Estimating Measurement Uncertainty

Most analytical methods are relative and hence require calibration. Calibration measurements are normally performed with reference materials or calibration standards. Typically least squares methods have been employed in such a manner as to ignore uncertainties associated with the calibration standards. However different authors [3, 4, 11] have shown that by taking into account the uncertainties associated with the calibration standards, better approximations to the model are possible. We show a complete, numerically stable and fast way to compute the Maximum Likelihood (fitting of a) Functional Relationship (MLFR), and we show how the model described in [11] is a unnecessarily restricted case of our fully general model. 1 Calibration and Measurement Since measurement instruments obey physical laws, they can be modelled by mathematical functions. These functions describe only the qualitative behaviour unless all free parameters p are quantified. Calibration is the process of quantifying these parameters. For example, consider the measurement of a weight using a spring balance. Here we denote weights with x ∈ R and stretches of the spring with y ∈ R. Let xcs be a known weight of a calibration standard and ycs be the stretch of the spring balance for this calibration standard. For a given calibrated balance p the following holds true: ycs = f(xcs,p). This f is called the calibration function. The art of calibrating consists of adjusting the parameters p (e.g. spring constant) using several calibrating pairs xi and yi such that yi ≈ f(xi,p) holds for all i as well as possible. For an unknown mass x̂, weighing with a calibrated spring balance requires the measurement of a spring stretch ŷ and subsequent application of the measurement function g(ŷ,p). In other words, x̂ ≈ g(ŷ,p). Note that g is the inverse function of f . In the case that the pairs (xi, yi) were exact this would be a standard problem. But in practice every measurement is subject to uncertainties. Therefore, not only the calibrating pairs (xi, yi) but also their respective uncertainties (αi, βi) must be known and taken into consideration. Throughout this paper the error random variables e i respectively e (y) i are treated as e (x) i ∼ N (0, αi) respectively e (y) i ∼ N (0, βi). This fully conforms with the “Guide to the Expression of Uncertainty in Measurement” (GUM) [10]. With respect to our spring balance example, this implies that the real calibration weights are not xi but xi + e (x) i . Analogously the correct values for the stretch are no longer yi but yi + e (y) i . Note that the uncertainties of x and y are uncorrelated. That is, the error of the calibration standard is independent of the error of the reading.