implicitly assumed to be dimensionless. A proper an AR( p) model to the data and then use

the derived auto-correlation function correlation

dimensional de¬nition would be

function in (17.4). A third approach is based on

„ D = ( t)„ D

˜ (17.6) the observation that

x x (0)

where t is the time increment.

Var X =

Equation (17.5) is the appropriate de¬nition n

of decorrelation time when we use the sample in the limit as n ’ ∞, where

x x (ω) is the

mean to make inferences about the population spectral density function of X . Therefore

t

mean. However, its arbitrariness in de¬ning a

σX 2

characteristic time scale becomes obvious when

we reformulate our problem by replacing the mean n ≈ n

x x (0)

in (17.2) with, for instance, the variance or the

correlation of two processes Xt and Yt . When this and

x x (0)

is done, the appropriate characteristic times are

„D ≈ .

given by (Trenberth [368])

σX 2

∞

(17.7) Thus „ D can also be estimated by estimating the

„ =1+2 ρ 2 (k)

spectral density at frequency zero.4

k=1

Thi´ baux and Zwiers [363] examined various

e

and

approaches and found that the true n values are

∞ dif¬cult to estimate accurately. In particular, the

„ =1+2 ρ X (k)ρY (k) (17.8) ¬rst approach performed very poorly. The second

k=1

approach (see also Zwiers and von Storch [454])

respectively. Thus the de¬nition of the character- is the best of the three methods when samples are

istic time is strongly dependent on the statistical large, and the spectral approach produces better

problem under consideration. In general, these estimates when the samples are moderate to small.

Prior knowledge can sometimes be used to

numbers do not correspond directly to the im-

portant physical time scales of the process under improve the estimate of n . For example, we know

that n < n when the observed process is ˜red™

study.

3 To be precise we must assume that the auto-correlation

17.1.2 Calculation of the „ D . We now prove function is absolutely summable. This is frequently called a

(17.4) by deriving Var X . Without loss of ˜mixing condition™ in the time series literature (see [10.3.0] and

generality we assume that E(Xt ) = 0 so that also texts such as [323] or [66]).

4 The usual approach is to use a good spectral estimator

Var(Xt ) = E X2 . Then, for an arbitrary time t,

t (cf. Section 10.3) to estimate the spectral density at a frequency

near zero and then to extrapolate this estimate to ω =

n’1

1 0. Madden [263] calls this the ˜low-frequency white noise™

2

EX =2 E Xt+i Xt+ j extension of the estimated spectral density. This approach

n works because the spectral density functions of many weakly

i, j=0

stationary ergodic processes are continuous and symmetric

2 See also [6.6.8], where this number comes up in the context about the origin, and therefore have zero slope at the origin.

of testing hypotheses about the mean. These processes are approximately white at long time scales.

17.1: The Decorrelation Time 373

because then x x (0) > σX , as in the top panel

2

of Figure 17.1. Thus it would be reasonable to Red

truncate any estimate n > n to n. Similarly, we

know that n > n when the observed process

is ˜blue™ as in the lower panel of Figure 17.1.5

Knowledge that the process tends to oscillate near

a given frequency in the interior of the frequency

Blue

interval (0, 1/2) is less useful for isolating the

possible range of values for n because we then can

not be sure about whether x x (0)/σX < 1.

2

17.1.4 The Decorrelation Time of AR( p) 0 0.5

Frequency

Processes. The decorrelation times for AR( p)

processes with p = 0, 1 and 2 are easily computed.

• For p = 0, the ˜white noise™ process without Figure 17.1: A schematic illustration of the spectra

any memory, the auto-correlation function of ˜red™ and ˜blue™ noise processes. The horizontal

ρ(k) is zero for nonzero k so that line indicates the variance of the process. The ˜red™

process has n < n and „ D > 1. The ˜blue™ process

„ D = 1. (17.10) has n > n and „ < 1.

D

• For an AR(1) process the decorrelation time

The AR(2) process with ±1 = ±2 = 0.3 is

is

of this type. It has y1 = 1.39, y2 = ’2.39,

∞

a1 = 0.74, and a2 = 0.26. Thus „ D =

„D = 1 + 2 ±1

k

4.53 + 0.11 = 4.64.

k=1

1 + ±1 When (10.11) has complex roots, the auto-

= . (17.11)

1 ’ ±1 correlation function is of the form

The decorrelation time for the two ˜red™ noise 1 + 4 Im(a)2

ρ(k) = cos(kφ + ψ)

examples discussed in Chapters 10 and 11 are

rk

„ D = 1.9 for ±1 = 0.3 and „d = 19 for ±1 =

0.9. Note that lim±1 ’+1 „ D = +∞. That is, where constants a, r, φ and ψ are determined

the decorrelation time becomes in¬nite when as in [11.1.9]. The corresponding decorrela-

the process becomes non-stationary. tion time is

• Recall from [11.1.9] that the auto-correlation ∞

cos(kφ + ψ)

„D = 1 + 2 1 + 4 Im(a)2 .

function of an AR(2) process can take one of

rk

two different forms, depending upon whether k=1

the characteristic polynomial (10.11) has real

The AR(2) process with (±1 , ±2 ) =

or complex roots.

(0.9, ’0.8) has a = 0.5 ’ i 0.032,

When (10.11) has real roots y1 and y2 , the r = 1.12, φ ≈ π/3, and ψ ≈ ’π/50

auto-correlation function is given by so that „ D = 0.33.

’|k| ’|k|

ρ(k) = a1 y1 + a2 y2 ,

17.1.5 The ˜Decorrelation Time™: a ˜Character-

where a1 and a2 are given by (11.7). Then, istic Time Scale?™

since the decorrelation time is linear in ρ, we We now brie¬‚y discuss the extent to which

„ D can be interpreted as a physical time scale in

have

AR( p) processes.

y1 + 1 y2 + 1

„ D = a1 + a2 .

• The decorrelation time (17.10) for an AR(0)

y1 ’ 1 y2 ’ 1

process makes physical sense since these

5 ˜Blue™ noise processes tend to oscillate about the mean

processes are devoid of temporal continuity.

more frequently than white noise processes, so they produce

observations that ˜bracket™ the mean much more quickly than

• Decorrelation time (17.11) has a reasonable

˜white™ or ˜red™ noise processes. Intuitively, then, it makes sense

that n > n. physical interpretation as an indicator of the

17: Speci¬c Statistical Concepts

374