of A2 . range that contains most of the variation is not

• To have appropriate units in (11.19) we always readily identi¬ed.

Another alternative is to plot ω (ω) on a log-

multiply by a constant factor carrying the unit

1/δ in the de¬nition of the Fourier transform log scale so that the units on the ordinate are

F so that the spectrum is expressed in units independent of time.

of A2 δ. In any case, it is advisable to clarify the plotting

format and units of a spectrum before making

• The frequency ω is expressed in units of 1/δ.

physical interpretations. Alleged inconsistencies

If, for example, we consider a process that is given are sometimes entirely due to the use of different

in metres, with a time step of months, then the display formats.

11: Parameters of Univariate and Bivariate Time Series

228

11.3 The Cross-covariance Function of the bivariate process satis¬es a mixing

condition such as

11.3.1 De¬nition. Let (Xt , Yt ) represent a pair

∞

of stochastic processes that are jointly weakly

|γab („ )| < ∞

stationary. Then the cross-covariance function γx y

„ =’∞

is given by

for ab = x x, x y, and yy.

γx y („ ) = E (Xt ’ µ X )(Yt+„ ’ µY )—

where µx is the mean of Xt and µ y is the mean of 11.3.3 Some Simple Examples. Let us consider

Yt . a few cases in which Yt is a simple function of a

Note that if Xt = Yt , then the cross-covariance zero mean weakly stationary process X .

t

function is simply the auto-covariance function

γx x . • Suppose Yt is a multiple of Xt ,

The cross-correlation function ρx y is the

Yt = ±Xt .

normalized cross-covariance function (11.28)

γx y („ )

ρx y („ ) = , (11.27) Then, the cross-covariance function

σX σY

where σX and σY are the standard deviations γx y („ ) = ±γx x („ )

√ (11.29)

γx x (0) and γ yy (0) of processes {Xt } and {Yt },

respectively. is proportional to the auto-covariance func-

tion of Xt .

11.3.2 Assumption. We list here the as-

• We make equation (11.28) slightly more

sumptions that are needed to ensure that the

complex by adding some independent white

cross-correlation function exists and is absolutely

noise Zt so that

summable. Speci¬cally, we assume that {(Xt , Yt ) :

t ∈ Z} is an ergodic weakly stationary bivariate

Yt = ±Xt + Zt . (11.30)

process. Hence we have the following results.

• The means µx and µ y are independent of The noise is assumed to be independent

of Xt+„ for all lags „ . Then the auto-

time.

covariance function of Y and the cross-

• The auto-covariance functions γx x and γ yy

covariance function of X and Y are

depend only on the absolute time difference:

± 2 γx x (0) + σ 2 if „ = 0

γ yy („ ) =

E (Xt ’ µx )(Xs ’ µx ) = γx x (|t ’ s|)

± 2 γx x („ ) if „ = 0

E (Yt ’ µ y )(Ys ’ µ y ) = γ yy (|t ’ s|).

γx y („ ) = ±γx x („ ). (11.31)

• The cross-covariance functions γx y and γ yx Thus, the addition of the noise changes

depend only on the time difference: the variance of process Y but not its

auto-covariance or its cross-covariance with

E (Xt ’ µx )(Ys ’ µ y ) = γx y (s ’ t) process X. It does, however, change its auto-

E (Yt ’ µ y )(Xs ’ µx ) = γ yx (s ’ t). correlation and its cross-correlation with X.

• Now suppose Yt is obtained by shifting Xt by

Note that

a ¬xed lag ζ ,

γx y („ ) = E (Xt ’ µx )(Yt+„ ’ µx )

Yt = Xt+ζ . (11.32)

= E (Yt+„ ’ µx )(Xt ’ µx )

= γ yx (’„ ). The resulting cross-covariance function is

a shifted version of the auto-covariance

• The process has limited memory. That is, the function of X,

auto-covariance function

γx y („ ) = E Xt Xt+ζ +„

γx x („ ) γx y („ )

(„ ) = = γx x (ζ + „ ). (11.33)

γ yx (’„ ) γ yy („ )

11.3: The Cross-covariance Function 229

• We could assume that Yt is the discretized Expressions (11.33) and (11.35, 11.36) are

special cases of (11.38, 11.39), which may be

time derivative of Xt , such that

re-expressed as

d

Yt = Xt ’ Xt’1 ≈ Xt (11.34)

γx,F(x) = F (—) {γx x } (11.40)

dt

γ F(x),x = F (r ) {γx x }

Then

γ F(x),F(x) = F (r ) {F (—) {γx x }} (11.41)

γx y („ ) = E(Xt Xt+„ ’ Xt Xt’1+„ )

by de¬ning

= γx x („ ) ’ γx x („ ’ 1) (11.35)

∞

(—) —

{γ }(„ ) = ak γ („ + k)

F

which one might loosely think of as

k=’∞

∞

d

F (r ) {γ }(„ ) = a’k γ („ + k).

γx y ≈ γx x („ ). (11.36)

d„ k=’∞

Similarly,

• Relationships (11.40, 11.41) can be general-

ized to two processes Xt and Yt that are

γ yy („ ) = 2γx x („ ) passed through two linear ¬lters F and G

’ γx x („ ’ 1) + γx x („ + 1) with coef¬cients ak and bk , k = ’∞, ∞,

respectively. Then

d2

≈ ’ 2 γx x .

d„

γ F(x),G(y) = F (r ) {G (—) {γx y }}.

A model such as (11.34) is often appropriate

11.3.4 Properties. We note the following.

for conservative quantities. For example,

the atmospheric angular momentum (Y) 1 The cross-covariance function is ˜Hermitian.™

has time-variability that is determined by That is,

the globally integrated torques (X). Thus,

γ yx („ ) = γx—y (’„ ).

the cross-covariance function between the (11.42)

atmospheric angular momentum and the

torques is of the form (11.36). 2 We have

• The last two examples are special cases of the

|γx y („ )| ¤ γx x (0)γ yy (0). (11.43)

situation in which Yt is a (linearly) ¬ltered

version of Xt . We showed in [11.1.13] that

Therefore |ρx y („ )| ¤ 1, where ρx y („ )

the auto-covariance function of process

is the cross-correlation function de¬ned by

∞ equation (11.27).

F(X)t = ak Xt+k (11.37)

3 The cross-covariance function is bi-linear.

k=’∞

That is,

is