## 3 Aggregate Measures of Uncertainty  
A central issue in design of experiment analysis is to maximise some measure of the information gain subject to constraints on resources. In a metrology context, in particular, it is natural to express the information gain in terms of uncertainties associated with the fitted parameters; i.e., related to the \(n\times n\) variance matrix \(V_{\mathbf{a}}\). Given an aggregate measure of uncertainty derived from \(V_{\mathbf{a}}\), one experiment will be judged to be more informative than another if it is associated with a lower aggregate uncertainty. There are a number of aggregate measures that are commonly used. In this paper, we are concerned with two: \(A\)-optimality and \(D\)-optimality. A full rank variance matrix \(V\) is a symmetric positive definite matrix and has an eigenvalue decomposition with eigenvalues \(\lambda_{j},j=1,\ldots n\). The A-measure is the trace of \(\mathrm{Tr}(V)\) of \(V\), the sum of the diagonal elements of \(V\), as well as the sum of the eigenvalues of \(V\):  
\[\mathrm{Tr}(V)=\sum_{j=1}^{n}\lambda_{j}.\]  
We note that \(\mathrm{Tr}(\sigma^{2}V)=\sigma^{2}\mathrm{Tr}(V)\). The D-measure is the determinant \(|V|\) of \(V\), and also the product of the eigenvalues of \(V\):  
\[|V|=\prod_{j=1}^{n}\lambda_{j}.\]  
We also evaluate  
\[\tilde{d}(V)=|V|^{1/n}=\left(\prod_{j=1}^{n}\lambda_{j}\right)^{1/n}, \tag{4}\]  
the geometric mean of the eigenvalues, noting that \(\tilde{d}(\sigma^{2}V)=\sigma^{2}\tilde{d}(V)\). Arguments can be made for and against using each of these two measures. Given \(V\), the A-measure can be evaluated in \(O(n)\) steps, while the D-measure will usually involve \(O(n^{3})\) steps, using the Cholesky factorisation of the variance matrix to evaluate its determinant. The A-measure has the apparent advantage of being most easily interpreted as an aggregate uncertainty, being the sum of the variances \(\sum_{j}u^{2}(a_{j})\) associated with each of the parameter estimates \(a_{j}\) or, more precisely, the numerical values of these estimates. From a dimensional analysis point of view [24], the A-measure can only be meaningful if all the parameters in \(\mathbf{a}\) have the same dimension, length, mass, etc. Otherwise, the trace involves adding quantities of different dimensions. The \(D\)-measure is consistent with a dimensional analysis. The A-measure can also be regarded as the variance associated with the sum \(\sum_{j}a_{j}\) of parameter estimates, _assuming_ these estimates as statistically independent. In general, they will not be independent, and one disadvantage of the A-measure is that it takes no account of correlation, being defined solely in terms of the diagonal elements of \(V\). The D-measure does take into account correlation. Consider the \(2\times 2\) variance matrix  
\[V=\left[ \begin{array}{cc}1&\rho\\ \rho&1\end{array} \right],\quad\mathrm{Tr}(V)=2,\quad|V|=1-\rho^{2},\]  
associated with parameters \(a_{1}\) and \(a_{2}\). The A-measure is independent of the correlation coefficient \(\rho\) while the D-measure depends directly on \(\rho^{2}\). If the result of an experiment is to change \(\rho\) from \(0\) to near \(1\), there is a gain of information, since \(a_{1}-a_{2}\) is now much more accurately estimated. The \(D\)-measure reflects this information gain, the A-measure does not. The D-measure is invariant to re-parametrizations of the model, while the A-measure is not. Suppose \(V_{\mathbf{a}}=(C^{\top}C)^{-1}\) and let \(\mathbf{b}=B\mathbf{a}\), where \(B\) is \(n\times n\) and full rank, so that \(\mathbf{b}\) canbe regarded as an alternative parametrization of the problem. For the same experimental design, the variance matrix associated with the estimates \(\mathbf{b}\) is given by  
\[V_{\mathbf{b}}=BV_{\mathbf{a}}B^{\top}.\]  
We have  
\[|V_{\mathbf{b}}|=|BV_{\mathbf{a}}B^{\top}|=|B|^{2}|V_{\mathbf{a}}|,\]  
so that a strategy that is optimal for the D-measure for parametrization \(\mathbf{a}\) will also be optimal for parametrization \(\mathbf{b}\).