A new family of constrained principal component analysis (CPCA)

Several decompositions of the orthogonal projector P_X = X(X′X)⁻X′ are proposed with a prospect of their use in constrained principal component analysis (CPCA). In CPCA, the main data matrix X is first decomposed into several additive components by the row side and/or column side predictor variables G and H. The decomposed components are then subjected to singular value decomposition (SVD) to explore structures within the components. Unlike the previous proposal, the current proposal ensures that the decomposed parts are columnwise orthogonal and stay inside the column space of X. Mathematical properties of the decompositions and their data analytic implications are investigated. Extensions to regularized PCA are also envisaged, considering analogous decompositions of ridge operators.