### Table of Contents

I Preliminaries

Chapter 1 Introduction

1.1 What Is Statistics?

1.2 Descriptive and Inferential Statistics

1.3 Uncertainty about the Atmosphere

Chapter 2 Review of Probability

2.1 Background

2.2 The Elements of Probability

2.3 The Meaning of Probability

2.4 Some Properties of Probability

2.5 Exercises

II Univariate Statistics

Chapter 3 Empirical Distributions and Exploratory Data Analysis

3.1 Background

3.2 Numerical Summary Measures

3.3 Graphical Summary Devices

3.4 Reexpression

3.5 Exploratory Techniques for Paired Data

3.6 Exploratory Techniques for Higher-Dimensional Data

3.7 Exercises

Chapter 4 Parametric Probability Distributions

4.1 Background

4.2 Discrete Distributions

4.3 Statistical Expectations

4.4 Continuous Distributions

4.5 Qualitative Assessments of the Goodness of Fit

4.6 Parameter Fitting Using Maximum Likelihood

4.7 Statistical Simulation

4.8 Exercises

Chapter 5 Frequentist Statistical Inference

5.1. Background

5.2 Some Commonly Encountered Parametric Tests

5.3 Nonparametric Tests

5.4 Multiplicity and "Field Significance"

5.5. Exercises

Chapter 6 Bayesian Inference

6.1 Background

6.2 The Structure of Bayesian Inference

6.3 Conjugate Distributions

6.4 Dealing With Difficult Integrals

6.5 Exercises

Chapter 7 Statistical Forecasting

7.1 Background

7.2 Linear Regression

7.3 Nonlinear Regression

7.4 Predictor Selection

7.5 Objective Forecasts Using Traditional Statistical Methods

7.6 Ensemble Forecasting

7.7 Ensemble MOS

7.8 Subjective Probability Forecasts

7.9 Exercises

Chapter 8 Forecast Verification

8.1 Background

8.2 Nonprobabilistic Forecasts for Discrete Predictands

8.3 Nonprobabilistic Forecasts for Continuous Predictands

8.4 Probability Forecasts for Discrete Predictands

8.5 Probability Forecasts for Continuous Predictands

8.6 Nonprobabilistic Forecasts for Fields

8.7 Verification of Ensemble Forecasts

8.8 Verification Based on Economic Value

8.9 Verification When the Observation is Uncertain

8.10 Sampling and Inference for Verification Statistics

8.11 Exercises

Chapter 9 Time Series

9.1 Background

9.2 Time Domain-I. Discrete Data

9.3 Time Domain-II. Continuous Data

9.4 Frequency Domain-I. Harmonic Analysis

9.5 Frequency Domain-II. Spectral Analysis

9.6 Exercises

III Multivariate Statistics

Chapter 10 Matrix Algebra and Random Matrices

10.1 Background to Multivariate Statistics

10.2 Multivariate Distance

10.3 Matrix Algebra Review

10.4 Random Vectors and Matrices

10.5 Exercises

Chapter 11 The Multivariate Normal (MVN) Distribution

11.1 Definition of the MVN

11.2 Four Handy Properties of the MVN

11.3 Assessing Multinormality

11.4 Simulation from the Multivariate Normal Distribution

11.5 Inferences about a Multinormal Mean Vector

11.6 Exercises

Chapter 12 Principal Component (EOF) Analysis

12.1 Basics of Principal Component Analysis

12.2 Application of PCA to Geophysical Fields

12.3 Truncation of the Principal Components

12.4 Sampling Properties of the Eigenvalues and Eigenvectors

12.5 Rotation of the Eigenvectors

12.6 Computational Considerations

12.7 Some Additional Uses of PCA

12.8 Exercises

Chapter 13 Canonical Correlation Analysis (CCA)

13.1 Basics of CCA

13.2 CCA Applied to Fields

13.3 Computational Considerations

13.4 Maximum Covariance Analysis (MCA)

13.5 Exercises

Chapter 14 Discrimination and Classification

14.1 Discrimination vs. Classification

14.2 Separating Two Populations

14.3 Multiple Discriminant Analysis (MDA)

14.4 Forecasting with Discriminant Analysis

14.5 Alternatives to Classical Discriminant Analysis

14.6 Exercises

Chapter 15 Cluster Analysis

15.1 Background

15.2 Hierarchical Clustering

15.3 Nonhierarchical Clustering

15.4 Exercises

Appendix A Example Data Sets

Appendix B Probability Tables

Appendix C Answers to Exercises

References

Index