## 3 Overall Architecture of the Proposed Method  
The proposed prediction method development process is shown in Figure 4. It includes some stages. The first stage is the processing of input data. For the field data measured by crystal-growing equipment on site, the grey correlation analysis (GRA) method [11] is used to perform correlation analysis on data with a significant impact on the tail diameter. For the temperature and pulling speed, the MIV algorithm [12] is used to select delaying orders that have a significant impact on the tail diameter, and then the data are randomly divided into training and testing datasets. The second stage is the establishment of a prediction model. The deep belief network is employed to extract the feature information in the data. Then, the extracted feature information is used as an input of the SVR. The ant colony optimization (ACO) algorithm [13] is employed to automatically select the optimal parameter of the SVR, and apply it to the soft measurement model. The final stage is to predict the tail diameter based on the DBN-ACO-SVR model, compare the predicted results with those of other methods, and validate the data of different lots based on this model. Figure 3: Capture device images. In the tail process of Czochralski crystal growth, the variables affecting the crystal diameter include temperature, pulling speed, etc. Other parameters such as the crystal rotation speed and crucible rotation speed are mostly kept constant at this stage. Therefore, in order to ensure that the data used in modeling reflect the actual growth status of the tail diameter as much as possible, it is necessary to adopt a method to determine the signals associated with the tail diameter. This article adopts the grey correlation analysis method, which refers to the geometric comparison of data sequences that reflect the changing characteristics of various factors. In the gray correlation analysis method, the reference series and the comparison sequence are respectively \(X_{0}=[X_{0}(k)|k=0,1,2,\cdots,n]\) and \(X_{i}=[X_{i}(k)|k=0,1,2,\cdots,n]\). The calculation formula for the correlation degree of each series is Formula (1):  
\[r_{i}=\frac{1}{n}\sum\frac{\min\limits_{i=1,nk=1,n}|X_{0}(k)-X_{i}(k)|+\rho \underset{i=1,nk=1,n}{\max}|X_{0}(k)-X_{i}(k)|}{|X_{0}(k)-X_{i}(k)|+\rho \underset{i=1,nk=1,n}{\max}|X_{0}(k)-X_{i}(k)|} \tag{1}\]  
where \(\rho\) is the resolution coefficient, the value range is 0-1, and the general value is 0.5. When predicting the tail crystal diameter, due to the delaying time of the temperature and pulling speed, the current crystal diameter cannot be directly trained and predicted by using the temperature and pulling speed at the current moment. It is necessary to determine the delaying order of the temperature and pulling speed on the diameter. In this paper, the MIV algorithm [14] is used to choose the significant input items from the temperature and pulling speed. Input variables that significantly affect the output  
Figure 4: The overall architecture of the proposed soft measurement method. are retained, and those that do not play a significant role are excluded. Finally, the above temperature and pulling speed input items are used to become the DBN-ACO-SVR network inputs. The delaying of temperature and pulling speed needs to be analyzed independently. Taking the temperature delaying time as an example, we establish a three-layer multi-input and single-output temperature-diameter dynamic neural network model [14]. The model is shown in Figure 5, and can be expressed as Formula (2):  
\[D(k)=f(T(k),T(k-1),\cdots T(k-n_{max})) \tag{2}\]  
where \(D(k)\) is the crystal diameter at moment \(k\), \(T(k-i)\) is the temperature at moment \(k-i\), \(n_{max}\) is the maximum delaying order of the temperature input, and \(f(\cdot)\) is a nonlinear function. First, the neural network shown in Figure 5 is trained by using the dataset. When the accuracy of the training results is satisfied, the network training parameters are saved.