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E-book: Statistical Methods for Climate Scientists

(George Mason University, Virginia), (Columbia University, New York)
  • Format: PDF+DRM
  • Pub. Date: 24-Feb-2022
  • Publisher: Cambridge University Press
  • Language: eng
  • ISBN-13: 9781108653008
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Statistical Methods for Climate ScientistsLarger Image
  • Format: PDF+DRM
  • Pub. Date: 24-Feb-2022
  • Publisher: Cambridge University Press
  • Language: eng
  • ISBN-13: 9781108653008
Other books in subject:

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"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"--

Reviews

'This text will be useful for teaching advanced undergraduates and graduate students about the applications of statistical methods to climate data analysis. It is filled with many relevant examples informed by the authors' long experience in the field. I am sure that I will frequently use it as a reference in the coming years for my own research.' Tom Hamill, National Oceanic and Atmospheric Administration 'This book is essential for any climate scientist and is ideally suited for an introductory graduate course in climate analysis. The material covered includes a comprehensive sample of classical and modern multi-variate techniques that are widely used in the peer-reviewed literature. The step-by-step examples are clearly based on years of hands-on teaching experience by the authors and are easily implementable and, importantly, highlight interpretation and limitations - a must for any climate analysist.' Ben Kirtman, University of Miami 'An appealing book written by outstanding authors, with basic to advanced topics in every chapter, including some unusual topics for a statistics book such as data assimilation and the most predictable modes. The iteration with several years of students to produce an understandable and logical text (which is formal and analytical in nature) is unique and worthy for consideration as a textbook for graduate courses or as refresher for any geophysical scientist.' Huug van den Dool, NOAA 'Includes both the mathematics and the intuition needed for climate data analysis.' Dennis L. Hartmann, University of Washington ' this comprehensive presentation of statistical techniques will be helpful to many graduate students and researchers in climate science Highly recommended.' S. G. Decker, Choice ' Timothy DelSole and Michael Tippett aim to streamline students' mathematical training by collecting the most important methods into a single textbook As more climate scientists venture into such subtle problems as whether to attribute extreme events like prolonged heat waves to climate change, the critical statistical thinking skills fostered in Statistical Methods for Climate Scientists will be of increasing importance.' Brad Marston, Physics Today ' the critical statistical thinking skills fostered in Statistical Methods for Climate Scientists will be of increasing importance.' Brad Marston, Physics Today

