# Genetic Multi-Objective Optimization of Sensor Placement for SHM of Composite Structures
## 3 Multi-Objective Optimization
### Data Preparation  
The process of multi-objective optimization requires preparing a high-accuracy trained classifier which allows the evaluation of the value of the multi-objective function for all individuals. In this study, the classifiers are learned and validated during every iteration of the optimization procedure for the known number of inputs (sensor numbers) and selected strain modes. Due to the stochastic nature of the selection of the subset of the data sets for learning and validation for the same set of design variables, small differences can be observed for the same value of the input variable vector as a result of this stochastic nature. To decrease the variability of the results of the type-I and type-II errors due to the mentioned stochastic nature, a ten-fold cross-validation procedure was used to evaluate the classification statistics.  
The proper evaluation of the objective function using type-I and type-II errors requires also a balanced data set preparation. For this purpose, the learning data set was augmented using replication of the learning examples with additive noise with normal distribution. The additive noise of the percentage \(k\) is defined as of 95% probability with the value \(\varepsilon\) falling within the range of \(\varepsilon\in[-a,+a]\), which can be written as follows:  
\[\Phi_{\mu,\sigma^{2}}(a)-\Phi_{\mu,\sigma^{2}}\left(-a\right)=95\%, \tag{7}\]where \(\Phi_{\mu,\sigma^{2}}\) is a cumulative normal distribution with the mean \(\mu\) and the standard deviation \(\sigma^{2}\). This requires the estimation of the standard deviation \(\sigma^{2}\) of the zero-mean normal distribution based on the following equality equation:  
\[\Phi^{-1}\Big{(}97.5\%;\;\sigma^{2}\Big{)}-\Phi^{-1}\Big{(}2.25;\;\sigma^{2} \Big{)}=2a \tag{8}\]  
where \(\Phi^{-1}\) is the inverse cumulative normal distribution of the zero mean estimated at the boundaries of the 95% probability interval, and \(\sigma^{2}\) is seeking standard deviation. Solving Equation (8) provides the required value for the standard deviation. In this case, for normalized values of strain maps, a 5% normal additive noise has a standard deviation of \(2.551\times 10^{-2}\).  
The noise level was established on the basis of the initial evaluation of an accuracy of the damage recognition on the basis of a random subset of input variables with a lower number of sensors and a lower number of modes. The goal of the selection of this noise level was to select the highest value of the noise level to obtain the highest possible generalization properties of the classifier, which still guarantee obtaining a satisfactory level of damage recognition. For this purpose, a five-fold cross-validation procedure was used for the selected subset of input variables with different numbers of sensors, modal shape numbers, and different noise levels from 2 up to 5%. As a satisfactory level of the classifier validation accuracy, at least 95% of classification accuracy [25] was used. It has been observed that for the first three mode shapes and two sensors with a level of 5% of additive random noise, the validation accuracy leads to 95.6%. For other cases and lower noise levels, the validation outcome resulted in higher accuracy. Due to this, the noise level of 5% was attributed as a noise level in the augmentation process of a leaning dataset. The idea of using a high possible noise level results from using numerous numerical cases in the learning examples to maintain high generalization properties of the used classifiers.