## 3 Machinery Monitoring
### Using the Neural Networks to Identify the Status of Machineries  
In this section, we discuss the use of neural networks to identify the operating status of machinery. We introduce the structure of the proposed neural network, outline the training equations, and illustrate the process of identifying the operating status. Figure 5 presents the structure of the proposed neural network, which includes three layers: an input layer, a hidden layer, and an output layer. The input layer receives the features selected by the LDA, which are then relayed to the hidden layer. Thus, the number of nodes in this layer must equal the number of selected features. The hidden layer merges the features and then uses a set of activation functions to formulate the relationship between the features and the status of the machinery. Note that in this study, we selected the hyperbolic tangent sigmoid function as  
Figure 4: Example of dimensionality reduction. an activation function, due to the fact that it features dual polarity signals, which have proven highly effective in neural networks [42]. We opted for two nodes in this layer, based on the recommendation of [PERSON] and [PERSON] [15]. The output layer has only one node, which is responsible for determining the status of the machinery. A value is closer to 1 indicates that the machinery is working well, whereas a value closer to \(-1\) is an indication of abnormal operating status. In the following, we introduce the equations used in the proposed neural network [15]. Equations (8)-(10) are used for the input, hidden, and output layers, respectively. To facilitate this discussion, have also included \(z\) to represent the \(z\)th set of features. \[out_{i}^{(\text{input layer})}(z)=input_{i}^{(\text{input layer})}(z) \tag{8}\]  
\[out_{i}^{(\text{hidden layer})}(z)=\frac{\left( \begin{array}{c}\exp(\sum \limits_{j=1}^{r}w^{1}_{ij}out_{j}^{(input layer)}(z)+b_{i})\\ -\exp(-\sum\limits_{j=1}^{r}w^{1}_{ij}out_{j}^{(input layer)}(z)+b_{i})\\ \hline\exp(\sum\limits_{j=1}^{r}w^{1}_{ij}out_{j}^{(input layer)}(z)+b_{i})\\ +\exp(-\sum\limits_{j=1}^{r}w^{1}_{ij}out_{j}^{(input layer)}(z)+b_{i})\\ \end{array} \right)} \tag{9}\]  
\[y(z)=\sum\limits_{j=1}^{2}w^{2}joint_{j}^{(hidden layer)}(z) \tag{10}\]  
From the above equations, we derived the equations for training the neural network based on the concept of back propagation [15]. We first assume that the target function intended for the training network should be as follows:  
\[Error(\mathbf{w},z)=\
icefrac{{1}}{{2}}(y_{d}(z)-y(z))^{2}=\
icefrac{{1}}{{ 2}}error(z)^{2} \tag{11}\]  
where \(error(z)\) is the error value between the ideal output \(y_{d}(z)\) and actual network output \(y(z)\). Based on the theorem of back propagation, all parameter in the neural network can be adjusted using the following function:  
\[w(z)=w(z-1)+\xi(-\frac{\partial^{+}Error}{\partial w}) \tag{12}\]  
Figure 5: The structure of the proposed neural network. where \(w\) is used as \(w^{1}\), \(b\), and \(w^{2}\) in Equations (8)-(10), respectively. Thus, we obtain the training functions of this neural network, as follows:  
\[w^{1}(z)=w^{1}(z-1)+\xi\Bigg{[} \begin{array}{c}error(z)\textbf{w}^{2}\times \textbf{input}^{(\textbf{input layer})}(z)\times\\ 4/\Big{(}\begin{array}{c}\exp(\textbf{out}^{(\textbf{input layer})}(z))\\ +\exp(-\textbf{out}^{(\textbf{input layer})}(z))\end{array}\Big{)}^{2}\end{array}  \Bigg{]} \tag{13}\]  
\[b(z)=b(z-1)+\xi\Bigg{[} \begin{array}{c}error(z)\textbf{w}^{2}\times\\ 4/\Big{(}\begin{array}{c}\exp(\textbf{out}^{(\textbf{input layer})}(z))\\ +\exp(-\textbf{out}^{(\textbf{input layer})}(z))\end{array}\Big{)}^{2}\end{array}  \Bigg{]} \tag{14}\]  
\[w^{2}(z)=w^{2}(z-1)+\xi(e(z)\textbf{out}^{(\textbf{hidden layer})}(z)), \tag{15}\]  
After training the neural network using Equations (13)-(15), the feature set in input to enable the identification of machinery status. For example, if a signal collected from the machinery can be divided into 100 windows, then these windows are transformed into a set of 100 features.