Generalized weighted rules for principal components tracking

We investigate a general class of weighted subspace (WS) rules for principal component analysis (PCA) in order to show the difference of the existing rules. We focus in this paper on the well-known weighted principal components tracking rules that are developed by Oja and Xu. We unify these rules to more generalized form that is parameterized by a scalar. It is then proved that the generalized rules are stable at only the fixed point from which the principal components are extracted. We moreover find the parameter of the rules that gives the dynamics preserving orthogonality of estimated principal components most strongly during the tracking. Finally, toy examples and application in adaptive image compression are illustrated to understand the theoretical analysis of the stability.