# Surrogate-Based Robust Design for a Non-Smooth Radiation Source Detection Problem
## 4 Robust Design in the Average Sense
### Optimal Sensor Locations  
We focus on the statistical properties of the maximum likelihood estimator whose estimate was derived in Equation (20). Based on these properties, optimal design theory provides solutions for the optimal placement of sensors problem. We first note that the estimator  
\[\tilde{\mathbf{\theta}}=\mathbf{M}^{-1}\sum_{i=1}^{N}\frac{1}{\sigma_{i}^{2}}\frac{ \partial\tilde{F}_{i}}{\partial\mathbf{\theta}}(\mathbf{\theta}^{0})\tilde{\mathrm{Y}}_ {i}, \tag{22}\]  
is unbiased since  
\[E[\tilde{\mathbf{\theta}}] =\mathbf{M}^{-1}\sum_{i=1}^{N}\frac{1}{\sigma_{i}^{2}}\frac{\partial \tilde{F}_{i}}{\partial\mathbf{\theta}}(\mathbf{\theta}^{0})E[\tilde{\mathrm{Y}}_{i}]= \mathbf{M}^{-1}\sum_{i=1}^{N}\frac{1}{\sigma_{i}^{2}}\frac{\partial\tilde{F}_{i}}{ \partial\mathbf{\theta}}(\mathbf{\theta}^{0})E\left[\frac{\partial\tilde{F}_{i}}{ \partial\mathbf{\theta}}(\mathbf{\theta}^{0})^{T}\mathbf{\theta}_{0}+\tilde{\varepsilon}_{ i}\right],\] \[=\mathbf{M}^{-1}\sum_{i=1}^{N}\frac{1}{\sigma_{i}^{2}}\frac{\partial \tilde{F}_{i}}{\partial\mathbf{\theta}}(\mathbf{\theta}^{0})\frac{\partial\tilde{F}_ {i}}{\partial\mathbf{\theta}}(\mathbf{\theta}^{0})\mathbf{\theta}_{0}=\mathbf{\theta}_{0},\]  
and \(\mathbf{\theta}_{0}\) is the true source location and intensity of the radiation source. Please note that we employed the relation \(E[\tilde{\varepsilon}_{i}]=0,\ i=1,\ldots N\), and Formula (21). The covariance of the estimator is given by  
\[ \begin{split}\mathrm{cov}[\tilde{\mathbf{\theta}}]&=E[( \tilde{\mathbf{\theta}}-\mathbf{\theta}_{0})(\tilde{\mathbf{\theta}}-\mathbf{\theta}_{0})^{T}] \\ &=E\left[(\mathbf{M}^{-1}\sum_{i=1}^{N}\frac{1}{\sigma_{i}^{2}}\frac{ \partial\tilde{F}_{i}}{\partial\mathbf{\theta}}(\mathbf{\theta}^{0})\tilde{\mathrm{Y} }_{i}-\mathbf{\theta}_{0})(\mathbf{M}^{-1}\sum_{j=1}^{N}\frac{1}{\sigma_{i}^{2}}\frac{ \partial\tilde{F}_{j}}{\partial\mathbf{\theta}}(\mathbf{\theta}^{0})\tilde{\mathrm{Y} }_{j}-\mathbf{\theta}_{0})^{T}\right]\\ &=\mathbf{M}^{-1}\sum_{i=1}^{N}\sum_{j=1}^{N}\frac{1}{\sigma_{i}^{2}} \frac{\partial\tilde{F}_{i}}{\partial\mathbf{\theta}}(\mathbf{\theta}^{0})E\left[ \tilde{\varepsilon}_{i}\tilde{\varepsilon}_{j}\right]\frac{1}{\sigma_{j}^{2}} \frac{\partial\tilde{F}_{j}}{\partial\mathbf{\theta}}(\mathbf{\theta}^{0})^{T}\mathbf{M}^ {-T}\\ &=\mathbf{M}^{-1}\sum_{i=1}^{N}\frac{1}{\sigma_{i}^{2}}\frac{\partial \tilde{F}_{i}}{\partial\mathbf{\theta}}(\mathbf{\theta}^{0})E\left[\tilde{ \varepsilon}_{i}^{2}\right]\frac{1}{\sigma_{i}^{2}}\frac{\partial\tilde{F}_{i} }{\partial\mathbf{\theta}}(\mathbf{\theta}^{0})^{T}\mathbf{M}^{-T}=\mathbf{M}^{-1}.\end{split}  \tag{23}\]  
The result is based on the independence of the errors and symmetry of \(\mathbf{M}\). The Fisher information matrix does not depend on the pseudo-measurements \(\tilde{\mathbf{v}}_{i}\) but instead on the sensor locations \(\mathbf{D}_{i},\ i=1,\ldots,N\), and the prior estimate \(\mathbf{\theta}^{0}\). As such, one can adjusts the sensor locations \(\mathbf{D}_{i}\) to minimize uncertainty in the estimator; i.e., minimize \(\mathrm{cov}[\tilde{\mathbf{\theta}}]\). As detailed in [26], an optimal configuration exists only for specific cases. This is the reason for the introduction of scalar metrics depending on all possible Fisher information matrices [26]. The most popular metrics are the D-optimality, E-optimality, A-optimality, and sensitivity criteria based on the determinant, smallest eigenvalue, and trace of the Fisher information matrix and its inverse. D-optimality is invariant under scale changes in the parameters and linear transformations of the output in contrast to A-optimality and E-optimality criteria.