# Predicting Ceramic Wool Diameter by Motor Frequency Using Improved BP Neural Network
## 4 The Calibration of Measurement Model
### Improved BP Neural Network  
BP neural network is a supervised learning algorithm which is trained by the error back propagation algorithm. It is widely utilized due to its perfect theoretical system, clear algorithm flow and powerful function of data recognition and simulation compared to other evaluation methods [27; 28; 29; 30]. BP neural network consisted of three layers including input layer, hidden layer, and output layer as shown in Figure 6. In this model, a series of motor frequencies formed the inputs and the corresponding outputs were the mean ceramic diameter.  
The input of the hidden layer was as follows.  
\[u_{i}=\sum_{i=1}^{M}(w_{ij}x_{j})+\theta_{i}, \tag{9}\]  
where \(u_{i}\) was the input of the hidden layer, \(w_{ij}\) and \(\theta_{i}\) were the connection weight and threshold between input layer and hidden layer where \(i=1,2,\ldots,q\) and \(j=1,2,\ldots,M\), respectively.  
The output of the hidden layer was as follows.  
\[y_{i}=\phi(u_{i}), \tag{10}\]  
where \(y_{i}\) was the output of the hidden layer and \(\phi\) was the transfer function of the hidden layer.  
The input of the output layer could be expressed as follows.  
\[v=\sum_{i=1}^{q}(\omega_{i}y_{i})+a, \tag{11}\]  
where \(v\) is the input of the output layer, \(\omega_{i}\) was the weight of the output layer to the corresponding hidden layer, \(a\) was the threshold of the output layer.  
Therefore, the output of the output layer was calculated as follows:  
\[o=\varphi(v)=\varphi\Big{\{}\sum_{i=1}^{q}\omega_{i}\varphi\Big{[}\sum_{i=1}^{ M}\big{(}w_{ij}x_{j}\big{)}+\theta_{i}\Big{]}+a\Big{\}}, \tag{12}\]  
where \(o\) was the output of the output layer and \(\varphi\) was the transfer function of the output layer.  
However, BP neural network also had its shortcomings as it sometimes fell into a local anomaly. It could not be guaranteed to converge to a global minimum. In addition, the BP algorithm took many training times, but the learning result was not satisfying. PSO was one of the optimization techniques and modern heuristic algorithms. It was found to be robust in solving nondifferentiability and nonlinearity problems which was applied for solving complex problems within stable convergence property and shorter computing time [31; 32; 33]. PSO first initialized a set of particles, and each particle represented a potential optimal solution in the search space. Each particle was treated as a volumeless particle with three indicators: location, velocity, and fitness value. Each particle kept track of its  
Figure 6: Topological structure of BP neural network model.  
coordinates and adjusted the current position through calculating the fitness value between the personal and group best. The value was dynamically adjusted according to its own flying experience. The algorithm flow of PSO is shown in Figure 7.  
The data in Table 1 were used as the training sample and BP neural network based on PSO algorithm was achieved. The maximum number of iterations and the number of particles swarm was set to 800, the maximum training times of BP neural network was 200,000, the iterative weight and the accuracy were 0.035 and 0.004, respectively. The calculated model based on PSO-BP neural network was indicated in Figure 8. Results of comparison of Figures 5 and 8 indicate that the measuring model based on PSO-BP neural network was obviously better than that of LSM.