10108 IUPAC Gold Book - Procrustes analysis 10.1351/goldbook.10108 10108 current 1 Comparison of shapes of multi-dimensional objects by a series of geometrical transformations to minimise the sum of squared distances between the transformed and target structures while maintaining the internal structure of the objects. For two objects defined by \(\boldsymbol{X}\) and \(\boldsymbol{Y}\) the manipulation orthogonal rotation/reflection matrix \(\boldsymbol{R}\) Procrustes analysis minimises: \(\parallel\! \boldsymbol{Y} - \boldsymbol{X}\boldsymbol{R}\parallel^{2}\) subject to \(\boldsymbol{R}^{\rm{T}} \boldsymbol{R} = \boldsymbol{R} \boldsymbol{R}^{\rm{T}} = 1\). Ordinary, or classical, Procrustes analysis is when an object is compared to one other object, which may be a reference shape. Generalized Procrustes analysis compares three or more shapes to an optimally-determined mean shape. PAC, 2016, 88, 407. 'Vocabulary of concepts and terms in chemometrics (IUPAC Recommendations 2016)' on page 423 (https://doi.org/10.1515/pac-2015-0605) https://goldbook.iupac.org/terms/view/10108/html https://goldbook.iupac.org/terms/view/10108/json https://goldbook.iupac.org/terms/view/10108/plain Citation: 'Procrustes analysis' in IUPAC Compendium of Chemical Terminology, 5th ed. International Union of Pure and Applied Chemistry; 2025. Online version 5.0.0, 2025. 10.1351/goldbook.10108 The IUPAC Gold Book is licensed under Creative Commons Attribution-ShareAlike CC BY-SA 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) for individual terms. If you are interested in licensing the Gold Book for commercial use, please contact the IUPAC Executive Director at executivedirector@iupac.org . The International Union of Pure and Applied Chemistry (IUPAC) is continuously reviewing and, where needed, updating terms in the Compendium of Chemical Terminology (the IUPAC Gold Book). Users of these terms are encouraged to include the version of a term with its use and to check regularly for updates to term definitions that you are using. 2026-08-06T08:03:44+00:00