18 - Predictable Component Analysis

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.