# 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
The standard BP neural network modifies the weights and bias according to the negative gradient of the error function, and the backward propagation process of the errors is represented as follow:  
\[E= \frac{1}{2}\sum_{k=1}^{m}(z_{k}^{*}-z_{k})^{2} \tag{11}\]  
\[\Delta W_{jk} =-\eta\frac{\partial E}{\partial W_{jk}}=\eta(z_{k}^{*}-z_{k})y_{j} f^{\prime}\left(\sum_{j=1}^{1}W_{jk}y_{j}-b_{k}\right) \tag{12}\]  
\[\Delta b_{k} =-\eta\frac{\partial E}{\partial b_{k}}=\eta(z_{k}-z_{k}^{*})f^{ \prime}\left(\sum_{j=1}^{1}W_{jk}y_{j}-b_{k}\right) \tag{13}\]  
\[\Delta W_{ij} =-\eta\frac{\partial E}{\partial W_{ij}}=\eta x_{i}f^{\prime}\left( \sum_{i=1}^{n}W_{ij}x_{i}-b_{j}\right)\sum_{k=1}^{m}W_{jk}f^{\prime}\left( \sum_{j=1}^{1}W_{jk}y_{j}-b_{k}\right)(z_{k}^{*}-z_{k}) \tag{14}\] Figure 3.0  
Once the adjustments for the weights and bias are found, the network weights and bias can be updated, and then the process of forward propagation of information, backward propagation of errors, and adjustments of weights and bias calculated again until the iteration termination condition is met. Neural networks have received increasing interest from academics and have been used in the field of equipment condition monitoring as artificial intelligence algorithms have advanced. The condition information of a device is the carrier of condition diagnosis, and different types of condition information respond to different degrees of the device's condition, so to achieve condition monitoring, scholars have tried to train ANN models with different condition information, which can be vibration, noise, temperature, pressure, acoustic emission, and so on [93; 94; 95; 96; 97; 98; 99; 100; 101; 102]. Vibration signals are used more frequently. In [103], [PERSON] et al. used the wavelet packet transform to extract statistical features from the original bearing vibration signal and developed a bearing damage index (BID) to select the dominant wavelet packet transform to further extract statistical features from the vibration signal features and used the extracted features as inputs to an ANN model. The results show that ANN is more effective in fault detection and prediction than features extracted from the original signal when data from higher BDI vibration signals are selected. [PERSON] et al. [40] decomposed the vibration signal of a centrifugal pump impeller using time analysis (TA), frequency analysis (FA) and wavelet packet transform (WPT) respectively to obtain the time domain statistical parameters, frequency domain peaks and time frequency domain wavelet packet energy presented as feature datasets. ANN fault diagnosis models were established for each of these three data sets. After adjusting the network nodes and transfer functions, the results show that the feature datasets obtained after WPT analysis can better represent the fault information and have higher diagnostic performance for pump impellers under different operating conditions. In [41], a non-linear auto-regression (NLAR) model was developed based on the time series of pump casing vibration signals under non-fault conditions, with a multilayer neural network as the non-linear estimator. Compared to the Fast Fourier Transform (FFT) based diagnostic method, the ANN based NLAR model can effectively detect faults at the early stage and is successfully applied to pump cavitation fault detection. [PERSON] et al. [104] obtained the frequency domain characteristics of the vibration signals of three-phase asynchronous induction motor ball bearings under different conditions by frequency domain analysis. These features were used to train an ANN classifier model, which ultimately enabled fault diagnosis of defects in the inner and outer rings of the bearings. In addition to the use of vibration signals, current signals and temperature signals are often used as state information for the training of ANN models. In [89], [PERSON] et al.