The results of an analysis that depends on a calibration curve is only as good
as the curve.  Selecting the curve that best matches the calibration points is
crucial to obtaining reasonable results.
Most instruments have an approximately linear response.  Therefore the user may
be tempted to use a straight line to fit the calibration data.  However, this
may result in undesirable behavior such as negative concentrations or a finite
concentration for no instrument response.
This paper explores alternatives to the simple straight line calibration curve.
Linear regression is a more general technique than simply fitting straight
lines.  Linear regression can be used to find optimal coefficients for any set
of linearly independent basis functions.  Examples of such basis functions are
monomials.  Therefore, linear regression can be used to fit quadratic or higher
order polynomials to the data.
More exotic basis functions can also be used, such as exponential or other
transcendental functions.  The only requirement is that the basis functions be
linearly independent.  The paper explores examples of where such functions are
appropriate.
In some circumstances, a nonlinear function can be transformed such that
transformed function consists of basis functions that are linearly independent.
Linear regression can then be used to find optimal coefficients to this
nonlinear function.
In addition, nonlinear regression can be used to find optimal coefficients for
curves where linear regression cannot be used.  The paper explores examples of
using nonlinear regression to fit calibration data.
All methods for finding optimal coefficients rely on a measure that combines the
discrete calibration points into a cost function which is then minimized.  The
contribution to this cost function can in general be weighted to assign a
greater or smaller importance to particular calibration points.  The paper
explores different methods for assigning weights to the calibration points and
the effects this has on the calibration curve.