Var(Xt )

annual cycle of the mean. We apply the expectation

„

= ±1 = 0, (10.16) operator E(·) to (10.17) for all „ to obtain

for all lags „ . Thus „ M = ∞ for an AR(1) p

process. This statement holds for all AR processes. µ„ = ±0,„ + ±k„ µ„ ’k . (10.18)

Thus de¬nition (10.1) is not useful for such k=1

processes. We suggest an alternative de¬nition in

This is a closed linear system since both µ„

Section 17.1.

and ±k„ are periodic in „ with period N . It can

therefore be re-expressed in matrix-vector form

10.3.8 Seasonal AR Processes. The ˜station-

and solved using standard techniques.

ary™ AR( p) process de¬ned by (10.6) can be

Calculation of the seasonal cycle of the variance

easily generalized to seasonal or cyclo-stationary

is more complicated. First, the past states Xt,„ ’1 ,

[10.2.5] AR( p) processes. However, before giving

Xt,„ ’2 , . . . in (10.18) are replaced with linear

a de¬nition we need to establish some notation.

combinations of previous states by recursive

First, we assume that there exists an external

application of (10.17). This recursion yields

deterministic ˜cycle™ that is indexed by time „ =

an in¬nite series (an ˜in¬nite moving average

1, . . . , N . This index may count months within a

process™; see [10.5.2])

year or hours in a day. We then express an arbitrary

time as a pair (t, „ ), where t counts repetitions of ∞

the external cycle, so that (t, „ + N ) ≡ (t + 1, „ ). Xt„ = β0,„ + β j„ Zt,„ ’ j+1 . (10.19)

Then, {Xt„ : t ∈ Z, „ = 1, . . . , N } is said to be a j=1

cyclo-stationary AR( p) process if

The βs are functions of the seasonal AR( p)

1 there are constants ±k„ , k = 0, 1, . . . , p such parameters and the cyclo-stationarity conditions

that ±k,„ +N = ±k„ for all „ and ± p„ = 0 for alluded to above ensure that this sum converges

some „ , in a suitable manner. The noise contributions

10: Time Series and Stochastic Processes

210

Zt„ have zero expectation and are mutually

independent so that

E(Xt„ ) = β0,„ (10.20)

∞

„ ’ j+1

Var(Xt„ ) = β 2 σ Z ,„ ’ j+1 .

j„

j=1

10.3.9 Example: A Seasonal Model of the SST

Index of the Southern Oscillation. A seasonal

AR(2) process can be used to model the SST index

of the Southern Oscillation [453]. A segment of

the full monthly time series is shown in Figure 1.4

Figure 10.13: A 50-year random realization of

(dashed curve). The model was ¬tted to seasonal

the seasonal AR(2) process which models the SST

means so that one ˜seasonal cycle™ comprises N =

index of the SO. Compare with Figure 1.4.

4 time steps, namely FMA, MJJ, ASO and NDJ.

The estimated process parameters ± k„ and the

standard deviation of the driving noise σ Z „ , which

with the largest SST anomalies (NDJ); weakest

¬t the data best, are:

variability occurs in northern summer (MJJ). Note

that the NDJ variance is 2.7 times greater than the

Season „ ± 0,„ ± 1,„ ± 2,„ σ Z ,„ MJJ variance.

FMA 0.39 0.571 0 0.332 A simulated 200 time step realization of the

’0.17 ’0.368

MJJ 1.032 0.374 ¬tted process is displayed in Figure 10.13. The

’0.471

ASO 2.55 1.436 0.362 character of the time series is similar to that of

NDJ 3.56 1.172 0 0.271 the original displayed in Figure 1.4. It resembles

the output of an ordinary AR(2) process with

When we examine the four sub-models for FMA,

frequent occurrences of positive (or negative)

MJJ, ASO, and NDJ separately using (10.13) to

anomalies extending over four and more seasons.

determine whether they satisfy the stationarity

The preference for maxima to occur in NDJ

condition of an AR(2) process, we ¬nd that

distinguishes the ¬tted process from an ordinary

the FMA, MJJ, and ASO processes satisfy the

AR(2) process. A non-seasonal process does not

condition but that the NDJ process lies outside

have a preferred season for generating extremes.

the ˜admissible™ triangle of Figure 10.11. The

This preference is indeed a characteristic feature

transition from NDJ to FMA, with ± 1,FMA =

of the SO.

0.571, is connected with substantial damping. On

the other hand, the step from ASO to NDJ, with

± 1,NDJ = 1.172, is associated with ampli¬cation 10.3.10 Bivariate and Multivariate AR Proces-

ses. The ˜univariate™ de¬nition (10.6) or (10.8)

of the process. Despite this, the full process is

of an AR process can be easily generalized to a

cyclo-stationary.

The estimated annual cycle of the means, µ X „ , multivariate setting. A sequence of -dimensional

random vectors {Xt : t ∈ Z} is said to be a

and standard deviations, σ X „ , derived from the

¬tted model are displayed in the following table:9 multivariate AR( p) process if Xt satis¬es a vector

difference equation of the form

µ X „ (—¦ C) σ X „ (—¦ C)

Season „

p

FMA 0.058 0.621

X t = A0 + Ak Xt’k + Zt (10.21)

MJJ 0.033 0.554

k=1

ASO 0.046 0.743

NDJ 0.091 0.911 for all t where

The overall mean value, as well as the expected 1 A0 is an -dimensional vector of constants,

values for the four seasons, are slightly positive.

2 Ak , for k = 1, . . . , p, are — matrices of

The standard deviation varies strongly with the

constants such that A p = 0, and

season. Maximum variability occurs in the season

9 The estimated means are different from zero because the

3 {Zt : t ∈ Z} is a sequence of iid zero mean

seasonal AR process was ¬tted to anomalies computed relative

-dimensional random vectors.

to a reference period that was shorter than the full record.

10.4: Stochastic Climate Models 211

Bivariate AR( p) processes that describe the Thus (10.22) may be reformulated as

joint evolution of two processes and multivariate

dy

= V — (y) + f — .

AR(1) processes are of particular interest. For (10.23)

dt

example, a multivariate AR(1) process (i.e., Ai =

0 for i ≥ 2) is ¬tted in Principal Oscillation The modi¬ed operator V — includes the effect of av-

Pattern analysis (see Chapter 15). eraging and, in particular, the constant contribution

from the ˜fast™ component x. The modi¬ed forcing

f — represents the slow component of the forcing.

10.4 Stochastic Climate Models Equation (10.23) is a ˜dynamical™ model

because the dynamics are explicitly accounted

10.4.1 Historic Excursion. What are the for by the function V — . It is also called

physical processes that excite slow climate a ˜statistical™ model because the averaging

variations such as the Ice Ages, the Medieval operator has embedded the moments of the

Warm Time, or the Little Ice Age? The noisy component x into function V — . However,

early scienti¬c mainstream opinion was that this nomenclature is somewhat misleading since

such variability stems exclusively from external (10.23) does not contain random components, but

forcings, such as variations in the Earth™s orbital rather describes the deterministic evolution of the

parameters. It was argued that the weather moments of a random variable. Equation (10.23)

¬‚uctuations were irrelevant because their in¬‚uence is fully deterministic and may, at least in principle,