Statistical Methods in the Atmospheric Sciences (Volume 100) (International Geophysics, Volume 100)

Statistical Methods in the Atmospheric Sciences, Third Edition, explains the latest statistical methods used to describe, analyze, test, and forecast atmospheric data. This revised and expanded text is intended to help students understand and communicate what their data sets have to say, or to make sense of the scientific literature in meteorology, climatology, and related disciplines. In this new edition, what was a single chapter on multivariate statistics has been expanded to a full six chapters on this important topic. Other chapters have also been revised and cover exploratory data analysis, probability distributions, hypothesis testing, statistical weather forecasting, forecast verification, and time series analysis. There is now an expanded treatment of resampling tests and key analysis techniques, an updated discussion on ensemble forecasting, and a detailed chapter on forecast verification. In addition, the book includes new sections on maximum likelihood and on statistical simulation and contains current references to original research. Students will benefit from pedagogical features including worked examples, end-of-chapter exercises with separate solutions, and numerous illustrations and equations. This book will be of interest to researchers and students in the atmospheric sciences, including meteorology, climatology, and other geophysical disciplines. Cover......Page 2 Copyright......Page 3 Preface to the Third Edition......Page 4 Preface to the Second Edition......Page 5 Preface to the First Edition......Page 6 Descriptive and Inferential Statistics......Page 7 Uncertainty about the Atmosphere......Page 8 Basics of CCA......Page 10 Contrasts between Multivariate and Univariate Statistics......Page 455 The Sample Space......Page 11 Frequency Interpretation......Page 12 Some Properties of Probability......Page 13 Domain, Subsets, Complements, and Unions......Page 14 DeMorgan's Laws......Page 15 Conditional Probability......Page 16 Conditional Relative Frequency......Page 17 Persistence as Conditional Probability......Page 18 Combining Conditional Probabilities Using the Law of Total Probability......Page 19 Bayes' Theorem from a Subjective Probability Standpoint......Page 20 Exercises......Page 21 Robustness and Resistance......Page 23 Quantiles......Page 24 Computation of Common Quantiles......Page 25 Inference and the Posterior Distribution......Page 190 Spread......Page 26 Simulating Multivariate Time Series......Page 27 Chapter 11......Page 626 Multivariate Central Limit Theorem......Page 499 Boxplots......Page 29 Construction of a Schematic Plot......Page 31 Histograms......Page 33 Kernel Density Smoothing......Page 34 Kernel Density Estimates for the Guayaquil Temperature Data......Page 36 Cumulative Frequency Distributions......Page 39 The Metropolis-Hastings Algorithm......Page 206 Power Transformations......Page 42 Standardized Anomalies......Page 46 Expressing Climatic Data in Terms of Standardized Anomalies......Page 48 Exploratory Techniques for Paired Data......Page 49 Pearson (Ordinary) Correlation......Page 50 Some Limitations of Linear Correlation......Page 54 Spearman Rank Correlation and Kendall's τ......Page 55 Serial Correlation......Page 57 Autocorrelation Function......Page 59 The Star Plot......Page 60 The Glyph Scatterplot......Page 61 The Correlation Matrix......Page 63 The Scatterplot Matrix......Page 66 Correlation Maps......Page 67 Exercises......Page 70 Parametric versus Empirical Distributions......Page 71 Time-Domain versus Frequency-Domain Approaches......Page 394 Chapter 5......Page 72 Sampling Distributions of the Regression Coefficients......Page 73 Binomial Distribution and the Freezing of Cayuga Lake, I......Page 75 Geometric Distribution......Page 76 Discrimination......Page 308 Negative Binomial Distribution, and the Freezing of Cayuga Lake, III......Page 79 Poisson Distribution......Page 80 Poisson Distribution and Annual U.S. Hurricane Landfalls......Page 81 Maximum Covariance Analysis (MCA)......Page 575 Expected Value of a Random Variable......Page 82 Expected Value of a Function of a Random Variable......Page 83 Expected Value of a Function of a Binomial Random Variable......Page 84 Distribution