namical systems with different degrees of com-

governed by the same dynamics as the full system,

plexity have been presented by Achatz and col-

is satis¬ed when the noise is turned off.

leagues [1, 2], Kwasniok [236, 237], and Sel-

15.5.4 Principal Interaction Patterns. Since ten [345, 344].

k ¤ m, the time coef¬cients Zt, j of a pattern p j

at a time t are not uniquely determined by Xt . 15.5.5 POPs as Simpli¬ed PIPs. The Principal

Thus the time coef¬cients are determined by least Oscillation Patterns can be understood as a kind

squares as of simpli¬ed Principal Interaction Patterns. For

that assume m = n. Then, the patterns P span

Zt = (P T P)’1 P T Xt . (15.33)

the full X-space, and their choice does not affect

When ¬tting the state space model from equa- [P; ±]. Also, let F be a linear model F[Zt , ±] =

tions (15.31) and (15.32) to a time series, the AZt , where the parameters ± are the entries of A.

following must be speci¬ed: the class of models Then the dynamical equation (15.31) is identical

F, the patterns P, the free parameters ± and the to (15.10). The constraint mentioned above results

dimension of the reduced system k. The class of in PIPs (of the admittedly simpli¬ed state space

models F, must be selected a priori on the basis model) that are given by the eigenvectors of A.

14 See, for example, Priestley [323, Section 10.4.4].

This Page Intentionally Left Blank

16 Complex Eigentechniques

16.1 Introduction but this usage is ambiguous since it also applies

to the eigenvectors of any general complex vector

16.1.1 Modelling the State and the ˜Momen- process. Similar ambiguity occurs when the POPs

tum.™ The purpose of EOF analysis (Chapter 13) is of the complexi¬ed process are called ˜Complex

simply to identify patterns that ef¬ciently charac- POPs™ or ˜CPOPs™ (see, e.g., B¨ rger [75]).

u

terize variations in the current ˜state™ or ˜location™ Therefore, for conceptual clarity, we revive a

of a vector ¬eld X. Consequently, the technique suggestion ¬rst made by Rasmusson et al. [329]

completely ignores the time evolution of the anal- in 1981; we refer to the EOFs of the complexi¬ed

ysed ¬eld. process as Hilbert EOFs, and to the corresponding

POPs as Hilbert POPs.1

POP analysis (Chapter 15) accounts for patterns

that evolve in time by representing the observed

¬eld as a vector AR(1) process, so that information 16.1.3 Outlook. The Hilbert transform is

about the present state is transferred to the next introduced in Section 16.2 and we de¬ne the

state. Such a system can describe oscillatory Hilbert EOFs in Section 16.3, where we also deal

behaviour since any m-dimensional system of brie¬‚y with Hilbert POPs.

¬rst-order difference equations is equivalent to one Canonical Correlation Analysis, rotated EOFs,

mth order difference equation. redundancy analysis, and other pattern analysis

A generalization of this approach is to model techniques can all be extended to complexi¬ed

not only the ˜state™ Xt but also an indicator processes. Attempts in this respect are currently

of its tendency δ Xt (Wallace and Dickinson underway, but no applications seem to have been

[408]). Such an approach is related to the published in the geophysical literature so far.2

Hamiltonian principle in mechanics that the future

of a system is described by a set of ¬rst-order

16.2 Hilbert Transform

differential equations for the location (state) and

the momentum.

16.2.1 Motivation and Heuristic Introduction.

The Hilbert transform XH (see Section 16.2) is

t

If X t is a real time series with Fourier

a reasonable measure of ˜momentum™ δ Xt when

decomposition

variations in Xt are con¬ned to a relatively narrow

ζ (ω)e’2πi ωt

time scale. Then the conventional eigentechniques, Xt = (16.1)

such as EOFs and POPs, are applied to the ω

complexi¬ed time series Xt + i XH .t

then its Hilbert transform is

ζ H (ω)e’2πi ωt

X tH =

16.1.2 Confusing Names. There is some (16.2)

ω

confusion in the literature about what to call the

EOFs or POPs of the complexi¬ed process.

where ζ H (ω) is de¬ned to be

The EOFs of the complexi¬ed process are

i ζ (ω) for ω ¤ 0

sometimes called ˜frequency domain EOFs™ or

ζ H (ω) = (16.3)

’i ζ (ω) for ω > 0.

˜FDEOFs™, since they may be understood as

eigenvectors of the cross-spectral matrix averaged

over some frequency interval (see below). When The Hilbert transform X H is identical to original

applied to narrowly band-pass ¬ltered data this time series X t except for a π/2 phase-shift of ξ

name makes sense, but the technique may also be 1 Rasmusson et al. [329] used the expression ˜Hilbert

used for broad-band features. Singular Decomposition™ (HSD).

The term ˜complex EOFs™ or ˜CEOFs™ is also 2 Brillinger [66] deals with the CCA of complexi¬ed

sometimes used to refer to the EOFs of Xt + i XH , processes, and Horel [181] discusses rotated Hilbert EOFs.

t

353

16: Complex Eigentechniques

354

time series Xt and its Hilbert transform XH into a

t

new complex vector time series

Yt = Xt + i XH . (16.6)

t

Conventional techniques, such as EOFs or POPs

(see Sections 16.3 and 16.2) are then applied to

these ˜complexi¬ed™ time series (16.6).

We complete this section by introducing the

Hilbert transform in mathematically rigorous

terms and describing its estimation. The ˜Hilbert

EOFs™ and ˜Hilbert POPs™ will be introduced in

Figure 16.1: A schematic illustration of the effect Section 16.3 and the former will be discussed in

of the Hilbert transform. The solid curves depict terms of examples.

the input time series, and the dashed curves depict

the corresponding Hilbert transforms. After Horel 16.2.2 Derivation of the Hilbert Transform.

[181, Fig. 1, p. 1662]. The motivation behind the Hilbert EOF and POP

analysis is the creation of a process XH that is

t

something like ˜momentum™. Physical arguments

that is performed separately at each frequency ω. tell us that the ˜momentum™ process XH should be

t

For instance, if related to the original process through a linear ¬lter

operator, that is,

X t = 2 cos(2π ω0 t) (16.4)

∞

= h δ Xt+δ .

XH

for some ¬xed ω0 , then ζ (±ω0 ) = 1, ζ H (±ω0 ) = (16.7)

t

δ=’∞

i, and

Also it should be out-of-phase by π/2 for all

= ’2 sin(2π ω0 t).

X tH (16.5) frequencies ω with the ˜change™ XH leading the

t

˜state™ Xt , that is,

That is, the Hilbert transform shifts X t a quarter of

x Hx (ω) = π/2 for ω > 0. (16.8)

a period to the right. Another interpretation, in this

H provides information about the

example, is that X t

To construct the ¬lter (16.7) we note that

rate of change of X t at time t. H

To illustrate, Figure 16.1 depicts two idealized the cross-spectrum (11.74) between Xt and Xt

input time series and their Hilbert transforms. satis¬es

If the input is monochromatic, the transform

x H x (ω) = H (ω) x x (ω). (16.9)

produces the same output, only advanced by a

quarter period. When the input is not monochro- Since the autospectrum x x is real, the phase

matic, there is a quarter period advance at ev- spectrum satis¬es (16.8) if and only if H (ω) is

ery frequency, with the result that the Hilbert imaginary and anti-symmetric, with a negative

transform can appear to be quite different from imaginary component for positive frequencies, as