Book contents
- Frontmatter
- Contents
- Preface
- 1 Basic Concepts in Probability and Statistics
- 2 Hypothesis Tests
- 3 Confidence Intervals
- 4 Statistical Tests Based on Ranks
- 5 Introduction to Stochastic Processes
- 6 The Power Spectrum
- 7 Introduction to Multivariate Methods
- 8 Linear Regression: Least Squares Estimation
- 9 Linear Regression: Inference
- 10 Model Selection
- 11 Screening: A Pitfall in Statistics
- 12 Principal Component Analysis
- 13 Field Significance
- 14 Multivariate Linear Regression
- 15 Canonical Correlation Analysis
- 16 Covariance Discriminant Analysis
- 17 Analysis of Variance and Predictability
- 18 Predictable Component Analysis
- 19 Extreme Value Theory
- 20 Data Assimilation
- 21 Ensemble Square Root Filters
- Appendix
- References
- Index
8 - Linear Regression: Least Squares Estimation
Published online by Cambridge University Press: 03 February 2022
- Frontmatter
- Contents
- Preface
- 1 Basic Concepts in Probability and Statistics
- 2 Hypothesis Tests
- 3 Confidence Intervals
- 4 Statistical Tests Based on Ranks
- 5 Introduction to Stochastic Processes
- 6 The Power Spectrum
- 7 Introduction to Multivariate Methods
- 8 Linear Regression: Least Squares Estimation
- 9 Linear Regression: Inference
- 10 Model Selection
- 11 Screening: A Pitfall in Statistics
- 12 Principal Component Analysis
- 13 Field Significance
- 14 Multivariate Linear Regression
- 15 Canonical Correlation Analysis
- 16 Covariance Discriminant Analysis
- 17 Analysis of Variance and Predictability
- 18 Predictable Component Analysis
- 19 Extreme Value Theory
- 20 Data Assimilation
- 21 Ensemble Square Root Filters
- Appendix
- References
- Index
Summary
Some variables can be modeled by a linear combination of other random variables, plus random noise. Such models are used to quantify the relation between variables, to make predictions, and to test hypotheses about the relation between variables. After identifying the variables to include in a model, the next step is to estimate the coefficients that multiply them, called the regression parameters. This chapter discusses the least squares method for estimating regression parameters. The least squares method estimates the parameters by minimizing the sum of squared differences between the fitted model and the data. This chapter also describes measures for the goodness of fit and an illuminating geometric interpretation of least squares fitting. The least squares method is illustrated on various routine calculations in weather and climate analysis (e.g., fitting a trend). Procedures for testing hypotheses about linear models are discussed in the next chapter.
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- Chapter
- Information
- Statistical Methods for Climate Scientists , pp. 185 - 209Publisher: Cambridge University PressPrint publication year: 2022