so that the cross-correlation matrix is really a ˜¬eld signi¬cance™ of the resulting map of

lagged cross-correlation matrix (see, for example, reject decisions can then be assessed using the

Horel and Wallace [182], who correlated the techniques described in Section 6.8 (see also the

Southern Oscillation Index with 500 hPa height related discussion in [6.5.2]). Note that the local

rejection rate will tend to be greater than the

throughout the Northern Hemisphere).

nominal level (often 5%) because correlations near

Thus the basic idea is very general and can

the base point will be large. Care must be exercised

be applied in many different settings. A key

to account for this phenomenon when determining

limitation, however, is that the method can only

whether H0 can be rejected globally. Note also

be used to diagnose linear relationships. The

that this problem is ampli¬ed in teleconnection

term ˜teleconnection™ is usually reserved for cases

analysis because many maps are screened. Despite

in which a correlation (rather than a cross-

these dif¬culties, local signi¬cance tests are useful

correlation) matrix is analysed.

because they identify important features in the

Teleconnection patterns are closely related to teleconnection maps.

associated correlation patterns derived for a single

index ([17.3.4]). If we normalize the base point

˜

time series zt , then the teleconnection is a point i 17.5 Time Filters

given by Cov(˜ t , Xit )/σ X it (cf. equation (17.29)).

z

17.5.0 General. We have often used the concept

that the variability of a time series may be caused

17.4.6 Assessing ˜Signi¬cance.™ As with EOFs

by different processes that are characterised by

and other patterns, there is a tendency to

their ˜time scales™. It is therefore useful to split a

confuse physical and statistical signi¬cance of

time series into certain components, such as

teleconnection patterns. In general, the patterns

are worthy of physical interpretation when the

Tt = TtF + TtS (17.38)

basic structure is not strongly affected by sampling

variability (i.e., when there is reproducibility).

where T F and T S represent stationary components

The best way to assess reproducibility is to with ˜fast™ and ˜slow™ variability, respectively. The

ensure that the pattern reappears in independent time ¬lters described in this section are designed

data sets and with other analysis techniques. for this purpose.

Wallace and Gutzler used this approach by

keeping part of the data to assess the stability

of their patterns in a second step. Barnston 17.5.1 Time Filters”Concepts. One of the

and Livezey [27] subsequently reproduced the simplest ¬ltering operations is to smooth a

results of Wallace and Gutzler using rotated EOFs time series by computing its running mean.

(Section 13.5). There is little doubt of the reality For example, Figure 17.12 shows smoothed and

of the teleconnections discussed so far. unsmoothed versions of an AR(1) time series with

17.5: Time Filters 385

±1 = 0.9. The smoothed version (dashed curve) is Substituting (17.44) into (17.43), and changing the

order of summation, we ¬nd

given by

∞

K K

e’2πi„ ω

5

yy (ω) =

1 ak al

yt = xt+k . (17.39) „ =’∞

k=’K l=’K

11 k=’5

— γx x („ ’ l + k)

K K

The large, slow variations remain but the small,

al e’2πilω

= 2πikω

ak e

fast variations have almost been eliminated.

k=’K l=’K

Running mean (17.39) is an example of a digital ∞

e’2πi(„ ’l+k) γx x („ ’ l + k)

¬lter given by

„ =’∞

K

= |c(ω)|2 x x (ω),

Yt = ak Xt+k , (17.40)

k=’K thus proving (17.41) and (17.42).

Now suppose the input contains a monochroma-

where {a’K , . . . , a K } is a set of 2K + 1

tic signal, say cos(2πωt), and that the purpose of

real weights. The weights can be tailored so

the ¬ltering is to isolate this signal. Certainly we do

that the ¬lter retains variation on long, short,

not want the ¬lter to shift the signal™s phase. That

or intermediate time scales. Filters with these

is, if cos(2π ωt) is input to the ¬lter, we require that

characteristics are known as low-, high-, and band-

the output be of the form r cos(2π ωt). Substituting

pass ¬lters.

Suppose now that we have a digital ¬lter of cos(2π ωt) = 1 e2πiωt + e’2πiωt

2

the form (17.40). It is then easily shown that the

spectral density function of the output Yt is related into (17.40) we ¬nd

to that of the input Xt by

K

ak cos 2π ω(t + k)

yy (ω) = |c(ω)|2 x x (ω) (17.41) k=’K

1K

ak e2πiω(t+k) + e’2πiω(t+k)

where c(ω) is the frequency response function =

2 k=’K

K

1 2πiωt

c(ω) + e’2πiωt c— (ω) .

c(ω) = ak e2πikω =

(17.42) e

2

k=’K

The latter is again a zero phase cosine

of the ¬lter. This can be proved as follows. c(ω) cos(2π ωt) only when c(ω) is real. Thus

Beginning with (11.8), we express the spectral the weights ak must be symmetric in k, that is,

density function of Yt in terms of its auto- ak = a’k . Therefore

covariance function:

K

c(ω) = a0 + 2 ak cos(2πkω). (17.45)

∞

2πi„ ω

yy (ω) = γ yy („ )e . (17.43) k=1

„ =’∞

A ¬nal detail that is important in some

applications is that it may be necessary to preserve

But the auto-covariance function of the output is

the time average of the input, in which case the

related to that of the input by

weights should also be constrained so that c(0) =

K

a0 + 2 k=1 ak = 1. On the other hand, the time

K K

γ yy („ ) = Cov ak Xt+k , average (i.e., the zero frequency component) of the

al Xt+„ ’l

input can be removed by selecting weights such

k=’K l=’K

that c(0) = 0.

K K

= ak al Cov(Xt+k , Xt+„ ’l ) The remainder of this section is laid out as

follows. We explore the so-called ˜(1-2-1)-¬lter™

k=’K l=’K

and further examine the running mean ¬lter in

K K

= ak al γx x („ ’ l + k). [17.5.2]. Then, in [17.5.3], we consider the effect

of adding ¬lters, and applying them in sequence.

k=’K l=’K

(17.44) The latter is a technique that is frequently