Cross-Validatory Estimation of the Number of Components in Factor and Principal Components Models

Abstract

By means of factor analysis (FA) or principal components analysis (PCA) a matrix Y with the elements y _ik is approximated by the model

Here the parameters α, β and θ express the systematic part of the data yik, “signal,” and the residuals ∊ _ik express the “random” part, “noise.”

When applying FA or PCA to a matrix of real data obtained, for example, by characterizing N chemical mixtures by M measured variables, one major problem is the estimation of the rank A of the matrix Y, i.e. the estimation of how much of the data y _ik is “signal” and how much is “noise.”

Cross validation can be used to approach this problem. The matrix Y is partitioned and the rank A is determined so as to maximize the predictive properties of model (I) when the parameters are estimated on one part of the matrix Y and the prediction tested on another part of the matrix Y.

KEYWORDS:

• Factor analysis
• Principal components
• Eigenvector decomposition
• Singular value decomposition
• Rank determination