transform, which essentially operates on an in¬nite

series that is abruptly ˜turned on™ at t = 1 and T /2 were applied to the simulated data used

abruptly ˜turned off™ again beyond t = T . That to produce Figure 12.10. The effectiveness of

is, the observed time series can be thought of as tapering in reducing the effects of variance leakage

the product of the in¬nite series {xt : t ∈ Z} and a can be seen in Figure 12.12 where we show the

data window periodogram (scaled by 1/U2 ; see (12.33)) of the

tapered data. We see that spectral line appears as

a narrow peak with increasing m as the amount of

if 1 ¤ t ¤ T

1

ht = leakage decreases.

0 otherwise.

There are some costs to pay for reducing

variance leakage by means of tapering. Smooth

This data window is sometimes called the box tapers have squared window functions with

car taper. The result is that the periodogram is wider central peaks than the box car taper (see

an unbiased estimator of the convolution of the Figure 12.11). Thus, while contamination of the

true spectrum (ω) with the square of the Fourier periodogram from remote frequencies is reduced,

transform HT of the data window (12.32). The data information from adjacent frequencies tends to

window h t and corresponding spectral window be ˜smeared™ together making it more dif¬cult

2

function HT are shown as the solid curves in to discriminate between adjacent spectral peaks

Figure 12.11. and lines in the sample spectrum. Also, while

the asymptotic properties of the periodogram

2

The large side lobes of HT can be reduced by

described above still hold, larger samples are

using a data window or data taper that turns on

needed to achieve distributional approximations of

and off more smoothly. Frequently used tapers are

the same quality when the data are tapered.

the Hanning or cosine bell taper that has nonzero

weights 15 It is easily shown that the window function for the Hanning

taper is given by HT (ω) = (HT (ω ’ π/ T ) + 2HT (ω) +

H

HT (ω+π/ T ))/4. The Hanning taper is thus constructed so that

(2t ’ 1)π

1 side lobes are destroyed by destructive interference. Bloom¬eld

ht = 1 ’ cos , 1 ¤ t ¤ T, [49] gives details.

2 T

12: Estimating Covariance Functions and Spectra

270

length. Since we generally have little control over

10

the chunk length in climatological applications,

8

little can be done to reduce bias. Fortunately, in

6

most applications the true spectrum is smooth and

bias is therefore not a big issue.

4

Variance leakage from spectral lines is a

2

potential problem in high-frequency data sets that

0

resolve, for example, the diurnal cycle or semi-

0.0 0.1 0.2 0.3 0.4 0.5

Frequency

diurnal tidal signals. In this case each chunk can be

tapered (cf. [12.3.8]) separately to control variance

10

leakage, or, if the frequency and shape of the signal

are known, it can be removed before performing

8

the spectral analysis.

6

4

12.3.10 The ˜Chunk™ Spectral Estimator: De-

2

tails. We assume, for consistency with spectral

0

estimators described later in this section, that we

0.0 0.1 0.2 0.3 0.4 0.5

have a single, contiguous time series x1 , . . . , xT of

Frequency

length T . The chunk estimator is then computed as

follows.

Figure 12.12: As Figure 12.10, except the

periodogram has been computed after tapering 1 Divide the time series into m chunks of length

M = m. T

with a split cosine bell.

Top: m = T /4.

2 Compute a periodogram

Bottom: m = T /2.

()

IT j , j = 0, . . . , q, q = M

2

12.3.9 The ˜Chunk™ Spectral Estimator:

from each chunk = 1, . . . , m.

General. A spectral estimator frequently used

in climatology is the ˜chunk™ estimator, ¬rst

3 Estimate the spectrum by averaging the

described by Bartlett [33] in 1948.16 The idea

periodograms:

is to divide the time series into a number of

chunks of equal length, compute the periodogram m

1 ()

(ω j ) = IT j .

of each chunk, and then estimate the spectrum by (12.34)

m =1

averaging the periodograms.

This estimator is frequently used in climatology

The result is an estimator with approximately

because of the cyclo-stationary nature of the

2m degrees of freedom at each frequency ω j

processes that are analysed. Typically, the annual

(except 0 and 1/2). The estimate at each frequency

cycle is removed from daily observations and it