## 3 Fault Diagnosis
### FD Methods
#### 3.1.3 Data Driven-Based Method
The implementation process is shown in Figure 6.  
The signal processing methods could cost a lot of human and financial resources since they require expert experience, extensive computational and storage resources, and robustness of the selected features, limiting their practical applications in PHM. (2) Multivariate statistical analysis  
Typically, the applications of multivariate statistical analysis methods in MSAE mainly include PCA, independent component analysis (ICA). * PCA can reduce the dimension of the dataset and retain the essential information of original measurement parameters to the greatest extent. This is very important for complex MSAE with many correlated variables. For a given data matrix with \(N\) observation samples and \(M\) variables, PCA identifies the sub-space with the largest variance in the \(M\)-dimensional variable space and linearly  
Figure 6: FD of diesel engine based on signal processing method. transforms the original variables into the principal components (PCs). Select the first \(A\) components from PCs and capture most of the percentage of variance. Its model expressed as  
\[\mathbf{X}=\mathbf{T}_{A}\cdot\mathbf{P}_{A}^{\prime}+\mathbf{E}\, \tag{5}\]  
where \(\mathbf{T}_{A}\) is the score matrix of size \(N\times A\), \(\mathbf{P}_{A}\) is the loading matrix of size \(M\times A\), and \(\mathbf{E}\) is the residual matrix of size \(N\times M\), corresponding to the variance not captured by the PCA model. The scores for a new observation, \(\mathbf{t}_{new}\), are computed by projecting the row vector corresponding to that observation, \(\mathbf{x}_{new}\), onto the model subspace:  
\[\mathbf{t}_{new}=\mathbf{x}_{new}\cdot\mathbf{P}_{A}. \tag{6}\]  
Once the scores have been computed, the residuals, \(\mathbf{e}_{new}\), are calculated:  
\[\mathbf{e}_{new}=\mathbf{x}_{new}-\mathbf{t}_{new}\cdot\mathbf{P}_{A}^{\prime}. \tag{7}\]  
[PERSON] et al. presented the use of a multi-class SVM for the FD of marine diesel engine cylinder covers, based on vibration analysis and PCA [95]. PCA as a foremost step of integration fault detection method of marine current turbines was proposed by [PERSON] et al. [37]. The PCA-based approach was used for building on-board sensor classifiers for water contaminant detection [96]. * ICA  
ICA is capable of separating independent sources contained in the observations and, therefore, suitable for MSAE with multiply sensors data. For a given sample set \(\mathbf{x}\in R^{d}\) can be expressed as linear combination of \(m\) unknown independent components \(\mathbf{s}=[s_{1},s_{2},\cdots,s_{m}]\in R^{m}\), that is,  
\[\mathbf{x}=\mathbf{A}\mathbf{s}, \tag{8}\]  
where \(\mathbf{A}\in R^{d\times m}\) is the mixing matrix. ICA tries to estimate \(\mathbf{A}\) and \(\mathbf{s}\) only from the known \(\mathbf{x}\). Therefore, it is necessary to find a de-mixing matrix \(\mathbf{W}\) which is given as:  
\[\mathbf{\overset{\mathbf{\wedge}}{s}}=\mathbf{W}\mathbf{x}, \tag{9}\]  
such that the reconstructed vector \(\mathbf{\overset{\mathbf{\wedge}}{s}}\) becomes as independent as possible. For convenience, we assume \(d=m\), and \(E(\mathbf{s}\mathbf{s}^{\mathrm{T}})=\mathbf{I}\). The whitening transformation is expressed as  
\[\mathbf{z}\mathbf{=}\mathbf{Q}\mathbf{x}\mathbf{=}\mathbf{Q}\mathbf{s}\mathbf{=}\mathbf{B}\mathbf{s}, \tag{10}\]  
where whitening matrix \(\mathbf{Q}=\Lambda^{-1/2}\mathbf{U}^{\mathrm{T}}\mathbf{x}\), \(\mathbf{B}\) is an orthogonal matrix. The relationship between \(\mathbf{W}\) and \(\mathbf{B}\) is as  
\[\mathbf{W}=\mathbf{B}^{\mathrm{T}}\mathbf{Q}. \tag{11}\]  
Hence, Equation (9) can be rewritten as  
Figure 6.0  
According to Equation (11), the ICA problem can be reduced to find an orthogonal matrix \(\mathbf{B}\). ICA-based scheme decomposes these acoustic signals for diesel engine fuel injection FD [97]. [PERSON] et al. applied a new ICA to identify the characteristic of the engine vibration signals to improve the fault detection performance for marine diesel engines [98]. [PERSON] et al.