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