cycles, the progression between warm and cold El Ni˜ o/Southern Oscillation in terms of monthly

n

phases is irregular. SST along the equatorial Paci¬c.

16.3: Complex and Hilbert EOFs 365

2

1

0

1950 1960 1970 1980 1990

3

1

-1

-3

1950 1960 1970 1980 1990

Figure 16.8: As Figure 16.7, except in polar-coordinate form.

a) Amplitude (top). Units: —¦ C. b) Phase (bottom). Units: radians.

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Part VI

Other Topics

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369

Overview

The last part of the book features aspects of applications of statistical concepts that are speci¬c to

climate research and, with the exception of a section on time ¬lters, are usually not found in other ¬elds

of statistical applications.

Chapter 17 contains those aspects that could not be logically included in the more systematically

designed earlier parts of the book. In fact, many concepts in Chapter 17 overlap with material presented

earlier. The so-called decorrelation time is related to the distribution of mean values calculated from

serially correlated data; the potential predictability may be considered a special variant of ANOVA;

teleconnections are a special representation of spatial correlations; associated correlation patterns are an

offspring of regression analysis. We tried, however, to write this chapter such that the material may be

understood without in-depth study of the previous chapters.

Most of the techniques in Chapter 17 were developed by climatologists while struggling with

speci¬c problems; as such, many of them are based on ad-hoc heuristic ideas with interpretations

that may or may not hold in real world situations. We have presented two cases of such heuristically

motived approaches, namely the frequency“wavenumber analysis and the quadrature EOFs. A typical

case to this end is ˜potential predictability.™ We try to clarify the methodical basis of the various

techniques, so that the reader may use them in a more objective manner without using tacitly inadequate

intuitive interpretations (such as the misconception that the decorrelation time is the time between two

independent observations).

Chapter 18 describes a classical problem in meteorology, namely a variety of techniques designed to

measure the relative advantages and disadvantages of (weather) forecasts.

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17 Speci¬c Statistical Concepts in Climate

Research

17.0.0 Overview. In this chapter we review or ˜associated correlation pattern™ analysis and

several additional topics that are important in ˜composite pattern™ or ˜epoch™ analysis.

atmospheric, oceanic or other geo-environmental A popular and simple method for identifying

sciences. These topics are as follows. dynamical links between well-separated areas is

In Section 17.1 we discuss the ˜decorrelation ˜teleconnection™ analysis (Section 17.4), which is

time™, a concept that is often misunderstood, be- essentially the mapping of ¬elds of correlations.

cause of its confusing name. The term suggests Digital ¬lters (Section 17.5) are tools that can be

that it is a physical time scale that represents the used to remove variation on time scales unrelated

interval between consecutive, uncorrelated obser- to the phenomenon under study. This is useful

vations. In fact, it is a statistical measure that because the climate varies on many time scales,

compares the information content of correlated ob- from day-to-day weather variability to the ˜slow™

servations with that of uncorrelated observations. variations connected with the coming and going

If a sample of n uncorrelated observations gives a of the Ice Ages. Depending upon the researcher™s

particular amount of information about the popu- goals, much of the variability in the observed

lation mean then n = n — ˜decorrelation time™ is record may be regarded as ˜noise™ that obscures the

the number of correlated observations required to ˜signal™ of interest. Filters can remove much of this

obtain the same amount of information about the noise.

population mean. Similarly, other ˜decorrelation

times™ can be derived for other parameters such

17.1 The Decorrelation Time

as the population variance or the lag-1 correlation

(cf. Trenberth [368]) by comparing the information

contained about the parameters in samples of 17.1.1 Motivation and De¬nition. We de¬ned

the characteristic time „ M in [10.2.1] as the time

independent and dependent observations. Not only

is the nomenclature confusing, but its meaning is that is required for a system to forget its current

highly dependent upon the parameter of interest. state. This has meaning for some processes, such

as MA(q) processes (cf. [10.5.2]) for which „ M =

We describe a concept called potential pre-

dictability in Section 17.2. Measures of potential q, but not for others, such as AR( p) processes, for

which „ M = ∞.

predictability determine whether the variation in

seasonal mean climate variables is caused by In this section we introduce another ˜character-

istic time,™ labelled „ D . The basic idea originates

anything other than daily weather variations. If

seasonal means have more variance than can be from the observation that the mean of n iid random

variables X1 , . . . , Xn has variance

accounted for by weather noise, then part of the

seasonal mean variance may be predictable from

σX2

slowly varying external sources.

Var X = , (17.1)

Processes, such as El Ni˜ o/Southern Oscillation

n n

or the Madden-and-Julian Oscillation, are often

while the mean of n identically distributed but

described by an index. It is therefore often of

correlated random variables has variance

interest to describe how ¬eld variables, such as

the sea-surface temperature distribution or the

σX2

oceanic ˜meridional overturning stream function™,1 Var X = (17.2)

n

evolve with the indexed process (Section 17.3).

Two techniques are frequently used: ˜regression™

where n = n depends upon the correlations

between X1 , . . . , Xn . We call n the equivalent

1 A measure of the strength of the deep ocean circulation.

371

17: Speci¬c Statistical Concepts

372

n’1

sample size.2 The decorrelation time is then 1

=2 γ (i ’ j) (17.9)

de¬ned as n i, j=0

n

„ D = lim . (17.3) n’1

|k|

1

n’∞ n

= 1’ γ (k).

n n

We will show in [17.1.2] that k=’n+1

n

(17.4) The last expression is obtained by gathering terms

n=

in (17.9) with identical differences i ’ j. Equation

n’1

1+2 1 ’ ρ(k) k

k=1 n

(17.5) follows by taking the limit as n ’ ∞.3

∞

„D = 1 + 2 ρ(k), (17.5)

k=1 17.1.3 Estimation of the Decorrelation Time.

where ρ(·) is the auto-correlation function of Xt . A straightforward way to estimate n is to

The decorrelation time de¬ned in (17.5) substitute the estimated auto-correlation function