2

positive imaginary part of the cross-spectrum. It is also

sometimes de¬ned as the negative imaginary part. This choice 3 It can be shown that

is arbitrary, but may cause a great deal of confusion in the

de¬nition of the frequency“wavenumber spectra, for example

0 ¤ κx y (ω) ¤ 1.

(see Section 11.5).

11: Parameters of Univariate and Bivariate Time Series

236

6 Since γ yx („ ) = γx y (’„ ) (equation (11.42)),

11.4.3 Properties of the Cross-spectrum of

Real Processes. Let Xt and Yt be a pair of we have

real-valued stochastic processes that are jointly

—

yx (ω) = x y (ω), (11.70)

weakly stationary. Then the following additional

yx (ω) = x y (ω)

properties hold.

yx (ω) = ’ x y (ω)

1 The co-spectrum is the Fourier transform of

A yx (ω) = A x y (ω)

the symmetric part of the cross-covariance

function, γxsy („ ), and the quadrature spectrum yx (ω) = ’ x y (ω)

is the Fourier transform of the anti-symmetric

κ yx (ω) = κ yx (ω).

part of the cross-covariance function, γx y („ ).

a

That is,

Thus it is suf¬cient to consider, and to plot, spectra

(11.68) only for positive ω, if the processes are real.

x y (ω) = γx y (0)

∞

+2 γxsy („ ) cos(2π„ ω) 11.4.4 Some Simple Examples. We described

„ =1 the cross-covariance functions of a number of

∞

simple processes in [11.3.3]. We present the cross-

x y (ω) = ’2 γx y („ ) sin(2π „ ω).

a

spectra of these processes here.

„ =1

• Yt = ±Xt . From (C.7) and because γ±x,x =

The symmetric and anti-symmetric parts of

±γx x (see (11.29)), the cross-spectrum is a

the cross-covariance function are given by

simple function of the spectrum of X:

1

γxsy („ ) = γx y („ ) + γx y (’„ ) x y (ω) =± x x (ω) (11.71)

2

yy (ω) = ± 2 x x (ω)

1

γx y („ ) = γx y („ ) ’ γx y (’„ ) .

a

x y (ω) = ± x x (ω)

2

x y (ω) = 0

2 Therefore, the co-spectrum is symmetric

A x y (ω) = ± x x (ω)

x y (ω) = x y (’ω)

x y (ω) = 0

κx y (ω) = 1.

and the quadrature spectrum is

anti-symmetric

These are intuitively reasonable results.

x y (ω) =’ x y (’ω). All events in the two time series occur

synchronously, thus the phase spectrum is

3 When the cross-covariance function is sym- zero everywhere and the coherency spectrum

metric (i.e., γx y = γ yx ), the quadrature and is one for all ω.

phase spectra are zero for all ω.

• Recall that we also considered the slightly

When the cross-covariance function is anti-

more complex case in which Y is composed

symmetric (i.e., γx y = ’γ yx ), the co-

of a scaled version of X plus white noise Z,

spectrum vanishes and the phase spectrum is

as

π

x y (ω) = ’ 2 sgn( x y (ω)), where sgn(·) is

the sign function.

Yt = ±Xt + Zt .

4 The amplitude spectrum is positive and

Equations (11.30) and (11.31) show that the

symmetric, and the phase spectrum is anti-

cross-, co-, quadrature, amplitude, and phase

symmetric, that is,

spectra are unaffected by the added noise.

A x y (ω) = A x y (’ω) ≥ 0 (11.69) However, the power spectrum of Y, and

x y (ω) = ’ x y (’ω).

therefore the coherency spectrum, do change.

Speci¬cally,

5 It follows from (11.67) and (11.69) that the

yy (ω) = ±2 x x (ω) + σ Z

2

coherency spectrum is symmetric,

± 2 x x (ω)

κx y (ω) = < 1.

κx y (ω) = κx y (’ω). σ Z + ± 2 x x (ω)

2

11.4: The Cross-spectrum 237

The coherency is now less than 1 at all

time scales, indicating that knowledge of the

sequence of the events in X is no longer

enough to completely specify the sequence of

events in Y. The impact of the noise is small

if its variance is small relative to that of ±Xt

(and vice versa).

• When we shifted Xt by a ¬xed lag ζ so that

Yt = Xt+ζ

we found (11.32) that γx y („ ) = γx x (ζ + „ ).

Using (C.8), we ¬nd that

Figure 11.9: Spectra x x and yy of an AR(1)

process Xt with ± = 0.3 and the differenced

ei2π ζ ω x x (ω)

x y (ω) = (11.72) process Yt = Xt ’ Xt’1 . Note that the differencing

yy (ω) = x x (ω) acts as a high-pass ¬lter.

x y (ω) = cos(2π ζ ω) x x (ω)

x y (ω) = sin(2π ζ ω) x x (ω) Thus, again using (C.8),

A x y (ω) = x x (ω)

= (1 ’ e’2πiω )

x y (ω) x x (ω)

x y (ω) = 2π ζ ω (11.73)

κx y (ω) = yy (ω) = 2(1 ’ cos(2π ω)) x x (ω)

1.

x y (ω) = (1 ’ cos(2πω)) x x (ω)

When we shift Xt a ¬xed number of lags we

x y (ω) = sin(2πω) x x (ω)

obtain the same coherency spectrum as when

Xt is simply scaled. It is 1 for all time scales A2 y (ω) = 2(1 ’ cos(2π ω)) x x (ω)

2

x

meaning that the sequence of events in Y is

= x x (ω) yy (ω)

completely determined by X. In contrast, the

phase spectrum has changed from being zero sin(2πω)

= tan’1

x y (ω)

for all ω to a linear function of ω. This type 1 ’ cos(2π ω)

of linear dependency is characteristic of shifts

= tan’1 (cot(π ω))

that are independent of the time scale.

1

= π ω’ ¤ 0 for ω ≥ 0

Note that if the process X lags the process

2