## 4 Choosing a \(D\)-Optimal Subset of \(n\) Measurements
### Set
Given \(\sigma_{R}\), \(\sigma_{V}\) and \(\sigma_{N}\), we can assign \(\sigma_{i}\) according to

\[\sigma_{i}^{2}=\sigma_{R}^{2}+n_{i,2}\sigma_{N}^{2}+v_{i}^{2}\sigma_{V}^{2}, \quad n_{i,2}=\max(n_{i}-2,0),\quad i=2,\ldots,n. \tag{16}\]

Table 5 presents the optimal designs calculated using the SSQR-GE algorithm with \(\sigma_{i}\) calculated as in (16) for the four different values of \(\sigma_{R}\), \(\sigma_{N}\) and \(\sigma_{V}\) shown in Figure 7:0 The third design in Table 5 is the result penalising experiments with a large number of artefacts, while the fourth design in the table is the result of penalising experiments with a larger value of \(v_{i}\). Table 6 gives the uncertainties \(u(a_{i})\) associated with \(a_{i}\) for four different assignments of \(\sigma_{i}\), \(i=2,\ldots,n=9\) for the designs in Tables 4 and 5. Also shown in the table is the aggregate measure \(d=|V_{\mathbf{A}}|^{1/n}\) of uncertainty, the total number \(\sum_{i}n_{i}\) of artefacts and a measure \(\sum_{i}v_{i}\) of the total nominal values involved in each set of experiments. The last three rows give the values of \(\sigma_{R}\), \(\sigma_{N}\) and \(\sigma_{V}\) used to calculate \(\sigma_{i}\), \(i=2,\ldots,n\). In all cases, the SSQR-GE algorithm leads to significant improvements over the expert judgement design. ## 6 Concluding Remarks

In this paper, we have looked at the experimental design problem: given \(m\) potential observations to determine \(n\) parameters, \(m>n\), what is the best choice of \(n\) observations. In the context of least squares estimation, the problem was formulated as finding the \(n\times n\) submatrix of the complete \(m\times n\) observation matrix that has maximum determinant, corresponding to the D-optimality criterion. We described two algorithms, the SSQR and GE algorithms, to address this problem. Both were adapted from numerical linear algebra algorithms associated with the QR factorisation of a matrix. We also described algorithms for updating estimates of the problem parameters that are sequentially optimal with the respect to the D-optimality and A-optimality criteria. We illustrated the behaviour of the algorithms on a number of applications drawn from the field of metrology. These algorithms are straightforward to implement. The SSQR algorithm, in particular, is a minor adaption of the standard QR factorisation algorithm with pivoting. The algorithms enable computationally efficient implementations, exploiting rank-one matrix updating techniques. The algorithms are descent-type algorithms that will converge to a local minimum that is not necessarily a global minimum. In the examples considered, the algorithms determined effective experimental designs, often with much better performance than 'hand-crafted' designs based on expert judgement. **Funding:** This work was supported by the UK's National Measurement System programme for Data Science. **Data Availability Statement:** No new data were created or analyzed in this study. Data sharing is not applicable to this article

**Acknowledgments:** The author is grateful to the anonymous referees for their helpful comments, and to the Czech Metrology Institute for permission to use their photograph of the hyperbolic paraboloid, Figure 7:1

**Conflicts of Interest:** The author declares no conflict of interest.