m real eigenvalues and m complex eigenvectors the conventional EOFs e r , then the coef¬cient

e k . The real eigenproblem (16.47) has 2m real ± c of the Hilbert EOF e c is the dot product

eigenvalues »1 , »1 , . . . , »m , »m , where »1 , . . . , »m of the complexi¬ed process X + i XH with the

are the eigenvalues of the complex eigenproblem. conventional real EOF e r :

The corresponding set of 2m eigenvectors is given

± c = X, e r ’ i XH , e r

by

= ±r ’ i±r H

’e I

k k

eR

k , e k : k = 1, . . . , m .

where ±r = X, e r is the conventional EOF

eI R

coef¬cient.

Equation (16.49) can be used to verify that these In summary, Hilbert EOF analysis has no

vectors are orthonormal. advantages over the conventional EOF analysis

when Xt consists of uncorrelated processes. Note

16.3.9 Real Eigenproblems for the Determina- also that neither of these approaches can provide

tion of Hilbert EOFs. We may use the result information that is useful for characterizing the

of the preceding subsection to characterize the temporal correlation of the time series that

frequency domain EOFs as eigenvectors of a real, comprise Xt .

frequency-integrated matrix. Let “x x denote the

integral of the spectral matrix over a frequency 16.3.11 Example: The POP Case. Another

band ω0 ± δω: situation occurs when the two processes are linked

through a lag relationship. A prototype of this

ω0 +δω

“x x = “x x (ω) dω. situation is the bivariate POP case discussed in

ω0 ’δω [11.3.8] and [11.4.10]. We will consider a bivariate

AR(1) process (cf. (11.45)) of the form

The complex m — m matrix “x x corresponds to

the 2m — 2m real matrix ’v

u

Xt = r Xt’1 + Zt

v u

’ Ψx x

Λx x

. (16.50)

Ψx x Λx x

where |r | < 1, u 2 + v 2 = 1, and Zt is a bivariate

where Λx x and Ψx x are the corresponding white noise process with covariance matrix

integrated m — m co-spectrum and quadrature

10

spectrum matrices. The frequency band ω0 ± δω Σzz = σ 2 .

01

could encompass all or part of [0, 1/2].

Equation (16.37) shows that there is also a real The system generates oscillatory behaviour with

2m — 2m real counterpart to the m — m complex X1t leading X2t when v is positive. Note also that

covariance matrix Σ yy of the complexi¬ed processes X1t and X2t are uncorrelated at lag zero.

process: In fact,

’Σx H x

Σx x σ2

. 10

(16.51) Σx x (0) = .

Σx H x Σx x 1 ’ r2 01

Thus we see that both the Hilbert and frequency 6 But note that Hilbert EOFs may be multiplied by

domain EOFs depend upon the cross-covariances any complex number whereas ordinary EOFs may only be

of the input series and its Hilbert transform. multiplied by real numbers.

16: Complex Eigentechniques

362

Therefore, the conventional EOFs e k of Xt are so that, consistent with (16.41),

degenerate; speci¬c choices of e k are the two unit H

Im(±1 e 1 ) = Re(±1 e 1 ).

vectors (0, 1)T and (1, 0)T (cf. [13.1.9]).7 (16.57)

Recalling equation (11.81), we ¬nd that the

For the second EOF we ¬nd

spectral matrix of Xt is

X1t ’ XH

Re(±2 e 2 ) =

11 (ω) 12 (ω)

2t (16.58)

(X1t ’ XH )H

“x x (ω) =

21 (ω) 22 (ω)

2t

and

11 (ω) i 12 (ω)

= .

’i 12 (ω) 11 (ω) (X1t ’ XH )H

Im(±2 e 2 ) = .

2t

’(X1t ’ XH )

The Hilbert EOFs are the solutions of the 2t

eigenproblem Thus, for both ˜signal™ time series, the second

element is the Hilbert transform of the ¬rst.

1

2

“x x (ω) dω e k = »k e k . We showed in [11.4.11] that the speci¬c

(16.52)

system considered here tends to form ˜typical™ xt

0

sequences of the type (11.83),

The eigenvalues are

’1

1 0

1

··· ’ ’ ’

»1 = 02 11 (ω) ’ 12 (ω) dω 0 1 0

(16.53)

1

(16.59)

»2 = 02 11 (ω) + 12 (ω) dω, 0 1 0

’ ’ ’ ’ ···

’1 0 1

and the corresponding eigenvectors are

when v is positive (cf. Figure 15.1). Therefore,

1 1

e1 = and e 2 = . (16.54) since these are oscillatory processes, it is

’i

i

reasonable to interpret the Hilbert transform as

The larger of the two eigenvalues is »1 since 12 a rate of change. Our system tends to generate

is negative for positive v. xH -sequences identical to (16.59) but shifted in

t

Note that the Hilbert EOFs are markedly time by a quarter of a period so that xt =

(1, 0)T and xH = (0, 1)T appear together. (The

different from the conventional EOFs. t

The time coef¬cients ±k of the Hilbert EOFs are ˜change™ XH leads the ˜state™ Xt .) The ˜state™ of the

t

second component equals the ˜change™ of the ¬rst

±1 = (Xt + i XH )† e 1

t component, and the ˜state™ of the ¬rst component

= X1t + XH + i (X2t ’ XH ) is the reversed ˜change™ of the second.

2t 1t

= X1t + X2t ’ i (X2t + X1t )

H HH The two ˜signals™ represented by the two

(16.55)

Hilbert EOFs, (16.56) and (16.58), may then be

which is consistent with equations (16.40) and characterized by sequences of the type (16.59) as

(16.41). Similarly well. The sequence (16.59) implies

±2 = X1t ’ XH + i (X2t ’ XH )H . XH ≈ X1 (16.60)

2t 1t 2

The ˜signal™ represented by the ¬rst Hilbert EOF so that

is

X1

Re(±1 e 1 ) ≈ 2 (16.61)

±1 e = Re(±1 e 1) + i 1)

1

XH

Im(±1 e

1

where and

X1t + XH 0

Re(±1 e 1 ) = 2t

Re(±2 e 2 ) ≈ .

(16.56) (16.62)

(X1t + XH )H 0