Statistical Methods in the Atmospheric Sciences

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