## 4 Choosing a \(D\)-Optimal Subset of \(n\) Measurements
### Set
Thus, there are disjoint subsets \(L_{i}\), \(R_{i}\subset I=\{1,2,\dots,n\}\), \(L_{i}\cap R_{i}=\emptyset\), \(i=1,\dots,m\), such that

\[\sum_{q\in L_{i}}a_{q}\approx\sum_{q\in R_{i}}a_{q}.\]

An example considered further below involves a set of artefacts having nominal values 1.00, 0.50, 0.50, 0.20, 0.20, 0.10, 0.10, 0.05, and 0.05. The fact that the two subsets are associated

Figure 10: Hyperbolic paraboloid calibration points: values of \(t_{q}\), \(q=1,\dots,100\), and its analytical approximation \(\tilde{I}(q)\) given by (12), determined by the sequential D-optimal Algorithm 4, with no repeat measurements, upper curve, and with repeat measurements, lower curve. with the same nominal value allows a comparator measurement to be made, with the associated observation equation

\[0\approx y_{i}=\sum_{q\in L_{i}}a_{q}-\sum_{q\in R_{i}}a_{q}+\epsilon_{i},\quad \epsilon_{i}\in\mathcal{N}(0,\sigma_{i}^{2}),\]

where \(\epsilon_{i}\) represents a random effect associated with the comparator measurement. The design problem is to define measurement strategies encoded by \(L_{i}\) and \(R_{i}\), \(i=1,\ldots,n\), that provide the most accurate calibration of the standards. For the example considered here, there are almost 400 viable experiments. Tables 4 and 5 show five designs for the calibration of the \(n=9\) standards. A '1' in column \(j\) indicates that \(j\in L_{i}\), while a '\(-1\)' indicates that \(j\in R_{i}\). Each array of \(9\times 9\) elements is in fact the unweighted observation matrix \(C\) associated with the particular design. The first row in each design represents the observation associated with the calibration of the first standard using an absolute measurement system. This first observation is the only source of information about the first standard and must be included in the optimal solution. The uncertainties associated with the remaining standards scale with the uncertainty associated with the first observation. The first design, given in Table 4, is one defined by 'expert judgement', and represents a typical set of experiments that would be performed in practice. (In fact, for this set of standards, it is not entirely obvious how to choose a design that will calibrate all of the standards.) The next design, indicated by the first nine rows in Table 5, is that determined by the SSQR-GE algorithm with \(\sigma_{1}=\sigma_{C}=1.0\) and \(\sigma_{i}=\sigma_{R}=0.5\), \(i=2,\ldots,n\). The most striking and, at first sight, surprising feature of this second design is that almost all the artefacts are involved in experiments 2 to \(n\). The reason for this is that, in some sense, the comparator uncertainties represented by \(\sigma_{i}=\sigma_{R}\) are partitioned amongst the artefacts involved, and more artefacts being included in an experiment means that each artefact is assigned a smaller uncertainty. The two designs considered so far are based on the assumption that all the comparator measurements are associated with the same uncertainty \(\sigma_{R}\), irrespective of the number of artefacts and the nominal values of the artefacts. A more plausible assignment of uncertainties would take into account both the number and nominal values. For a mass balance, it would be usual for the associated uncertainty to have a component that varied in proportion to the mass. Similarly, a length measuring device will usually have one or more influencing factors, e.g., refractive index effects for the case of a laser interferometric comparator [41], that vary in proportion to length. The use of multiple artefacts in each experiment will also likely introduce effects that will add to the uncertainties. For each experiment, let \(n_{i}\) be the total number of artefacts involved and let

\[v_{i}=\sum_{q\in L_{i}\cup R_{i}}a_{q}^{\sharp},\]

be a measure of the total value of the quantities involved in the \(i\)th experiment.