Statistical Methods in the Atmospheric Sciences

Daniel S. Wilks9780127519661, 0127519661

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.* Presents and explains techniques used in atmospheric data summarization, analysis, testing, and forecasting* Chapters feature numerous worked examples and exercises* Model Output Statistic (MOS) includes an introduction to the Kalman filter, an approach that tolerates frequent model changes* Detailed section on forecast verification, including statistical inference, diagrams, and other methods

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Table of contents :
Statistical Methods in the Atmospheric Sciences……Page 4
Contents……Page 6
Preface to the First Edition……Page 16
Preface to the Second Edition……Page 18
Preliminaries……Page 20
Descriptive and Inferential Statistics……Page 22
Uncertainty about the Atmosphere……Page 23
Events……Page 26
The Sample Space……Page 27
The Meaning of Probability……Page 28
Bayesian (Subjective) Interpretation……Page 29
Domain,Subsets,Complements,and Unions……Page 30
Conditional Probability……Page 32
Independence……Page 33
Law of Total Probability……Page 35
Bayes’ Theorem……Page 36
Exercises……Page 37
Univariate Statistics……Page 40
Robustness and Resistance……Page 42
Quantiles……Page 43
Numerical Summary Measures……Page 44
Spread……Page 45
Graphical Summary Techniques……Page 47
Stem-and-Leaf Display……Page 48
Boxplots……Page 49
Schematic Plots……Page 50
Histograms……Page 52
Kernel Density Smoothing……Page 54
Cumulative Frequency Distributions……Page 58
Power Transformations……Page 61
Standardized Anomalies……Page 66
Scatterplots……Page 68
Pearson (Ordinary)Correlation……Page 69
Spearman Rank Correlation and Kendall ’s tou……Page 74
Serial Correlation……Page 76
Autocorrelation Function……Page 77
The Star Plot……Page 78
The Glyph Scatterplot……Page 79
The Rotating Scatterplot……Page 81
The Correlation Matrix……Page 82
The Scatterplot Matrix……Page 84
Correlation Maps……Page 86
Exercises……Page 88
Parametric vs.Empirical Distributions……Page 90
Parameters vs.Statistics……Page 91
Binomial Distribution……Page 92
Geometric Distribution……Page 95
Negative Binomial Distribution……Page 96
Poisson Distribution……Page 99
Expected Value of a Random Variable……Page 101
Expected Value of a Function of a Random Variable……Page 102
Distribution Functions and Expected Values……Page 104
Gaussian Distributions……Page 107
Gamma Distributions……Page 114
Beta Distributions……Page 121
Extreme-Value Distributions……Page 123
Mixture Distributions……Page 128
Superposition of a Fitted Parametric Distribution and Data Histogram……Page 130
Quantile-Quantile (Q –Q)Plots……Page 132
The Likelihood Function……Page 133
The Newton-Raphson Method……Page 135
The EM Algorithm……Page 136
Statistical Simulation……Page 139
Uniform Random Number Generators……Page 140
Nonuniform Random Number Generation by Inversion……Page 142
Nonuniform Random Number Generation by Rejection……Page 143
Box-Muller Method for Gaussian Random Number Generation……Page 145
Simulating from Mixture Distributions and Kernel Density Estimates……Page 146
Exercises……Page 147
Parametric vs.Nonparametric Tests……Page 150
The Elements of Any Hypothesis Test……Page 151
Error Types and the Power of a Test……Page 152
One-Sided vs.Two-Sided Tests……Page 153
Confidence Intervals:Inverting Hypothesis Tests……Page 154
One-Sample t Test……Page 157
Tests for Differences of Mean under Independence……Page 159
Tests for Differences of Mean for Paired Samples……Page 160
Test for Differences of Mean under Serial Dependence……Page 162
Goodness-of-Fit Tests……Page 165
Likelihood Ratio Test……Page 173
Classical Nonparametric Tests for Location……Page 175
Introduction to Resampling Tests……Page 181
Permutation Tests……Page 183
The Bootstrap……Page 185
Field Significance and Multiplicity……Page 189
The Multiplicity Problem for Independent Tests……Page 190
Field Significance Given Spatial Correlation……Page 191
Exercises……Page 195
Background……Page 198
Simple Linear Regression……Page 199
Distribution of the Residuals……Page 201
The Analysis of Variance Table……Page 203
Goodness-of-Fit Measures……Page 204
