Latent models for cross-covariance
Cross-covariance problems arise in the analysis of multivariate data that can be divided naturally into two blocks of variables, X and Y, observed on the same units. In a cross-covariance problem we are interested, not in the within-block covariances, but in the way the Ys vary with the Xs.In the current work several approaches to the cross-covariance problem are discussed, including Reduced-Rank Regression (RRR), Canonical Correlation Analysis (CCA), Partial Least Squares (PLS, also called Projection to Latent Structures), Structural Equation Models (SEM), and Graphical Markov Models (GMM).A family of latent models for cross-covariance, called paired latent models, is specified. It is shown that the set of covariance matrices which can be modeled under the rank-r paired latent model is the same as those which can be modeled under rank-r Reduced-Rank Regression. The degree to which the parameters of the rank-one paired latent model are underidentified is precisely characterized, and a natural convention is proposed which makes the model identifiable. This result has implications for the estimation of correlation between the latent variables.It is shown that symmetric and asymmetric versions of the paired latent model are covariance equivalent, and that this equivalence fails when the within-block covariance is constrained to be diagonal.
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