Statistical Methods for Climate Scientists

✍ Scribed by Timothy DelSole, Michael Tippett

• Publisher: Cambridge University Press
• Year: 2022
• Tongue: English
• Leaves: 545
• Edition: New
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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