# 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
Therefore, pooling of the feature map is required to reduce the data dimensionality. The pooling operation is a process of further abstraction of information, similar to the process of extracting features from a convolutional layer, by sliding a sliding window over the feature map, taking the statistical values of the local region corresponding to the sliding window as the sampled values of that region, and then concatenating the values extracted from these local regions to form a new feature map. The pooling operation preserves significant feature information while reducing dimensionality, and its operation can be expressed as:  
\[\mathrm{fea_{i,j}^{L}}=\mathrm{f}\left(\mathrm{fea_{i,j}^{L-1}}\right) \tag{35}\] \[\mathrm{p(v,h;\mathrm{\theta})}=\frac{\exp(-\mathrm{E(v,h;\mathrm{\theta})})}{Z} \tag{30}\]0  
where \(\mathrm{fea_{i,j}^{L-1}}\) denotes the \(\mathrm{j^{th}}\) feature map of size i input at layer L, f represents the method of pooling processing. And d denotes the pooling function size, when \(\mathrm{d}\ <\ \mathrm{i}\), it denotes local pooling of the feature map, and when \(\mathrm{d}=\mathrm{i}\), it represents pooling of the entire feature map. s denotes the pooling function move step, and \(\rm{fe}_{i^{\prime}j}^{L}\) is its corresponding output feature map of size \(i^{\prime}\) after the pooling operation. A fully connected layer is used to integrate local features extracted from a convolutional or pooling layer, and it connects each neuron in the previous layer to each neuron in the next layer, thus a fully connected structure. For a one-dimensional input \(x\) of length M, the fully connected layer has N neurons, and the output of each neuron can be expressed as:  
\[\mathrm{p(v,h;\mathrm{\theta})}=\frac{\exp(-\mathrm{E(v,h;\mathrm{\theta})})}{Z} \tag{30}\]1  
where \(w_{jj}^{L}\) is the connection weight of the \(i^{th}\) neuron in layer L \(-1\) to the \(j^{th}\) neuron in layer L, \(b_{j}^{L}\) denotes the bias of the \(j^{th}\) neuron in layer L, \(z_{i}^{L-1}\) and \(z_{j}^{L}\) are the input and output of the \(j^{th}\) neuron in layer L, M and N denote the number of neurons in layers L \(-1\) and L, respectively, and \(\varphi\) is the non-linear activation function. The output layer outputs the recognition results of the model in the form of categories or probabilities through the classifier and calculates the difference between the actual output and the ideal output. The error back propagation algorithm then passes the error layer by layer and finally the parameters of each layer are updated using gradient descent. Initially, CNNs were mostly employed for two-dimensional image processing. However, because to its powerful feature learning and pattern recognition capabilities, it has been recently used to equipment condition monitoring to enhance the analysis and diagnosis of non-linear, multi-source, high-dimensional data. Depending on the size of the convolutional kernel, its condition monitoring applications can be divided into one-dimensional convolutional models and two-dimensional convolutional models. CNNs for condition monitoring were first primarily based on a two-dimensional convolutional strategy [63; 145; 146; 147; 148; 149; 150]. [PERSON] et al. [151] constructed a unique LeNet-5 based CNN for fault diagnostics. The raw 1-dimensional device signal data is transformed into a 2-dimensional picture input, avoiding the need to manually choose features, and allowing the CNN to classify the 2-dimensional images directly. The approach was verified on rolling bearing, centrifugal pump, and hydraulic pump datasets, and the results indicated that the achieved diagnostic accuracy are 99.79%, 99.481%, and 100%, respectively. [PERSON] et al. [64] combined wavelet packets with image-space reconstruction to reconstruct wavelet packet energy (WPE) information in the frequency domain.