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
18 - Predictable Component Analysis
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
The previous chapter discussed Analysis of Variance (ANOVA), a procedure for deciding if populations have identical scalar means. This chapter discusses the generalization of this test to vector means, which is called Multivariate Analysis of Variance, or MANOVA. MANOVA can detect predictability of random vectors and decompose a random vector into an sum of components ordered such that the first maximizes predictability, the second maximizes predictability subject to being uncorrelated with the first, and so on. This decomposition is called Predictable Component Analysis (PrCA) or signal-to-noise maximizing EOF analysis. A slight modification of this procedure can decompose forecast skill. The connection between PrCA, Canonical Correlation Analysis, and Multivariate Regression is reviewed. In typical climate studies, the dimension of the random vector exceeds the number of samples, leading to an ill-posed problem. The standard approach to this problem is to apply PrCA on a small number of principal components. The problem of selecting the number of principal components can be framed as a model selection problem in regression.
- Type
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- Information
- Statistical Methods for Climate Scientists , pp. 418 - 445Publisher: Cambridge University PressPrint publication year: 2022