B C C

B

A B B

A A A A

1 2 3 4 5 6

2 4 6 8 10 12

Figure 13.11: Eigenspectra obtained with window

length m = 6 for AR(1) processes with a = Figure 13.12: First six time EOFs of an AR(2)

process with ± = (0.3, 0.3) obtained using

0.99, 0.90, 0.80, 0.60, 0.40, 0.20, and 0.05. The

window length m = 12. The patterns are

spectra are normalized by the variance of the

process. normalized with the square root of the eigenvalue.

The eigenvalues are given at the bottom.

When m = 2, ΣY Y has eigenvalues (1 ’

a) √ (1 + a) and corresponding time EOFs

and √ √ √

(1/ 2, ’1/ 2 )T and (1/ 2, 1/ 2 )T . The order [10.3.3]. The auto-covariance function is either the

sum of the auto-covariance functions of two red

of the eigenvalues and EOFs depends upon the sign

noise processes (11.4), or it is a damped oscillatory

of a. Given a speci¬c window length m, the same

function (11.9) (for details, see [11.1.9]).

EOFs are obtained for all AR(1) processes Xt .

The process with coef¬cients a1 = a2 =

The ¬rst four AR(1) time EOFs for window

length m = 6 are shown in Figure 13.10. The 0.3 was found to belong to the former ˜non-

patterns are multiplied by the square root of the oscillatory™ group. The ¬rst six time EOFs

obtained using window length m = 12 are

eigenvalue, as in equation (13.22). The kth pattern

crosses the zero line k ’ 1 times. Thus, the shown in Figure 13.12. Similar to the AR(1)

time EOFs are ordered by time scale, with most process, all eigenvectors have different patterns,

with the kth eigenvector having (k ’ 1) zeros. All

variance contributed by the variability with longest

time scales. eigenvalues are well separated. Consistent with the

discussion in [11.1.7] and [11.2.6], an oscillatory

One characteristic of the time EOFs is that no

mode is not identi¬ed. Note the similarity between

two patterns have the same number of zeros. Thus

the patterns in Figure 13.12 and the AR(1)

oscillatory behaviour, such as that described in

patterns shown in Figure 13.10. (The patterns in

[13.6.4], is not possible. This is consistent with

Figure 13.12 are not sensitive to the choice of

the discussion in [11.1.2], when we also found

m.)

no indication of oscillatory behaviour in AR(1)

processes. The other AR(2) process considered previously

has ±1 = 0.9, ±2 = ’0.8. This process has

Figure 13.11 shows the eigenspectra of several

AR(1) processes for the same window length

oscillatory behaviour with a ˜period™ of 6 time

m. The larger the ˜memory™ a, the steeper the

steps (see [10.3.3], [11.1.7] and [11.2.6]). The

spectrum. In the extreme case with a = 0.99,

time EOFs of this process, obtained using window

length m = 12, are shown in Figure 13.13. The

almost all variance is contributed by the ˜almost

constant™ ¬rst time EOF. At the other end of the

¬rst two time EOFs are sinusoidal, with a period

memory scale (a = 0.05) all time EOFs contribute

of 6 time steps, and phase-shifted by 1 to 2 time

about the same amount of variance. steps (a quarter of a period, 1.5 time steps, can

not be represented in time steps of 1). The two

time EOFs share similar eigenvalues (4.2 and 4.1)

13.6.7 SSA of an AR(2) process. An AR(2) and obviously represent an oscillatory mode as

process has the form described in [13.6.4]. The higher index time EOFs

are reminiscent of the time EOFs obtained for

Xt = a1 Xt’1 + a2 Xt’2 + Zt AR(1) processes.

13: Empirical Orthogonal Functions

316

13.6.9 Estimation. We conclude with a brief

15.6 3.67

comment on the estimation of eigenvalues and

3

15.1 1.91

4.29 1.47 time EOFs in SSA. The same applies, by

2

extension, to the eigenvalues and space-time EOFs

in MSSA.

1

SSA is applied to a ¬nite sample of observations

{x1 , . . . , xn } with n m by ¬rst forming Y-

0

vectors,

-1 y1 = (x1 , . . . , xm )T

y2 = (x2 , . . . , xm+1 )T

2 4 6 8 10 12

.

.

.

Figure 13.13: First six time EOFs of an AR(2)

= (xn’m+1 , . . . , xn )T .

process with a = (0.9, ’0.8) obtained using yn’m+1

window length m = 12. The patterns are

normalized with the square root of the eigenvalue. Conventional EOF analysis is applied to the

resulting sample of n ’ m + 1 Y-vectors. The

The eigenvalues are given at the bottom.

estimated eigenvalues and EOFs can be computed

from either the estimated covariance matrix of Y

Multichannel Singular Spectrum (see [13.2.4]) or by means of SVD (see [13.2.8]).

13.6.8

Note that neither North™s Rule-of-Thumb

Analysis. MSSA (see Vautard [380]) differs

from SSA only in the dimension of the basic time [13.3.5] nor Lawley™s formulae (13.38, 13.39) can

series, which is now m -dimensional rather than be used to assess the reliability of the estimate

one-dimensional. The derived random vector Yt directly because consecutive realizations of Yt

is therefore mm -dimensional. Thus MSSA is Ex- are auto-correlated (see (13.33)). The effects of

tended EOF analysis [13.1.8] in which m consec- temporal dependence must be accounted for (see

utively observed ¬elds are concatenated together. Section 17.1) when using these tools.

Allen and co-workers [8, 9, 10] discuss the

The number of ¬elds m is usually small compared

with the ¬eld dimension m in EEOF analysis. The problem of discriminating between noisy compo-

opposite, m > m , is often true in MSSA. nents and truly oscillatory modes in detail.

14 Canonical Correlation Analysis

14.0.0 Overview. Just as EOF analysis (Chap- seasonal means and daily standard deviations of

ter 13) is used to study the variability of a random

the daily mean, minimum, maximum, and range

vector X, Canonical Correlation Analysis (CCA) of temperature, precipitation, wind speed, relative

is used to study the correlation structure of a pair

humidity, and relative sunshine duration. The

of random vectors X and Y. large-scale state of the atmosphere was represented

by a vector Y consisting of the near-surface

CCA and EOF analyses share similar objectives

temperature and sea-level pressure (SLP) ¬elds

and similar mathematics. One interpretation of the

¬rst EOF e 1 of X is that XT e 1 is the linear over Europe and the Northeast Atlantic Ocean.

combination of elements of X with the greatest CCA was used to analyse the joint variability of

variance (subject to e 1 = 1). The second EOF X and Y. As noted above, this technique ¬nds pairs