use of rotation, namely i) the arbitrary choice of

the subject of rotation. Part of the community

the rotation criterion, ii) the sensitivity of the result

advocates the use of rotation fervently, arguing that

to the normalization of the EOFs (see [13.5.3]),

it is a means with which to diagnose physically

iii) the need to redo the entire calculation if the

meaningful, statistically stable patterns from data.

number of EOFs is changed (see [13.5.4]), and iv)

Several arguments are raised in favour of the

the loss of information about the dominant sources

rotated EOFs.

of variation in the data.

• The technique produces compact patterns that

can be used for ˜regionalization,™ that is,

to divide an area in a limited number of 13.5.3 The Mathematics of ˜Rotation.™ ˜Rota-

tion™ consists of the transformation of a set of

homogeneous sub-areas.

˜input vectors™ P = ( p 1 | · · · | p K ) into another

• Rotated EOFs are less sensitive to the set of vectors Q = (q 1 | · · · |q K ) by means of an

distribution of observing locations than invertible K — K matrix R = (ri j ):

conventional EOFs.

Q = PR (13.46)

• Rotated EOFs are often statistically more i

stable then conventional EOFs (see, e.g., or, for each vector q :

Cheng, Nitsche, and Wallace [82]). That is, K

the sampling variance of rotated EOFs is q i = ri j p j . (13.47)

often less than that of the input vectors. j=1

The matrix R is chosen from a class of matrices,

Others in the scienti¬c community are less

such as orthonormal matrices, subject to the

convinced because of the heuristic arguments that

13: Empirical Orthogonal Functions

308

constraint that a functional V (Q) is minimized. An re-normalized as in equations (13.21) and (13.22)

+

so that p j = e j and Var ± + = 1, then the

example of such a functional is described in the

j

next subsection. rotated patterns are no longer orthogonal but the

Under some conditions, operation (13.46) can coef¬cients remain pairwise uncorrelated.

be viewed as a rotation of the ˜input vectors.™ Since Thus two important conclusions may be drawn.

these are often the ¬rst K EOFs, the resulting

vectors q i are called ˜rotated EOFs.™ • The result of the rotation exercise depends on

When matrix R is orthonormal, the operation is the lengths of the input vectors. Differently

said to be an ˜orthonormal rotation™; otherwise it is scaled but directionally identical sets of

said to be ˜oblique.™ input vectors lead to sets of rotated patterns

Now let X be a random vector that takes values that are directionally different from one

in the space spanned by the input vectors. That is another. Jolliffe [199] demonstrates that the

differences can be large.

X = P± (13.48)

The rotated vectors are a function of the input

where ± is a k-dimensional vector of random vectors rather than the space spanned by the

expansion coef¬cients. Then, because of operation input vectors.

(13.46)

• After rotating EOF patterns, the new patterns

’1

X = (PR)(R ±) = Qβ (13.49) and coef¬cients are not orthogonal and

uncorrelated at the same time. When the

where β = R’1 ± is the k-dimensional vector coef¬cients are uncorrelated, the patterns are

of random expansion coef¬cients for the rotated not orthogonal, and vice versa. Thus, the

patterns. percentage of variance represented by the

Let us assume for the following that the matrix individual patterns is no longer additive.

R is orthonormal so that β = RT ±.23

• When the input vectors are orthogonal, the 13.5.4 The ˜Varimax™ Method. ˜Varimax™ is a

scalar products between all possible pairs of widely used orthonormal rotation that minimizes

the ˜simplicity™ functional

rotated vectors are given by the matrix

K

QT Q = RT P T PR = RT DR, (13.50) 1, . . . , q K ) = f V (q i )

V (q (13.52)

i=1

1T p 1, . . . , K T p K ).

where D = ( p p Thus i

the rotated vectors are orthogonal only if D = where q is given by equation (13.47) and f V is

I, or, in other words, if the input vectors are de¬ned by

normalized to unit length.

1 m qi 4 m

qi 2

1

f V (q ) = ’2 .

• Similarly, if the expansion coef¬cients of m i=1 si m i=1 si

the input vectors are pairwise uncorrelated, (13.53)

so that Σ±± = diag(σ1 , . . . , σ K ), then the

2 2

coef¬cients of the rotated patterns are also The constants si are chosen by the user. The

pairwise uncorrelated only if coef¬cients ± j raw varimax rotation is obtained when si = 1,

i = 1, . . . , K , and the normal varimax rotation

have unit variance. Then

j2

K

is obtained by setting si = j=1 ( pi ) . Another

Σββ = Cov RT ±, RT ± option is to de¬ne si as the standard deviation of

the ith component of

= RT Σ±± R. (13.51)

K

(K )

= ±j p j,

Equations (13.50) and (13.51) imply that rotated X

j=1

patterns derived from normalized EOFs, as de¬ned

in [13.1.2,3] so that p j = e j , are also which is the projection of the original full random

orthonormal, but their time coef¬cients are not vector X onto the subspace spanned by the K

uncorrelated. If, on the other hand, the EOFs are vectors { p 1 . . . p K }.

Note that f V (q ) (13.53) can be viewed as

23 All matrices and vectors in this section are assumed to

be real valued. Thus orthonormal matrices satisfy RRT = the spatial variance of the normalized squares

(qi /si )2 . That is, f V (q ) measures the ˜weighted

RT R = I.

13.5: Rotation of EOFs 309

Figure 13.6: January (left) and July (right) versions of the North Atlantic Oscillation pattern derived by

Barnston and Livezey [27] by applying varimax rotation to the ¬rst 10 normalized EOFs of January and

July mean 700 hPa height, respectively. Courtesy R. Livezey.

square amplitude™ variance of q . Therefore, the North Atlantic Oscillation (NAO, Figure 13.6)

minimizing function (13.52) is equivalent to is evident in every month of the year. Barnston and

¬nding a matrix R such that the sum of the Livezey estimate that it represents between 15.4%

total weighted square amplitude variance of the K (March) and 7.4% (October) of the total variance.

patterns (q 1 | · · · |q k ) = PR is minimized. See The NAO is the dominant circulation pattern in

Richman [331] for further details. the solstitial seasons (DJFM and MJJAS). The

NAO is characterized by a ˜high™ (this adjective is

arbitrary since the sign of the pattern is arbitrary)

13.5.5 Example: Low-frequency Atmospheric

that is centred, roughly, over Greenland and a low

Circulation Patterns. Barnston and Livezey

pressure band to the south. Figure 13.6 displays

[27] argued extensively that rotated EOF analy-

˜typical™ con¬gurations in winter and summer. The

sis is a more effective tool for the analysis of

Greenland centre is located at about 70—¦ N and

atmospheric circulation patterns than the ˜telecon-

40“60—¦ W in winter, and has a zero line at about

nection™ analysis (Wallace and Gutzler [409]; see

50—¦ N. This centre retreats northward in summer

also Section 17.4). They used a varimax rotation

and a second zero line appears at about 30“35—¦ N.

of re-normalized EOFs (13.21, 13.22) to isolate

Another pattern extracted by Barnston and

the dominant circulation patterns in the Northern

Livezey that has been studied by many others