# Root Cause Tracing Using Equipment Process Accuracy Evaluation for Looper in Hot Rolling
## 3 Root-Cause-Tracing Algorithm
### Construction of Root-Cause-Tracing Algorithm
(5) Neural Network  
In this paper, due to the unclear functional relationship between the characteristic values of each factor of the looper and its score, a back-propagation (BP) neural network is employed [26]. The characteristic values of each factor of the looper are used as inputs, while the looper score is the output; an appropriate activation function is selected to fit the unknown function to facilitate subsequent analysis. It can be seen from Figure 2 that material factors, equipment factors, and control factors directly affect the parameters of the looper itself: looper angle and looper tension. Therefore, the above three types of factors are used as the input of the first neural network, and the looper angle and looper tension are used as outputs. Then the second neural network is constructed by utilizing the looper angle and looper tension as inputs and the looper score as the output. By combining the above two neural networks and using the output of the first neural network as the input of the second neural network, the relationship between each factor of the looper and the looper score can be obtained. (6) Calculate weight  
The state of the looper involves many factors. In order to facilitate subsequent analysis, we express the dimensionless eigenvalues and scores as \(x_{1}\dots x_{n}\) and \(y\), respectively. These factors directly act on the state of the looper, that is, the looper angle \(angle=f_{1}(x_{1}\dots x_{n})\)and the looper tension \(force=f_{2}(x_{1}\ldots x_{n})\). The status of the looper directly affects the evaluation score of the looper, that is, \(y=u(angle,force)\). Therefore, the relationship between the looper score and the eigenvalue can be directly expressed as  
\[y=f(x_{1}\ldots x_{n}) \tag{22}\]  
Then, a one-stage Taylor expansion of \(f\) is performed:  
\[f(x_{1}\ldots x_{n})=p_{1}x_{1}+p_{2}x_{2}+\ldots+p_{n}x_{n}+o(x_{1}\ldots x_{ n}) \tag{23}\]  
where \(o(x_{1}\ldots x_{n})\) is the higher-order term of each eigenvalue. Since each eigenvalue has been dimensionally processed, \(p_{i}\) can represent the weight of each eigenvalue. A larger \(p_{i}\) means the factor has a greater impact on the looper score. In the actual production process, \(f\) is a nonlinear function, and it is difficult to explain its specific expression form. Therefore, it is necessary to use existing data and use the BP neural network [27] to fit \(f\). The dimensionless characteristic deviation and fractional deviation are put into the above formula to obtain  
\[\Delta y=f(\Delta x_{1}\cdots\Delta x_{n}) \tag{24}\]  
\[f(\Delta x_{1}\cdots\Delta x_{n})=\sum_{i=1}^{n}p_{i}\Delta x_{i}+o(\Delta x_ {i}) \tag{25}\]  
In order to fully consider the contingency of the new sample data, it is necessary to select a value for \(\Delta y_{m}\) that is not much different from the score deviation \(\Delta y\) in the existing database as the reference data, that is, \(\Delta y_{m}=f(\Delta x_{m1}\ldots\Delta x_{mn})\). Since \(\Delta y\) and \(\Delta y_{m}\) are not much different, the form of each characteristic deviation is basically similar, so it can be used as a reference to establish the following system of equations:  
\[f(\Delta x_{11}\ldots\Delta x_{1n}) =\sum_{i=1}^{n}p_{i}\Delta x_{1i}+o(\Delta x_{1i}) \tag{26}\] \[f(\Delta x_{m1}\ldots\Delta x_{1n}) =\sum_{i=1}^{n}p_{i}\Delta x_{mi}+o(\Delta x_{mi})\]  
Since \(\Delta x\in[-1,1]\), \(o(\Delta x_{m1}\ldots\Delta x_{mn})\) is a bounded minimum quantity, which can be specified based on the lower limit of the actual data. Think of \(p_{i}\) as the weight of each factor, \(0\leq p_{i}\leq 1\). At the same time, new samples exert a more significant influence on the results, so their proportion in the solution process should be appropriately increased. (7) Particle Swarm Optimization Algorithm  
Since each eigenvalue has been dimensionally processed, the coefficient \(p_{i}\) in front of each factor can represent the weight of each factor.