{"term":{"id":"10093","title":"factor analysis","longtitle":"IUPAC Gold Book - factor analysis","doi":"10.1351\/goldbook.10093","code":"10093","status":"current","definitions":[{"id":1,"text":"Matrix decomposition of a data matrix (\\(\\boldsymbol{X}\\)) into the product of a scores matrix (\\(\\boldsymbol{T}\\)) and the transpose of the loadings matrix (\\(\\boldsymbol{P}^{\\rm{T}}\\)).","notes":{"1":"Hence \\(\\boldsymbol{X} = \\boldsymbol{TP}^{\\rm{T}} + \\boldsymbol{E}\\), where \\(\\boldsymbol{E}\\) is a residual matrix.","2":"Factor analysis methods include common factor analysis (also called ‘factor analysis’), principal component analysis, and multivariate curve resolution.","3":"The number of factors selected in factor analysis is smaller than the rank of the data matrix.","4":"Factor analysis is equivalent to a rotation in data space where the factors form the new axes. This is not necessarily rotation that maintains orthogonality except in the case of PCA.","5":"The residual matrix contains data that are not described by the factor analysis model, and is usually assumed to contain noise."},"links":[{"term":"common factor analysis","url":"https:\/\/goldbook.iupac.org\/\/terms\/view\/10089"},{"term":"data matrix","url":"https:\/\/goldbook.iupac.org\/\/terms\/view\/10047"},{"term":"loadings","url":"https:\/\/goldbook.iupac.org\/\/terms\/view\/10096"},{"term":"multivariate curve resolution","url":"https:\/\/goldbook.iupac.org\/\/terms\/view\/10100"},{"term":"noise","url":"https:\/\/goldbook.iupac.org\/\/terms\/view\/10041"},{"term":"principal component analysis","url":"https:\/\/goldbook.iupac.org\/\/terms\/view\/10106"},{"term":"scores","url":"https:\/\/goldbook.iupac.org\/\/terms\/view\/10112"}],"sources":["PAC, 2016, 88, 407. 'Vocabulary of concepts and terms in chemometrics (IUPAC Recommendations 2016)' on page 420 (https:\/\/doi.org\/10.1515\/pac-2015-0605)"]}],"altoutputs":{"html":"https:\/\/goldbook.iupac.org\/terms\/view\/10093\/html","xml":"https:\/\/goldbook.iupac.org\/terms\/view\/10093\/xml","plain":"https:\/\/goldbook.iupac.org\/terms\/view\/10093\/plain"},"citation":"Citation: 'factor 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.10093","license":"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.","collection":"If you are interested in licensing the Gold Book for commercial use, please contact the IUPAC Executive Director at executivedirector@iupac.org .","disclaimer":"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.","accessed":"2026-07-16T02:26:04+00:00"}}