exercise also identi¬es an equivalent bandwidth,

more weight on low lag covariances and less on

namely that of the ˜equivalent™ Daniell estimator.

lags near the cutoff lag M.

Therefore, when the smoothed periodogram

estimator has r equivalent degrees of freedom, its

12.3.17 Equivalent Degrees of Freedom and

equivalent bandwidth is r/2 T .

Bandwidth of Smoothed Periodogram Spectral

Estimators. The bandwidth and degrees of 21 This method of ¬nding an approximating distribution is

freedom of the Daniell estimator [12.3.11,13] also used in [9.4.9].

12: Estimating Covariance Functions and Spectra

276

12.3.18 Bias, Variance and Variance Leakage. series. They can be represented as a discrete

Again, we emphasize the point made in [12.3.12] convolution

about the tradeoff that the practitioner must j+(n’1)/2

make between bias and variance when estimating (ω j ) = W (ωk ’ ω j )IT k

spectra. It is always good to be aware of the k= j’(n’1)/2

dichotomy

of the periodogram with a spectral window W (ω),

and as the Fourier transform of the estimated auto-

Low variance Low bias

covariance function weighted by a lag window

w(„ ),

High bandwidth Low bandwidth

T ’1

w(„ )c(„ )e’2πiω j „ .

High bias High variance (ω j ) =

„ =’(T ’1)

between bias and variance in spectral estimation.

The spectral and lag windows form a Fourier

However, its is also important to remember that

transform pair.

estimators with the same equivalent bandwidths

Note that the chunk estimator can be similarly

and degrees of freedom (see [12.3.19]) are not

represented as either a discrete convolution or

created equal. These are asymptotic concepts that

as the Fourier transform of a windowed auto-

hold in the limit as the sample length becomes

covariance function estimate. The convolution

large and the equivalent bandwidth becomes small

form of the chunk spectral estimator is

enough so that the spectral density function is

approximately constant within any bandwidth.

(ω j ) = WC (ωk ’ ω j )I Mk

Variance leakage, that is, the contamination

k

of the spectral estimate by contributions from

where

periodogram ordinates at frequencies far removed

±

from the frequency of interest, is also an important 1

ω=0

consideration in selecting a good estimator when

WC (ω) = m

samples are ¬nite. 0 otherwise

12.3.19 Summary. For easy reference, we and where I

Mk is the mean of the periodograms

()

now brie¬‚y summarize the periodogram derived

I Mk computed from the individual chunks. The

spectral estimators described above.

Fourier transform form of the chunk estimator is

The periodogram (see [12.3.1“7]) of a time

M’1

series of length T is de¬ned as

wC („ )c(„ )e’2πiω j „

(ω j ) =

T „ =’(M’1)

IT j = |Z T j |2 ,

4 where

where Z T j is the Fourier transform

wC („ ) = 1 for |„ | ¤ M ’ 1

T

2 and c(„ ) is the mean of the auto-covariance

xt e’2πiω j t

=

ZT j

function estimates c( ) („ ) computed from the

T t=1

individual chunks.

of the time series. Asymptotic p — 100% con¬dence intervals (see

˜

The ˜chunk™ spectral estimator is constructed [12.3.10,11]) for the spectral density function have

by dividing the time series into m chunks of the form

length M (see [12.3.9,10]), separately computing

r

+log (ω j ) ¤ log (ω j )

the periodogram of each chunk, and then averaging log

br

the periodograms. Because climate processes are r

¤ log +log (ω j )

cyclo-stationary, many climate problems present

ar

the practitioner with disjoint chunks of length one

where ar and br are the (1 ’ p)/2 and (1 +

˜

season at yearly intervals such that it is possible

p)/2 critical values of the χ

˜ 2 (r ) distribution (see

to assume that the process is roughly stationary

within chunks. Appendix E) and r is the equivalent degrees

Smoothed periodogram spectral estimators (see of freedom (see [12.3.17]) of the periodogram

[12.3.11“18]) are computed from contiguous time derived spectral estimator.

12.3: Estimating the Spectrum 277

This lag window is applied to the average

Lag Window Spectral Window

of the auto-covariance function estimates

1.0

computed from the individual chunks.

0.12

0.8

• Daniell (n odd)

0.10

0.6

sin(πn„/ T )

0.08

π n„/ T

0.4

0.06

• Bartlett

0.2