CPCA: a comprehensive theory

Constrained principal component analysis (CPCA) incorporates external information into principal component analysis (PCA). CPCA first decomposes the matrix according to the external information (external analysis) and then applies PCA to decomposed matrices (internal analysis). The external analysis amounts to projections of the data matrix onto the spaces spanned by matrices of external information, while the internal analysis involves the generalized singular value decomposition (GSVD). Since its original proposal (Takane and Shibayama, 1991), CPCA has evolved both conceptually and methodologically; it is now founded on firmer mathematical ground, allows a greater variety of decompositions, and includes a wider range of interesting special cases. In this paper we present a comprehensive theory and various extensions of CPCA. We also discuss four special cases of CPCA; 1) CCA (canonical correspondence analysis) and CALC (canonical analysis with linear constraints), 2) GMANOVA, 3) Lagrange's theorem, and 4) CANO (canonical correlation analysis) and related methods.