Procrustes Problems

Let X1 and X2 be given matrices. The simplest Procrustes problem is to find a matrix T that minimizes . In analogy with the story of Procrustes, son of Poseidon, we can imagine that X1 is the unfortunate traveller, X2 is the Procrustean bed, and T is the treatment (racking, hammering, or amputation). Under classical conditions, minimization of is the multivariate multiple regression problem with the solution T=(X1’X1)-1X1’X2. The two-sided form of the Procrustes problem is to minimize under some conditions. A further variant consists of the double Procrustes problem concerning minimization of . The solutions of the mentioned problems depend on the set of matrices T, which are allowed to be used. In some cases T is constrained to be an orthogonal matrix, which represents a generalized rotation, in other problems T must be a permutation matrix and so on. Generally we can say that Procrustean methods are used to transform one set of data to represent another set of data as closely as possible. The book also contains oblique Procrustes problems, weighting, scaling, and missing values. Further topics are accuracy and stability, and some applications. The authors inform that the process of writing the book was similar to the machinations of Procrustes, the material seemed to be continuously stretching and so it was necessary to chop things. However, this is probably the fate of many mathematical papers and books.

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ISBN 0-19-851058-6

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