## 4 Choosing a \(D\)-Optimal Subset of \(n\) Measurements  
Let \(C\) be an \(m\times n\) observation matrix, \(m>n\), representing \(m\) measurements that could potentially be made. If we are constrained only to use \(n\) measurements, which \(n\) should we choose, in order to minimise an aggregate measure of the uncertainties associated with the fitted parameters? It is assumed that there is at least one \(n\times n\) submatrix constructed from the rows of \(C\) that is full rank, allowing all the parameters \(\mathbf{a}\) to be estimated. In this section, we consider two algorithms described below in Sections 4.1 and 4.2, for addressing the issue of subset selection with a view to providing a D-optimal solution. Both algorithms were designed to address other problems but can be repurposed for design of experiment applications.  
Let \(C_{k}\) be a full rank \(n\times n\) submatrix of \(C\) defined by a subset of the rows of \(C\) indexed by row indices \(I_{k}=\{i_{1},\ldots,i_{n}\}\), and set \(V_{k}=H_{k}^{-1}\), where \(H=C_{k}^{\top}C_{k}\). Then,  
\[|V_{k}|=|H_{k}|^{-1}=|C_{k}|^{-2},\]  
so that choosing \(I_{k}\) to minimise \(|V_{k}|\) amounts to choosing \(I_{k}\) to maximise \(|C_{k}|\). For any square \(n\times n\) matrix \(A\) with rows \(\mathbf{a}_{i}\), \(|A|\) is the volume of the parallelopiped spanned by the vectors \(\mathbf{a}_{i}\) in \(\mathbb{R}^{n}\). This volume will depend on the Euclidean lengths \(\|\mathbf{a}_{i}\|_{2}\) of the vectors and also their mutual independence. For vectors of equal length, \(|A|\) is maximised if the vectors \(\mathbf{a}_{i}\) are orthogonal to each other (and can therefore be rotated or reflected to align with the coordinate axes). In terms of the observation matrix \(C\), the lengths of the row vectors \(\mathbf{c}_{i}\) reflect the accuracy associated with the measurement \(y_{i}\)--more accurate measurements will be accorded a greater weight and therefore larger row vector \(\mathbf{c}_{i}\) in terms of Euclidean length. The orthogonality of the row vectors relates to how independent the sets of information represented by the measurements are from each other. Consider  
\[\left[\begin{array}{c}y_{1}\\ y_{2}\end{array}\right]=C\left[\begin{array}{c}a_{1}\\ a_{2}\end{array}\right]+\left[\begin{array}{c}\mathbf{c}_{1}\\ \mathbf{c}_{2}\end{array}\right],\quad C=C(\theta)=\left[\begin{array}{cc}1&0\\ \cos\theta&\sin\theta\end{array}\right]\quad\mathbf{c}_{1},\mathbf{c}_{2}\in\mathcal{ N}(0,1).\]  
Both measurements \(y_{1}\) and \(y_{2}\) are associated with the equal information gain, since \(\|\mathbf{c}_{1}\|=\|\mathbf{c}_{2}\|=1\). The data point \(y_{1}\) provides information about \(a_{1}\) alone, while the data point \(y_{2}\) potentially provides information about both \(a_{1}\) and \(a_{2}\). For \(\theta\approx 0\), the observation equation involving \(y_{2}\) only serves to provide more information about \(a_{1}\), and \(a_{2}\) remains poorly estimated with \(|C(\theta)|=|\sin\theta|\approx 0\). For \(\theta\approx\pi/2\), the information provided by \(y_{2}\) provides significant new information about \(a_{2}\) and \(|C(\theta)|=|\sin\theta|\approx 1\). In this way, amongst a set of measurements of comparable accuracy, we look for subsets that are mutually maximally independent. Both of the algorithms described below involve trying to select a set of measurements that are sufficiently independent.