# 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 neurons in the adjacent layers are connected and the weights indicate the strength of the connections between the nodes. It abstracts the human nervous system from the perspective of data processing by building individual neuron models and forming various types of networks according to different connections and weights. ANNs mimic the human nervous system in that they can learn from example data by changing their structure and connection weights so that they can learn the relationship between the input and output variables. Common ANNs include back propagation (BP) neural networks, radial basis function (RBF) neural networks, self-organizing competitive (SC) neural networks, etc. This paper only introduces BP neural networks in detail as they are the most widely used. A BP neural network is a multi-layer feed-forward neural network consisting of nonlinear units with an input layer, implicit (hidden) layers, and an output layer, with full connectivity between the adjacent layers and no interconnection between units in the same layer. A typical 3-layer BP neural network structure is given in Figure 3. The BP algorithm is the fundamental approach for training ANNs, which is simply a problem of finding the minimum of the error function and allows for the weight modification of a multi-layer feed-forward neural network. The BP network training process contains two stages: the forward propagation of information and the backward propagation of error. Forward propagation means that the neurons in the input layer receive incoming stimuli and transfer them, after interacting with the weights and biases, to the neurons in the hidden layer, which in turn process the received information and then pass to the output layer neurons through the connection weights. The calculated network outputs are compared with the corresponding target values to work out the neural network prediction errors. The neural network prediction errors are then processed backwards layer by layer to retrieve the deviation of each layer's weights and biases, which are subsequently modified. Figure 3: Structure of a 3-layer BP neural network. The whole procedure consists of alternating forward propagation of information and backward propagation of error until the output errors are sufficiently small. Take the BP neural network layout with a single hidden layer as an example and suppose that \(n\) is the number of neurons in the input layer (i is the index number of neurons), \(l\) is the number of neurons in the hidden layer (j is the index number of neurons), and \(m\) is the number of neurons in the output layer (k is the index number of neurons). \(W_{ij}\) represents the weight between the input layer and the hidden layer neurons, whereas \(W_{jk}\) represents the weight between the hidden layer and the output layer neurons. \(b_{j}\) represents the bias of neurons in the hidden layer, whereas \(b_{k}\) represents the bias of neurons in the output layer. The forward propagation of information is represented as follow:  
\[y_{j} =t\left(\sum_{i=1}^{n}W_{ij}x_{i}+b_{j}\right) \tag{9}\] \[z_{k} =t\left(\sum_{j=1}^{1}W_{jk}y_{j}+b_{k}\right) \tag{10}\]  
where \(x=(x_{1},x_{2},\cdots,x_{n})\) represents the input vector of the input layer, \(y=(y_{1},y_{2},\cdots,y_{1})\) represents the output vector of the hidden layer, and \(z=(x_{1},x_{2},\cdots,x_{m})\) represents the output vector of the output layer. Let \(x^{*}=(x_{1}^{*},\)\(x_{2}^{*},\cdots,\)\(x_{m}^{*})\) be the desired output vector, \(E\) represents the error function which is typically half of the sum of squared errors, and \(n\) represents the learning rate in BP neural networks.