# A Review on Data-Driven Condition Monitoring of Industrial Equipment
## 2 Data-Driven Condition Monitoring of Industrial Equipment
### Fault Diagnosis and Prediction
#### 2.2.2 Method Based on Deep Learning
[144] proposed a DBN model consisting of improved condition restrict Boltzmann Machines (ICRBMs) to predict the remaining useful life of a hydraulic pump. Adding timing linkage factors and constraint variables to nodes on the same layer enhanced the original RBM. Following Bispectral analysis, the normalized data of the vibration signal's features were utilized for the training and testing of the DBN model. Comparative experiments show that the improved DBN model can achieves more prediction accuracy than the original DBN, BP neural network, and SVM models. The relevant references for the DBN methods discussed in this section are summarized in Table 17.  
The Convolutional Neural Network Approach  
A convolutional neural network is a typical supervised feed-forward neural network whose training goal is to learn abstract features by alternating and superimposing convolu  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  
tional kernels and pooling operations. It is structured with an input layer, a convolutional layer, a pooling layer kernel, a fully connected layer, and an output layer. The CNN model is a supervised learning model that requires learning with sample labels. Therefore, the input consists of sample X and the corresponding label Y. For example, for a classification problem, the inputs to the model are as follows:  
\[\{\mathrm{X},\mathrm{Y}\}=\ \{\mathrm{x_{i}},\mathrm{y_{i}}\}^{\mathrm{N}} \tag{33}\]  
where N denotes the number of samples, \(\mathrm{x_{i}}\) denotes the \(\mathrm{i^{th}}\) input sample and \(\mathrm{y_{i}}\) denotes the category label corresponding to the \(\mathrm{i^{th}}\) input sample. The convolutional layer is the core component of the CNN model. The idea of local connectivity and weight sharing is achieved by means of convolutional kernels, which slide vertically along the coordinate transverse kernels of the input feature map and perform convolutional operations with the data in the receptive field, thereby extracting the structural features hidden inside the data. The convolutional layers are organized according to three dimensions: depth, width, and height, with width and height referring to the width and height of the convolutional kernel respectively, i.e., the size of the local receptive field. Depth is the number of convolution kernels. To extract different features on the input feature map, the convolution layer performs a convolution operation by a certain number of convolution kernels, each with a different weight, corresponding to a feature extraction. The convolution operation extracts feature from the input feature map based on the convolution kernel size, and shift compensation. The process of feature extraction by convolution kernels is described as follows:  
\[\mathrm{fea_{i,j}^{L}}=\sum_{q=1}^{Q}\sum_{j=1}^{J}w_{i,j}^{L} \mathrm{efa_{M,q}^{L-1}+b_{j}^{L}} \tag{34}\] \[\mathrm{i}= \lfloor\frac{M+2p-k}{s}\rfloor+1\]  
where \(w_{i,j}^{L}\) and \(b_{j}^{L}\) denote the weight and bias of the \(\mathrm{j^{th}}\) convolutional kernel in the \(\mathrm{L^{th}}\) convolutional layer, respectively, J denotes the number of convolutional kernels, s is the distance that convolutional kernel slides over the input feature map (step size), p denotes the fill size, \(\mathrm{fea_{M,q}^{L-1}}\) represents the \(\mathrm{q^{th}}\) feature map of size M at the output of the \(\mathrm{(L-1)^{th}}\) layer, * denotes the discrete convolution operator, and \(\mathrm{fea_{i,j}^{L}}\) is the \(\mathrm{j^{th}}\) convolutional kernel in the \(\mathrm{L^{th}}\) convolutional layer extracted over the output feature map to generate feature map of size i. Each convolution kernel finds a specific feature at each position in the feature map, and the type of feature learned is determined dynamically by the algorithm. The pooling layer Is the network layer that implements the pooling operation. After the features are extracted from the convolutional layer, direct use for classification can lead to excessive computational effort and thus overfitting.