## 3 Fault Diagnosis
### Current New Fault Diagnosis Algorithm
#### 3.2.2 Intelligent FD Methods Using Transfer Learning Theories  
Sufficient labeled data are necessary for data-driven methods to train diagnosis models. However, in marine engineering scenarios, obtaining appropriate data and further labeling them will be hard to implement. Previous test results or relevant data of the same or similar MSAE can be used as diagnostic knowledge. For example, the diagnosis knowledge from manufacturer's test, such as bench test and sea trial, may help recognize the health condition. Transfer learning could use the diagnosis knowledge of existing systems and equipment to realize the diagnosis of new equipment [118]. Feature-based transfer learning approaches which could realize correcting serious across-domain discrepancy are widely used. Transfer component analysis (TCA) is a typical feature-based approach and its optimization objective is  
\[\min_{W}\text{trace}\Big{(}\mathbf{W}^{\text{T}}\mathbf{K}\mathbf{L}\mathbf{K}\mathbf{W}\Big{)}+ \mu\cdot\text{trace}\Big{(}\mathbf{W}^{\text{T}}\mathbf{W}\Big{)},\text{ s.t. }\mathbf{W}^{\text{T}}\mathbf{K}\mathbf{H}\mathbf{K}\mathbf{W}=\mathbf{I}, \tag{20}\]  
where \(\mathbf{K}=\big{[}K_{i,j}\big{]}\in R^{(n_{s}+n_{t})\times(n_{s}+n_{t})}\) is the kernel matrix of the input cross-domain samples and \(K_{i,j}=k\big{(}\mathbf{x}_{i},\mathbf{x}_{j}\big{)}\), \(\mathbf{W}=\mathbf{K}^{-1/2}\mathbf{W}\in R^{(n_{s}+n_{t})\times m}\) maps the cross-domain samples from the space \(R^{n_{s}+n_{t}}\) to the m-dimensional space \(R^{m}\) and \(n_{s}+n_{t}>m\),\(\mu\) is the tradeoff parameter, \(\mathbf{H}=\mathbf{I}_{n_{s}+n_{t}}-1/(n_{s}+n_{t})11^{\text{T}}\) is the centering matrix, and \(\mathbf{L}=\big{[}L_{i,j}\big{]}\geq 0\) can be calculated as  
\[L_{i,j}=\left\{ \begin{array}{ll}\frac{1}{n_{s}^{2}},&\mathbf{x}_{i},\mathbf{x}_{j} \in X^{s}\\ \frac{1}{n_{t}^{2}},&\mathbf{x}_{i},\mathbf{x}_{j}\in X^{t}\\ -\frac{1}{n_{s}n_{t}},&\left\{\begin{array}{l}\mathbf{x}_{i}\in X^{s},\mathbf{x}_{j} \in X^{t}\\ \mathbf{x}_{i}\in X^{t},\mathbf{x}_{j}\in X^{s}\end{array}\right.\end{array} \right.. \tag{21}\]  
The optimal feature mapping \(\mathbf{W}^{\text{*}}\) obtained by solving Equation (20) can be further used to calculate the cross-domain features \(\mathbf{W}^{\text{*}}\mathbf{K}\) subject to a similar distribution. As described in Figure 8, such approaches could be classified into feature mapping, feature extraction, feature adaptive update, and fault identification. Figure 7: FD method of MSAE based on 1D CNN. The development of intelligent ship technologies promotes the sharing of data between different ships. The effective extraction, transfer, and application of knowledge in the later stage will become the critical research content of the ship intelligent O&M. FD model based on transfer learning, shown in Figure 9. The primary calculation process of the algorithm includes the construction of input data, empirical and depth feature extraction, feature transfer learning, fault pattern recognition. (1) The input data  
On the one hand, making full use of the test bench provided by various MSAE manufacturers, a large number of test data collected from the factory, ship mooring, and sea trial data, and the massive operation data, a state knowledge transfer learning model was established to reduce the distribution difference of working conditions and operating environment in the migration characteristic layer. The problems of non-convergence and low recognition accuracy caused by insufficient marking information in the existing MSAE state diagnosis model could be effectively solved. On the other hand, knowledge extraction, transferring, and adaptation are carried out for the status data of different ship service life, operation process information of different working conditions and environment, and data samples of different fault types, to promote the transferring and sharing of characteristic knowledge among monitoring data.