# 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
Let the connection weights between the input layer to the hidden layer and the hidden layer to the output layer be \(w\) and \(v\) respectively:  
\[w= \begin{bmatrix}w_{1}\\ w_{2}\\ \cdots\\ w_{1}\end{bmatrix} = \begin{bmatrix}w_{11}&w_{12}&\cdots&w_{1n}\\ w_{21}&w_{22}&\cdots&w_{2n}\\ \cdots&\cdots&\ddots&\vdots\\ w_{11}&w_{12}&\cdots&w_{1n}\end{bmatrix} _{1\times n} \tag{21}\]  
\[v= \begin{bmatrix}v_{1}\\ v_{2}\\ \cdots\\ v_{1}\end{bmatrix} = \begin{bmatrix}v&v_{12}&\cdots&v_{1n}\\ v_{21}&v_{22}&\cdots&v_{2n}\\ \cdots&\cdots&\ddots&\vdots\\ v_{11}&v_{12}&\cdots&v_{1m}\end{bmatrix} _{1\times m} \tag{22}\]  
The hidden layer bias is given as:  
\[b= \begin{bmatrix}b_{1}\\ b_{2}\\ \cdots\\ b_{1}\end{bmatrix}  \tag{23}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**References** & **Application** &
\begin{tabular}{c} **Type of** \\ **Equipment** \\ \end{tabular} & **Signal** & **Fault Type** \\ \hline
[46] & Fault detection & Rolling & Vibration & \\ detection & bearing & signal & \\
[73] & Fault detection & Bearing in main coolant & Vibration & \\ and diagnosis & pump and feed water & signal & Inner race; outer race and ball faults \\
[79] & Fault & Rolling & Vibration & \\ diagnosis & bearing & signal & Outer race, inner race, and ball faults \\  & Fault & Rolling & Vibration & \\ diagnosis and & bearing & signal & Outer race, inner race, and ball faults \\  & prognosis & & \\
[113] & Fault & Bearing form induction & Vibration & \\  & diagnosis & motor & signal & Ball, Inner race, and Outer race faults \\  & Fault & Rolling & Vibration & \\  & diagnosis & bearing & signal & Ball, Inner race, and Outer race faults \\  & & A run-to-failure test & \\  & conducted by & & \\
[120] & Early fault detection & Intelligent Maintenance & Vibration & \\  & Systems, University of & & \\  & Cincinnati, USA & & \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Bearing condition monitoring using support vector machine. The activation function is \(\mathrm{g(x)}\), then for an output \(\mathrm{Y}\) with \(\mathrm{N}\) samples can be expressed as:  
\[\mathrm{Y}=\mathrm{H}\mathrm{v} \tag{24}\]  
where, \(\mathrm{H}\) is the hidden layer output matrix:  
\[\mathrm{H}= \begin{bmatrix}\mathrm{g(w_{1}\cdot x_{1}+b_{1})}&\mathrm{g(w_{2} \cdot x_{2}+b_{2})}&\cdots&\mathrm{g(w_{1}\cdot x_{1}+b_{1})}\\ \mathrm{g(w_{1}\cdot x_{2}+b_{1})}&\mathrm{g(w_{2}\cdot x_{2}+b_{2})}&\cdots& \mathrm{g(w_{1}\cdot x_{2}+b_{1})}\\ \cdots&\cdots&\ddots&\vdots\\ \mathrm{g(w_{1}\cdot x_{N}+b_{1})}&\mathrm{g(w_{2}\cdot x_{N}+b_{2})}&\cdots& \mathrm{g(w_{1}\cdot x_{N}+b_{1})}\end{bmatrix} _{\mathrm{N}\times 1} \tag{25}\]  
The following theorem was proposed by [PERSON] et al. :  
Given a single hidden layer forward neural network (SLFN) with \(\mathrm{N}\) hidden layer neurons, \(\mathrm{N}\) distinct samples \(\mathrm{(x_{i},\,y_{i})}\), where \(\mathrm{x_{i}}{=[\,x_{i1},\,x_{i2},\,\cdots,x_{in}]^{T}\in R^{n}}\) and \(\mathrm{y_{i}}{=[\,y_{i1},\,y_{i2},\,\cdots,y_{im}]^{T}\in R^{m}}\), and an infinitely differentiable function \(\mathrm{g(x)}\), then for any assignment \(\mathrm{w_{i}}\in R^{n}\) and \(\mathrm{b_{i}}\in R\), there is a hidden layer output matrix \(\mathrm{H}\) that is invertible and satisfies \(\left\|\mathrm{H}\mathrm{v}-\mathrm{Y}\right\|=0\). Given \(\mathrm{N}\) distinct samples \(\mathrm{(x_{i},\,y_{i})}\), where \(\mathrm{x_{i}}{=[\,x_{i1},\,x_{i2},\,\cdots,x_{in}]^{T}\in R^{n}}\) and \(\mathrm{y_{i}}{=[\,y_{i1},\,y_{i2},\,\cdots,y_{im}]^{T}\in R^{m}}\), and an infinitely differentiable function \(\mathrm{g(x)}\). Given any small error \(\varepsilon(\varepsilon\,>\,0)\), there exists a SLFN with \(\mathrm{M}\,(\mathrm{M}\,\leq\,\mathrm{N})\) hidden layer neurons that has a hidden layer output matrix \(\mathrm{H}\) that is invertible and satisfies \(\left\|\mathrm{H}\mathrm{v}-\mathrm{Y}\right\|<\,\varepsilon\) for any assignment \(\mathrm{w_{i}}\in R^{n}\) and \(\mathrm{b_{i}}\in R\).