# A Review on Data-Driven Condition Monitoring of Industrial Equipment
## 2 Data-Driven Condition Monitoring of Industrial Equipment
### Fault Diagnosis and Prediction
#### 2.2.1 Method Based on Shallow Machine Learning

Table 10: Motor condition monitoring using artificial neural networks. The Support Vector Machine (SVM) Approach  
The initial objective of SVM theory was to devise a method for dealing with classification issues by locating an optimum classification hyperplane that meets the classification requirements, and by maximizing the blank area on both sides of the hyperplane. Theoretically, SVM is capable of obtaining optimum classification for two-dimensional linearly divisible data. Consider the classification of two data types as an example, given a training sample set \(\mathrm{D_{i}}=(\mathrm{x_{i}},\mathrm{y_{i}})\), \(\mathrm{i=1,2,\cdots,l}\), \(\mathrm{x\in R^{n}}\), \(\mathrm{y\in\{\pm 1\}}\), where \(\mathrm{x}\) is the input sample, \(\mathrm{y}\) is the category value, and \((\mathrm{\omega\cdot x})+\mathrm{b=0}\) (\(\mathrm{\omega\in R^{n}}\), \(\mathrm{b\in R}\)) is the hyperplane, in order for the classification plane to accurately classify all samples and have a classification interval, it is important to verify that the hyperplane meets the constraint \(\mathrm{y_{i}}[(\mathrm{\omega x_{i}+b})]\geq 1\), \(\mathrm{i=1,2,\cdots l}\). Therefore, the task of generating the optimal hyperplane becomes the problem of obtaining the minimal value within the restrictions. \[\min\frac{1}{2}\|\mathrm{\omega}\|^{2} \tag{16}\]  
The Lagrange function is introduced to resolve this restricted optimization issue. \[(\mathrm{\omega,b,\alpha})=\frac{1}{2}\|\mathrm{\omega}\|^{2}-\sum_{\mathrm{i=1 }}^{1}\alpha_{\mathrm{i}}(\mathrm{y_{i}}(\mathrm{\omega x_{i}+b})-1) \tag{17}\]  
where, \(\alpha_{\mathrm{i}}>\ 0\) is the Lagrange multiplier for each sample. Letting the partial derivatives of Equation (18) be zero for \(\mathrm{b}\) and \(\mathrm{\omega}\) respectively, it can be derived that:  
\[\mathrm{\omega=\sum_{i=1}^{1}\alpha_{i}y_{i}x_{i}} \tag{18}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**References** & **Application** &
\begin{tabular}{c} **Type of** \\ **Equipment** \\ \end{tabular} & **Signal** & **Fault Type** \\ \hline
[47] & Fault diagnosis & Rolling bearings & Vibration signal & Inner race and outer race faults \\  & & & & & \\  & & & & & \\  & & & & & \\  & & & & & \\  & & & & & \\  & Remaining & Rolling bearing & Vibration signals & Inner race, ball, and outer race fault \\
[97] & useful life prediction & Rolling bearing & Vibration signals & Local spalls fault and Pits or distributed surface wear fault \\
[98] & Fault diagnosis & bearing signal & Local spalls fault and Pits or distributed surface wear fault \\
[99] & Fault & Rolling vibration & Inner race, ball, and outer race fault \\  & prognosis & bearing & Signal & Inner race, ball, and outer race fault \\  & Fault detection and prognosis & Beaming signal & Inner race, ball, and outer race fault \\
[104] & Fault diagnosis & Rolling bearing & Vibration signal & Inner race and outer race faults \\  & & & & \\  & & & Temperature & \\
[106] & Fault detection & Rolling bearing & measurement for five bearings & Damaged due to spalling in the bearing. \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Bearing condition monitoring using artificial neural networks. \[\sum_{i=1}^{1}\alpha_{i}y_{i}=0 \tag{19}\]  
The solution vector's expansion consists of a subset of training sample vectors, none of which have zero Lagrange multipliers, i.e., the support vector. A sample vector with zero Lagrange multipliers contributes zero and is irrelevant for selecting a classification hyperplane. Therefore, a decision function describing the optimal classification hyperplane, also known as a support vector machine, is obtained from the training set, and its classification function is determined by the support vector. \[\text{f(x)}=\text{sgn}\left(\sum_{i=1}^{m}\alpha_{i}y_{i}(x\cdot x_{i})+b\right) \tag{20}\]  
In the field of condition monitoring SVM is mainly used for classification problems [109; 110; 111; 112; 113; 114; 115; 116; 117]. Compared to ANN, SVM is more suitable for small sample classification and is more efficient and accurate.