Generalized joint Procrustes analysis
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Generalized joint Procrustes analysis Kohei Adachi
Received: 15 October 2012 / Accepted: 14 March 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract In this paper, we propose a generalized version of Adachi’s (Psychometrika 74:667–683, 2009) Joint Procrustes Analysis (GJPA), in order to transform the principal component score and loading matrices obtained for multiple data sets of the same size. The transformation is made so that multiple score and loading matrices are optimally matched to two unknown target matrices, respectively, without affecting the fit of the score and loading matrices to data sets, and without any constraint imposed on the transformation, except for its being nonsingular. The resulting transformed score and loading matrices can reasonably be compared across data sets. A simulation study is performed for assessing an alternate least-squares algorithm for GJPA. Additional procedures for interpreting GJPA solutions are also presented and they are illustrated with a real data example. Finally, GJPA is reconsidered in the contexts of three-way data analysis and canonical correlation analysis. Keywords Procrustes analysis · Principal component analysis · Nonsingular transformation · Canonical correlation analysis · Multiple data sets 1 Introduction Principal component analysis (PCA) for an n-objects × m-variables data matrix X can be formulated as minimizing X − GB 2 over component matrices G (n × p) and B (m × p) with the number of components, say p, not exceeding min(n, m) (e.g., Krzanowski and Marriott 1994, p. 85). In this formulation, the solution is not uniquely determined with GB = GSS−1 B ; matrices GS and BS−1 , transformed by
K. Adachi (B) Graduate School of Human Sciences, Osaka University, 1-2 Yamadaoka, Suita, Osaka 565-0871, Japan e-mail: [email protected]
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an arbitrary p× p nonsingular matrix S, can also be regarded as the optimal component matrices. To reduce this indeterminacy, identification conditions are imposed on G or B. Alternatively, Adachi (2009) proposed a method for fixing S at the matrix optimally matching GS and BS−1 to target matrices F and A, respectively; that is, for minimizing φ0 (S) =
2 1 1 GS − F2 + BS−1 − A n m
(1)
over S. Though Adachi (2009) described the minimization of (1) divided by p, it is equivalent to the minimization of (1). The name for this procedure has been coined joint Procrustes analysis (JPA), since GS and BS−1 are jointly matched to target matrices, in contrast to ordinary Procrustes analysis in which either matching GS to F or matching BS−1 to A is performed (Gower and Dijksterhuis 2004; Green 1952; Mosier 1939). A feature unique to JPA is that (1) is a function of a parameter matrix and its inverse. In this paper, we generalized JPA toward cases where PCA is performed separately for n-objects × m-variables data matrix Xk obtained from source k = 1, . . ., K . In this separate PCA, Xk −Gk Bk 2 is minimized over Gk and Bk . Since its solution is not uniquely determined with Gk Bk = Gk
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