Some aspects of principal components analysis as measurement error model

Abstract

In this paper , Principal Component Analysis (PCA) is formulated within a likelihood framework, based on a specific form of Gaussian model. We suggest how the principal axes of observed data vectors may be determined through maximum-likelihood estimation of parameters in a latent variable model closely related to a Factor Analysis and Error-in-Variables Models. We consider the properties of the associated likelihood function and discuss the advantages conveyed by the definition of a probability density function for PCA. The numeric/graphical quantities of the usual output now represent the maximum likelihood estimates of the parameters of a probabilistic model, rather than approximations of an algebraic-geometric algorithm. The use of PCA as method of analysis communicated by the interpretation of the principal components: we determine the subdivision a posteriori predictors of redundant variables. We investigate the proposed model in a case-control study of cardiovascular disease.