Statistical Methods for Climate Scientists

"A comprehensive introduction to the most commonly used statistical methods relevant in atmospheric, oceanic and climate sciences. Each method is described step-by-step using plain language, and illustrated with concrete examples, with relevant statistical and scientific concepts explained as needed. Particular attention is paid to nuances and pitfalls, with sufficient detail to enable the reader to write relevant code. Topics covered include hypothesis testing, time series analysis, linear regression, data assimilation, extreme value analysis, Principal Component Analysis, Canonical Correlation Analysis, Predictable Component Analysis, and Covariance Discriminant Analysis. The specific statistical challenges that arise in climate applications are also discussed, including model selection problems associated with Canonical Correlation Analysis, Predictable Component Analysis, and Covariance Discriminant Analysis. Requiring no previous background in statistics, this is a highly accessible textbook and reference for students and early-career researchers in the climate sciences"-- Provided by publisher Front matter Copyright Contents Preface 1 Basic Concepts in Probability and Statistics 1.1 Graphical Description of Data 1.2 Measures of Central Value: Mean, Median, and Mode 1.3 Measures of Variation: Percentile Ranges and Variance 1.4 Population versus a Sample 1.5 Elements of Probability Theory 1.6 Expectation 1.7 More Than One Random Variable 1.8 Independence 1.9 Estimating Population Quantities from Samples 1.10 Normal Distribution and Associated Theorems 1.11 Independence versus Zero Correlation 1.12 Further Topics 1.13 Conceptual Questions 2 Hypothesis Tests 2.1 The Problem 2.2 Introduction to Hypothesis Testing 2.3 Further Comments on the t-test 2.4 Examples of Hypothesis Tests 2.5 Summary of Common Significance Tests 2.6 Further Topics 2.7 Conceptual Questions 3 Confidence Intervals 3.1 The Problem 3.2 Confidence Interval for a Difference in Means 3.3 Interpretation of the Confidence Interval 3.4 A Pitfall about Confidence Intervals 3.5 Common Procedures for Confidence Intervals 3.6 Bootstrap Confidence Intervals 3.7 Further Topics 3.8 Conceptual Questions 4 Statistical Tests Based on Ranks 4.1 The Problem 4.2 Exchangeability and Ranks 4.3 The Wilcoxon Rank-Sum Test 4.4 Stochastic Dominance 4.5 Comparison with thet-test 4.6 Kruskal–Wallis Test 4.7 Test for Equality of Dispersions 4.8 Rank Correlation 4.9 Derivation of the Mean and Variance of the Rank Sum 4.10 Further Topics 4.11 Conceptual Questions 5 Introduction to Stochastic Processes 5.1 The Problem 5.2 Stochastic Processes 5.3 Why Should I Care if My Data Are Serially Correlated? 5.4 The First-Order Autoregressive Model 5.5 The AR(2) Model 5.6 Pitfalls in Interpreting ACFs 5.7 Solutions of the AR(2) Model 5.8 Further Topics 5.9 Conceptual Questions 6 The Power Spectrum 6.1 The Problem 6.2 The Discrete Fourier Transform 6.3 Parseval’s Identity 6.4 The Periodogram 6.5 The Power Spectrum 6.6 Periodogram of Gaussian White Noise 6.7 Impact of a Deterministic Periodic Component 6.8 Estimation of the Power Spectrum 6.9 Presence of Trends and Jump Discontinuities 6.10 Linear Filters 6.11 Tying Up Loose Ends 6.12 Further Topics 6.13 Conceptual Questions 7 Introduction to Multivariate Methods 7.1 The Problem 7.2 Vectors 7.3 The Linear Transformation 7.4 Linear Independence 7.5 Matrix Operations 7.6 Invertible Transformations 7.7 Orthogonal Transformations 7.8 Random Vectors 7.9 Diagonalizing a Covariance Matrix 7.10 Multivariate Normal Distribution 7.11 Hotelling’s T-squared Test 7.12 Multivariate Acceptance and Rejection Regions 7.13 Further Topics 7.14 Conceptual Questions 8 Linear Regression: Least Squares Estimation 8.1 The Problem 8.2 Method of Least Squares 8.3 Properties of the Least Squares Solution 8.4 Geometric Interpretation of Least Squares Solutions 8.5 Illustration Using Atmospheric CO2 Concentration 8.6 The Line Fit 8.7 Always Include the Intercept Term 8.8 Further Topics 8.9 Conceptual Questions 9 Linear Regression: Inference 9.1 The Problem 9.2 The Model 9.3 Distribution of the Residuals 9.4 Distribution of the Least Squares Estimates 9.5 Inferences about Individual Regression Parameters 9.6 Controlling for the Influence of Other Variables 9.7 Equivalence to“Regressing Out” Predictors 9.8 Seasonality as a Confounding Variable 9.9 Equivalence between the Correlation Test and Slope Test 9.10 Generalized Least Squares 9.11 Detection and Attribution of Climate Change 9.12 The General Linear Hypothesis 9.13 Tying UpLoose Ends 9.14 Conceptual Questions 10 Model Selection 10.1 The Problem 10.2 Bias–Variance Trade off 10.3 Out-of-Sample Errors 10.4 Model Selection Criteria 10.5 Pitfalls 10.6 Further