Functions and Expected Values......Page 85 Gaussian Distributions......Page 87 Evaluating Gaussian Probabilities......Page 90 Gamma Distributions......Page 95 Evaluating Gamma Distribution Probabilities......Page 98 Gamma Distribution in Operational Climatology, I. Reporting Seasonal Outcomes......Page 99 Gamma Distribution in Operational Climatology, II. The Standardized Precipitation Index......Page 101 Beta Distributions......Page 103 Extreme-Value Distributions......Page 105 Ensemble Forecasts......Page 110 Qualitative Assessments of the Goodness of Fit......Page 112 Hedging, and Strictly Proper Scoring Rules......Page 344 Interpolation of the Annual Cycle to Average Daily Values......Page 431 Quantile-Quantile (Q-Q) Plots......Page 115 The Likelihood Function......Page 116 Algorithm for Maximum-Likelihood Estimation of Gamma Distribution Parameters......Page 118 The EM Algorithm......Page 119 Fitting a Mixture of Two Gaussian Distributions with the EM Algorithm......Page 120 Statistical Simulation......Page 122 Uniform Random-Number Generators......Page 123 Generation of Exponential Variates Using Inversion......Page 125 Nonuniform Random-Number Generation by Rejection......Page 126 Simulating from Mixture Distributions and Kernel Density Estimates......Page 128 Simulation from the Kernel Density Estimate in Figure3.8b......Page 129 Exercises......Page 130 Background......Page 132 Four Handy Properties of the MVN......Page 133 Fisher's Linear Discriminant for Multivariate Normal Data......Page 134 One-Sided versus Two-Sided Tests......Page 135 Confidence Intervals: Inverting Hypothesis Tests......Page 136 A Hypothesis Test Involving the Binomial Distribution......Page 137 One-Sample t Test......Page 140 Tests for Differences of Mean under Independence......Page 141 Calculating CCA through Direct Eigendecomposition......Page 143 Exercises......Page 610 Why Truncate the Principal Components?......Page 147 Eigenvalues and Eigenvectors of a Covariance Matrix Using SVD......Page 148 Comparing Gaussian and Gamma Distribution Fits Using the χ2 Test......Page 149 Maximum Covariance Analysis of the January 1987 Temperature Data......Page 576 Filliben Q-Q Correlation Test for Gaussian Distribution......Page 154 Likelihood Ratio Tests......Page 155 Testing for Climate Change Using the Likelihood Ratio Test......Page 156 Nonparametric Tests......Page 157 Classical Nonparametric Tests for Location......Page 158 Evaluation of a Cloud-Seeding Experiment Using the Wilcoxon-Mann-Whitney Test......Page 160 Comparing Thunderstorm Frequencies Using the Signed Rank Test......Page 163 A Set of Classical Statistical Forecast Equations......Page 165 Testing for Climate Change Using the Mann-Kendall Test......Page 166 Introduction to Resampling Tests......Page 167 Permutation Tests......Page 168 Two-Sample Permutation Test for a Complicated Statistic......Page 169 The Bootstrap......Page 171 One-Sample Bootstrap: Confidence Interval for a Complicated Statistic......Page 172 Probability Forecasts for Multiple-Category Events......Page 173 A BCa Confidence Interval: Example 5.11 Revisited......Page 175 Higher Harmonics......Page 433 Full Continuous Forecast Probability Distributions......Page 349 Illustration of the Livezey-Chen Approach to the Multiplicity Problem......Page 178 Field Significance and the False Discovery Rate......Page 179 Field Significance and Spatial Correlation......Page 180 Resampling to Respect Spatial Correlation in Multiple Testing......Page 181 Exercises......Page 184 Background......Page 186 Organization of Data and Basic Notation......Page 187 Iterative Use of Bayes' Theorem......Page 188 Simulation from the Multivariate Normal Distribution......Page 191 Definition of Conjugate Distributions......Page 193 Binomial Data-Generating Process......Page 194 Eigenvalues and Eigenvectors of a Square Matrix......Page 196 Nucleated Agglomerative Clustering......Page 198 Poisson Mean for U.S. Landfalling Hurricanes......Page 200 Gaussian Data-Generating Process......Page 202 Poisson Regression......Page 240 Markov Chain Monte Carlo (MCMC) Methods......Page 205 Gaussian Inference Without a Conjugate Prior Distribution......Page 207 The Gibbs Sampler......Page 209 Hierarchical Bayesian Model for Hurricane Occurrences......Page 210 Exercises......Page 212 Linear