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Statistical Analysis in 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 find 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 specifically 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. 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 The Statistical Description 1.2 Some Typical Problems and Concepts I Fundamentals 17 2 Probability Theory 19 2.1 Introduction 20 2.2 Probability 21 2.3 Discrete Random Variables 23 2.4 Examples of Discrete Random Variables 25 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 3 Distributions of Climate Variables 51 3.1 Atmospheric Variables 52 3.2 Some Other Climate Variables 4 Concepts in Statistical Inference 69 4.1 General 69 4.2 Random Samples 74 4.3 Statistics and Sampling Distributions 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 II Confirmation and Analysis 95 Overview 97 6 The Statistical Test of a Hypothesis 6.1 The Concept of Statistical Tests 6.2 The Structure and Terminology of a Test 6.3 Monte Carlo Simulation 6.4 On Establishing Statistical Significance 6.5 Multivariate Problems 6.6 Tests of the Mean 6.7 Test of Variances 6.8 Field Significance Tests 6.9 Univariate Recurrence Analysis 6.10 Multivariate Recurrence Analysis 7 Analysis of Atmospheric Circulation Problems 7.1 Validating a General Circulation Model 7.2 Analysis of a GCM Sensitivity Experiment 7.3 Identification of a Signal in Observed Data 7.4 Detecting the ‘CO2 Signal’ III Fitting Statistical Models Overview 89 8 Regression 8.1 Introduction 8.2 Correlation 8.3 Fitting and Diagnosing Simple Regression Models 8.4 Multiple Regression 8.5 Model Selection 8.6 Some Other Topics 9 Analysis of Variance 9.1 Introduction 9.2 One Way Analysis of Variance 9.3 Two Way Analysis of Variance 9.4 Two Way ANOVA with Mixed Effects 9.5 Tuning a Basin Scale Ocean Model IV Time Series Overview 107 10 Time Series and Stochastic Processes 10.1 General Discussion 10.2 Basic Definitions and Examples 10.3 Auto-regressive Processes 10.4 Stochastic Climate Models 10.5 Moving Average Processes 11 Parameters of Univariate and Bivariate Time Series 11.1 The Auto-covariance Function 11.2 The Spectrum 11.3 The Cross-covariance Function 11.4 The Cross-spectrum 11.5 Frequency–Wavenumber Analysis Contents 99 100 104 106 108 111 118 121 122 126 137 139 145 145 146 150 160 166 168 V Eigen Techniques Overview 290 13 Empirical Orthogonal Functions 13.1 Definition of Empirical Orthogonal Functions 13.2 Estimation of Empirical Orthogonal Functions 13.3 Inference 13.4 Examples 13.5 Rotation of EOFs 13.6 Singular Systems Analysis 14 Canonical Correlation Analysis 14.1 Definition of Canonical Correlation Patterns 14.2 Estimating Canonical Correlation Patterns 14.3 Examples 14.4 Redundancy Analysis 15 POP Analysis 15.1 Principal Oscillation Patterns 15.2 Examples INDEX 260 bias of, 86 covariance, 83 cross-correlation function, 281 cross-covariance function, 281 cross-spectrum, 284 distribution function, 81, 82 eigenvalues, 316 EOFs, 300 estimator variance, 88 interval, 90 jth moment, 83 L-moment, 84, 86 INDEX false alarm rate, 403 FDEOF, 353 feedback, negative, positive field significance test, 14–15, 99, 121–122, 176 Finley’s tornado forecast, 403 Finley, J.P., 403 First GARP Global Experiment (FGGE), 69 first moment, see mean Fisher’s information, 114 Fisher, R.A., 88, 143 Folland, C., 395, 396 forecast categorical, climatological conditionally unbiased damped persistence persistence POP technique probabilistic quantitative random reference forecast reference tornado unbiased forecast skill annual cycle of skill scores anomaly correlation coefficient artifical conditional bias LEPS score mean squared error Murphy–Epstein decomposition of POP forecast proportion of explained variance unconditional bias forecast verification West Glacier rainfall example forward selection Fourier analysis Fourier coefficients covariance structure of Fourier transform, properties Fraedrich, K., 41, 242, 245, 293 Fram Strait Frankignoul, C., 111, 212, 233 freeboard frequency domain Frequency Domain EOF frequency domain EOF examples Hayashi’s standing wave variance Pratt’s standing wave variance the steps travelling wave variance variance of the waves frequency–wavenumber analysis, 241–242 examples, 245–246 Hayashi’s standing wave variance, 247–249 Pratt’s standing wave variance, 246–247 the steps, 242–243 travelling wave variance, 245–246 variance of the waves, 243–244 frequency–wavenumber spectrum, 244 frequentist statistics versus Bayesian freshwater flux anomalies gappy data Gaussian distribution, see normal distribution General Circulation Model (GCM) and conformationary analysis artifact downscaling the response experiment perpetual mode spin-up period intercomparison sensitivity experiment validation generalized normal equations geopotential height, 3, 32 geostatistics, ix geostrophic wind, 56 global null hypothesis, 108, 109, 121, 122 global test, 109, 121 global warming detecting the greenhouse signal Goodman, N.R., 284 goodness-of-fit, 81 goodness-of-fit statistic, 81 goodness-of-fit test, 81–82 grid point tests, 14 gridded data, 52 guess pattern, 110, 132–133 hierarchies, 111 optimal rotated Gumbel (EV-I) distribution density function return values Gumbel, E.J., 46, 49 Gutzler, D., 60, 383 Gyalistras, D., 318 Hadley cell, 6, 125 Ключевые слова: mx, wave-amplitude indicator, skill, true, tr y, interval, yt, null hypothesis, function, bias, xi, sample, observation, nx ny, xt, blocking, eki, coef?cient, estimate, xk, y y, estimator, variation, lag, space, ei jl, p rt, general, point, complex, time series, fy, spectrum, maximum, process, signi?cance, basic precept, nina, oscillation, z xt, event, probability, case, zu, input vectors, kz, data, noise, component, term, variability, scale, number, sst, atmos, statistical, ph, auto, ?eld, covariance, climate, frequency, ln, assume, difference, model, van andel, ?rst, jk, equation, xz, ha, level, observed, random variable, zl, pr, akxx, cc, north, eof, size, xt x, problem, correlation, approach, zwiers, result, eigenvalue, v vu, statistic, press, zt, p-quantile, y j, change, fx, potsdam germany, anomaly, spectral, wc, ar, length, nino, tx, con?dence, ei j, ak, lagrange multiplier, wa, agree broadly, right, parameter, real, density, pattern, xy, sci, time scale, ai, analysis, drake passage, time, x y, ct st, property, large, signicance level, discussed, xx, wind, signal, xt yt, cks, yy, forecast, fw, day, yi, temperature, xi i, hypothesis, error, greenhouse gas, linear, heads, variance, phase, yi jl, note, wd, independent, df, nin, regression, test, ai yi, coef?cients, meteor, ny, yj, xn, dt nt, fn, dx, wave, tr, height, realization, suburban airport, experiment, vice versa, p xn, rev, var, variable, mixing condition, estimated, xq, series, normal, vector, pop, ei, wea, matrix, eofs, soc, random, null, assumption, sable island, mutually exclusive, wr, set, distribution, sum, xt ha, von storch, ocean, method