Topics 10.7 Conceptual Questions 11 Screening: A Pitfall in Statistics 11.1 The Problem 11.2 Screening iid Test Statistics 11.3 The Bonferroni Procedure 11.4 Screening Based on Correlation Maps 11.5 Can You Trust Relations Inferred from Correlation Maps? 11.6 Screening Based on Change Points 11.7 Screening with a Validation Sample 11.8 The Screening Game: Can You Find the Statistical Flaw? 11.9 Screening Always Exists in Some Form 11.10 Conceptual Questions 12 Principal Component Analysis 12.1 The Problem 12.2 Examples 12.3 Solution by Singular Value Decomposition 12.4 Relation between PCA and the Population 12.5 Special Considerations for Climate Data 12.6 Further Topics 12.7 Conceptual Questions 13 Field Significance 13.1 The Problem 13.2 The Livezey–Chen Field Significance Test 13.3 Field Significance Test Based on Linear Regression 13.4 False Discovery Rate 13.5 Why Different Tests for Field Significance? 13.6 Further Topics 13.7 Conceptual Questions 14 Multivariate Linear Regression 14.1 The Problem 14.2 Review of Univariate Regression 14.3 Estimating Multivariate Regression Models 14.4 Hypothesis Testing in Multivariate Regression 14.5 Selecting X 14.6 Selecting Both X and Y 14.7 Some Details about Regression with Principal Components 14.8 Regression Maps and Projecting Data 14.9 Conceptual Questions 15 Canonical Correlation Analysis 15.1 The Problem 15.2 Summary and Illustration of Canonical Correlation Analysis 15.3 Population Canonical Correlation Analysis 15.4 Relation between CCA and Linear Regression 15.5 Invariance to Affine Transformation 15.6 Solving CCA Using the Singular Value Decomposition 15.7 Model Selection 15.8 Hypothesis Testing 15.9 Proof of the Maximization Properties 15.10 Further Topics 15.11 Conceptual Questions 16 Covariance DiscriminantAnalysis 16.1 The Problem 16.2 Illustration: Most Detectable Climate Change Signals 16.3 Hypothesis Testing 16.4 The Solution 16.5 Solution in a Reduced-Dimensional Subspace 16.6 Variable Selection 16.7 Further Topics 16.8 Conceptual Questions 17 Analysis of Variance and Predictability 17.1 The Problem 17.2 Framing the Problem 17.3 Test Equality of Variance 17.4 Test Equality of Means: ANOVA 17.5 Comments about ANOVA 17.6 Weather Predictability 17.7 Measuresof Predictability 17.8 What Is the Difference between Predictability and Skill? 17.9 Chaos and Predictability 17.10 Conceptual Questions 18 Predictable Component Analysis 18.1 The Problem 18.2 Illustration of Predictable Component Analysis 18.3 Multivariate Analysis of Variance 18.4 Predictable Component Analysis 18.5 Variable Selection inPrCA 18.6 PrCA Basedon Other Measures of Predictability 18.7 Skill Component Analysis 18.8 Connection to Multivariate Linear Regression and CCA 18.9 Further Properties of PrCA 18.10 Conceptual Questions 19 Extreme Value Theory 19.1 The Problem and a Summary of the Solution 19.2 Distribution of the Maximal Value 19.3 Maximum Likelihood Estimation 19.4 Nonstationarity: Changing Characteristics ofExtremes 19.5 Further Topics 19.6 Conceptual Questions 20 Data Assimilation 20.1 The Problem 20.2 A Univariate Example 20.3 Some Important Properties and Interpretations 20.4 Multivariate Gaussian Data Assimilation 20.5 Sequential Processing of Observations 20.6 Multivariate Example 20.7 Further Topics 20.8 Conceptual Questions 21 Ensemble Square Root Filters 21.1 The Problem 21.2 Filter Divergence 21.3 Monitoring the Innovations 21.4 Multiplicative Inflation 21.5 Covariance Localization 21.6 Further Topics 21.7 Conceptual Questions Appendix A.1 Useful Mathematical Relations A.2 Generalized Eigenvalue Problems A.3 Derivatives of Quadratic Forms and Traces References Index An introduction to the most commonly used statistical methods relevant in atmospheric, oceanic, and climate sciences. Each method is described step-by-step using plain language, and illustrated with concrete examples, with relevant statistical and scientific concepts explained as needed. Particular attention is paid to nuances and pitfalls, with sufficient detail to enable the reader to write relevant code. Topics covered include hypothesis testing, time series analysis, linear regression, data assimilation, extreme value analysis, Principal Component Analysis, Canonical Correlation Analysis, Predictable Component Analysis, and Covariance Discriminant Analysis. The specific statistical challenges that arise in climate applications are also discussed, including model selection problems associated with Canonical Correlation Analysis, Predictable Component Analysis, and Covariance Discriminant Analysis. Requiring no previous background in statistics, this is an accessible textbook and reference for students and early-career researchers in the climate sciences. -- Adapted from publisher's description