## Joint Procrustes analysis for simultaneous nonsingular transformation of component score and loading matrices.(English)Zbl 1179.62084

Summary: In component analysis solutions, post-multiplying a component score matrix by a nonsingular matrix can be compensated by applying its inverse to the corresponding loading matrix. To eliminate this indeterminacy on nonsingular transformation, we propose Joint Procrustes Analysis (JPA) in which component score and loading matrices are simultaneously transformed so that the former matrix matches a target score matrix and the latter matches a target loading matrix. The loss function of JPA is a function of the nonsingular transformation matrix and its inverse, and is difficult to minimize straightforwardly. To deal with this difficulty, we reparameterize those matrices by their singular value decomposition, which reduces the minimization to alternately solving quartic equations and performing the existing multivariate procedures. This algorithm is assessed in a simulation study. We further extend JPA for cases where targets are linear functions of unknown parameters. We also discuss how the application of JPA can be extended in different fields.

### MSC:

 62H25 Factor analysis and principal components; correspondence analysis 62H99 Multivariate analysis 62P15 Applications of statistics to psychology 65C60 Computational problems in statistics (MSC2010) 15A99 Basic linear algebra
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