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An accessible introduction to statistical methods for students in the climate sciences.
Preface xiii
1 Basic Concepts in Probability and Statistics
1(29)
1.1 Graphical Description of Data
2(2)
1.2 Measures of Central Value: Mean, Median, and Mode
4(2)
1.3 Measures of Variation: Percentile Ranges and Variance
6(2)
1.4 Population versus a Sample
8(1)
1.5 Elements of Probability Theory
8(3)
1.6 Expectation
11(2)
1.7 More Than One Random Variable
13(3)
1.8 Independence
16(2)
1.9 Estimating Population Quantities from Samples
18(2)
1.10 Normal Distribution and Associated Theorems
20(7)
1.11 Independence versus Zero Correlation
27(1)
1.12 Further Topics
28(1)
1.13 Conceptual Questions
29(1)
2 Hypothesis Tests
30(22)
2.1 The Problem
31(2)
2.2 Introduction to Hypothesis Testing
33(7)
2.3 Further Comments on the r-test
40(3)
2.4 Examples of Hypothesis Tests
43(6)
2.5 Summary of Common Significance Tests
49(1)
2.6 Further Topics
50(1)
2.7 Conceptual Questions
51(1)
3 Confidence Intervals
52(17)
3.1 The Problem
53(1)
3.2 Confidence Interval for a Difference in Means
53(2)
3.3 Interpretation of the Confidence Interval
55(2)
3.4 A Pitfall about Confidence Intervals
57(1)
3.5 Common Procedures for Confidence Intervals
57(7)
3.6 Bootstrap Confidence Intervals
64(3)
3.7 Further Topics
67(1)
3.8 Conceptual Questions
68(1)
4 Statistical Tests Based on Ranks
69(25)
4.1 The Problem
70(1)
4.2 Exchangeability and Ranks
71(2)
4.3 The Wilcoxon Rank-Sum Test
73(5)
4.4 Stochastic Dominance
78(1)
4.5 Comparison with the t-test
79(2)
4.6 Kruskal--Wallis Test
81(2)
4.7 Test for Equality of Dispersions
83(2)
4.8 Rank Correlation
85(3)
4.9 Derivation of the Mean and Variance of the Rank Sum
88(4)
4.10 Further Topics
92(1)
4.11 Conceptual Questions
93(1)
5 Introduction to Stochastic Processes
94(32)
5.1 The Problem
95(5)
5.2 Stochastic Processes
100(5)
5.3 Why Should I Care if My Data Are Serially Correlated?
105(4)
5.4 The First-Order Autoregressive Model
109(8)
5.5 The AR(2) Model
117(2)
5.6 Pitfalls in Interpreting ACFs
119(2)
5.7 Solutions of the AR(2) Model
121(1)
5.8 Further Topics
122(2)
5.9 Conceptual Questions
124(2)
6 The Power Spectrum
126(30)
6.1 The Problem
127(2)
6.2 The Discrete Fourier Transform
129(4)
6.3 Parseval's Identity
133(1)
6.4 The Periodogram
134(1)
6.5 The Power Spectrum
135(3)
6.6 Periodogram of Gaussian White Noise
138(1)
6.7 Impact of a Deterministic Periodic Component
139(1)
6.8 Estimation of the Power Spectrum
140(4)
6.9 Presence of Trends and Jump Discontinuities
144(2)
6.10 Linear Filters
146(4)
6.11 Tying Up Loose Ends
150(2)
6.12 Further Topics
152(3)
6.13 Conceptual Questions
155(1)
7 Introduction to Multivariate Methods
156(29)
7.1 The Problem
157(2)
7.2 Vectors
159(1)
7.3 The Linear Transformation
160(3)
7.4 Linear Independence
163(3)
7.5 Matrix Operations
166(2)
7.6 Invertible Transformations
168(2)
7.7 Orthogonal Transformations
170(2)
7.8 Random Vectors
172(3)
7.9 Diagonalizing a Covariance Matrix
175(3)
7.10 Multivariate Normal Distribution
178(1)
7.11 Hotelling's T-squared Test
179(2)
7.12 Multivariate Acceptance and Rejection Regions
181(1)
7.13 Further Topics
182(1)
7.14 Conceptual Questions
183(2)
8 Linear Regression: Least Squares Estimation
185(25)
8.1 The Problem
186(2)
8.2 Method of Least Squares
188(4)
8.3 Properties of the Least Squares Solution
192(4)
8.4 Geometric Interpretation of Least Squares Solutions
196(3)
8.5 Illustration Using Atmospheric CO2 Concentration
199(6)
8.6 The Line Fit
205(1)
8.7 Always Include the Intercept Term
206(1)
8.8 Further Topics
207(2)
8.9 Conceptual Questions
209(1)
9 Linear Regression: Inference
210(27)
9.1 The Problem
211(1)
9.2 The Model
212(1)
9.3 Distribution of the Residuals
212(1)
9.4 Distribution of the Least Squares Estimates
213(2)
9.5 Inferences about Individual Regression Parameters
215(1)
9.6 Controlling for the Influence of Other Variables
216(2)
9.7 Equivalence to "Regressing Out" Predictors
218(4)
9.8 Seasonality as a Confounding Variable
222(2)
9.9 Equivalence between the Correlation Test and Slope Test
224(1)
9.10 Generalized Least Squares
225(1)
9.11 Detection and Attribution of Climate Change
226(7)
9.12 The General Linear Hypothesis
233(1)
9.13 Tying Up Loose Ends
234(2)
9.14 Conceptual Questions
236(1)
10 Model Selection
237(18)
10.1 The Problem
238(2)
10.2 Bias-Variance Trade off
240(3)
10.3 Out-of-Sample Errors
243(2)
10.4 Model Selection Criteria
245(4)
10.5 Pitfalls
249(4)
10.6 Further Topics
253(1)
10.7 Conceptual Questions
254(1)
11 Screening: A Pitfall in Statistics
255(18)
11.1 The Problem
256(3)
11.2 Screening iid Test Statistics
259(3)
11.3 The Bonferroni Procedure
262(1)
11.4 Screening Based on Correlation Maps
262(3)
11.5 Can You Trust Relations Inferred from Correlation Maps?
265(1)
11.6 Screening Based on Change Points
265(3)
