# 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  
As an extension and evolution of machine learning, deep learning is based on neural networks but is distinct from them. It solves the inherent problems of traditional neural networks by learning the data representation layer by layer through a multi-hidden layer network structure. It has been widely implemented in the fields of image processing and text recognition, but industrial equipment condition monitoring is still in the research  
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phase. Deep belief networks (DBN), convolutional neural networks (CNN), and recurrent neural networks (RNN) are the most common deep learning models at present. The Deep Belief Network Approach  
As shown in Figure 5, a DBN is multi-hidden layer probabilistic generative model consisting of multiple restricted Boltzmann machines (RBM) and one output layer (usually a classification layer) combined to build a joint distribution between observed data and labels by training layer by layer. Unlike the directed/unidirectional connectivity of the hidden layers in DBNs, RBMs are multi-hidden Boltzmann networks in which the hidden layers both convey information and allow for top-down feedback adjustment. The RBM can derive potential feature representations from the training data distribution. It is assumed that the activation conditions of each hidden unit are independent for a given input data again, and conversely, the activation conditions of the visible units are independent for a given hidden unit state. The model structure of the RBM consists of the weight matrix \(\mathrm{W}\) of size \(\mathrm{m}~{}\times~{}\mathrm{n}\), each element \(\mathrm{w_{ij}}\) in this matrix is associated with the corresponding visible layer neuron state vector \(\mathrm{v_{i}}\) and hidden layer neuron state vector \(\mathrm{h_{j}}\). Also, each layer has a corresponding offset coefficient \(\mathrm{a_{i}}\) (for visible units) and \(\mathrm{b_{j}}\) (for hidden units). When given the model parameters \(\mathrm{\theta}=[\mathrm{u},~{}\mathrm{a},~{}\mathrm{b}]\), the energy function of the RBM can be written as:  
\[\mathrm{E(v,~{}h;~{}\mathrm{\theta})}=-\sum_{i}^{m}\sum_{j}^{n}\mathrm{w_{ij} }\mathrm{v_{i}}\mathrm{h_{j}}-\sum_{i=1}^{m}\mathrm{a_{i}}\mathrm{v_{i}}- \sum_{j=1}^{n}\mathrm{b_{j}}\mathrm{h_{j}} \tag{29}\]  
where \(\mathrm{m}\) and \(\mathrm{n}\) represent the number of visible units and hidden units respectively. The joint distribution of all units can be obtained from the calculation of the energy function \(\mathrm{E}\):  
\[\mathrm{p(v,h;\mathrm{\theta})}=\frac{\exp(-\mathrm{E(v,h;\mathrm{\theta})})}{Z} \tag{30}\]  
where, \(Z=\sum_{h,v}\exp(-\mathrm{E(v,h;\mathrm{\theta})})\) is the normalization factor. Figure 5: (a) Restricted Boltzmann machines; (b) Structure of RBM-based deep belief network. From the above it follows that the conditional probability of a hidden layer neuron being activated is:  
\[\mathrm{p}\big{(}\mathrm{h}_{\mathrm{j}}|\mathrm{v};\theta\big{)}=\delta\left( \mathrm{b}_{\mathrm{j}}+\sum_{i=1}^{m}\mathrm{w}_{\mathrm{ij}}\mathrm{v}_{ \mathrm{i}}\right) \tag{31}\]  
Since it is a bidirectional connection, the explicit layer neuron can be equally activated by the implicit layer neuron with the conditional probability of:  
\[\mathrm{p}(\mathrm{v}_{\mathrm{i}}|\mathrm{h};\theta)=\delta\left(\mathrm{a}_{ \mathrm{i}}+\sum_{j=1}^{n}\mathrm{w}_{\mathrm{ij}}\mathrm{h}_{\mathrm{j}}\right) \tag{32}\]  
A deep belief network can be formed by stacking multiple RBMs, as shown in Figure 5b, where the outputs of layer i (the hidden layer) are used as inputs to layer i + 1 (the visible layer). The joint distribution between the observed data and the labels is established by training layer by layer, the DBN can learn a deep representation of the training data. [PERSON] et al. [129] introduced DBN to the field of defect diagnostics after the deep learning theory was developed. First, a pre-set health status is created, followed by the pre-processing of the acquired sensor data.