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: \[y = f\left(a_{0},a_{1},\,...,a_{K};x_{0},x_{1},\,...\right)\] where a0, a1, ..., aK 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 variable S, defined by \[S = \sum_{i}(w_{i}\ (y_{i}- f_{i}))^{2}\] and solving the equations obtained by writing \[\frac{\partial S}{\partial a_{j}}\ \overset{˜}{a}_{j}=0\] a(tilde)j = all a 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.
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)