## 2 Mathematical Methods
### Sensor Placement Optimization Algorithm  
As previously mentioned, once the measurement matrix \(\mathbf{M}\) is determined and \(\mathbf{W}\) is extracted from the simulation results, the reconstruction solution is determined. However, in the sensor placement problem, the measurement matrix \(\mathbf{M}\) is unknown.  
Equation (9) can be written as follows:  
\[\mathbf{P}\mathbf{x}_{\rm R}=\mathbf{Q}, \tag{11}\]  
where \(\mathbf{P}=(\mathbf{M}\mathbf{W}){}^{\rm T}\mathbf{M}\mathbf{W}\) and \(\mathbf{Q}=(\mathbf{M}\mathbf{W}){}^{\rm T}\mathbf{t}_{\rm M}\), \(\mathbf{P}\) and \(\mathbf{Q}\) are associated with the measurement matrix \(\mathbf{M}\), so \(\mathbf{M}\) has a significant impact on the reconstruction result.  
The condition number is used to show the morbidity of the matrix; furthermore, it is involved with the linear dependence of vectors [35]. The condition number of \(\mathbf{P}\) is defined as follows:  
\[\text{cond}(\mathbf{P})=\|\mathbf{P}\|\cdot\|\mathbf{P}^{-1}\|. \tag{12}\]Mathematically, the condition number reflects the sensitivity of the output relative to the changes in the input. In Equation (11), a large condition number of \(\mathbf{P}\) increases the sensitivity of solution \(x_{\text{R}}\) to the measurement error, and leads to a poor reconstruction solution. As a result, a small condition number of \(\mathbf{P}\) is needed.  
On the other hand, as mentioned above, when using n sensors, the measurement matrix can be denoted as \(\mathbf{I}_{\text{nxn}}\), as \(\mathbf{W}\) is constituted by eigenvectors, \(\mathbf{P}=(\mathbf{M}\mathbf{W})^{\text{T}}\mathbf{M}\mathbf{W}=\mathbf{I}_{\text{kxl}}\). When using k (k \(<\) n) sensors in the reconstruction process, the orthogonality between the vectors in \(\mathbf{P}\) no longer exists. For the purpose of retaining the ability to distinguish the principal components and to avoid an unstable reconstruction process, the diagonal entries of \(\mathbf{P}\) must be large enough to retain the vectors' orthogonality. Moreover, a small \(\text{cond}(\mathbf{P})\) represents less linear dependence between these vectors, and it is helpful for the vectors in \(\mathbf{P}\) to inherit the features from the original eigenvectors.  
Above all, the objective of the sensor placement optimization algorithm is to find the measurement matrix \(\mathbf{M}\) that can keep orthogonality and minimize \(\text{cond}(\mathbf{P})\). Using a greedy algorithm [23; 37] can achieve this goal, as follows:  
1. Traverse all of the possible measurement sites and update matrix \(\mathbf{P}\). In order to keep the orthogonality, select the location that can maximize the summation of the diagonal elements, minus off-diagonal elements of \(\mathbf{P}\) as the first optimal sensor placement.
2. Traverse all of the remaining measurement sites and update matrix \(\mathbf{P}\) with the determined optimal sensor placement; after that, select the location with minimum \(\text{cond}(\mathbf{P})\) as the second optimal sensor placement.
3. Repeat step 2 until all of the sensor placements have been determined [34].  
Although this optimization algorithm can only achieve a local optimization solution, it can be used in practice, especially when there are too many possible measurement sites.