Regression......Page 213 Simple Linear Regression......Page 214 Distribution of the Residuals......Page 216 The Analysis of Variance Table......Page 218 Goodness-of-Fit Measures......Page 219 A Simple Linear Regression......Page 222 Fitting a Two-State, First Order Markov Chain......Page 401 Divisive Methods......Page 607 Derived Predictor Variables in Multiple Regression......Page 231 A Multiple Regression with Derived Predictor Variables......Page 233 The Finley Tornado Forecasts......Page 312 Probabilistic Classification with G = 3 Groups......Page 235 Logistic Regression......Page 236 Comparison of REEP and Logistic Regression......Page 238 A Poisson Regression......Page 241 An Overfit Regression......Page 242 Screening Predictors......Page 245 Equation Development Using Forward Selection......Page 246 Stopping Rules......Page 247 Cross Validation......Page 250 Protecting against Overfitting Using Cross Validation......Page 251 Classical Statistical Forecasting......Page 253 Perfect Prog and MOS......Page 255 PCA via SVD......Page 262 Stochastic Dynamical Systems in Phase Space......Page 265 Choosing Initial Ensemble Members......Page 269 Ensemble Average and Ensemble Dispersion......Page 271 Graphical Display of Ensemble Forecast Information......Page 273 Effects of Model Errors......Page 280 Why Ensembles Need Postprocessing......Page 282 Regression Methods......Page 284 Kernel Density (Ensemble ``Dressing ́ ́) Methods......Page 288 The Nature of Subjective Forecasts......Page 290 The Subjective Distribution......Page 291 Central Credible Interval Forecasts......Page 292 Assessing Discrete Probabilities......Page 294 Assessing Continuous Distributions......Page 295 Exercises......Page 296 Forecast Verification......Page 299 Background......Page 393 Chapter 4......Page 621 Example 11.2 Three-Dimensional MVN Distributions as Cucumbers......Page 490 Chapter 6......Page 303 Nonprobabilistic Forecasts for Discrete Predictands......Page 304 Correlation- versus Covariance-Based PCA for Arbitrarily Scaled Variables......Page 521 Accuracy......Page 306 Skill Scores for 2 x 2 Contingency Tables......Page 309 Which Score?......Page 313 Interpretation of Multivariate Statistical Significance......Page 314 Extensions for Multicategory Discrete Predictands......Page 316 A Set of Multicategory Forecasts......Page 317 Gerrity Skill Score for a 3 x 3 Verification Table......Page 320 Nonprobabilistic Forecasts for Continuous Predictands......Page 321 Conditional Quantile Plots......Page 322 Scalar Accuracy Measures......Page 323 Skill Scores......Page 325 Skill of the Temperature Forecasts in Figure8.6......Page 326 The Joint Distribution for Dichotomous Events......Page 327 The Brier Score......Page 329 Algebraic Decomposition of the Brier Score......Page 330 The Reliability Diagram......Page 332 The Discrimination Diagram......Page 338 The Logarithmic, or Ignorance Score......Page 339 The ROC Diagram......Page 340 Two Example ROC Curves......Page 341 Illustration of the Mechanics of the Ranked Probability Score......Page 348 Comparison of CRPS and Ignorance for 2 Gaussian forecast PDFs......Page 351 Central Credible Interval Forecasts......Page 352 General Considerations for Field Forecasts......Page 353 The S1 Score......Page 355 Mean Squared Error......Page 357 Anomaly Correlation......Page 362 Field Verification Based on Spatial Structure......Page 365 Characteristics of a Good Ensemble Forecast......Page 367 The Verification Rank Histogram......Page 369 Minimum Spanning Tree (MST) Histogram......Page 373 Shadowing, and Bounding Boxes......Page 374 Optimal Decision Making and the Cost/Loss Ratio Problem......Page 375 The Value Score......Page 377 Connections with Other Verification Approaches......Page 379 Verification When the Observation is Uncertain......Page 380 Sampling Characteristics of Contingency Table Statistics......Page 381 Inferences for Selected Contingency Table Verification Measures......Page 383 ROC Diagram Sampling Characteristics......Page 384 Confidence and Significance Statements about a ROC Diagram......Page 385 Brier Score and Brier Skill Score Inference......Page 386 Reliability Diagram Sampling Characteristics......Page 387 Resampling Verification Statistics......Page 388 Exercises......Page 389 Markov Chains......Page 395 Two-State, First-Order