Sampling Distributions of the Regression Coefficients……Page 206
Examining Residuals……Page 208
Prediction Intervals……Page 213
Multiple Linear Regression……Page 216
Derived Predictor Variables in Multiple Regression……Page 217
Logistic Regression……Page 220
Poisson Regression……Page 224
Why Is Careful Predictor Selection Important?……Page 226
Screening Predictors……Page 228
Stopping Rules……Page 231
Cross Validation……Page 234
Classical Statistical Forecasting……Page 236
Perfect Prog and MOS……Page 239
Operational MOS Forecasts……Page 245
Stochastic Dynamical Systems in Phase Space……Page 248
Ensemble Forecasts……Page 251
Choosing Initial Ensemble Members……Page 252
Ensemble Average and Ensemble Dispersion……Page 253
Graphical Display of Ensemble Forecast Information……Page 255
Effects of Model Errors……Page 261
Statistical Postprocessing:Ensemble MOS……Page 262
The Nature of Subjective Forecasts……Page 264
The Subjective Distribution……Page 265
Central Credible Interval Forecasts……Page 267
Assessing Discrete Probabilities……Page 269
Assessing Continuous Distributions……Page 270
Exercises……Page 271
Purposes of Forecast Verification……Page 274
The Joint Distribution of Forecasts and Observations……Page 275
Scalar Attributes of Forecast Performance……Page 277
Forecast Skill……Page 278
The 2 ×2 Contingency Table……Page 279
Scalar Attributes Characterizing 2 ×2 Contingency Tables……Page 281
Skill Scores for 2 ×2 Contingency Tables……Page 284
Which Score?……Page 287
Conversion of Probabilistic to Nonprobabilistic Forecasts……Page 288
Extensions for Multicategory Discrete Predictands……Page 290
Nonprobabilistic Forecasts of Continuous Predictands……Page 295
Conditional Quantile Plots……Page 296
Scalar Accuracy Measures……Page 297
Skill Scores……Page 299
The Joint Distribution for Dichotomous Events……Page 301
The Brier Score……Page 303
Algebraic Decomposition of the Brier Score……Page 304
The Reliability Diagram……Page 306
The Discrimination Diagram……Page 312
The ROC Diagram……Page 313
Hedging,and Strictly Proper Scoring Rules……Page 317
Probability Forecasts for Multiple-Category Events……Page 318
Full Continuous Forecast Probability Distributions……Page 321
Central Credible Interval Forecasts……Page 322
General Considerations for Field Forecasts……Page 323
The S1 Score……Page 325
Mean Squared Error……Page 326
Anomaly Correlation……Page 330
Characteristics of a Good Ensemble Forecast……Page 333
The Verification Rank Histogram……Page 335
Recent Ideas in Verification of Ensemble Forecasts……Page 338
Optimal Decision Making and the Cost/Loss Ratio Problem……Page 340
The Value Score……Page 343
Connections with Other Verification Approaches……Page 344
Sampling Characteristics of Contingency Table Statistics……Page 345
ROC Diagram Sampling Characteristics……Page 348
Reliability Diagram Sampling Characteristics……Page 349
Exercises……Page 351
Stationarity……Page 356
Time-Series Models……Page 357
Markov Chains……Page 358
Two-State,First-Order Markov Chains……Page 359
Test for Independence vs.First-Order Serial Dependence……Page 363
Some Applications of Two-State Markov Chains……Page 365
Multiple-State Markov Chains……Page 367
Higher-Order Markov Chains……Page 368
Deciding among Alternative Orders of Markov Chains……Page 369
First-Order Autoregression……Page 371
Higher-Order Autoregressions……Page 376
The AR(2)Model……Page 377
Order Selection Criteria……Page 381
The Variance of a Time Average……Page 382
Autoregressive-Moving Average Models……Page 385
Simulation and Forecasting with Continuous Time-Domain Models……Page 386
Cosine and Sine Functions……Page 390
Representing a Simple Time Series with a Harmonic Function……Page 391
Estimation of the Amplitude and Phase of a Single Harmonic……Page 394
Higher Harmonics……Page 397
The Harmonic Functions as Uncorrelated Regression Predictors……Page 400
The Periodogram,or Fourier Line Spectrum……Page 402
Computing Spectra……Page 406
Aliasing……Page 407
Theoretical Spectra of Autoregressive Models……Page 409
Sampling Properties of Spectral Estimates……Page 413
Exercises……Page 418
Multivariate Statistics……Page 420
Contrasts between Multivariate and Univariate Statistics……Page 422
