rules. We outline a better approach, based on

separate the physically relevant EOFs from those

North™s Rule-of-Thumb, in the next subsection.

that are not.19 One popular procedure of this

type is ˜Rule N™ [321]. The basic supposition

is that the full phase space can be partitioned 13.3.5 North™s Rule-of-Thumb. Using a scale

into one subspace that contains only noise and argument, North et al. [296] obtained an approx-

another that contains dynamical variations (or imation for the ˜typical™ error of the estimated

˜signals™). It is assumed that the signal-subspace EOFs

is spanned by well-de¬ned EOFs while those in 2m c

i≈ ej

e (13.43)

the noise-subspace are degenerate. Thus, the idea

n j=1 » j ’ »i

is to attempt to identify the signal-subspace as the

j=i

space spanned by the EOFs that are associated with

where c is a constant and n is the number of

large, well-separated eigenvalues.

The selection rules compare the eigenspec- independent samples. There are three things to

trum20 estimated from the sample with distribu- notice about this equation.

tions of sample eigenspectra that are obtained

• The ¬rst-order error e i is of the order of

under the assumption that all or the smallest m

1

n . Thus convergence to zero is slow.

true eigenvalues are equal. The number m is either

speci¬ed a priori or determined recursively. All

• The ¬rst-order error e i is orthogonal to the

estimated eigenvalues that are larger than, say, the

true EOF e i .

95% percentile of the (marginal) distribution of the

reference ˜noise™ spectra, are identi¬ed as being

• The estimate of the ith EOF e i is most

˜signi¬cant™ at the 5% level.

strongly contaminated by the patterns of

One problem with this approach is that this

those other EOFs e j that correspond to the

selection rule is mistakenly understood to be a

eigenvalues » j closest to »i . The smaller the

statistical test of the null hypothesis that EOFs

difference between » j and »i , the more severe

˜

e 1 , . . . , e m , for m < m, span noise against the

the contamination.

alternative hypothesis that they span the signal-

subspace. The connection between this alternative Lawley™s formulae (13.38, 13.39) yield a ¬rst-

and the determination of a ˜signal-subspace™ is order approximation of the ˜typical error™ in »i :

vague. Also, the approach sketched above does not

2

consider the reliability of the estimated patterns »i ≈ »i . (13.44)

n

since the selection rules are focused only on the

eigenvalues. Combining this with a simpli¬ed version of

The other problem with the ˜selection rule™ approximation (13.44), North et al. [296] ¬nally

approach is that there need not be any connection obtain

between the shape of the eigenspectrum on the one

c »i j

hand and the presence or absence of ˜dynamical ei ≈ e (13.45)

» j ’ »i

structure™ on the other. To illustrate, suppose that a

process Xt = Dt + Nt , containing both dynamical

where c is a constant and » j is the the closest

19 See, for example, Preisendorfer, Zwiers, and Barnett [321]. eigenvalue to »i . North™s ˜Rule-of-Thumb™ follows

20 An eigenspectrum is the distribution of variance (i.e., from approximation (13.45): ˜If the sampling error

eigenvalues), with EOF index. The eigenspectrum is an

21 We would need to also analyse at least part of the lagged

analogue of the power spectrum (see Section 11.2) since both

describe the distribution of variance across the coef¬cients of covariance structure of Xt to reveal the ˜dynamics™ in this

orthonormal basis functions. example.

13: Empirical Orthogonal Functions

304

True Estimated (n=300) Estimated (n=1000

Figure 13.2: North et al.™s [296] illustration of North™s Rule-of-Thumb [13.3.5]. From [296].

Left: The ¬rst four true eigenvalues and EOFs.

Middle: Corresponding estimates obtained from a random sample of size n = 300.

Right: As middle column, except n = 1000.

of a particular eigenvalue » is comparable to a reasonable guess is demonstrated in the middle

or larger than the spacing between » and a and right hand columns of Figure 13.2, which

neighbouring eigenvalue, then the sampling error displays EOFs estimated from random samples of

size n = 300 and n = 1000, respectively.

e of the EOF will be comparable to the size of

the neighbouring EOF™.

13.4 Examples

13.3.6 North et al.™s Example. North et al.

[296] constructed a synthetic example in which

13.4.1 Overview. We will present two examples

the ¬rst four eigenvalues are 14.0, 12.6, 10.7 and

of conventional EOF analysis in this section. This

10.4 to illustrate North™s Rule-of-Thumb [13.3.4].

¬rst case, on the globally distributed SST, is most

The ¬rst four (true) EOFs are shown in the

straight forward. The second example involves a

left hand column of Figure 13.2. According to

data vector that is constructed by combining the

approximation (13.44) the typical error for the

same variable at several levels in the vertical.

¬rst four estimated eigenvalues is »i ≈ ±1 for

n = 300 and »i ≈ ±0.6 for n = 1000. Since

»1 ’ »2 = 1.4, »2 ’ »3 = 2 and »3 ’ »4 = 0.3, 13.4.2 Monthly Mean Global Sea-surface

one would expect the ¬rst two EOFs to be mixed22 Temperature. The ¬rst two EOFs of monthly

when n = 300 and the third and fourth EOF to be mean sea-surface temperature (SST) of the global

mixed for both n = 300 and n = 1000. That this is ocean between 40—¦ S and 60—¦ N are shown in

Figure 13.3. They represent 27.1% and 7.9% of the

22 That is, we expect the ¬rst two EOFs to be a combination

total variance, respectively.

of the EOFs that correspond to nearby eigenvalues.

13.5: Rotation of EOFs 305

manageable by performing the analysis in two

steps. Separate EOF analyses were ¬rst performed

at each level. In each analysis, the coef¬cients

representing 90% of the variance were retained.

A combined vector, composed of EOF coef¬cients

selected for the ¬ve levels, is used as input for

the eventual EOF analysis of the three-dimensional

zonal wind ¬eld.

The ¬rst two EOFs are shown in Figure 13.5,

and their coef¬cient time series are shown as

traces ˜B™ and ˜C™ in Figure 13.4. The ¬rst EOF,

representing 11% of the total monthly variance,

is mostly barotropic, not only in the extratropics

but also in the tropics. Its coef¬cient time series

exhibits a trend parallel to that found in the

coef¬cient of the second SST EOF. The mean

Figure 13.3: EOFs 1 (top) and 2 (bottom) of

westerly winds in the Southern Hemisphere were

monthly mean sea-surface temperature (SST).

analysed as being weaker in the 1970s than in the

Units: 10’2 . Courtesy Xu.

mid 1980s (negative sign indicates easterly wind

anomalies). At the same time the mean low-level

easterlies along the equatorial Paci¬c were weaker

The ¬rst EOF, which is concentrated on

in the early 1970s and stronger in the mid 1980s

the Paci¬c Ocean, represents ENSO. Its time

(positive anomalies represent anomalous westerly

coef¬cient, shown as curve ˜D™ in Figure 13.4,

winds). The results of the EOF analysis of the SST

is highly correlated with the two Southern