## 4 Choosing a \(D\)-Optimal Subset of \(n\) Measurements
### Set
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(n\) & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** \\ \hline \(\tilde{d}_{0}\) & 0.4871 & 0.4152 & 0.3748 & 0.3511 & 0.3379 & 0.3316 & 0.3304 & 0.3332 \\ \(\tilde{d}_{\text{AS}}\) & 0.4714 & 0.3789 & 0.3175 & 0.2734 & 0.2403 & 0.2143 & 0.1935 & 0.1763 \\ \(\tilde{d}_{\text{QR}}\) & 0.4682 & 0.3746 & 0.3130 & 0.2691 & 0.2362 & 0.2107 & 0.1901 & 0.1733 \\ \(\tilde{d}_{\text{GE}}\) & 0.4673 & 0.3735 & 0.3119 & 0.2682 & 0.2354 & 0.2099 & 0.1894 & 0.1726 \\ \hline \(n_{\text{GE}}\) & 3 & 6 & 7 & 9 & 12 & 17 & 19 & 21 \\ \hline \end{tabular}
\end{table} 
Table 2: For \(n=4,5,\ldots,11\), rows two to five give the computed geometric mean measure \(\tilde{d}(V_{\mathbf{A}})\) given in (4) for evenly spaced calibration points, the arcsine solutions, the SSQR solutions, and the SSQR-GE solutions given in Table 1. The final row gives the number of GE exchanges undertaken to improve the SSQR solutions. Figure 1: Polynomial calibration points: values of \(t_{q}\), \(q=1,\ldots,100\), and its analytical approximation \(\tilde{t}(q)\) given by (12), determined by the sequential D-optimal Algorithm 4, for the cases \(n=4\), upper curve, and \(n=11\), lower curve. ### Tensor Product of Polynomials

Our second example arose from a problem relating to the estimation of heat distribution on an plate [32]. The application involved modelling the heat distribution as a Gaussian process model [33] involving tensor product polynomials, along with spatially correlated effects. A sub-problem is related to choosing a set of calibration points from a \(10\times 14\) grid of possible sensor locations in order to determine the tensor product mean function. The tensor product model is of the form

\[f(x,y,\mathbf{a})=\sum_{i=1}^{n_{x}}\sum_{j=1}^{n_{y}}a_{ij}f_{i}(x)g_{j}(y),\]

where \(f_{i}(x)\) and \(g_{j}(y)\) are Chebyshev polynomial basis functions. Figure 3 shows the solution calibration points for \(n_{x}=n_{y}=5\) constructed from polynomial basis functions up to degree 4. Two solutions are shown: the first in which the calibration points are selected from a \(131\times 91\) finer grid of 11,921 points, the second in which the calibration points are selected from a \(14\times 10\) grid of 140 points. The solutions found for the fine grid coincide with a grid \(5\times 5\) grid of points \((x_{k}^{*},y_{l}^{*})\), where the \(x_{k}^{*},k=1,\ldots,5\) are the D-optimal solutions derived from the solution of (13) on the interval [0, 20] and the \(y_{l}^{*}\) are the D-optimal solutions on the interval [0, 10]. For this arrangement, the observation matrix \(C\) associated with the polynomial fit is the tensor product \(C=C_{x}\otimes C_{y}\). For square, full rank matrices \(A\) and \(B\) of size \(n_{A}\times n_{A}\) and \(n_{B}\times n_{B}\)

\[|A\otimes B|=|A|^{n_{A}}|B|^{n_{B}}.\]

Hence, it is consistent that the D-optimal points for the tensor product are constructed from the D-optimal points for the \(x\)- and \(y\)-basis functions. The solution points for the coarse grid are seen to approximate the solution for the fine grid, given the constraints imposed by the coarse grid. The coarse grid solution points do not represent a regular grid in this case. Figure 2: As Figure 1, but for the case where repeated measurements are permitted. ### Coordinate Metrology

In coordinate metrology, the size and shape of a geometric surface \(\mathcal{S}\) is determined from the measurement of the coordinates \(\mathbf{x}_{i}\) of points lying on the surface.