{"term":{"id":"10152","title":"multilinear least squares regression","longtitle":"IUPAC Gold Book - multilinear least squares regression","doi":"10.1351\/goldbook.10152","code":"10152","status":"current","initialism":"<em>initialisms<\/em>: MLR, MLSR","definitions":[{"id":1,"text":"Multivariate calibration in which the coefficients of the regression are calculated directly from the indications and values of standards.","notes":{"1":"In \\(\\boldsymbol{X} \\boldsymbol{b} = \\boldsymbol{c} + \\boldsymbol{e}\\) (where \\(\\boldsymbol{X}\\) is a vector of indications, \\(\\boldsymbol{b}\\) are coefficients of the model, \\(\\boldsymbol{c}\\) is a vector of values of the quantity of interest and \\(\\boldsymbol{e}\\) a vector of errors), \\(\\boldsymbol{b} = \\boldsymbol{X^{+}} \\boldsymbol{c}\\), where \\(\\boldsymbol{X^{+}}\\) is the pseudo-inverse of \\(\\boldsymbol{X}\\), calculated as \\(\\boldsymbol{X^{+}} = (\\boldsymbol{X}^{\\rm{T}} \\boldsymbol{X})^{-1} \\boldsymbol{X}^{\\rm{T}}\\)","2":"It is assumed that \\(\\boldsymbol{X}^{\\rm{T}} \\boldsymbol{X}\\) has full rank, i.e. there are more objects than variables, and the variables are independent.","3":"If \\(\\boldsymbol{X}^{\\rm{T}} \\boldsymbol{X}\\) has full rank only because of noise, the solution can become unstable."},"links":[{"term":"Multivariate calibration","url":"https:\/\/goldbook.iupac.org\/\/terms\/view\/10153"},{"term":"noise","url":"https:\/\/goldbook.iupac.org\/\/terms\/view\/10041"}],"sources":["PAC, 2016, 88, 407. 'Vocabulary of concepts and terms in chemometrics (IUPAC Recommendations 2016)' on page 434 (https:\/\/doi.org\/10.1515\/pac-2015-0605)"]}],"altoutputs":{"html":"https:\/\/goldbook.iupac.org\/terms\/view\/10152\/html","xml":"https:\/\/goldbook.iupac.org\/terms\/view\/10152\/xml","plain":"https:\/\/goldbook.iupac.org\/terms\/view\/10152\/plain"},"citation":"Citation: 'multilinear least squares regression' 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.10152","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":"2026-08-07T09:49:37+00:00"}}