γx y (0) to 1/e. This characteristic time is referred to as the

0

e-folding time.

where

Auto- and cross-covariance functions for two

«2

v ’2uv

2

processes with rotation time 1/· ≈ 20 time

u

2 .

B = I ’r v +2uv

2 2

u units are shown in Figure 11.6. In both processes,

uv ’uv u 2 ’ v2 X leads Y. The functions displayed in the left

panel belong to a process that is only weakly

Since γx x = γ yy and u 2 + v 2 = 1, the solution of damped. Its e-folding time ’1/ ln r ≈ 100 time

equation (11.52) is units. Oscillatory behaviour is clearly apparent on

the 20-unit time scale. A large proportion of the

σz2 10

Σx y (0) = . information that is carried by component X (or Y)

1 ’ r2 01

is returned to that component in approximately 20

To obtain the lagged cross-covariance matrix time steps.

Σx y („ ) by means of equation (11.50), we need to In contrast, the functions displayed in the right

„

calculate the powers A . To do this we let · be the panel belong to a process that is strongly damped.

Its e-folding time ’1/ ln r ≈ 1.4 time units. The

angle for which

peak in γx y („ ) that occurs for „ = 1, 2 indicates

u = cos(2π ·) and v = sin(2π ·),

that the process is attempting to convey some

and then note that information from the leading component X to the

lagging component Y. However, because damping

cos(2π „ ·) ’ sin(2π „ ·)

„ |„ |

A =r . is strong, not enough information is transferred to

sin(2π „ ·) cos(2π „ ·)

initiate oscillatory behaviour.

Then The characteristics of two processes with a

much shorter rotation time of 1/· ≈ 3.2 time units

σz2r |„ | cos(2π „ ·)

γx x („ ) = γ yy („ ) = are shown in Figure 11.7. Again, X leads Y in both

1 ’ r2

processes. The e-folding times for these processes

and are 4.5 and 2.8 time units for left and right panels

2 r |„ | sin(2π „ ·)

σ respectively. The main difference between the two

γx y („ ) = γ yx (’„ ) = ’ z . processes is that the auto- and cross-covariance

1 ’ r2

functions decay more quickly in the right panel.

10 POPs are ˜Principal Oscillation Patterns™ (see Chapter 15).

On average, information transfer in both processes

Pairs of POP time coef¬cients are represented by a bivariate

is suf¬cient for oscillatory behaviour to develop.

AR(1) process with a rotation matrix such as (11.51).

11 Note that except for its sign, v is completely determined by We will revisit these four examples in [11.4.6]

u, and vice versa, i.e., v = ± 1 ’ u 2. when we calculate the spectra of bivariate AR(1)

11: Parameters of Univariate and Bivariate Time Series

232

cross

-40 -20 0 20 40

r=0.5

1.0

auto u=0.95

0.5

cross

0.0

r=0.99

auto u=0.95

0 2 4 6 8 10

0 20 40 60 80 100

Figure 11.6: Auto- and cross-covariance functions γx x (solid) and γx y (dashed) for two bivariate AR(1)

processes with parameter matrix (11.51). The rotation time 1/· is approximately 20 time units. The

e-folding times are approximately 100 time units (left) and 1.4 time units (right). The corresponding

power spectra are shown in Figure 11.10.

0.5 1.0 1.5 2.0

auto auto

2

cross

cross

1

0

r=0.8 r=0.7

-0.5

-1

u=-0.4 u=-0.4

0 5 10 15 20 0 5 10 15 20

Figure 11.7: As Figure 11.6, except the rotation time 1/· is now approximately 3.2 time units. The

e-folding times are approximately 4.5 time units (left) and 2.8 time units (right). The corresponding

power spectra are shown in Figure 11.10.

processes that have rotation matrices as their where Nt = 0 and Mt = Zt+1 . Then,

parameters.

00

ΣZ = ,

0 σz2

11.3.9 Example: Cross-correlation Between an

AR(1) Process and its Driving Noise. The auto- where σz2 = Var(Zt ). Using (11.49), we see that

the covariance matrix Σx y (0) satis¬es

covariance function between an AR(1) process

« «2 « «

γx x (0)

±1 1 2±1

Xt = ±1 Xt’1 + Zt 0

(11.53)

I ’ 0 0 γ yy (0)= σz2

0

γx y (0)