## 3 Fault Diagnosis
### FD Methods
#### 3.1.3 Data Driven-Based Method
proposed the ICA and SVM for diesel engine condition monitoring and FD [99]. (3) Traditional Machine learning  
In recent years, these traditional machine learning theories, including ANN and SVM, are applied to MSAE fault diagnosis. * ANN-based FD model  
ANN-based FD model has strong self-learning ability. It can obtain diagnosis knowledge from input data and realize fault pattern recognition. Given the training dataset \(\{\mathbf{x}_{i},\mathbf{y}_{i}\}_{i=1}^{m}\) with \(m\) samples, where \(\mathbf{x}_{i}\in R^{d}\) includes \(d\) features and \(\mathbf{y}_{i}\in R^{l}\) includes \(l\) health states, the output of the \(h\) th hidden layer is expressed as  
Figure 6.1  
where \(\left(\mathbf{x}_{i}^{h}\right)_{j}\) is the output of the \(j\) th neuron in the \(h\) th hidden layer, and \(\mathbf{x}_{i}^{0}=\mathbf{x}_{i}\), \(n_{h}\) is the number of neurons in the \(h\) th hidden layer, \(\sigma^{h}\) represents the activation function of the \(h\) th hidden layer, \(n_{h-1}\) is the number of neurons in the \((h-1)\) th hidden layer, \(\omega_{j}^{h}\) is the weights between the neurons in the previous layer and the \(j\) th neuron in the \(h\) th hidden layer, and \(b_{j}^{h}\) is the bias of the \(h\) th hidden layer. The predicted output is  
Figure 6.2  
where \(\left(\mathbf{y}_{i}^{\star}\right)_{k}\) is the predicted output of the \(k\) th neuron in the output layer, \(\sigma^{out}\) is the activation function of the output layer, \(\omega_{j}^{out}\) and \(b_{j}^{out}\) are respectively the weights and bias of the output layer. ANN is widely used in FD of the rotor system, hydraulic unit, bearing, and gearbox. [PERSON] et al. applied a hierarchical ANN model for the rotating machines, which decompose large pattern space into several subspaces [100]. This method could be used to diagnose a variety of faults. [PERSON] et al. presented an FD model using the Kohonen neural network for the rotary machinery [101]. The probabilistic neural network was employed for efficiently FD of hydraulic units [102]. [PERSON] et al. applied the growing neural network to establish an FD model for bearings, which obtained a better diagnostic effect than traditional methods. [PERSON] et al. realize marine gearbox fault condition monitoring by using a neural network to analyze its vibration signal [103]. A feed-forward ANN for the condition monitoring of the air intake and fuel injection system of a medium-speed marine engine had been proposed [104]. [PERSON] et al. proposed an ANNs fusion SOM method for the HCM of a marine diesel engine [61]. Although many ANN methods have been successful in mechanical FD, this method also has disadvantages. One is that the complexity of the model increases significantly with the growth of input condition parameters. The increasing input data affect training efficiency and may lead to overfitting of the model. On the other hand, the ANN-based FD models are based on the black box principle and lack the relationship mapping of physical processes. Some results could not be reasonably explained. * SVM-based FD model  
SVM is a supervised learning method with the input vectors \(\mathbf{x}_{i}\in R^{d}(i=1,2,\cdots,n)\) and the corresponding labels \(\mathbf{y}_{i}\in\{-1,+1\}\). There exists a separating hyperplane and expressed as  
Figure 6.3where \(\delta\in R^{n}\) is a normal vector, the \(b\) is a scale, and \(\frac{|b|}{\|\delta\|}\) represents the perpendicular distance from the separating hyperplane to the origin. Two parallel hyperplanes can be represented as  
Figure 6.4  
SVM tries to maximize the margin between two classes, where the margin width between the two parallel hyperplanes equals to \(\frac{2}{\|\delta\|}\). Therefore, the optimization objective of the linear SVM is  
\[ \begin{array}{l}Min\ \frac{1}{2}\|\delta\|^{2}\\ s.t.\ y_{i}(\delta\cdot\mathbf{x}_{i}+b)\geq 1\end{array}