MULTIVARIATE LATENT VARIABLE REGRESSION: MODELLING AND ESTIMATION

Abstract

<p>The topic of the multivariate latent variable regression model as it applies to practical problems in applied science is investigated. This model is used for sets of data that have been divided into two groups, X and Y, both containing multiple variables and linked by a common latent structure. A detailed statistical model is described for this application. This model is based on models and descriptions found in the literature but is unique in its clear definitions for all of the components. The wide applicability of this model is clearly demonstrated on four real examples taken from both chemistry and chemical engineering. A set of frameworks has been developed for the objective functions satisfied by the estimates of the latent variables in the model. The frameworks encompass the majority of the methods currently used or proposed for use for the multivariate latent variable regression problems. This work provides a basis for research into when each method would be appropriate. Finally, a new method of parameter estimation is developed using the maximum likelihood method on the model with the added assumptions of multivariate normal errors in both spaces with known covariance matrices. This method involves a constraint on the maximization due to the practical nature of the division of the data into X and Y spaces. A special case of this method is used to form a continuum regression that spans several of the existing methods and in particular is shown to be very close to the method used most often in practice, multivariate partial least squares (PLS). This provides some statistical justification for the use of PLS and some guidelines regarding its implementation.</p>