and σZ is the estimated noise variance.

2

the differences in window properties in this

example are much less severe than the effect of The theoretical justi¬cation for AR spectral

oversmoothing the periodogram (Figure 12.13). estimation is that any ergodic weakly stationary

The appearance of the chunk estimate is different process can be approximated arbitrarily closely by

from that of the other estimators because it only an AR process.

has one value for every equivalent bandwidth. This approach to spectral estimation is attractive

The Daniell, Bartlett, and Parzen estimators are because it describes the distribution of variance

de¬ned at every Fourier frequency. However, only with time scale using a model of the time series

points separated by at least one bandwidth can that has a dynamical interpretation (cf [10.3.1]). It

be considered roughly independent. The chunk also produces spectral estimates that are generally

estimator seems to have some dif¬culty with smoother than those made by smoothing the

variance leakage in this example; the peak appears periodogram. Periodic features of the process can

to be spreading slightly and the spectrum is be identi¬ed if the practitioner is willing to use AR

overestimated at high frequencies. This behaviour models of high enough order.22

is to be expected since the chunks are very short On the other hand, interpretation of the

(M = 21). estimated spectrum is more dif¬cult. Spectral

In this case one of the smoothed periodogram estimates at well separated frequencies may not

estimators would de¬nitely be preferred over the be approximately independent, as they are when

chunk estimator. However, if for some physical made with a smoothed periodogram estimator, and

reason our sample consisted of disjoint junks con¬dence intervals are dif¬cult to construct.

of length 21, we would have no choice but to Maximum entropy spectral estimation (see

use the chunk estimator. In such circumstances Burg [73, 74], Lacoss [239], Priestley [323])

its properties can be improved somewhat by is a particular form of AR-spectral estimation.

using the cosine bell data taper [12.3.8]. The Suppose we have available estimated auto-

covariances c(0), . . . , c(M). Then the maximum

difference between Figure 12.20 and the top panel

entropy spectral estimator (ω) is the non-

of Figure 12.19 gives an indication of the type of

improvement that can be obtained in this way. negative function that maximizes the entropy

1

2

ln (ω) dω (12.52)

’1

12.3.21 Auto-Regressive Spectral Estimation 2

and Maximum Entropy. Two closely related 22 Tillman et al. [366], for example, use models of

spectral estimation methods that are also occa- successively higher order to estimate the spectrum of a Martian

sionally used in climatology are maximum entropy surface pressure time series.

12: Estimating Covariance Functions and Spectra

280

12.3.22 Aliasing. The time series objects that

Maximum Entropy, M=2

we have considered have a discrete time index,

20

but they presumably represent processes that take

place in continuous time. Do we need to worry

10

about how the sampling interval is chosen?

0

Some years ago, a 500 hPa height time series

-10

was analysed at a tropical location. The time

series was obtained from a 20-year climate

-20

simulation performed with an atmospheric General

0.0 0.1 0.2 0.3 0.4 0.5

Circulation Model. The model had been sampled

Frequency

Maximum Entropy, M=20 at 18-hour intervals because it was felt that this

20

would produce better long-term statistics than a

12- or 24-hour sampling interval. It was argued

10

that monthly and seasonal means would be more

representative of the diurnal cycle since the 18-

0

hour sampling strategy views the globe with the

-10

sun in four different positions. When the spectrum

-20

was analysed, a spectral line was discovered at

0.0 0.1 0.2 0.3 0.4 0.5

the highest resolved frequency, one cycle per

Frequency

two observing times (36 hours). The source of

this line was not a physical process taking place

Figure 12.21: Maximum entropy estimates of an at the 36-hour time scale, but rather, one with

AR(2) time series of length 240 plotted on the a characteristic period of 12 hours, namely the

solar“thermal tide.24 This oscillation has a 12-

decibel scale. The true spectrum is dashed.

Top: Using an AR(2) model. hour period because the atmosphere is not deep

Bottom: Using an AR(20) model. enough to propagate the fundamental diurnal wave

effectively.

The phenomenon that leads to the spectral

subject to the constraint that line at the half sampling interval frequency is

called aliasing; Figure 12.22 shows a schematic

1

2

example.25 The upper panel shows a wave with

(ω)e2πiω„ dω = c(„ ) (12.53)

period 4 /3 (solid curve) that is sampled every

’1

2

time intervals ( = 1 1 ). The resulting time

for „ = 0, . . . , M. Lacoss [239] shows that (12.52) 2

series appears to contain a wave with period 4

and (12.53) have a unique solution that is given

(dashed curve). We say that the variation taking

by an AR-spectral estimator (12.51) in which an

place at frequency 3/(4 ) has been aliased onto

AR-model of order p = M is ¬tted using the Burg

the 1/(4 ) frequency.

procedure.23

The highest frequency that can be resolved

Maximum entropy spectral estimates for our

with an observing interval of time units is

familiar AR(2) time series are shown in Fig-

1/(2 ). The middle panel in Figure 12.22 shows

ure 12.21. The spectral estimate constructed with

that frequencies greater than 1/(2 ) are, in

M = 2 very closely approximates the true density,