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least-squares technique

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 ...
where a 0 , a 1 , ... , a K 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 a 0 , a 1 , ... , a K are found by constructing a variable S, defined by
S = ∑ i ( w i ( y i − f i ) ) 2,
and solving the equations obtained by writing
∂ S ∂ a j a ˜ j = 0
a ˜ j = all a except a j. 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.
Source:
PAC, 1981, 53, 1805 (Assignment and presentation of uncertainties of the numerical results of thermodynamic measurements (Provisional)) on page 1822
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IUPAC. Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). Compiled by A. D. McNaught and A. Wilkinson. Blackwell Scientific Publications, Oxford (1997). XML on-line corrected version: http://goldbook.iupac.org (2006-) created by M. Nic, J. Jirat, B. Kosata; updates compiled by A. Jenkins. ISBN 0-9678550-9-8. doi:10.1351/goldbook.
Last update: 2014-02-24; version: 2.3.3.
DOI of this term: doi:10.1351/goldbook.L03492.
Original PDF version: http://www.iupac.org/goldbook/L03492.pdf. The PDF version is out of date and is provided for reference purposes only. For some entries, the PDF version may be unavailable.
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