8.4 Multiple Regression 160

8.5 Model Selection 166

8.6 Some Other Topics 168

9 Analysis of Variance 171

9.1 Introduction 171

9.2 One Way Analysis of Variance 173

9.3 Two Way Analysis of Variance 181

9.4 Two Way ANOVA with Mixed Effects 184

9.5 Tuning a Basin Scale Ocean Model 191

IV Time Series 193

Overview 195

10 Time Series and Stochastic Processes 197

10.1 General Discussion 197

10.2 Basic De¬nitions and Examples 199

10.3 Auto-regressive Processes 203

10.4 Stochastic Climate Models 211

10.5 Moving Average Processes 213

11 Parameters of Univariate and Bivariate Time Series 217

11.1 The Auto-covariance Function 217

11.2 The Spectrum 222

11.3 The Cross-covariance Function 228

11.4 The Cross-spectrum 234

11.5 Frequency“Wavenumber Analysis 241

CONTENTS vii

12 Estimating Covariance Functions and Spectra 251

12.1 Non-parametric Estimation of the Auto-correlation Function 252

12.2 Identifying and Fitting Auto-regressive Models 255

12.3 Estimating the Spectrum 263

12.4 Estimating the Cross-correlation Function 281

12.5 Estimating the Cross-spectrum 282

V Eigen Techniques 289

Overview 291

13 Empirical Orthogonal Functions 293

13.1 De¬nition of Empirical Orthogonal Functions 294

13.2 Estimation of Empirical Orthogonal Functions 299

13.3 Inference 301

13.4 Examples 304

13.5 Rotation of EOFs 305

13.6 Singular Systems Analysis 312

14 Canonical Correlation Analysis 317

14.1 De¬nition of Canonical Correlation Patterns 317

14.2 Estimating Canonical Correlation Patterns 322

14.3 Examples 323

14.4 Redundancy Analysis 327

15 POP Analysis 335

15.1 Principal Oscillation Patterns 335

15.2 Examples 339

15.3 POPs as a Predictive Tool 345

15.4 Cyclo-stationary POP Analysis 346

15.5 State Space Models 350

16 Complex Eigentechniques 353

16.1 Introduction 353

16.2 Hilbert Transform 353

16.3 Complex and Hilbert EOFs 357

VI Other Topics 367

Overview 369

17 Speci¬c Statistical Concepts in Climate Research 371

17.1 The Decorrelation Time 371

17.2 Potential Predictability 374

17.3 Composites and Associated Correlation Patterns 378

17.4 Teleconnections 382

17.5 Time Filters 384

18 Forecast Quality Evaluation 391

18.1 The Skill of Categorical Forecasts 392

18.2 The Skill of Quantitative Forecasts 395

18.3 The Murphy“Epstein Decomposition 399

18.4 Issues in the Evaluation of Forecast Skill 402

18.5 Cross-validation 405

CONTENTS

viii

VII Appendices 407

A Notation 409

B Elements of Linear Analysis 413

C Fourier Analysis and Fourier Transform 416

D Normal Density and Cumulative Distribution Function 419

E The χ 2 Distribution 421

F Student™s t Distribution 423

G The F Distribution 424

H Table-Look-Up Test 431

I Critical Values for the Mann“Whitney Test 437

J Quantiles of the Squared-ranks Test Statistic 443

K Quantiles of the Spearman Rank Correlation Coef¬cient 446

L Correlations and Probability Statements 447

M Some Proofs of Theorems and Equations 451

References 455

Preface

• The concept of the statistical model. Such a

The tools of mathematical statistics ¬nd wide

application in climatological research. Indeed, model is implicit in every statistical analysis

climatology is, to a large degree, the study of the technique and has substantial implications for

statistics of our climate. Mathematical statistics the conclusions drawn from the analysis.

provides powerful tools which are invaluable for

• The differences between parametric and non-

this pursuit. Applications range from simple uses

parametric approaches to statistical analysis.

of sampling distributions to provide estimates

• The estimation of ˜parameters™ that describe

of the uncertainty of a climatological mean to

sophisticated statistical methodologies that form the properties of the geophysical process

the basis of diagnostic calculations designed being studied. Examples of these ˜parame-

to reveal the dynamics of the climate system. ters™ include means and variances, temporal

However, even the simplest of statistical tools and spatial power spectra, correlation coef-

has limitations and pitfalls that may cause the ¬cients, empirical orthogonal functions and

climatologist to draw false conclusions from Principal Oscillation Patterns. The concept of

valid data if the tools are used inappropriately parameter estimation includes not only point

and without a proper understanding of their estimation (estimation of the speci¬c value

conceptual foundations. The purpose of this of a parameter) but also interval estimation

book is to help the climatologist understand which account for uncertainty.

the basic precepts of the statistician™s art and

• The concepts of hypothesis testing, signi¬-

to provide some of the background needed

cance, and power.

to apply statistical methodology correctly and

usefully. We do not deal with:

We do not claim that this volume is in any

• Bayesian statistics, which is philosophically

way an exhaustive or comprehensive guide to the

quite different from the more common