„

Tt = TtS + Tt , realizations of a weakly stationary time series of

(17.15)

length „ .

„

where X indicates averaging over an interval of The ˜inter-chunk™ variability is used to estimate

„

length „ . Thus the „ -mean of Tt is controlled the variance of the „ -means of T, Var Tt . A

by two mechanisms: the slow process T S and

˜chunk-mean™

the integrated fast process T F . In terms of the

temperature example, we would typically set „ to „

1

„

=

tj t jk (17.18)

90 days, interpret T F as the weather noise, and „

assume that T S is the slow variability from the k=1

is ¬rst computed from each chunk, and these,

7 That is, forecasts of the monthly or seasonal mean

in turn, are used to estimate the ˜inter-chunk™

conditions made at leads of up to about a year.

8 The methods described here are appropriate for quantita-

variance

tive values that vary continuously in time, such as near-surface

n

temperature or sea-level pressure. Wang and Zwiers [414] 1

„ „ „2

= t j ’ t—¦

Var Tt

describe methods suitable for use with quantities that vary

n’1

episodically, such as precipitation. j=1

17: Speci¬c Statistical Concepts

376

where 2 ¬t a low-order AR model using the Yule“

Walker method [12.2.2]. The order can be de-

n

1

„ „

= tj . termined either from physical considerations

t—¦

n or by means of an objective criterion such as

j=1

the AIC [12.2.10] or BIC [12.2.11], and

The next step is to understand the properties

of this estimator. Our assumptions about model 3 substitute the auto-correlation function de-

„

rived from the ¬tted AR model into (17.4) to

(17.15) can be used to show that Var Tt is an

obtain „ .

unbiased estimate of the sum of the variances of

the slow process TtS and the integrated fast process

„ The problem with this approach is that the

TtF . That is, distributional properties of

„

„

E Var Tt = Var(TtS ) + Var(TtF ). „ 1

=

Var TtF Var TtF (17.19)

„

If we also make the distributional assumptions that

„ are not well known.

TtS and TtF are normal,9 then we can also show

An alternative approach is based on the

that

observation that

„

(n ’ 1)Var Tt „ 1

Var TtF ≈ F F (0),

„

„

∼ Var(TtS ) + Var(TtF ) χ 2 (n ’ 1).

where F F (ω) is the spectral density function

To test the null hypothesis that potential

of the weather noise. As discussed in [17.1.3],

predictability is absent (i.e., to test H0 : S„ = 1

F F (0) can be estimated by assuming that

or, equivalently, H0 : Var(TtS ) = 0), it is necessary

the spectrum is white near the origin. Thus a

to obtain a statistically independent estimator of „

„ reasonable estimator of Var TtF is

Var Tt . This is done by using the ˜intra-chunk™

„

variations t jk = t jk ’ t j to infer the inter-chunk „ 1 1

=

Var TtF

„ FF

„ „

variance of TtF .

„

Several methods can be used to infer Var TtF F F (1/„ ) is the chunk estimator of

where

F F (1/„ ) (see [12.3.9,10]). An advantage of

from the intra-chunk variations. One approach is

based on the observation that this approach is that the asymptotic distributional

„

„ properties of Var TtF are well known. In fact,

1

= Var Tt

F F

Var Tt

„ asymptotically

where „ is the ˜equivalent chunk length™. Thus „ „

∼ Var TtF χ 2 (2n).

2n Var TtF

„

Var TtF can be estimated as

See Madden [263] and Zwiers [440] for more

„ 1 discussion.

= Var TtF

Var TtF

„

17.2.4 Testing the Null Hypothesis H0 : S„ = 1.

where

„

Now that we have estimates of Var(Tt ) and

„

n

„

1

= (t jk )2

TtF Var(TtF ), we can estimate S„ with

Var

n(„ ’ 1) j=1 k=1

„

Var Tt

S„ =

and „ is estimated by using one of the methods (17.20)

„

TtF

Var

discussed in [17.1.3]. A suitable method is to

1 compute a ˜pooled™ estimate of the intra- and use S „ to test H0 : S„ = 1. The test

is performed at the (1 ’ p) signi¬cance level

˜

chunk auto-correlation function

by rejecting H0 when S „ is greater than the

„ ’l

n

k=1 t jk t j (k+l) appropriate critical value S„,˜ .

p

j=1

ρ(l) = ,

„ The method used to estimate the variance of

n