“x x = F {Σx x }. 2

(16.26) Σx H x = ’Σx x H = 2 Ψx x (ω) dω. (16.30)

0

The complex m — m matrix “x x is called the

spectral matrix. The element in the jth row and

16.3 Complex and Hilbert EOFs

lth column is the cross-spectrum x j xl between the

jth and the lth components of X. Thus the matrix 16.3.1 Outline. EOFs were de¬ned in Chap-

is Hermitian, that is, “† x (ω) = “x x (ω), and its

x ter 13 not only for real vectors X but also for

main diagonal contains the autospectra xk xk (ω). complex random vectors Y (although we showed

The lag covariance matrix can be recovered examples only for real vectors).3 In this section

from the spectral matrix by inverting the Fourier we introduce the Hilbert EOFs that are a special

transform. Thus case of complex EOFs, that is, EOFs derived

1

2

“x x (ω)e2πi„ ω dω

3 Here we use Y to denote complex vectors and reserve X

Σx x („ ) = (16.27)

for real vectors.

’1

2

16: Complex Eigentechniques

358

where ξ is an arbitrary angle. Thus the angles φ k

from complex random vectors. We ¬rst review the j

concept of complex EOFs in [16.3.2“4]. may be expressed relative to any a priori speci¬ed

angle.

The straightforward way to de¬ne Hilbert EOFs

is to complexify a random vector by adding This ambiguity with respect to the angle

its Hilbert transform as an arti¬cial imaginary of complex EOFs may be used to rotate the

component. Then the Hilbert EOFs are simply the EOFs in the complex domain so that either the

complex EOFs of this complexi¬ed random vector. imaginary and the real parts of each EOF are

kT

orthogonal (i.e., e R e Ik = 0), or the real and

This is discussed in [16.3.6]. The direct approach

is useful when most of the variability is con¬ned imaginary components of the EOF coef¬cients are

uncorrelated (i.e., Cov(Re(±k ), Im(±k )) = 0).

to a relatively narrow frequency. When this is not

the case, the approach described in [16.3.7] may The complex EOF coef¬cient may be written in

be useful. It involves computing eigenvectors from polar coordinates as

the spectral matrix after it has been averaged over

±k (t) = ak (t) exp iψk (t)

a frequency band. Some computational aspects of (16.35)

complex EOF analysis are explored in [16.3.8,9]

The part of the ¬eld or signal that is represented by

and examples are presented in [16.3.10,11]. Their

the kth EOF at time t is given by

interpretation and estimation is brie¬‚y considered

in [16.3.12,13] and a further example is presented

±k (t)e k = ak (t) Ak exp (iψk (t) + φ k )

in [16.3.15].

where Ak is the vector of amplitudes

(Ak , . . . , Ak )T and φ k is the corresponding

16.3.2 Reminder: Complex EOFs. We know

m

1

from the conventional EOF analysis, (Chapter 13) vector of angles (φ1 , . . . , φm )T . Thus the spatial

k k

that the eigenvectors e k of the covariance matrix distribution of a signal ±k (t)e k at a given time t

Σ yy of a complex random vector Y form a basis is obtained by rotating the elements of vector e k

such that Yt can be expanded as through a common angle ψk (t) and scaling the

elements with a common factor ak (t).

Yt = ±k (t)e k (16.31)

The eigenvalues obtained in an EOF analysis

k

indicate the variance of the input vector that is

with the ˜principal components™ carried by the corresponding principal component

(EOF coef¬cient). This statement is also valid for

±k (t) = Yt , e k = Y† e k . (16.32)

t complex input vectors Y. However, no general

statement can be made about the amount of

The basis is ˜optimal™ in the sense that, for every

K = 1, . . . , m, the expected error variance that is represented by just the real or

imaginary part of the principal component.4

K

We present an example of a complex EOF

2

= Yt ’ ±k (t)e k

K analysis in the next subsection.

k=1

K

= Var(Y) ’ »k (16.33) 16.3.3 An Example of a Complex EOF

Analysis: An Analysis of Velocities and Wind

k=1

Stress Currents at a Coastal Mooring. Several

is smaller for the EOFs than for any other basis.

moored sensors were used in an observational

The complex EOFs may be displayed as

campaign to measure surface variables such as

a pair of patterns, representing the real and

wind stress and sub-surface variables in the Santa

imaginary components e R and e Ik . An alternative

k

Barbara channel of the coast of California. The

representation uses polar coordinates:

observational campaign extended over 60 days,

e j = A j exp i φ j

k k k

(16.34) during which velocities were recorded every 7.5

min at ¬ve depths and wind stress was recorded

for each component j = 1, . . . , m of the m- hourly at two neighbouring locations (see Brink

dimensional vector e k . Thus, the kth complex and Muench [67]). Figure 16.5 shows the mooring

EOF may also be plotted as a pattern of two- location, the mean wind stress vectors, and the

dimensional vectors, with vector of Ak and angle mean current vectors. The wind stress is directed

j

φjk plotted at each point in much the same way

4 For example, it is easy to construct a complex random

that we plot the vector wind. Note that complex vector that has a ¬rst complex EOF e 1 such that Var(Re(±1 )) =

eigenvectors are unique only up to a constant e i ξ 0.

16.3: Complex and Hilbert EOFs 359

hand corner. As mentioned above, complex EOFs

have arbitrary base angles. Thus the orientation

of the velocity and wind stress EOFs was chosen

to maximize the correlation (0.62) between the

corresponding EOF coef¬cients.

The ¬rst velocity CEOF consists of a rather

uniform set of anomalies even though the mean

state varies considerably with depth in terms

of speed and direction. The most important

pattern of current variability is characterised by

a maximum current speed anomaly at the surface

and counterclockwise veering with increasing

depth. Thus positive current anomalies near

the surface tend to be associated with weaker

anomalies at depths related to the left of the

near-surface anomaly.

The ¬rst CEOF of the wind stress indicates that

it varies very similarly at the two locations. Current

anomalies near the surface tend to lie to the right

of the wind stress anomalies, and those at greater

depths tend to lie to the left.

Figure 16.5: Mean and ¬rst complex EOF of

16.3.4 Complex EOF Analysis and Propagating

currents (solid arrows; depth in metres given by

Waves. Horel [181] points out that under

numbers) at a mooring in the Santa Barbara

special circumstances, such as waves associated

Channel and wind stress at neighbouring buoys

with out-of-phase zonal and meridional currents,

(labelled S and C). The mean state is the time

propagating oscillating signals may be identi¬ed

average of the currents and the wind stress, and

through a complex EOF analysis by attributing the

the ¬rst complex EOF was calculated separately

zonal current to the real part of a complex vector

for the wind stress and for the currents.