14 - Multivariate Linear Regression

Summary

Multivariate linear regression is a method for modeling linear relations between two random vectors, say X and Y. Common reasons for using multivariate regression include (1) to predicting Y given X, (2) to testing hypotheses about the relation between X and Y, and (3) to projecting Y onto prescribed time series or spatial patterns. Special cases of multivariate regression models include Linear Inverse Models (LIMs) and Vector Autoregressive Models. Multivariate regression also is fundamental to other statistical techniques, including canonical correlation analysis, discriminant analysis, and predictable component analysis. This chapter introduces multivariate linear regression and discusses estimation, measures of association, hypothesis testing, and model selection. In climate studies, model selection often involves selecting Y as well as X. For instance, Y may be a set of principal components that need to be chosen, which is not a standard selection problem. This chapter introduces a criterion for selecting X and Y simultaneously called Mutual Information Criterion (MIC).