12.3.8 Tapering the Data. While the peri-

odogram is asymptotically an unbiased estimate

-10

of the spectral density, it can have poor bias

properties for ¬nite samples if the spectrum is not

0.0 0.1 0.2 0.3 0.4 0.5

very smooth or if periodic components cause lines

Frequency

in the spectrum (see [11.2.8]).

Equation (12.30) suggests how problems can

Figure 12.9: Periodograms computed from three

arise. It gives the expectation of IT j as

simulated time series. The dashed line shows the

true spectral density. 1

4 2

Top: The periodogram of a white noise time series E IT j = HT (ω ’ ω j )2 (ω) dω.

2

of length T = 120. T ’1 2

Middle: The periodogram of a white noise time (12.32)

series of length T = 240.

Bottom: The periodogram of a time series of length When (ω) is not smooth or when the spectral

T = 240 generated from the AR(2) process with density has a line, there can be substantial variance

(±1 , ±2 ) = (0.9, ’0.8). For this panel only, the leakage through the side lobes of spectral window

2

periodogram is plotted on the decibel scale (i.e., HT (see Figure 12.8).

ω j versus 10 log10 (IT j )). The problem is illustrated in Figure 12.10. It

shows the periodogram of a time series of length

240 generated from process

Yt = Xt + 10 sin(2π 0.3162t + 0.5763)

is subtracted from the time series before the

periodogram is computed. When this is not true, where Xt is an AR(2) process with (±1 , ±2 ) =

IT,0 is completely confounded with the sample (0.3, 0.3). The spectral density function of process

2

mean since IT,0 = T X . The zero frequency Yt , which is depicted by the dashed curve, has a

spectral line at frequency ω = 0.3162. Instead

periodogram ordinate is therefore useless as an

estimator of the variance of the sample mean. of being nearly unbiased as in Figure 12.9, the

Many people have considered the problem of periodogram now has substantial bias in a wide

band centred on ω = 0.3162 which is caused

estimating the variance of the sample mean

including Madden [263], Thi´ baux and Zwiers

e by variance leakage through the side lobes of the

2

[363], Zwiers and von Storch [454], and Wilks spectral window HT .

12.3: Estimating the Spectrum 269

and the split cosine bell that has weights

Data Window Window Transform

±

1 1 ’ cos (2t’1)π if 1 ¤ t ¤ m

2

1.0

2m

1.0

1 if m +1 ¤ t ¤ T ’m

0.8

ht =

0.8

1 1 ’ cos (2T ’2t+1)π

2

0.6

m

0.6

if T ’m +1 ¤ t ¤ T.

0.4

The number of non-unit weights 2m is typically

0.4

0.2

chosen so that 10%“20% of the data are tapered.

0.2 The window function corresponding to a data taper

0.0

{h t : t = 1, . . . , T } is15

0.0

-0.2

T

’(T +1)2πi/2

H (ω) = e h t e2πit .

0 10 20 30 40 50 -0.10 -0.05 0.0 0.05 0.10

Frequency

t=1

The Hanning taper has very strongly reduced side

lobes (see Figure 12.11). The split cosine bell taper

Figure 12.11: Characteristics of some popular

has side lobes that are intermediate between those

data tapers.

of the box car and Hanning tapers.

Left: The box car (solid), Hanning (dots) and split

Equation (12.32) shows us that tapering induces

cosine bell (dashed) data windows for a time series

of length T = 50. The split cosine bell uses m = bias in the periodogram if the weights are not

T /4 so that 25% of the data are tapered at each suitably normalized. Dividing the periodogram by

end of the time series. T

1

Right: Corresponding window functions. U2 = h2 (12.33)

t

T t=1

ensures that the result is an approximately

The problem is a side effect of the ¬nite Fourier unbiased estimator of the spectrum.