Model-based principal components of correlation matrices

Abstract

A model for principal components of correlation matrices is proposed. The model satisfies the correlation constraint (i.e., unit valued diagonal elements) as well as optional constraints on eigenvalues and/or eigenvectors. The model yields simplified principal components that retain both orthogonality and variance maximization properties. Inference procedures for eigenvalues, eigenvectors, and loadings on rotated or raw principal components are given. Multivariate normality is not required. A major issue in the modeling process is that the eigen-structure of the population correlation matrix can induce rank deficiencies in the submatrix of the constraint Jacobian matrix that is associated with the correlation constraint. This rank deficiency is a property of the population constraint Jacobian matrix; it is not necessarily a property of the sample Jacobian matrix evaluated at the solution to the estimating equation. Furthermore, if degenerate constraints are eliminated, then the fitted correlation matrix need not satisfy the correlation constraint. Procedures are proposed for detecting rank deficiencies, eliminating degenerate constraints, and constructing auxiliary constraints that ensure that the correlation constraint is satisfied. The procedures are illustrated on two real data sets.