Interpretation of the Probabilistic Principal Components Analysis with Anisotropic Gaussian Distribution of Latent Variables

Principal component analysis (PCA) is a well established technique for data analysis and processing. Recently, it has been shown that the principal axes of a set of observed data vectors might be determined trough maximum likelihood estimation of parameter in a specific form of latent variable model closely related to factor analysis. It is assumed that the latent variables have a unit isotropic Gaussian distribution. In view of this, in this study, we express some interpretation for covariance between PPCs, correlation between PPCs and variables, and covariance matrix between PPCs and PCs in common PCA case. Further, we consider more general case in which the latent variables are independent with different variances. We also investigate properties of the associated likelihood function.