about the propagating feature of this process.

One such bivariate index, derived through a POP 10 Composite analysis is also sometimes called epoch

analysis, is shown in Figure 10.3. analysis.

17.3: Composites and Associated Correlation Patterns 379

Figure 17.5: Composite analysis of the lati-

tude/height distribution of the zonal wind in JJA

averaged between 60—¦ E and 90—¦ E longitude as

simulated in a 20-year integration with an AGCM.

The difference between the six cases with strongest

Southeast Asian monsoon precipitation signal and

the six cases with weakest precipitation is shown. Figure 17.6: Division of the two-dimensional plane

Local t tests were performed to test the stability containing the bivariate MJO index Z into nine

of the difference. Rejection of the local zero differ- sectors. The composite sets j are labelled K j .

ence null hypothesis at the 5% signi¬cance level is Realizations z that fall into the inner white region,

indicated by cross-hatching. From [443]. such as the short heavy vector, are not classi¬ed.

Realizations that fall into the outer shaded sectors

are classi¬ed as belonging to classes K 1 to K 8 .

17.3.3 Examples of Composite Analyses. We

will discuss two examples in this subsection.

The ¬rst example, from [443], deals with a sets i and r , representing ˜intensi¬ed™ and

univariate index and demonstrates a test of the ˜reduced™ Southeast Asian rainfall, were formed:

null hypothesis (17.23). The second example, from r consisted of the six smallest values of zt ,

[389], is on the oscillatory MJO and features eight and i the six largest. The estimated composite

different composites which supposedly represent

difference V i ’ V r is shown in Figure 17.5.

canonical sequences of events.

The circulation differences seen in Figure 17.5

Zwiers [443] analysed the variability of the

are similar to differences between weak and strong

Asian summer monsoon simulated in a 20-year

monsoon years seen in the real atmosphere. The

GCM experiment. The study included a Canonical

t test indicates that the upper tropospheric anti-

Correlation Analysis (CCA, see Chapter 14) of

cyclonic ¬‚ow resulting from heating of the Tibetan

the surface heat ¬‚ux on the Tibetan Plateau and

Plateau is signi¬cantly stronger in strong monsoon

rainfall over Southeast Asia.11 The CCA exercise

years than in weak monsoon years. At the same

produced a univariate index zt that represented

time we also see some evidence of a signi¬cantly

a signi¬cant part of the simulated interannual

enhanced Somali jet near the near the surface

variability of the monsoon rainfall. Since 20 years

between 20—¦ N and 25—¦ N. Thus all three centres

were simulated, a sample of 20 indices z1 , . . . , z20

with statistically signi¬cant wind changes are part

were available.

of the same physical signal.

A composite analysis was performed to deter-

von Storch et al. [389] derived composites from

mine whether large-scale circulation changes that

an extended GCM simulation that are supposedly

are associated with the monsoon in the real atmos-

representative of different parts of the lifecycle of

phere also occur in the model.12 Thus V was set to

the MJO. The two-dimensional plane that contains

the latitude/height distribution of the zonal wind

averaged between 60—¦ E and 90—¦ W longitude. Two the bivariate MJO index z (see Figure 10.3

and [17.2.4]) was divided into nine regions (see

11 Heating of the Tibetan Plateau is thought to in¬‚uence the

Figure 10.5 and the sketch in Figure 17.6). The

strength of the Asian summer monsoon.

˜inner circle™ set, which covers all indices with

12 The westerly zonal jet shifts northward and an easterly jet

small amplitudes (i.e., |z|2 = (z1 )2 + (z2 )2 < L s )

develops to the south at the onset of the Asian summer monsoon

is disregarded in the analysis. The remaining eight

in response to the heating of the Tibetan Plateau.

17: Speci¬c Statistical Concepts

380

a linear statistical model which relates the index or

indices zi,t with the vector variable Vt :

˜

Vt = zi,t q i + noise, (17.24)

i

where Vt usually represents anomalies (i.e.,

˜

E(Vt ) = 0), and zi is usually a normalized index

˜

given by z = (z’µz )/σz . Patterns q i usually carry

the same units as Vt because of the normalization

˜

of the index time series zi,t .

We now brie¬‚y discuss the one and two index

versions of (17.24).

Only one pattern q = q 1 is obtained when a

single index, zt = zt , is used. This pattern is often

called the ˜regression pattern™ for obvious reasons.

Then

˜

Vt = zt q + noise. (17.25)

The interpretation of q is that we observe pattern

˜ ˜

q , on average, when z = 1 and ’q when z = ’1.

More precisely, (17.25) says that

E Vt |˜ = ± = ± q

z (17.26)

Figure 17.7: The composite mean tropical velocity for any number ±, provided that the noise has mean

zero.

potential anomalies Vk j derived from MJO indices

Pattern q must be estimated from data. This can

z ∈ K j . From [389].

be done by minimizing the expected mean squared

error,

sectors, which each represent a 45—¦ segment, are 2

˜

= E Vt ’ zq , (17.27)

labelled K 1 , . . . , K 8 .

Composite means of tropical velocity potential that is, by ¬nding the vector q such that

anomalies at 200 hPa were computed for each

‚

sector (Figure 17.7). All eight composites exhibit a = 0. (17.28)

‚q

zonal wavenumber 1 pattern with maximum values

on the equator. By differentiating, we ¬nd that q satis¬es

The POP model from which the index was

˜ ˜t

’2E zt Vt + 2E z2 q = 0.

derived shows that the index tends to rotate

counterclockwise in the two-dimensional plane (as

˜t ˜

Since E z2 = Var z = 1, the solution of (17.28)

indicated by the circular arrow in Figure 17.6).

is given by

We may therefore interpret the composites as a

sequence of patterns that appear consecutively in

˜

q = Cov zt , Vt . (17.29)

time. The main features propagate eastward, and

intensi¬cation occurs whenever a maximum or Thus the associated correlation pattern q

minimum enters a region with active convection. consists of the regressions between the index time

series and the components of the analysed vector

Note that composite V K 2 is almost a mirror image

time series.

of V K 6 .

Two patterns are obtained when Zt is a

bivariate index. Such indices arise naturally in

17.3.4 Associated Correlation Patterns. Com- many circumstances, including POP analysis (see