σ2

σ2 = (11.78)

(1 ’ ± yy ζ )(1 ’ ± yy ζ — )

x x (ω) = 1 + ± 2 ’ 2± cos(2πω)

2

D

σ2

+ b±x y ζ ζ —

2

yy (ω) = b 1 + ± 2 ’ 2± cos(2π ω)

2

D

σ2

= σ 2 b.

= 2 1 + ± 2 + b±x y 2 (11.79)

yy

D

’ 2± yy cos(2π ω) , All other spectra, such as the cross-spectrum and

the coherency spectrum, are zero. The results

σ 2

(ω) = 2 b + b±x x + ± 2

2

(11.78), (11.79) are, of course, identical to (11.22),

yy yx

D

(11.23).

’ 2b±x x cos(2π ω)

Luksch™s second example features two indepen-

dent AR(1) processes with the same parameter,

and that is,

±

σ2 0

± yx ζ — (1 ’ ± yy ζ ) A= ,

x y (ω) = ±

0

D2

+ b(1 ’ ±x x ζ — )±x y ζ

and with noise forcing of equal variance (i.e., b =

σ2 1). Then

= 2 ± yx ζ — + b±x y ζ

D

D 2 = 1 + 4± 2 + ± 4 ’ 4(1 + ± 2 )± cos(2πω)

’ (± yx ± yy + b±x x ±x y )ζ ζ —

+ 2± 2 cos(4πω)

= x y (ω) + i x y (ω),

2

= 1 + ± 2 ’ 2± cos(2πω)

where the co-spectrum and quadrature spectrum

are given by and

σ2

σ2 x x (ω) = 1 + ± 2 ’ 2± cos(2π ω)

x y (ω) = (b±x y + ± yx ) cos(2π ω) D2

D2

σ2

’ (± yx ± yy + b±x x ±x y ) =

1 + ± 2 ’ 2± cos(2πω)

σ2

= yy (ω).

x y (ω) = (b±x y ’ ± yx ) sin(2π ω).

D2

This result is identical to (11.23) since the bivariate

D2

Note that, in all of these expressions, is a process considered here is composed of two

function of ω. independent but identical AR(1) processes.

11: Parameters of Univariate and Bivariate Time Series

240

In practice, however, the oscillation frequency ·

11.4.8 Cross-spectra for the POP Process.

Luksch™s third example is an AR(1) process with is often diagnosed as the frequency with maximum

a rotational parameter matrix and with noise coherency ω0 . For the POP-case we ¬nd

components of equal variance so that b = 1. Recall

ω0 > · for u > 0 (· < 1/4)

that a 2 — 2 rotational parameter matrix has the

ω0 < · for u < 0 (· > 1/4).

form

That is, the coherency maximum underestimates

u ’v

A=r (11.80) the ˜deterministic period™ T when the determinis-

v u

tic frequency is low (i.e., · < 1/4) and it overes-

2 + v 2 = 1 and 0 < r < 1. timates T for high deterministic frequencies. The

where u

When matrix (11.80) is applied to a vector discrepancy between the deterministic period and

a, it rotates that vector through · radians into the frequency of maximum coherency increases as

b = Aa, where cos(2π ·) = u. The rotated the ˜damping™ coef¬cient r decreases. In the limit

vector is returned to its initial direction by applying as r tends to zero, the maximum of the coherency

the matrix (11.80) T = 1/· times. When the spectrum (which is also decreasing in magnitude)

˜damping™ and the noise are switched off (i.e., converges towards 1/4 independently of the value

r = 1 and σ = 0), the system oscillates with of u.

Power and coherency spectra are shown in

period

Figure 11.10 for processes with a number of

T = 2π/ cos’1 (u) combinations of r and u. Coherency spectra

or, equivalently, frequency · = 1/T . Note that (dashed curves) and power spectra (solid curves)

· < 1/4 (and T > 4) when u is positive are displayed for processes with oscillation

and that · > 1/4 (and T < 4) when u is frequencies · = 0.050 (u = 0.95; top row) and

negative. The direction of rotation is determined · = 0.315 (u = ’0.4; bottom row). The location

of the deterministic period · is indicated by the

by v (see [11.4.9]). ’1 ’1

The auto- and cross-covariance functions for vertical bar at · = (2π ) cos (u). The same

this process are given in [11.3.8]. The spectra are examples were discussed in [11.3.8].

Damping is almost absent in the r = 0.99, u =

given by

0.95 case. The power spectrum has a pronounced

σ 2

peak at · = cos(u)/2π and the coherency

x x (ω) = 1 + r 2 ’ 2r u cos(2πω)

spectrum peaks at about the same frequency.

2

D

= yy (ω) Both processes have maximum ˜energy™ and vary

’1

(11.81) coherently at the T = 2π/ cos (u) time scale.

x y (ω) = 0

The second example has the same u and thus

x y (ω) = ’2r vσ sin(2π ω)/D

2 2

has the same ˜period™ as the ¬rst case, namely

A x y (ω) = | x y (ω)| T ≈ 20. However, much more damping occurs

±

with r = 0.5. Neither spectrum has a maximum

’π/2 if v < 0

π/2 if v > 0

x y (ω) = at 2π/T . Instead the power spectrum is red with

unde¬ned if v = 0 a maximum at zero frequency, and the coherency

spectrum peaks, with a very small maximum, at

2

2r v sin(2π ω)

κx y (ω) = , about 0.1. The strong damping almost obliterates

1 + r 2 ’ 2r u cos(2π ω) the connection between the two components of the

process.

where

The lower two panels display spectra for two

D 2 = 1 + 4r 2 u 2 + r 4 ’ 4r u(1 + r 2 ) cos(2π ω) processes with a deterministic time scale T ≈ 3.2

+2r 2 cos(4π ω). (11.82) and slightly different damping rates. We see that

the power spectra are substantially affected by the

The coherency spectrum has a maximum at change in damping between the two processes but

that there are only subtle differences between the

1 2r u

’1

ω0 = .

cos coherency spectra. They show that the components

1 + r2

2π