11.7 Screening with a Validation Sample
268(1)
11.8 The Screening Game: Can You Find the Statistical Flaw?
268(3)
11.9 Screening Always Exists in Some Form
271(1)
11.10 Conceptual Questions
272(1)
12 Principal Component Analysis
273(25)
12.1 The Problem
274(2)
12.2 Examples
276(7)
12.3 Solution by Singular Value Decomposition
283(2)
12.4 Relation between PCA and the Population
285(4)
12.5 Special Considerations for Climate Data
289(6)
12.6 Further Topics
295(2)
12.7 Conceptual Questions
297(1)
13 Field Significance
298(16)
13.1 The Problem
299(4)
13.2 The Livezey--Chen Field Significance Test
303(2)
13.3 Field Significance Test Based on Linear Regression
305(5)
13.4 False Discovery Rate
310(1)
13.5 Why Different Tests for Field Significance?
311(1)
13.6 Further Topics
312(1)
13.7 Conceptual Questions
312(2)
14 Multivariate Linear Regression
314(21)
14.1 The Problem
315(2)
14.2 Review of Univariate Regression
317(3)
14.3 Estimating Multivariate Regression Models
320(3)
14.4 Hypothesis Testing in Multivariate Regression
323(1)
14.5 Selecting X
324(4)
14.6 Selecting Both X and Y
328(3)
14.7 Some Details about Regression with Principal Components
331(1)
14.8 Regression Maps and Projecting Data
332(1)
14.9 Conceptual Questions
333(2)
15 Canonical Correlation Analysis
335(31)
15.1 The Problem
336(1)
15.2 Summary and Illustration of Canonical Correlation Analysis
337(6)
15.3 Population Canonical Correlation Analysis
343(4)
15.4 Relation between CCA and Linear Regression
347(2)
15.5 Invariance to Affine Transformation
349(1)
15.6 Solving CCA Using the Singular Value Decomposition
350(7)
15.7 Model Selection
357(2)
15.8 Hypothesis Testing
359(3)
15.9 Proof of the Maximization Properties
362(2)
15.10 Further Topics
364(1)
15.11 Conceptual Questions
364(2)
16 Covariance Discriminant Analysis
366(33)
16.1 The Problem
367(3)
16.2 Illustration: Most Detectable Climate Change Signals
370(8)
16.3 Hypothesis Testing
378(4)
16.4 The Solution
382(6)
16.5 Solution in a Reduced-Dimensional Subspace
388(4)
16.6 Variable Selection
392(3)
16.7 Further Topics
395(3)
16.8 Conceptual Questions
398(1)
17 Analysis of Variance and Predictability
399(19)
17.1 The Problem
400(1)
17.2 Framing the Problem
401(2)
17.3 Test Equality of Variance
403(1)
17.4 Test Equality of Means: ANOVA
404(2)
17.5 Comments about ANOVA
406(1)
17.6 Weather Predictability
407(4)
17.7 Measures of Predictability
411(3)
17.8 What Is the Difference between Predictability and Skill?
414(2)
17.9 Chaos and Predictability
416(1)
17.10 Conceptual Questions
417(1)
18 Predictable Component Analysis
418(28)
18.1 The Problem
419(3)
18.2 Illustration of Predictable Component Analysis
422(2)
18.3 Multivariate Analysis of Variance
424(3)
18.4 Predictable Component Analysis
427(3)
18.5 Variable Selection in PrCA
430(2)
18.6 PrCA Based on Other Measures of Predictability
432(3)
18.7 Skill Component Analysis
435(2)
18.8 Connection to Multivariate Linear Regression and CCA
437(2)
18.9 Further Properties of PrCA
439(6)
18.10 Conceptual Questions
445(1)
19 Extreme Value Theory
446(22)
19.1 The Problem and a Summary of the Solution
447(6)
19.2 Distribution of the Maximal Value
453(6)
19.3 Maximum Likelihood Estimation
459(4)
19.4 Nonstationarity: Changing Characteristics of Extremes
463(3)
19.5 Further Topics
466(1)
19.6 Conceptual Questions
467(1)
20 Data Assimilation
468(21)
20.1 The Problem
469(1)
20.2 A Univariate Example
469(4)
20.3 Some Important Properties and Interpretations
473(2)
20.4 Multivariate Gaussian Data Assimilation
475(2)
20.5 Sequential Processing of Observations
477(1)
20.6 Multivariate Example
478(3)
20.7 Further Topics
481(6)
20.8 Conceptual Questions
487(2)
21 Ensemble Square Root Filters
489(21)
21.1 The Problem
490(7)
21.2 Filter Divergence
497(2)
21.3 Monitoring the Innovations
499(1)
21.4 Multiplicative Inflation
500(3)
21.5 Covariance Localization
503(4)
21.6 Further Topics
507(2)
21.7 Conceptual Questions
509(1)
Appendix
510(4)
A.1 Useful Mathematical Relations
510(1)
A.2 Generalized Eigenvalue Problems
511(1)
A.3 Derivatives of Quadratic Forms and Traces
512(2)
References 514(9)
Index 523
Timothy M. DelSole is Professor in the Department of Atmospheric, Oceanic, and Earth Sciences, and Senior Scientist at the Center for Oceanic Atmospheric, and Land Studies, at George Mason University, Virginia. He has published over one hundred peer-reviewed papers in climate science and served as co-Editor-in-Chief of the Journal of Climate. Michael Tippett is an Associate Professor at Columbia University. His research includes forecasting El Niño and relating extreme weather (tornadoes and hurricanes) with climate, now and in the future. He analyzes data from computer models and weather observations to find patterns that improve understanding, facilitate prediction, and help manage risk.
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