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 \(y_{i},x_{0i},x_{1i},\,...\) and it is decided to write a relation: \[y = f\left(a_{0},a_{1},\,...,a_{K};x_{0},x_{1},\,...\right)\] 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 @V06600@ \(S\), defined by \[S = \sum_{i}(w_{i}\ (y_{i}- f_{i}))^{2}\] and solving the equations obtained by writing \[\frac{\partial S}{\partial a_{j}}\ \overset{˜}{a}_{j}=0\] \(\overset{˜}{a}_{j}=\text{all}\:\, a\) 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.
PAC, 1981, 53, 1805. (Assignment and Presentation of Uncertainties of the Numerical Results of Thermodynamic Measurements) on page 1822 [Terms] [Paper]