Frameworks for latent variable multivariate regression

A set of frameworks for latent variable multivariate regression method is developed. The first two of these frameworks describe the objective functions satisfied by the latent variables chosen in canonical coordinates regression (CCR), reduced rank regression (RRR) and SIMPLS. These frameworks show the methods as a natural progression from CCR (maximizing correlation) to SIMPLS (maximizing covariance) via RRR (which is an intermediate method). These frameworks are unique in that they look at these methods in terms of latent variables in both the X‐ and Y‐spaces. This adds insight to the nature of the latent variables being chosen. These frameworks are then extended to include PLS for latent variables beyond the first component. This new framework provides a detailed description of the objective function satisfied by PLS latent variables for the multivariate case. It also includes CCR, RRR and SIMPLS, allowing comparisons between the methods. A further framework suggests a new method, undeflated PLS (UDPLS), which adds insight to the effect of the deflation process on PLS. The impact of the objective functions on each of the methods is illustrated on real data from a mineral sorting plant.