17 Assessing dimensionality in multivariate regression

Abstract

This chapter describes a certain generalization of the multivariate linear regression model, which also provides a unified approach to the classical multivariate techniques of principal component and canonical variate and correlation analysis. A true multivariate feature enters the model when it is known that the regression coefficient matrix C may not have full-rank so that a number of linear restrictions on the set of regression coefficients of the model may be present. The regression coefficient matrix C is called “full-rank” or “reduced-rank” as appropriate. The chapter focuses on the sense by which one multivariate regression can be close to another multivariate regression and discusses the main results concerning reduced-rank regression and its relationship to principal component and canonical variate analysis, the nature of the residuals from a specific (known rank) reduced-rank regression, and the problem of assessing the rank of matrix C . New types of graphical displays by which the dimensionality may be determined are also elaborated in the chapter.