Markov Chains......Page 396 Test for Independence versus First-Order Serial Dependence......Page 400 Some Applications of Two-State Markov Chains......Page 402 An Operational CCA Forecast System......Page 569 Hotelling's T2......Page 500 Deciding among Alternative Orders of Markov Chains......Page 406 Likelihood Ratio Test for the Order of a Markov Chain......Page 407 First-Order Autoregression......Page 408 A First-Order Autoregression......Page 411 Higher-Order Autoregressions......Page 412 The AR(2) Model......Page 413 Order Selection Criteria......Page 417 Example 9.4 Order Selection among Autoregressive Models......Page 418 The Variance of a Time Average......Page 419 Variances of Time Averages of Different Lengths......Page 420 Autoregressive-Moving Average Models......Page 421 Simulation and Forecasting with Continuous Time-Domain Models......Page 422 Statistical Simulation with an Autoregressive Model......Page 423 Forecasting with an Autoregressive Model......Page 425 Cosine and Sine Functions......Page 426 Representing a Simple Time Series with a Harmonic Function......Page 427 Example 9.8 Transforming a Cosine Wave to Represent an Annual Cycle......Page 429 Estimation of the Amplitude and Phase of a Single Harmonic......Page 430 A More Complicated Annual Cycle......Page 434 The Harmonic Functions as Uncorrelated Regression Predictors......Page 436 The Periodogram, or Fourier Line Spectrum......Page 438 Discrete Fourier Transform of a Small Data Set......Page 439 Another Sample Spectrum......Page 441 Computing Spectra......Page 442 Aliasing......Page 443 The Spectra of Autoregressive Models......Page 445 Smoothing a Sample Spectrum Using an Autoregressive Model......Page 447 Sampling Properties of Spectral Estimates......Page 448 Example 9.15 Statistical Significance of the Largest Spectral Peak Relative to a Red-Noise H0......Page 452 Exercises......Page 453 Multivariate Distance......Page 457 Euclidean Distance......Page 458 Mahalanobis (Statistical) Distance......Page 459 Vectors......Page 460 Matrices......Page 463 Computation of the Covariance and Correlation Matrices......Page 467 Multiple Discriminant Analysis with G = 3 Groups......Page 469 Square Roots of a Symmetric Matrix......Page 475 Square Roots of a Matrix and Its Inverse......Page 476 Singular-Value Decomposition (SVD)......Page 477 Expectations and Other Extensions of Univariate Concepts......Page 478 Partitioning Vectors and Matrices......Page 479 Linear Combinations......Page 481 Mean Vector and Covariance Matrix for a Pair of Linear Combinations......Page 482 Mahalanobis Distance, Revisited......Page 483 Exercises......Page 485 Definition of the MVN......Page 486 Example 11.1 Probability Ellipses for the Bivariate Normal Distribution......Page 488 Assessing Multinormality......Page 491 Example 11.3 Assessing Bivariate Normality for the Canandaigua Temperature Data......Page 493 Simulating Independent MVN Variates......Page 494 Example 11.4 Fitting and Simulating from a Bivariate Autoregression......Page 497 Domain Size Effects: Buell Patterns......Page 531 Simultaneous Confidence Statements......Page 506 Example 11.6 Comparison of Unadjusted Univariate, Bonferroni, and MVN Confidence Regions......Page 508 Example 11.7 Interpreting the New York and Boston Mean January Temperature Differences......Page 511 Discrimination and Classification Using Logistic Regression......Page 512 Definition of PCA......Page 514 PCA in Two Dimensions......Page 517 The Varied Terminology of PCA......Page 522 Fisher's Procedure for More Than Two Groups......Page 587 Connections to the Multivariate Normal Distribution......Page 525 Application of PCA to Geophysical Fields......Page 526 Simultaneous PCA for Multiple Fields......Page 528 Truncation of the Principal Components......Page 533 Rules Based on the Size of the Last Retained Eigenvalue......Page 534 Rules Based on Hypothesis-Testing Ideas......Page 536 Exercises......Page 537 Effective Multiplets......Page 539 The North et al. Rule of Thumb......Page 540 Why Rotate the Eigenvectors?......Page 542 Rotation Mechanics......Page 543 Sensitivity of Orthogonal Rotation to Initial Eigenvector Scaling......Page 546 Direct Extraction of Eigenvalues and Eigenvectors from [S]......Page 549 Singular Spectrum Analysis (SSA): Time-Series PCA......Page 