## 3 Fault Diagnosis
### FD Methods
#### 3.1.3 Data Driven-Based Method  
The data-driven FD method of MSAE originates from the condition monitoring technology and has attracted the attention of more and more engineers and researchers. The implementation procedure of data-driven methods simply involves two steps: a training model based on historical data and an online FD based on real-time data. In this section, the data-driven FD methods for MSAE will be summarized in detail. Data-driven FD methods mainly include signal processing, multivariate statistical analysis, and machine learning. These methods adaptively learn the diagnosis knowledge and automatically establish the relationship between the measurement parameters and the health condition of MSAE instead of expert experience and knowledge. (1) Signal processing methods  
Some of the collected and processed signals carry the fault information, which is presented as characteristics. Fault diagnosis can be realized by generating fault symptoms with appropriate signal processing methods [93]. Typical symptoms include time-domain features such as amplitude, arithmetic or square mean, limit value, partial derivative, the statistical moment of amplitude distribution or envelope, or frequency-domain features such as spectral power density, spectral line, and cepstrum analysis. Typically, the existing signal analysis-based FD methods for MSAE mainly include three categories [94], such as time-domain methods, frequency-domain methods, and time-frequency-domain methods. [PERSON] et al. proposed an automatic vibration-source extraction and feature visualization method fault detection of marine diesel engines where a time-frequency reference signal constructed by Stockwell transform (ST) method. Then, the t-distributed stochastic neighbor embedding (t-SNE) is used to extract fault features and realize visualization to intelligently identify diesel engine faults [42]. The ST method can be indicated as:  
\[y(\tau,f)=\int_{-\infty}^{\infty}x(t)|f|e^{-\pi(\tau-t)^{2}f^{2}}e^{-2\pi ft}dt, \tag{1}\]where \(x(t)\) is the sensor signal, \(\tau\) and \(f\) are the time and frequency terms, and the \(|f|e^{-\pi(\tau-t)^{2}f^{2}}\) is a Gaussian window. When the integration result of the Gaussian window is equal to 1 in the integration operation, we can obtain:  
\[ \begin{array}{l}Y=\int_{-\infty}^{\infty}y(\tau,f)d\tau=\int_{-\infty}^{ \infty}\int_{-\infty}^{\infty}x(t)\omega(\tau-t,f)e^{-\beta\pi ft}dtd\tau\\ =\int_{-\infty}^{\infty}x(t)e^{-\beta\pi ft}\Big{\{}\int_{-\infty}^{\infty} \omega(\tau-t,f)d\tau\Big{\}}dt=X(f)\end{array}  \tag{2}\]  
where \(X(f)\) is the Fourier transform of \(x(t)\). The original signal \(x(t)\) can be recovered by performing inverse Fourier transform on \(Y\) based on Equation (2). Then, t-SNE is used to select the most obvious fault characteristics. This method learns a map \(F\) by measuring the pairwise similarity of the elements in parameter set \(D\), and projects \(D\) into a low dimensional (usually 2 or 3) embedding \(E=\left[e_{1},e_{2},\cdots e_{l}\right]^{T}\), as shown in Equation (3). \[\left\{ \begin{array}{l}p_{j|i}=\frac{\exp\left(-F\left(d_{i},d_{j}\right)^{2 }/2\sigma_{i}^{2}\right)}{\sum_{k\
eq i}\exp\left(-F\left(d_{i},d_{k}\right)^ {2}/2\sigma_{i}^{2}\right)},and\;p_{j|i}=0\\ p_{ij}=\left(p_{i|j}+p_{j|i}\right)/2l\end{array} \right., \tag{3}\]  
where \(p_{ij}\) and \(p_{j|i}\) are the joint and the conditional probability between \(d_{i}\) and \(d_{j}\), \(\sigma_{i}\) is the bandwidth of the Gaussian kernels. The pairwise similarity of the elements in \(E\) is expressed as  
\[\widetilde{p}_{ij}=\frac{\left(1+\left\|e_{i}-e_{j}\right\|^{2}\right)^{-1}}{ \sum_{k\
eq h}\left(1+\left\|e_{k}-e_{h}\right\|^{2}\right)^{-1}},and\; \widetilde{p}_{ii}=0\;, \tag{4}\]  
where \(\widetilde{p}_{ij}\) is the joint probability between \(e_{i}\) and \(e_{j}\).Lastly, the location of embedding element \(e_{i}\) will be determined by optimizing the Kullback-Leibler divergence between the joint distributions \(p_{ij}\) and \(\widetilde{p}_{ij}\).