but that constructed with M = 20 is considerably

the spectrum across the 1/(2 ) line. Frequency

noisier. Note that the spectral estimate can never

1/(2 ) is called the Nyquist folding frequency.

have more than M/2 peaks when the spectrum

The solid curve in the bottom panel of Figure 12.22

is estimated in this way. The exact number of

shows the sum of the unaliased and aliased parts

peaks will depend upon the mix of AR(1) and

of the spectrum. This distorted version of the

AR(2) components in the AR model that is ¬tted.

real spectrum is the function that a good spectral

The spectral estimate shown in the lower panel of

24 A subsequent GCM experiment with a one-hour save

Figure 12.21 contains eight local peaks.

interval showed that the model simulated the solar“thermal tide

23 The Burg procedure chooses the AR coef¬cients that well, but that it did so for the wrong physical reasons (Zwiers

and Hamilton [447]).

minimize the sum of the forward and backward forecast

25 Another example of aliasing is described by Bohle-

squared errors (the ˜back™ forecast is described in [11.1.12]).

Priestley [323, pp. 604“606] shows that for contiguous time Carbonell [54] who demonstrates how a 14-day period appears

series, the Burg estimates are precisely the Yule“Walker in daily Cuxhaven salinity measurements that are affected by

estimates (cf. [12.2.2]). the M2 tide (12.5 hours).

12.4: Estimating the Cross-correlation Function 281

between ω = 1/18 hours and ω = 1/12 hours are

Aliasing Effect

folded onto frequencies between ω = 0 and ω =

•

1.0

1/36 hours, and so on. The line in the unaliased

spectrum at ω = 1/12 hours therefore appears at

• • •

0.0

the Nyquist folding frequency.

•

-1.0

0 2 4 6

12.4 Estimating the Cross-correlation

Time

Spectral Folding Function

01234

12.4.1 Estimating the Cross-covariance and

Cross-correlation Functions. Suppose that a

sample (xt , yt ), t = 1, . . . , T , is obtained from

an ergodic weakly stationary bivariate process

0.0 0.2 0.4 0.6

{(Xt , Yt ) : t ∈ Z}. Then an estimator of the

Frequency

Aliased Spectrum cross-covariance function γx y („ ) is

01234

1 T ’„

cx y („ ) = (xt ’ x)(yt+„ ’ y) for „ ≥ 0

T t=1

1T

0.0 0.1 0.2 0.3

= (xt ’ x)(yt+„ ’ y) for „ < 0

Frequency

T t=„ +1

= 0 for |„ | ≥ T, (12.54)

Figure 12.22: Top: This illustrates a wave of period

and γ yx („ ) is estimated by cx y (’„ ).

2 that is sampled once every 1.5 time intervals.

As with the auto-covariance function (see

The resulting collection of observations is periodic

[12.1.1] and [12.1.2]), these estimates are some-

with period 6. This sampling scheme has aliased

times in¬‚ated with a factor T /(T ’ |„ |). This

variation at frequency ω = 1/2 onto frequency

ω = 1/6. makes the estimator unbiased if the process has

zero mean and if the sample means are not sub-

Middle: This illustrates ˜folding™ of the spectral

density across the Nyquist folding frequency (ω = tracted in (12.54). However, this practice also in-

¬‚ates the variability of the estimator, particularly at

1/3 in this example) onto frequencies less than the

large lags where the true cross-covariance function

Nyquist folding frequency. The solid curve is the

is close to zero anyway, and it affects the properties

original spectrum and part that is folded back is

of weighted covariance spectral estimators (cf.

indicated by the dashed curve.

Section 12.3) by subtly changing the lag-window.

Bottom: The resulting aliased spectrum (solid

curve). The dashed curve indicates the real The cross-correlation function is estimated as

spectrum.

cx y („ )

r x y („ ) = .

1/2

cx x (0)c yy (0)

estimator would be able to estimate from a time

series sampled every time intervals. A poor 12.4.2 Properties of the Estimated Cross-corre-

choice of sampling interval can obviously lead to a lation Function. The types of problems that oc-

badly distorted spectrum and misleading physical cur when estimating the auto-correlation function

interpretation. also occur when estimating the cross-correlation

In the thermal tide example, we sample every function (see the discussion in [12.1.2]). Bias is a

18 hours, resulting in a Nyquist folding frequency dif¬culty, particularly at large lags and when the

of ω = 1/36 hours. Variation at shorter time magnitude of the true cross-correlation is near 1.

scales is folded accordion style onto the interval As with the auto-correlation function, Bartlett [34]

between ω = 0 and ω = 1/36 hours. Thus, also derived approximations for the covariance

variation at frequencies between ω = 1/36 hours between cross-correlation function estimates at

and ω = 1/18 hours is folded onto frequencies different lags (see Box and Jenkins [60, p. 376]).

between ω = 1/36 hours and ω = 0 (i.e., 18-hour Using this approximation, it can be shown that, if

ρx y („ ) is zero for all „ outside some range of lags

variations are aliased to the mean). Variations

12: Estimating Covariance Functions and Spectra

282

„1 ¤ „ ¤ „2 , then Also, as with univariate spectral analysis, small

equivalent bandwidth is associated with high

∞

1 variance. But, in contrast with univariate spectral

Var r x y („ ) ≈ ρx x ( )ρ yy ( )

T ’ |„ | estimators, insuf¬ciently smoothed periodograms

=’∞

tend to overestimate the coherency between time

(12.55)

series. Thus, in cross-spectral analysis one must

for all „ outside the range. This result, and be careful to balance the smoothing, or reduction

others similar to it that can be derived from of variance, against the biases that are associated

Bartlett™s approximation, can sometimes be used to with too much smoothing. These problems cannot

determine whether an estimated cross-correlation be avoided by using the chunk estimator: the

cx y („ ) is consistent with the null hypothesis that use of many chunks that are excessively short

γx y („ ) is zero. This is done by performing an is equivalent to oversmoothing the periodogram

appropriate test at the 5% signi¬cance level by of a contiguous time series, and the use of

declaring inconsistency if |r x y („ )| > 2s where only a few chunks, each of moderate or greater

s 2 is the estimated variance of r x y obtained length, is equivalent to insuf¬ciently smoothing

by substituting the estimated auto-correlation the periodogram.

functions of Xt and Yt into equation (12.55).26

12.5.1 Notation and Assumptions. Most of

12.5 Estimating the Cross-spectrum

the ideas discussed in this section apply equally

in bivariate and multivariate settings. However,

12.5.0 Introduction. Our purpose in this section

to keep concepts as concrete as possible, vector

is to give a brief introduction to cross-spectral

quantities, such as the random vector X, will