Uncertainty of functionals of calibration curves

摘要

Typically an estimate f(x) of an unknown response curve f(x) is obtainedwith an associated function u[f(x)] describing the standard uncertainty of the estimate at each value of x. Often the quantity of interest will be a functional of f(x), such as a derivative or integral. In such a case the standard uncertainty cannot be calculated without knowledge of the correlation between u[f(x_(i))] and u[f(x_(j))] for all relevant pairs of points (x_(i),x_(j)). This information might be stored as a two-dimensional function u[f(x_(i)), f(x_(j))] in the continuous case or as a matrix (u_(ij)) in the discrete case, but this will often be impractical. The difficulty can often be avoided by instead storing the 'random' and 'systematic' components of uncertainty, which is a concept that is familiar but out of favour. This step enables the calculation of standard uncertainty for many functionals of f(x) from numerical data and from a graphical representation. Three examples are given illustrating these concepts. The paper also discusses the issue of expressing the uncertainty associated with f(x) as a whole; that is, the simultaneous estimation of f(x) at every value of x.