Preferred Label : least-squares technique;
IUPAC definition : 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 \[\frac{ S}{ a_{j}}\ \overset{ }{a}_{j} 0\] 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.;
Origin ID : L03492;
See also
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 \[\frac{ S}{ a_{j}}\ \overset{ }{a}_{j} 0\] 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.
