂiП‰ |2

|1 в€’ =1 О±

in Figure 12.19.

Note that there is little difference between the

of the п¬Ѓtted AR process where, О± , =

Daniell (upper panel of Figure 12.13), Bartlett,

1, . . . , p, are the estimated AR parameters

стр. 160 |