A constrained PARAFAC method for positive manifold data

Abstract

A set of non-negatively correlated variables, referred to as positive manifold data, display a peculiar pattern of loadings in principal components analysis (PCA). If a small set of principal components is rotated to a simple structure, the variables correlate positively with all components, thus displaying positive manifold. However, this phenomenon is critically dependent on the freedom of rotation, as is evident from the unrotated loadings. That is, although the first principal component is without contrast (which means that all variables correlate either positively or negatively with the first component), subsequent components have mixtures of positive and negative loadings-which means that positive manifold is absent. PARAFAC is a generalization of PCA that has unique components, which means that rotations are not allowed. This paper examines how PARAFAC behaves when applied to positive manifold data. It is shown that PARAFAC does not always produce positive manifold solutions. For cases in which PARAFAC does not produce a positive manifold solution, a constrained PARAFAC method is offered that restores positive manifold by introducing non-negativity constraints. Thus, noncontrast PARAFAC components can be found that explain only a negligible amount of variance less than the PARAFAC components. These noncontrast components cannot be degenerate and cannot be partially unique in the traditional sense. Index terms: degenerate components; noncontrast components; non-negativity constraints; PARAFAC; positive manifold.