# Novel Methodology for Condition Monitoring of
## 2 Theoretical Considerations
### Linear Discriminant Analysis (LDA)  
In regard to the condition monitoring of gearbox transmission systems, the estimation of a characteristic and significative set of features plays an important role in determining the occurrence of faults. Yet, although a high-dimensional set of features may be estimated to assess the actual condition of transmission systems, the calculation of non-useful and correlated information is inevitable. Thus, dimensionality reduction techniques play an important role in reducing high-dimensional sets of features and also for removing such non-significative and correlated information that may lead to low performances during the condition assessment [34]. Since the LDA technique is a supervised technique, its consideration in condition monitoring schemes is suitable due to the fact that it can face multi-class problems; moreover, through LDA an original \(n\)-dimensional feature space is reduced, aiming to maximize as much as possible the linear separation between the considered classes.  
Indeed, such dimensional reduction is performed by means of a linear transformation, where the resulting low-dimensional space represents a linear combination containing different weights from the original features. Thus, to guarantee the maximum class separability, the ratio of the between-classvariance to the within-class variance is estimated [35]; thereby, by considering a multi-class problem with \(C\) classes of \(N\) number of samples, the between-class scatter matrix is calculated as follows:  
\[S_{b}=\sum_{j=1}^{C}N_{j}\Big{(}\mathbf{m_{j}}-\overline{\mathbf{m}}\Big{)}\big{(}\mathbf{m _{j}}-\overline{\mathbf{m}}\Big{)}^{T} \tag{1}\]  
where \(N_{j}\) belongs to the total number of samples for the \(j\)-th class \(C_{j}\); considering all the evaluated classes, \(\overline{\mathbf{m}}\) is the global mean of all data samples, and \(\mathbf{m_{j}}\) is the local mean of each class \(C_{j}\). Thus, the within-class scatter matrix is calculated as:  
\[S_{w}=\sum_{j=1}^{C}\sum_{i=1}^{N_{j}}\Big{(}x_{i}^{j}-\mathbf{m_{j}}\Big{)}\Big{(} x_{i}^{j}-\mathbf{m_{j}}\Big{)}^{T}=\sum_{j=1}^{C}S_{w_{j}} \tag{2}\]  
where \(x_{i}^{j}\) is the \(i\)-th sample that corresponds to each class \(C_{j}\); as a result, in \(S_{wj}\), the corresponding covariance matrix of class \(C_{j}\) is estimated.  
Accordingly, the optimum and resulting vector of projection \(\mathbf{W_{LDA}}\) chosen by the LDA allows us to perform a good separation of the evaluated classes, since the estimated transformation matrix has orthonormal columns that maximize the ratio of the determinant of the between-class matrix of the projected samples to the determinant of the within-class scatter matrix of the projected samples, that is:  
\[\mathbf{W_{LDA}}=\arg\max\frac{\left|W^{T}S_{b}W\right|}{\left|W^{T}S_{w}W\right|} =\left[\begin{array}{cccc}&\mathbf{w_{1}}&\mathbf{w_{2}}&\cdots&\mathbf{w_{m}}\\ \end{array}\right] \tag{3}\]  
where \(\{\mathbf{w_{i}}|i=1,2,\cdots,m\}\) is the set of generalized eigenvectors, also known as discriminant vectors, of \(S_{b}\) and \(S_{w}\) that correspond to the \(C\)-\(1\) largest generalized eigenvalues \(\{\mathbf{\lambda}_{i}|i=1,2,\cdots,m\}\).  
Thereby, the resulting extracted features represented in \(\mathbf{V}\) are calculated by means of projecting the original data set of features \(\mathbf{X}\) into the low-dimensional space \(\mathbf{W_{LDA}}\) as follows:  
\[\mathbf{V}=\mathbf{W_{LDA}}^{T}\mathbf{X} \tag{4}\]