Climate Research

Hans von Storch

Francis W. Zwiers

CAMBRIDGE UNIVERSITY PRESS

Climatology is, to a large degree, the study of

the statistics of our climate. The powerful tools of

mathematical statistics therefore ¬nd wide application

in climatological research, ranging from simple methods

for determining the uncertainty of a climatological mean

to sophisticated techniques which reveal the dynamics of

the climate system.

The purpose of this book is to help the climatologist

understand the basic precepts of the statistician™s art and

to provide some of the background needed to apply

statistical methodology correctly and usefully. The

book is self contained: introductory material, standard

advanced techniques, and the specialized techniques

used speci¬cally by climatologists are all contained

within this one source. There is a wealth of real-

world examples drawn from the climate literature to

demonstrate the need, power and pitfalls of statistical

analysis in climate research.

This book is suitable as a main text for graduate

courses on statistics for climatic, atmospheric and

oceanic science. It will also be valuable as a reference

source for researchers in climatology, meteorology,

atmospheric science, and oceanography.

Hans von Storch is Director of the Institute

of Hydrophysics of the GKSS Research Centre

in Geesthacht, Germany and a Professor at the

Meteorological Institute of the University of Hamburg.

Francis W. Zwiers is Chief of the Canadian Centre

for Climate Modelling and Analysis, Atmospheric

Environment Service, Victoria, Canada, and an Adjunct

Professor of the Department of Mathematics and

Statistics of the University of Victoria.

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Statistical Analysis in Climate Research

Hans von Storch

and Francis W. Zwiers

PUBLISHED BY CAMBRIDGE UNIVERSITY PRESS (VIRTUAL PUBLISHING)

FOR AND ON BEHALF OF THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge CB2 IRP

40 West 20th Street, New York, NY 10011-4211, USA

477 Williamstown Road, Port Melbourne, VIC 3207, Australia

http://www.cambridge.org

© Cambridge University Press 1999

This edition © Cambridge University Press (Virtual Publishing) 2003

First published in printed format 1999

A catalogue record for the original printed book is available

from the British Library and from the Library of Congress

Original ISBN 0 521 45071 3 hardback

Original ISBN 0 521 01230 9 paperback

ISBN 0 511 01018 4 virtual (netLibrary Edition)

Contents

Preface ix

Thanks x

1 Introduction 1

1.1 The Statistical Description 1

1.2 Some Typical Problems and Concepts 2

I Fundamentals 17

2 Probability Theory 19

2.1 Introduction 19

2.2 Probability 20

2.3 Discrete Random Variables 21

2.4 Examples of Discrete Random Variables 23

2.5 Discrete Multivariate Distributions 26

2.6 Continuous Random Variables 29

2.7 Example of Continuous Random Variables 33

2.8 Random Vectors 38

2.9 Extreme Value Distributions 45

3 Distributions of Climate Variables 51

3.1 Atmospheric Variables 52

3.2 Some Other Climate Variables 63

4 Concepts in Statistical Inference 69

4.1 General 69

4.2 Random Samples 74

4.3 Statistics and Sampling Distributions 76

5 Estimation 79

5.1 General 79

5.2 Examples of Estimators 80

5.3 Properties of Estimators 84

5.4 Interval Estimators 90

5.5 Bootstrapping 93

II Con¬rmation and Analysis 95

Overview 97

v

CONTENTS

vi

6 The Statistical Test of a Hypothesis 99

6.1 The Concept of Statistical Tests 99

6.2 The Structure and Terminology of a Test 100

6.3 Monte Carlo Simulation 104

6.4 On Establishing Statistical Signi¬cance 106

6.5 Multivariate Problems 108

6.6 Tests of the Mean 111

6.7 Test of Variances 118

6.8 Field Signi¬cance Tests 121

6.9 Univariate Recurrence Analysis 122

6.10 Multivariate Recurrence Analysis 126

7 Analysis of Atmospheric Circulation Problems 129

7.1 Validating a General Circulation Model 129

7.2 Analysis of a GCM Sensitivity Experiment 131

7.3 Identi¬cation of a Signal in Observed Data 133

7.4 Detecting the ˜CO2 Signal™ 136

III Fitting Statistical Models 141

Overview 143

8 Regression 145

8.1 Introduction 145

8.2 Correlation 146