## 4 Choosing a \(D\)-Optimal Subset of \(n\) Measurements
### Set
With \(\mathbf{f}_{i}\) and \(g_{i}\) as in (10), if

\[V_{i}=(R^{\top}R+\mathbf{c}_{i}\mathbf{c}_{i}^{\top})^{-1}=V-\frac{V\mathbf{c}_{i}(V\mathbf{c }_{i})^{\top}}{1+\mathbf{c}_{i}^{\top}V\mathbf{c}_{i}},\]7

then

\[V_{i}=(R^{\top}R+\mathbf{c}_{i}\mathbf{c}_{i}^{\top})^{-1}=V-\frac{V\mathbf{c}_{i}(V\mathbf{c }_{i})^{\top}}{1+\mathbf{c}_{i}^{\top}V\mathbf{c}_{i}},\]8

This last expression calculates the change in the A-measure for a given choice of one additional measurement. A sequential algorithm can therefore be designed which at each stage chooses the \(i\) that maximises \(\tau_{i}^{2}\). Each stage requires the calculation of the quantities \(f_{i}^{2}\) and \(g_{i}^{2}\), \(i=1,\dots,m\). If \(V\) is the current variance matrix, then the \(f_{i}^{2}\) are given by the sum of the squares of row elements of \(F=VC^{\top}\). Suppose measurement \(k\) maximises \(\tau_{i}^{2}\). Then, the update of \(F\) is given by

\[V_{i}=(R^{\top}R+\mathbf{c}_{i}\mathbf{c}_{i}^{\top})^{-1}=V-\frac{V\mathbf{c}_{i}(V\mathbf{c }_{i})^{\top}}{1+\mathbf{c}_{i}^{\top}V\mathbf{c}_{i}},\]9

and the updated of \(g_{i}^{2}\) is given by \(g_{i}^{2}-w_{i}^{2}\), with the latter evaluated as \((g_{i}-w_{i})(g_{i}+w_{i})\) for better numerical accuracy. These steps are summarised in (Algorithm 5):As for Algorithm 4, the computational requirement for each step is \(O(mn)\). ## 5 Numerical Examples

### Polynomial Calibration Curves

Many instrument response functions are modelled as polynomial curves, most commonly, of order 2 (degree 1), i.e., modelling a linear response, but higher orders arise, e.g., in the calibration of platinum resistance thermometers [28], or in the calibration of a stage motion [29] where polynomials of order 10 are involved. (Design problems for problems for polynomial models augmented by Gaussian process models are considered in [4].) Suppose a polynomial response function of order \(n\) is defined on the interval [\(-1\),\(1\)], and that it is desired to calibrate the response function using \(n\) calibration points \(x_{i}\in[-1,1]\). Which choice of \(n\) points is D-optimal? For a linear (quadratic) response \(n=2\) (\(n=3\)), it is intuitively clear that the end points \(x=\pm 1\) (and \(x=0\)) are optimal. For higher orders, the optimal choice is not obvious. In fact, the D-optimal set [30] for order \(n\) is given by solutions of

\[\mathbf{f}_{i}=V\mathbf{c}_{i},\quad g_{i}^{2}=\mathbf{c}_{i}^{\top}\mathbf{f}_{i}=\mathbf{c}_{i}^ {\top}V\mathbf{c}_{i}. \tag{10}\]0

where \(L_{n}(x)\) is the derivative of Legendre polynomial of order \(n\) (degree \(n-1\)). These solutions can be approximated by the so-called 'arcsine' points

\[\mathbf{f}_{i}=V\mathbf{c}_{i},\quad g_{i}^{2}=\mathbf{c}_{i}^{\top}\mathbf{f}_{i}=\mathbf{c}_{i}^ {\top}V\mathbf{c}_{i}. \tag{10}\]1

Table 1 gives the results of applying the SSQR algorithm and the combined SSQR-GE algorithm to estimate the D-optimal calibration points for the cases \(n=4,5,\ldots,11\) choosing from \(m=2001\) possible points equally spaced in the interval \([-1,1]\) in steps of \(0.001\). For these calculations, we have used Chebyshev polynomial basis functions [31] for numerical

stability (but the optimal choice of points is independent of the choice of basis functions.) The arcsine estimates are also given in the table, along with the D-optimal points derived from (13); see Appendix B. The results show that, in terms of closeness to the D-optimal points, the SSQR algorithm improves on the arcsine estimates, while the GE algorithm improves the SSQR solution and is accurate to three decimals in almost all cases (there is one exception for the case \(n=11\)).