{"term":{"id":"03492","doi":"10.1351\/goldbook.L03492","code":"L03492","status":"current","longtitle":"IUPAC Gold Book - least-squares technique","title":"least-squares technique","version":"2.3.3","lastupdated":"2014-02-24","definitions":[{"id":"1","text":"A procedure for replacing the discrete set of results obtained from an experiment by a continuous function. It is defined by the following. For the set of variables y , x 0 , x 1 , ... there are n measured values such as y i , x 0 i , x 1 i , ... and it is decided to write a relation: y = f a 0 a 1 ... a K x 0 x 1 .. are undetermined constants. If it is assumed that each measurement y i of y has associated with it a number w i-1 characteristic of the uncertainty, then numerical estimates of the a0, a1, ..., aK are found by constructing a variableS, defined by S = ∑ i ( w i ( y i- f i ) ) 2 except aj. If the relations between the a and y are linear, this is the familiar least-squares technique of fitting an equation to a number of experimental points. If the relations between the a and y are non-linear, there is an increase in the difficulty of finding a solution, but the problem is essentially unchanged.","links":[{"title":"variable","type":"goldify","url":"https:\/\/goldbook.iupac.org\/terms\/view\/V06600"}],"math":[{"alttext":"y , x 0 , x 1 , ...","latex":"y,x_{0},x_{1},\\,..."},{"alttext":"n","latex":"n"},{"alttext":"y i , x 0 i , x 1 i , ...","latex":"y_{i},x_{0i},x_{1i},\\,..."},{"alttext":"y = f a 0 a 1 ... a K x 0 x 1 ..","latex":"y = f\\left(a_{0},a_{1},\\,...,a_{K};x_{0},x_{1},\\,...\\right)"},{"alttext":"a0, a1, ..., aK","latex":"a_{0},a_{1},\\,...,a_{K}"},{"alttext":"y i","latex":"y_{i}"},{"alttext":"y","latex":"y"},{"alttext":"w i-1","latex":"w_{i}^{-1}"},{"alttext":"a0, a1, ..., aK","latex":"a_{0},a_{1},\\,...,a_{K}"},{"alttext":"S","latex":"S"},{"alttext":"S = ∑ i ( w i ( y i- f i ) ) 2","latex":"S = \\sum_{i}(w_{i}\\ (y_{i}- f_{i}))^{2}"},{"type":"math","alttext":"∂S\/∂aj.a~j = 0","latex":"\\frac{\\partial S}{\\partial a_{j}}\\ \\overset{˜}{a}_{j}=0"},{"alttext":"a(tilde)j = all a","latex":"\\overset{˜}{a}_{j}=\\text{all}\\:\\, a"},{"alttext":"aj","latex":"a_{j}"},{"alttext":"a","latex":"a"},{"alttext":"y","latex":"y"},{"alttext":"a","latex":"a"},{"alttext":"y","latex":"y"}],"sources":["PAC, 1981, 53, 1805. 'Assignment and Presentation of Uncertainties of the Numerical Results of Thermodynamic Measurements' on page 1822 (https:\/\/doi.org\/10.1351\/pac198153091805)"]}],"links":{"html":"https:\/\/goldbook.iupac.org\/terms\/view\/L03492\/html","json":"https:\/\/goldbook.iupac.org\/terms\/view\/L03492\/json","xml":"https:\/\/goldbook.iupac.org\/terms\/view\/L03492\/xml","plain":"https:\/\/goldbook.iupac.org\/terms\/view\/L03492\/plain","pdf":"https:\/\/goldbook.iupac.org\/terms\/view\/L03492\/pdf"},"citeas":"IUPAC. Compendium of Chemical Terminology, 2nd ed. (the \"Gold Book\"). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997). Online version (2019-) created by S. J. Chalk. ISBN 0-9678550-9-8. https:\/\/doi.org\/10.1351\/goldbook.","license":"Licensed under Creative Commons Attribution-NoDerivatives (CC BY-NC-ND) 4.0 International (https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/)","collection_reuse":"For parties interested in reusing the entire IUPAC Gold Book please contact IUPAC here https:\/\/www.cognitoforms.com\/IUPAC1\/ContactInformationForm","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":"2023-06-08T04:21:22+00:00"}}