2

j=1

the integrated weather noise affects the choice of

„

critical value. When the spectral approach is used,

9 The Central Limit Theorem [2.7.5] often ensures that T F

t

the numerator and denominator are asymptotically

is close to normal.

17.2: Potential Predictability 377

proportional to independent χ 2 random variables

under H0 with „ ’ 1 and 2n degrees of freedom

respectively. Thus S „ ∼ F(„ ’ 1, 2n) under H0 so

˜

that S„,˜ is the p-quantile of the F distribution

p

with „ ’ 1 and 2n degrees of freedom (Appendix

G). This test will be nearly unbiased (i.e., it will

operate at the speci¬ed signi¬cance level) when

the weather noise spectrum has a moderate peak or

trough at zero frequency. It will tend to be liberal

(i.e., reject H0 more frequently than speci¬ed)

when F F (ω) has a strong peak at zero frequency Figure 17.3: Horizontal distribution of the esti-

because the extrapolation of spectral estimate at mated S D J F potential predictability ratios ob-

frequency 1/„ to frequency zero will negatively tained from a 20-year run with an atmospheric

„

bias the estimate of Var(TtF ). This results in GCM. Stippled areas mark grid points where the

S „ -values that tend to be slightly larger than 1 local null hypothesis of no potential predictability

under H0 . The opposite happens when F F (ω) has is rejected at the 5% signi¬cance level. Hatched

a strong trough at the zero frequency. areas represent regions where the S D J F -ratios

It is more dif¬cult to determine an appropriate have values in the lower 5%-tail of the respective

critical value for S „ when the ˜equivalent chunk F distribution. From [440].

length™ approach is used. One solution is to argue

that both estimates in (17.19) have little sampling

F test was performed independently at the 5%

variability since they are obtained from a large

signi¬cance level at each of the 2080 grid points.

number („ n) of deviations t jk . This reasoning

As a guide to interpretation, a rejection rate of

allows us to ignore uncertainty in the denominator

approximately 10% would be ¬eld signi¬cant at

of (17.20), with the result that the test can be

the 5% signi¬cance level in a ¬eld with 100 spatial

conducted by comparing („ ’1) S „ with χ 2 („ ’ 1)

degrees of freedom (cf. [6.8.3] and Figure 6.12).

critical values (Appendix E). The resulting test

The results are shown in Figure 17.3. Several

will tend to be liberal because the variability

„ things can be noticed.

in Var TtF has been ignored. It may also be

• The variance ratio S D J F is of the order 2 over

adversely affected by bias in this estimator. A

better approach is to estimate the distribution large areas, particularly in the tropics and

of S „ under H0 by applying the moving blocks in the Southern Hemisphere, suggesting that

bootstrapping procedure (cf. [5.5.3]) to the only half of the interannual variability may

deviations t jk . An example can be found in [414]. be the integrated effect of weather variability.

The local null hypothesis of no potential

predictability is rejected at about 40% of

17.2.5 Example. Zwiers [440] analysed the

all grid points making it unlikely that all

potential predictability of the climate simulated by

rejections are due to chance.

a GCM in a 20-year run. Sea-surface temperature

and sea ice were speci¬ed from the same • There are other areas in which the variance

climatological annual cycle in each of the 20 ratio S D J F is less than 1. Although puzzling

years. Land surface conditions (snow cover, and at ¬rst glance, this is consistent with

soil temperature, moisture content and albedo) our model (17.14) since it is a natural

were computed interactively. Except for these consequence of the sampling variability of

land surface processes, the only other source of S D J F . The number of grid points with

interannual variability in the GCM simulation is S D J F ratios in the lower 5% tail of the F

˜internal™ variability. distribution is about 5%, indicating that our

Daily surface air pressure was gathered into 20 basic assumptions, which lead us to the F

„

chunks, one for each DJF season. Thus t j is the distribution, are approximately correct.

DJF-mean of surfaced pressure during the jth DJF

season at each grid point. The interannual variance Further analysis indicated that the large S D J F -

of these seasonal means is the estimated ˜inter- values were not related to the surface hydrology,

chunk™ variance that constitutes the numerator of soil moisture, and snow cover terms in the GCM.

S „ (17.20). The spectral approach was used to The potential predictability in this simulated

obtain the ˜intra-chunk™ variance. The resulting climate seems to arise from the occurrence of

17: Speci¬c Statistical Concepts

378

Further insight into the way a signal is expressed

in other variables can often be obtained with

composite analysis10 (discussed in [17.3.2,3])

and associated correlation or regression patterns

[17.3.4,5].

In the following we assume that we have either

a univariate index zt or a bivariate index zt =

(z1t , z2t )T . The variable in which we want to

identify the signal represented by the index is

labelled Vt .

Figure 17.4: Same as Figure 17.3 but for the SON

17.3.2 Composites. The general idea is to form

season. From [440].

sets of the index z and to estimate the expected

value of V conditional on z ∈ . Formally, the

a single large anomaly extending over a period composite V is given by

of about a season, during which atmospheric

mass is systematically shifted from the tropics to V = E Vt |zt ∈ . (17.21)

the high latitudes of the Southern Hemisphere. In practice, the expectation operator in (17.21) is

Such large extended anomalies have also been replaced by a sum to obtain an estimate of the

observed in other climate simulations and in the composite

real atmosphere.

1k

The same potential predictability analysis was

V= vt (17.22)

k j=1 j

conducted in other seasons. The SON (September-

October-November) map is shown in Figure 17.4.

In this season the areas with S S O N in the upper where the sum is taken over the observing times

5% tail of the F distribution are small and the t1 , . . . , tk for which zt j ∈ .

There are several things to note about this

˜signi¬cant areas™ cover roughly 5% of the globe.

Thus the data do not contradict the null hypothesis approach.

of no potential predictability. • It does not make any speci¬c assumptions

about the link between Z and V. This link

may be linear or nonlinear.

17.3 Composites and Associated

• The basic idea with composites is to construct

Correlation Patterns

˜typical™ states of V conditional on the value

17.3.1 Introduction. An important part of of the external index. It achieves this goal in

climate research deals with the identi¬cation, the sense that we obtain estimates of the mean

description and understanding of processes, such state. However, there may be considerable

as the El Ni˜ o/Southern Oscillation (ENSO) or the

n variability around each composite, and thus

Madden-and-Julian Oscillation (MJO). Univariate the composite may not be representative of

the typical state of V when Z ∈ (recall the

and bivariate indices are frequently used to identify

and characterize such signals. discussion in [1.2.1]).

For example, many aspects of the temporal

• One way to determine whether aspects of the

behaviour of ENSO are captured by the conven-

signal captured by Z are expressed in V is to

tional Southern Oscillation Index, the surface air-

test null hypotheses of the form

pressure difference between Darwin (Australia)

and Papeete (Tahiti) (see Figure 1.2). Wright [427]

=V

H0 : V (17.23)

1 2

found that many, roughly equivalent, ENSO in-

dices can be de¬ned (see, for example, Figure 1.4, for appropriately chosen disjoint subsets 1

which displays the SOI and a related tropical and 2 . This is often done with one of

Paci¬c SST index). the difference of means tests discussed in

Another example is the MJO. In this case a Section 6.6. An example is given in the next