= det( ’ »I) = (»i ’ »). (13.11)

by a fundamental constraint. While it is often

i=1

possible to clearly associate the ¬rst EOF with

a known physical process, this is much more

dif¬cult with the second (and higher-order) EOF 13.1.8 Degeneracy. As noted above, EOFs

because it is constrained to be orthogonal to are not uniquely determined. If »—¦ is a root of

the ¬rst EOF. However, real-world processes multiplicity 1 of p (») and e is a corresponding

do not need to have orthogonal patterns or (normalized) eigenvector, then e is unique up to

uncorrelated indices. In fact, the patterns that most sign, and either e or ’e is chosen as the EOF that

ef¬ciently represent variance do not necessarily corresponds to »—¦ . On the other hand, if »—¦ is a root

have anything to do with the underlying dynamical of multiplicity k, the solution space of

structure. Σe = »—¦ e

13.1.7 Vector Notation. The random vector X is of dimension k. The solution space is uniquely

may conveniently be written in vector notation by determined in the sense that it is orthogonal to

the space spanned by the m ’ k eigenvectors of

(13.8) Σ that correspond to eigenvalues »i = »—¦ . But

X = P±

any orthonormal basis e 1 , . . . , e k for the solution

where P is the m — m matrix space can be used as EOFs. In this case the

(13.9) EOFs are said to be degenerate. (An example is

P = e 1 |e 2 | · · · |e m

discussed in [13.1.9].)

that has EOFs in its columns, and ± is the m- Degeneracy can either be bad or good news. It is

dimensional (column) vector of EOF coef¬cients bad news if the EOFs are estimated from a sample

13.1: De¬nition of Empirical Orthogonal Functions 297

set of orthonormal vectors and associate them with

of iid realizations of X. Then degeneracy is mostly

a set of uncorrelated random variables.

a nuisance, because the patterns, which may

represent independent processes in the underlying The example may also be used to demonstrate

dynamics, can not be disentangled. the phenomenon of degeneracy. To do so, we

assume that all ±s have variance 1. Then, the

However, degeneracy may be good news if

the EOFs are estimates from a realization of a EOFs are degenerate and may be replaced by any

stochastic process Xt . Suppose, for example, that other set of orthonormal vectors. One such set of

p (») has a root of multiplicity 2. By construction, orthonormal vectors are the unit vectors u k with

the cross-correlation of the two corresponding a 1 in the kth row and zeros elsewhere. Then, the

representation (13.8), with P = ( p 1 | · · · | p m ), is

EOF coef¬cient time series will be zero at lag-0.

But this does not imply that the lagged cross- transformed as

correlations will be zero, and, in fact, they are often

X = P ± = PP T (P ±) = U β, (13.14)

nonzero. This means that a pair of EOFs and their

coef¬cient series could represent a signal that is

where the new EOFs are the columns of

propagating in space.

The representation of such a spatially propagat-

U = PP T = I = (u 1 | · · · |u m )

ing signal requires two patterns whose coef¬cients

vary coherently and are 90—¦ out-of-phase. The two and the EOF coef¬cients are given by

patterns representing a propagating signal are not

β = P ±.

uniquely determined; indeed if any two patterns

represent the signal, then any linear combination

These coef¬cients are uncorrelated as well because

of the two do so as well. Therefore, degeneracy is

of Var(±k ) = 1 for all k:

a necessary condition for the description of such

signals. Cov(β, β) = Cov(P ±, P ±)

= P Cov(±, ±)P T

13.1.9 Examples. To demonstrate the mathe-

= PIP T = I.

matics of EOF analysis and the phenomenon of

degeneracy we now consider the case of a random

Obviously, the only meaningful information the

vector

EOF analysis offers in this case is that there is

m no preferred direction in the phase space. The

X= ±k p k (13.12) only property that matters is the uniformity of the

k=1

variance in all directions.

where coef¬cients ±k are uncorrelated real

univariate random variables and p 1 , . . . , p m are 13.1.10 Coordinate Transformations. Let us

¬xed orthonormal vectors. For simplicity we consider two m-variate random vectors X and Z

assume that the ±s have mean zero. Then the that are related to each other by

covariance matrix of X is

Z = LX (13.15)

T

Σ=E ±k p k ±l p l

k l

where L is an invertible matrix so that X = L’1 Z.

T

= Var ± j p j p j . (13.13) Both vectors represent the same information but

j the data are given in different coordinates. The

It is easily veri¬ed that Var(± ) is an eigenvalue of covariance matrix of Z is

k

this covariance matrix with eigenvector p k : Σ Z Z = LΣ X X L† , (13.16)

jT

Var ± j p j p p k = Var(±k ) p k . for L† the conjugate transpose of L. Suppose the

transformation is orthogonal (i.e., L’1 = L† ),

j

Thus, the chosen orthonormal vectors are the EOFs and also let » be an eigenvalue of Σ XX and let

X

of the random vector (13.12). The ordering is e be the corresponding eigenvector. Then, since

determined by the variance of the uncorrelated Σ XX e = »e ,

X X

univariate random variables ±k .

Σ Z Z Le X = LΣ XX L† Le X

The example has two merits. First it may be used

= LΣ XX e X

as a recipe for constructing random vectors with a

= »Le X .

given EOF structure. To do so one has to select a

13: Empirical Orthogonal Functions

298

• The EOFs of some random vectors or random

Thus » is also an eigenvalue of Σ Z Z and the EOFs

functions are given by sets of analytic

of Z are related to those of X through

orthogonal functions. For instance, if the

e Z = Le X . (13.17) covariance structure of a spatial process is

independent of the location, then the EOFs

Thus eigenvectors are transformed (13.17) just as

on a continuous or regularly discretized

a random vector is transformed (13.15).

sphere (circle) are the spherical harmonics

Another consequence of using an orthogonal

(trigonometric functions). See North et

transformation is that the EOF coef¬cients are

al. [296].

invariant. To see this, let P X be the matrix

composed of the X-EOFs e X and let P Z the the • The analysed vector X may be a combina-

corresponding matrix of Z-EOFs. We see from tion of small vectors that are expressed on

equation (13.17) that different scales, such as temperature and pre-

cipitation or geopotential height at 700 and

P Z = LP X . (13.18)

200 hPa. Then the technique is sometimes

called Combined Principal Component Anal-

Using equation (13.10) and transformation

ysis (see, e.g., Bretherton, Smith, and Wal-

(13.15), the vector of X-EOF coef¬cients

lace [64]). Vector X might also consist of

† † †

± X = PX X = PX L† Z = (LP)† Z = PY Z smaller vectors representing a single ¬eld

observed at different times, in which case

= ±Z

the technique is called Extended EOF Anal-

ysis (EEOF; see Weare and Nasstrom [417])

is seen to be equal to the vector of Z-

or Multichannel Singular Spectrum Analysis

EOF coef¬cients. Thus the EOF coef¬cients are

(MSSA; see Section 13.6).

invariant under orthogonal transformations. They