Generalized Procrustes Analysis

1975/03/01 by J. C. Gower · 3,267 citations
Agricultural and Biological Sciences · Computer Science · Mathematics · #Applied mathematics #Blind Source Separation Techniques #Centroid #Computer science #Configuration space #Geometry #Goodness of fit #Leaf Properties and Growth Measurement #Mathematics #Multidimensional scaling #Physics #Procrustes analysis #Scaling #Set (abstract data type) #Statistics #Tensor decomposition and applications #Variance (accounting)

paper · doi:10.1007/bf02291478

published in Psychometrika 40(1), 33-51 (Springer Science+Business Media)

openalex publication_date 1975/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Suppose P i ( i ) ( i = 1, 2, ..., m, j = 1, 2, ..., n ) give the locations of mn points in p -dimensional space. Collectively these may be regarded as m configurations, or scalings, each of n points in p -dimensions. The problem is investigated of translating, rotating, reflecting and scaling the m configurations to minimize the goodness-of-fit criterion Σ i=1 m Σ i=1 n Δ 2 ( P j ( i ) G i ), where G i is the centroid of the m points P i ( i ) ( i = 1, 2, ..., m ). The rotated positions of each configuration may be regarded as individual analyses with the centroid configuration representing a consensus, and this relationship with individual scaling analysis is discussed. A computational technique is given, the results of which can be summarized in analysis of variance form. The special case m = 2 corresponds to Classical Procrustes analysis but the choice of criterion that fits each configuration to the common centroid configuration avoids difficulties that arise when one set is fitted to the other, regarded as fixed.

Citations

• Analysis of Individual Differences in Multidimensional Scaling Via an N-way Generalization of “Eckart-Young” Decomposition
• The Approximation of One Matrix by Another of Lower Rank

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