## 4 Choosing a \(D\)-Optimal Subset of \(n\) Measurements
### Set
If the geometric surface \(\mathcal{S}=\mathcal{S}(\mathbf{a})\) is parametrized by \(\mathbf{a}=(a_{1},\ldots,a_{n})^{\top}\) and \(d(\mathbf{x},\mathbf{a})\) is a measure of the distance from a point \(\mathbf{x}\) to \(\mathcal{S}(\mathbf{a})\), then for measurement points \(\mathbf{x}_{1:m}=\{\mathbf{x}_{i},i=1,\ldots,m\}\), \(m\geq n\), estimates of \(\mathbf{a}\) are determined by solving the least squares optimisation problem

\[\min_{\mathbf{a}}=\sum_{i=1}^{m}d^{2}(\mathbf{x}_{i},\mathbf{a}).\]

Figure 4: Tensor product 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 fine grid, with no repeat measurements, upper curve, and with repeat measurements, lower curve. Figure 5: Tensor product 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 coarse grid, with no repeat measurements, upper curve, and with repeat measurements, lower curve. If the variance matrix associated with the measured points coordinates is \(\sigma^{2}I\), then the variance matrix \(V_{\mathbf{a}}\) associated with the fitted parameters is approximated by

\[V_{\mathbf{a}}=\sigma^{2}\left(J^{\top}J\right)^{-1},\quad J_{ij}=\frac{\partial d} {\partial a_{j}}(\mathbf{x}_{i},\mathbf{a}),\]

involving the Jacobian matrix \(J\). Thus, it is possible to determine optimal measurement strategies, i.e., where to measure on the surface \(\mathcal{S}\), in terms of the determinant of the Jacobian matrix \(J\). Often, the case of choosing a minimal set of points with \(m=n\) is of interest. For simple geometries, an optimal set of points is straightforward to arrive at, e.g., three points equally spaced around a circle, but for more complex geometries, the choice of an optimal set may not be obvious, as considered below. Figure 6 shows a reference artefact designed and calibrated by the Czech Metrology Institute (CMI) [34; 35]. The form of the geometric surface is a hyperbolic paraboloid given by the equation

Table 1.0

A paraboloid with its axis approximately parallel to the \(z\)-axis is parametrized in terms of two rotation angles \(\alpha\) and \(\beta\) and six further parameters \(\mathbf{b}\) such that

Table 1.1

where \(R(\alpha,\beta)=R_{x}(\alpha)R_{y}(\beta)\) is the rotation matrix constructed from the plane rotation matrices

\[R_{x}(\alpha)=\left[ \begin{array}{ccc}1&0&0\\ 0&\cos\alpha&-\sin\alpha\\ 0&\sin\alpha&\cos\alpha\end{array} \right],\quad R_{y}(\beta)=\left[ \begin{array} []{ccc}\cos\beta&0&\sin\beta\\ 0&1&0\\ -\sin\beta&0&\cos\beta\end{array} \right].\]

For the surface (15), nominally \(\mathbf{b}^{\top}=(0,0,1/64,-1/8,-1/8,25)\). Figures 7-9 show the distribution of the \(x\)- and \(y\)-coordinates for three sets \(X_{k}\), \(k=1,2,3\), of eight calibration points. The first two are given by 'expert judgement'; that is, possible choices for a measurement strategy based on engineering practice. The first expert choice, \(X_{1}\), Figure 7, represents an approximately uniform distribution, a first guess at a good distribution. This distribution of points is sufficient to determine all the parameters, and the associated \(8\times 8\) Jacobian matrix is full rank. (If the \(x\)- and \(y\)- coordinates of the data points are rotated through 45 degrees about the \(z\)-axis, the associated Jacobian matrix is rank deficient by three.) The second set \(X_{2}\) came about through experimenting with moving the inner four points in dataset \(X_{1}\) closer or further from the origin of the \(xy\)-plane. It turned out that moving the four points further from the origin was better, resulting in dataset \(X_{2}\), Figure 8. At first glance, the distribution of points in \(X_{2}\) does not look a good choice. The third dataset is that obtained using the SSQR-GE algorithm. The GE algorithm performed 11 interchanges starting from the SSQR solution.