Principal component analysis with external information on both subjects and variables

Abstract

orthogonal projection operator, trace-orthogonality, generalized singular value decomposition (GSVD), QR decomposition, vector preference models, two-way CANDELINC, dual scaling, redundancy analysis, GMANOVA (growth curve models),

Analysis of Knowledge Representations in Cascade Correlation Networks

“…We may eliminate the effect of the summed contributions from the matrix of contributions and analyze the rest. More generally, the matrix of contributions may be decomposed into several components before each component is subjected to PCA (Takane & Shibayama, 1991). In general, partialing out known or trivial effects from the matrix of contributions is an effective way of extracting unique contributions of particular units.…”

Generalized Constrained Redundancy Analysis

“…First, data matrix Y is decomposed into the sum of several matrices according to the external information on rows and columns. If both row and column information matrices on Y are available, the data matrix can be decomposed into the sum of four matrices (Takane and Shibayama, 1991),…”

“…Takane and Shibayama (1991;also, see Takane and Hunter, 2001) proposed a comprehensive framework for various decompositions of the data matrix Y incorporating its row and column information matrices, G Y and H Y . Takane, Yanai, and Hwang (2006) also proposed a comprehensive framework for various decompositions of the orthogonal projector P [X,G X ] onto the range space of [X, G X ], incorporating the column information matrix H X on X.…”

Structural Equation Modeling by Extended Redundancy Analysis

“…I-Iere it is assumed that W'Z~Z2W = 1 for identification. Minimizing (2.3) computationally cornes down to calculating the generalized singular value decomposition (GSVO) of (Z~Z2)-lZ~Zl with metric matrices Z~Z2 and 1 (e.g., Takane & Shibayama, 1991). (For the computation of GSVO, refer to Greenacre, 1984, Appendix A.…”

“…Two kinds ofmetric matrices May be considered, one on the row side~and the other on the column side ofZ(l) (e.g., Takane & Shibayama, 1991). Let K denote an n by n row-side metric matrix.…”