∞

(11.44)

ak al— γx x („ ’ k + l).

γ F(x),F(x) („ ) =

k,l=’∞

for all processes X, Y, and Z.

(11.38)

11.3.5 Example: SST and SLP in the Tropics

Similarly, the cross-covariance function of Xt

and Extratropics. Frankignoul [131] estimated

and F(X)t is

cross-correlation functions for monthly means of

area averaged SST (St ) and turbulent heat ¬‚ux

∞

—

(11.39) (Ht ) for different areas of the Atlantic Ocean.

γx,F(x) („ ) = ak γx x („ + k).

Figure 11.5 shows the cross-correlation functions

k=’∞

11: Parameters of Univariate and Bivariate Time Series

230

The symmetry indicates that Ht and St+„ tend

0.4

to have the same sign for moderate lags „ .

0.3 30oN-26oN

Such behaviour often indicates that both quantities

0.2

are forced by the same external mechanism

0.1

0 or are coupled together by a positive feedback

-0.1 mechanism (see [11.3.11]).

-0.2

-0.3

11.3.6 Bivariate AR(1) Processes: Notation.

-0.4

The next few subsections focus on the bivariate

-20 -16 -12 -8 -4 0 4 8 12 16 20

LAG [month]

auto-regressive processes of ¬rst order. For

0.4

convenience, we represent these processes in

0.3 2oN-2oS

matrix-vector notation as

0.2

X X N

0.1

=A + (11.45)

Yt Y M

0

t’1 t

-0.1

where the coef¬cient matrix A is given by

-0.2

±x x ±x y

-0.3

A= . (11.46)

± yx ± yy

-0.4

-20 -16 -12 -8 -4 0 4 8 12 16 20

LAG [month]

We also use the corresponding component-wise

representation of (11.45)

Xt = ±x x Xt’1 + ±x y Yt’1 + Nt

Figure 11.5: Estimated cross-correlation functions (11.47)

ρhs for Ht , the monthly mean turbulent heat

Yt = ± yx Xt’1 + ± yy Yt’1 + Mt , (11.48)

¬‚ux into the atmosphere, and St , the monthly

where it is convenient.

mean sea-surface temperature (SST), averaged

The two components of the driving noise, N and

over different latitudinal bands in the Atlantic

ocean (top: 26—¦ “30—¦ N, bottom: 2—¦ S“2—¦ N). The M, are assumed to form a bivariate white noise

SST leads for negative lags „ . From Frankignoul process. This means that the lagged covariances

and cross-covariances of Nt and Mt are zero.

[131].

However, it is possible that the components of the

bivariate white noise processes are correlated at

for a sub-tropical belt and the equatorial belt. The zero lag (i.e., γnm (0) = 0).

lagged correlations are small in both cases.

The cross-correlation function in the subtropics 11.3.7 Bivariate AR(1) Process: Cross-

(top panel) is approximately anti-symmetric about covariance Matrix. The variances γ (0),

xx

the origin. The negative cross-correlation at „ = 1 γ (0) of the components of the bivariate process

yy

tells us that, on average, the SST is higher than and their lag zero cross-covariance γ (0)

xy

normal one month after a negative (downward are obtained by solving a 3 — 3 system of

into the ocean) heat ¬‚ux anomaly. Similarly, the linear equations. These equations are derived

positive cross-correlation at „ = ’1 indicates that by squaring equations (11.47) and (11.48),

a positive SST anomaly usually precedes a positive multiplying equations (11.47, 11.48) with each

(upward into the atmosphere) heat ¬‚ux anomaly. other, and taking expectations to obtain

« «

This suggests that there is a typical sequence of

γx x (0) γnn (0)

events of the form

B γ yy (0)= γmm (0) (11.49)

γx y (0) γnm (0)

· · · Ht’1 < 0 ’ St > 0 ’

where

’ Ht+1 > 0 ’ St+2 < 0 · · · «

±x x ±x y 2±x x ±x y

2 2

B = I ’ ± 2

The two quantities apparently interact with each ±2 2± yx ± yy

yx yy

other in such a way that an initial anomaly is ±x x ± yx ±x y ± yy ±x x ± yy +±x y ± yx

damped by means of a negative feedback process

and I is the 3 — 3 identity matrix. The cross-

(see [11.3.11]).

The cross-correlation function of the equatorial covariance matrix at nonzero lag „ ,

γx x („ ) γx y („ )

turbulent heat ¬‚ux and SST (Figure 11.5, bottom)

is more symmetric with a maximum at lag zero. Σx y („ ) = γ yx („ ) γ yy („ ) ,

11.3: The Cross-covariance Function 231

When · is positive (or equivalently, when v is

may be computed recursively as

positive), the cross-covariance γx y („ ) is positive

Σx y („ ) = AΣx y („ ’ 1) for lags 0 < „ < 1/(2·) and negative for lags

= A„ Σx y (0). (11.50) ’1/(2·) < „ < 0. Thus, although the variability

of the processes is uncorrelated at lag zero, the

10 Example. We now consider a correlation becomes positive when X leads Y

11.3.8 A POP t t

bivariate AR(1) process in which the coef¬cient (i.e., „ > 0) and negative when Xt lags Yt (i.e.,

„ < 0).

matrix A (11.46) is a rotation matrix

This interpretation can be veri¬ed by repeatedly

u ’v

A=r (11.51) applying matrix A to vector (1, 0)T . For example,

vu

we see that

with u 2 + v 2 = 1 and 0 ¤ r ¤ 1.11

The noise components N and M are assumed to A„ 1 = r „ 0

0 1

be uncorrelated and of equal variance. That is,

γnm (0) = 0 and γnn (0) = γmm (0) = σz2 . Thus the

after „ = 1/(4·) applications. The information

lag zero covariance matrix for the POP coef¬cients

that was contained in Xt is transferred to Yt+„ .

Xt and Yt , which is obtained by solving (11.49),

Thus, Xt leads Yt for positive vs. Furthermore, we

satis¬es

can interpret 1/· as a rotation ˜period™ and r as a

« « 2

σz

γx x (0) damping rate. Note that „ = ’1/ ln r applications