κx y (ω) = 1 for ω = 0.

is positive for positive frequencies.18

• We also considered the ¬rst difference Several things can be noted here.

i) The coherency is 1 at all time scales except

Yt = Xt ’ Xt’1 0. This is reasonable since integration can

undo differentiation up to a constant.

that approximates a discretized time deriva- ii) The spectrum of the differenced process

tive. Recall from (11.35) that Y has more short time scale variability than

the spectrum of the original process X.

γx y („ ) = γx x („ ) ’ γx x („ ’ 1) Indeed, differencing acts as a high-pass ¬lter

γ yy („ ) = 2γx x („ ) that dampens long time scale variability and

eliminates the time mean ( yy (0) = 0). For

’ (γx x („ ’ 1) + γx x („ + 1).

example, Figure 11.9 displays the spectrum of

an AR(1) process Xt with ± = 0.3 and that of

18 The de¬nition of the phase is arbitrary to some extent, and

the differenced process Yt = Xt ’ Xt’1 . The

thus some care is needed. We say that Y leads X when certain

˜events™ in Y are followed by similar events in X at a later time

X-spectrum is ˜red™ with a maximum at zero

(i.e., Xt+ζ ≈ Yt ). With this de¬nition, the phase difference

frequency whereas the Y-spectrum is ˜blue™

yx is positive. At the same time X lags Y, Xt ≈ Yt’ζ , and

with a maximum at frequency 1/2.

the phase difference x y is negative.

11: Parameters of Univariate and Bivariate Time Series

238

• Finally, the cross-spectrum of two ¬ltered

iii) ˜Physical reasoning™ suggests that the

processes F(Xt ) and G(Yt ) is given by

forcing should lead the response19 in the

sense that the phase lag yx between the

{b}— .

= F {a} xyF (11.75)

˜forcing™ Y and the ˜response™ X is π/2. This F(x)G(y)

is approximately the case for the long time

scales near ω = 0, since x y (0) = ’π/2. 11.4.5 The Spectrum of a Multivariate AR( p)

Process. The following general representation of

The phase converges towards zero on shorter

the spectrum of a multivariate AR( p) process will

time scales. This effect occurs because the

be useful when describing the spectra of a bivariate

time derivative is only approximated by the

AR process. Let Xt be a weakly stationary -

time difference, and the accuracy of this

dimensional AR( p) process

approximation increases with the time scale.

p

• Now consider again process (11.37) Xt = Ak Xt’k + Zt .

k=1

∞

— spectral matrix x x (ω) of Xt is

The

F(X)t = ak Xt+k .

constructed by placing the power spectra of the

k=’∞

elements of Xt on the diagonal and by placing

which is obtained by passing a weakly cross-spectra in the off-diagonal positions. Note

stationary stochastic process through a linear that for any two elements Xit and X jt of Xt ,

—

xi x j (ω) = x j xi (ω) (see equation (11.70)). Thus

¬lter. The cross-covariance function of the

the matrix function x x is Hermitian. It can be

two processes F(X)t and Xt is (11.39)

shown (see Jenkins and Watts [195, p. 474]) that

∞ ’1

= B(ω)’1 B(ω)—

x x (ω) zz (ω)

— (11.76)

γx,F(x) („ ) = ak γx x („ + k).

k=’∞ where

p

Ak eik2π ω

B(ω) = I ’

The cross-spectra are then (cf. (C.17))

k=1

{a}—

x,F(x) (ω) =F (ω) x x (ω) is the characteristic polynomial of the process

(11.74)

evaluated at ei2π ω , I is the — identity matrix,

F(x),x (ω) = F {a}(ω) x x (ω),

and zz (ω) = Σz is the spectral matrix of the

where F {a}(ω) is the Fourier transform multivariate white noise process that drives Xt .

of the sequence of ¬lter coef¬cients {ak : The ˜—™ is used to denote the conjugate transpose

k ∈ Z}. The examples discussed above and operation. We evaluate (11.76) for a bivariate

in [11.4.3] can all be cast in a linear ¬lter AR(1) process in the next subsection.

format. In particular, note the following.

i) When Yt = ±Xt , the sequence of ¬lter 11.4.6 Cross-spectrum of a Bivariate AR(1)

coef¬cients are a0 = ± and ak = 0 for k = 0. Process. We assume, in the following, that the

Thus F {a}— (ω) = k ak e2πikω = ± for all

— bivariate AR(1) process (Xt , Yt )T (11.45) has been

ω, and hence x y (ω) = ± x x (ω). transformed to coordinates in which the variance

covariance matrix of the driving noise Zt has the

ii) When Yt = ±Xt+ζ , the ¬lter is determined

form

by ak = 0 for all k = ζ , and aζ = 1. The

complex conjugate of the Fourier transform Σz = σ 2 1 0 .

of this series is F {a}— (ω) = e2πi„ ω (cf. 0b

For AR(1) processes, matrix function B(ω) is

(11.72)).

iii) When Y = X ’ X , a = 1, a = given by ’1

t t t’1 0

’1, and ak = 0 for all k = 0, ’1. The B(ω) = I ’ Aζ

complex conjugate of the Fourier transform i2π ω . Thus, from (11.76), we see that

’2πiω (cf. where ζ = e

—

of this ¬lter is F {a} (ω) = 1 ’ e

the spectral matrix of the process is given by

(11.73)).

σ2 10 —

(ω) = 2 I ’ A j ζ I ’ Ajζ

19 The ˜physical™ argument is as follows. Suppose d X/dt = xx

0b

D

Y where Y = A cos(ωt). Then X = A/ω cos(ωt + x y ) where

π

xy = ’ 2 . (11.77)

11.4: The Cross-spectrum 239

where D is the modulus of the determinant of B 11.4.7 Cross-spectra of Some Special AR(1)

(i.e., D = |det(B)|), and A j is the adjoint of the Processes. The spectra described above are

easily computed for a number of special AR(1)

coef¬cient matrix

processes, three of which are described by Luksch,

von Storch, and Hayashi [262]. These models are

± yy ’±x y

Aj = . brie¬‚y described here, and we revisit them in

’± yx ±x x

[11.5.5], [11.5.8], and [11.5.11].

In the ¬rst of Luksch™s examples, the two

After some manipulation, we ¬nd

components of the AR(1) process are not

connected. Also, one process is red, and the other

D = 1 + (±x x + ± yy )

2 2

is white. That is,

+ (±x x ± yy ’ ±x y ± yx )2 ’ 2(±x x + ± yy )

± 0

A= .

— (1 + ±x x ± yy ’ ±x y ± yx ) cos(2πω) 0 0

+ 2(±x x ± yy ’ ±x y ± yx ) cos(4π ω).

Then D 2 = 1 + ± 2 ’ 2± cos(2πω) and the spectra

The spectra are consequently derived from are

equation (11.77) as

x x (ω) = σ /D

2 2