# Optimization of Sensor Placement for Modal Testing Using Machine Learning
## 2 A Review of Traditional Sensor Placement Techniques for Modal Testing
### Effective Independence Method (EIM)  
The Effective Independence Method, also known as the effective independence algorithm, is one of the most popular sensor placement techniques, and it bases its analysis on the sum of the diagonal terms of the Fisher information matrix. The Fisher information matrix is constructed using the modal characteristics extracted by a finite element model. It is an iterative method that evaluates the contribution of all possible sensor locations (i.e., the nodes of the finite element method) to the linear independence of the mode shapes. Sensors with small contributions to the linear independence are progressively eliminateduntil the desired number of sensors remains. This final set of sensors maximizes the sum of the diagonal and the condition number of the Fisher information matrix. The method begins with a set of target mode shapes that encompass the set of candidate sensor locations generally derived from an FE model of the structure under analysis [20; 21]. The algorithm attempts to predict the independence of each node based on the expected measured mode shape, with higher values indicating increased independence [20]. For this method, it is necessary to know both the expected mode shape as well as the location of candidate sensors; therefore, it is well suited for use when a finite element model is available. The candidate locations are then ranked according to the algorithm, removing the lowest ranking sensor and recalculating. As potential locations are eliminated, the relative independence of the remaining solutions increases, and the process is repeated until the required number of sensor locations is reached. One of the challenges of EIM is that the optimal number of sensors must be defined a priori, and the potential locations of the sensors must be available [22]. A very fine grid of the finite element model allows the user to analyze all the possible locations of sensors, but the method then becomes very time-consuming owing to the iterative nature of EIM. Some research also suggests that more optimal results may be produced compared with kinetic energy methods, although at the cost of less ability to measure unexpected modes [20; 23]. Additionally, the EIM approach does not account for unknown modes that may occur in the real world but do not appear in FEA. At the same time, if there are specific modes in the FEA results that are of more interest, EIM can provide targeted sensor selection for those modes that may require fewer sensors than necessary to capture the full behavior of the structure. The EIM is derived by [PERSON] et al. [23] and is based on the concept that each sensor output \(u_{s}\) can be represented as a linear combination of the mode shapes of the system at a given sensor location \(\mathbf{\Phi}_{s}\) through the target modal coordinates \(\mathbf{q}\):  
\[\mathbf{u}_{s}=\mathbf{\Phi}_{s}\mathbf{q} \tag{7}\]  
The mode shapes of the system are obtained using finite element analysis. Each row in matrix \(\mathbf{\Phi}_{s}\) represents a possible sensor location, and each column is the corresponding mode shape. The linear independence of the mode shapes is defined through the Modal Assurance Criteria (_MAC_), as follows:  
\[\mathbf{MAC}=\mathbf{\Phi^{T}\Phi} \tag{8}\]  
which yields 1.0 on the diagonal when the modes are normalized as such. An effective independence score \(\mathbf{EI}\) is calculated for each possible sensor location, using the reduced modal content of the numerical model. \[\mathbf{EI}=\mathbf{\Phi}_{s}{\left(\mathbf{\Phi}_{s}^{T}\mathbf{\Phi}_{s}\right)}^{-1}\mathbf{ \Phi}_{s}^{T} \tag{9}\]  
This diagonal vector \(\mathbf{EI}\) yields a value that ranges between 0 and 1. A row with a value close to zero indicates that the sensor location is not able to sense the target modes, whereas a value close to one indicates that the sensor location is important to observe the target modes.