## least-squares technique

https://doi.org/10.1351/goldbook.L03492
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.
Source:
PAC, 1981, 53, 1805. (Assignment and Presentation of Uncertainties of the Numerical Results of Thermodynamic Measurements) on page 1822 [Terms] [Paper]