550 SSA for an AR(2) Series......Page 552 Principal-Component Regression......Page 554 The Biplot......Page 555 Exercises......Page 557 Overview......Page 558 Canonical Variates, Canonical Vectors, and Canonical Correlations......Page 559 Some Additional Properties of CCA......Page 560 CCA of the January 1987 Temperature Data......Page 562 CCA Applied to Fields......Page 566 Chapter 9......Page 602 Forecasting with CCA......Page 567 Computational Considerations......Page 571 CCA via SVD......Page 572 The Computations behind Example 13.1......Page 573 Equal Covariance Structure: Fisher's Linear Discriminant......Page 578 Minimizing Expected Cost of Misclassification......Page 584 Unequal Covariances: Quadratic Discrimination......Page 586 Minimizing Expected Cost of Misclassification......Page 590 Probabilistic Classification......Page 591 Forecasting with Discriminant Analysis......Page 592 Alternatives to Classical Discriminant Analysis......Page 594 Discrimination and Classification Using Kernel Density Estimates......Page 595 Exercises......Page 596 Distance Measures and the Distance Matrix......Page 598 Agglomerative Methods Using the Distance Matrix......Page 599 Ward's Minimum Variance Method......Page 601 A Cluster Analysis in Two Dimensions......Page 603 The K-Means Method......Page 609 Appendix A Example Data Sets......Page 612 Appendix B Probability Tables......Page 614 Chapter 7......Page 623 Chapter 8......Page 624 Chapter 14......Page 627 Chapter 15......Page 628 References......Page 629 Index......Page 654 Praise for the First Edition:
"I recommend this book, without hesitation, as either a reference or course text...Wilks' excellent book provides a thorough base in applied statistical methods for atmospheric sciences."--BAMS (Bulletin of the American Meteorological Society)

Fundamentally, statistics is concerned with managing data and making inferences and forecasts in the face of uncertainty. It should not be surprising, therefore, that statistical methods have a key role to play in the atmospheric sciences. It is the uncertainty in atmospheric behavior that continues to move research forward and drive innovations in atmospheric modeling and prediction.

This revised and expanded text explains the latest statistical methods that are being used to describe, analyze, test and forecast atmospheric data. It features numerous worked examples, illustrations, equations, and exercises with separate solutions. Statistical Methods in the Atmospheric Sciences, Second Edition will help advanced students and professionals understand and communicate what their data sets have to say, and make sense of the scientific literature in meteorology, climatology, and related disciplines.

Accessible presentation and explanation of techniques for atmospheric data summarization, analysis, testing and forecasting

Many worked examples

End-of-chapter exercises, with answers provided

"Praise for the First Edition: 'I recommend this book, without hesitation, as either a reference or course text ... Wilks' excellent book provides a thorough base in applied statistical methods for atmospheric sciences.'--BAMS (Bulletin of the American Meteorological Society). Fundamentally, statistics is concerned with managing data and making inferences and forecasts in the face of uncertainty. It should not be surprising, therefore, that statistical methods have a key role to play in the atmospheric sciences. It is the uncertainty in atmospheric behavior that continues to move research forward and drive innovations in atmospheric modeling and prediction. This revised and expanded text explains the latest statistical methods that are being used to describe, analyze, test and forecast atmospheric data. It features numerous worked examples, illustrations, equations, and exercises with separate solutions. Statistical Methods in the Atmospheric Sciences, Second Edition will help advanced students and professionals understand and communicate what their data sets have to say, and make sense of the scientific literature in meteorology, climatology, and related disciplines. Accessible presentation and explanation of techniques for atmospheric data summarization, analysis, testing and forecasting. Many worked examples End-of-chapter exercises, with answers provided"--EBL book details