Organization of Data and Basic Notation……Page 423
Multivariate Extensions of Common Univariate Statistics……Page 424
Euclidean Distance……Page 425
Mahalanobis (Statistical)Distance……Page 426
Matrix Algebra Review……Page 427
Vectors……Page 428
Matrices……Page 430
Eigenvalues and Eigenvectors of a Square Matrix……Page 439
Square Roots of a Symmetric Matrix……Page 442
Singular-Value Decomposition (SVD)……Page 444
Expectations and Other Extensions of Univariate Concepts……Page 445
Partitioning Vectors and Matrices……Page 446
Linear Combinations……Page 448
Mahalanobis Distance,Revisited……Page 450
Exercises……Page 451
Definition of the MVN……Page 454
Four Handy Properties of the MVN……Page 456
Assessing Multinormality……Page 459
Simulating Independent MVN Variates……Page 463
Simulating Multivariate Time Series……Page 464
Inferences about a Multinormal Mean Vector……Page 467
Hotelling ’s T2……Page 468
Simultaneous Confidence Statements……Page 475
Interpretation of Multivariate Statistical Significance……Page 478
Exercises……Page 481
Definition of PCA……Page 482
PCA Based on the Covariance Matrix vs.the Correlation Matrix……Page 488
The Varied Terminology of PCA……Page 490
Scaling Conventions in PCA……Page 491
Connections to the Multivariate Normal Distribution……Page 492
PCA for a Single Field……Page 494
Simultaneous PCA for Multiple Fields……Page 496
Scaling Considerations and Equalization of Variance……Page 498
Domain Size Effects:Buell Patterns……Page 499
Why Truncate the Principal Components?……Page 500
Subjective Truncation Criteria……Page 501
Rules Based on Hypothesis Testing Ideas……Page 503
Asymptotic Sampling Results for Multivariate Normal Data……Page 505
Effective Multiplets……Page 507
The North et al.Rule of Thumb……Page 508
Why Rotate the Eigenvectors?……Page 511
Rotation Mechanics……Page 512
Sensitivity of Orthogonal Rotation to Initial Eigenvector Scaling……Page 515
Direct Extraction of Eigenvalues and Eigenvectors from [S ]……Page 518
PCA via SVD……Page 519
Singular Spectrum Analysis (SSA):Time-Series PCA……Page 520
Principal-Component Regression……Page 523
The Biplot……Page 524
Exercises……Page 526
Overview……Page 528
Canonical Variates,Canonical Vectors,and Canonical Correlations……Page 529
Some Additional Properties of CCA……Page 531
Combining CCA with PCA……Page 536
Forecasting with CCA……Page 538
Calculating CCA through Direct Eigendecomposition……Page 541
Calculating CCA through SVD……Page 543
Maximum Covariance Analysis……Page 545
Exercises……Page 547
Discrimination vs.Classification……Page 548
Equal Covariance Structure:Fisher ’s Linear Discriminant……Page 549
Fisher ’s Linear Discriminant for Multivariate Normal Data……Page 553
Minimizing Expected Cost of Misclassification……Page 554
Unequal Covariances:Quadratic Discrimination……Page 556
Fisher ’s Procedure for More Than Two Groups……Page 557
Minimizing Expected Cost of Misclassification……Page 560
Probabilistic Classification……Page 561
Forecasting with Discriminant Analysis……Page 563
Discrimination and Classification Using Logistic Regression……Page 564
Discrimination and Classification Using Kernel Density Estimates……Page 565
Exercises……Page 566
Cluster Analysis vs.Discriminant Analysis……Page 568
Distance Measures and the Distance Matrix……Page 569
Agglomerative Methods Using the Distance Matrix……Page 570
Ward ’s Minimum Variance Method……Page 571
The Dendrogram,or Tree Diagram……Page 572
How Many Clusters?……Page 573
The K-Means Method……Page 578
Nucleated Agglomerative Clustering……Page 579
Exercises……Page 580
APPENDIX A Example Data Sets……Page 584
Table A.1 Daily precipitation and temperature data for Ithaca and Canandaigua,New York,for January 1987……Page 585
Table A.3 June climate data for Guayaquil,Ecuador,1951–1970……Page 586
APPENDIX B Probability Tables……Page 588
Table B.1 Cumulative Probabilities for the Standard Gaussian Distribution……Page 589
Table B.2 Quantiles of the Standard Gamma Distribution……Page 590
Table B.3 Right-tail quantiles of the Chi-square distribution……Page 595
APPENDIX C Answers to Exercises……Page 598
References……Page 606
Index……Page 630
International Geophysics Series……Page 647